III
NBS SPECIAL PUBLICATION 484

National Bureau of Standards

Library, E-01 Admin. Btdg.

OCT 1 1981

13i0i+8

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Flow Measurement in Open Channels

and Closed Conduits

Volume 1

A11103 0fia^7b

i

i 4
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards* was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and

technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to pro- mote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Institute for Computer Sciences and Technology, the Office for Information Programs, and the

Office of Experimental Technology Incentives Program.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consist- ent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry,

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Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Center for Radiation Research — Laboratory Astrophysics^ — Cryogenics'' — Electromagnetics — Time and Frequency'.

THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized materials needed by industry, commerce, educational insti- tutions, and Government; provides advisory and research services to other Government agencies; and develops, produces, and

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national Bureau of Standardi

'NOV 8 1977

Flow Measurement in Open Channels

and Closed Conduits

Volume 1

Proceedings of the Symposium on Flow Measurement

in Open Channels and Closed Conduits held at the

National Bureau of Standards in Gaithersburg,

Maryland on February 23-25, 1977

Lafayette K. Irwin, Editor

Mechanics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

0 .S. NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued October 1977
Library of Congress Cataloging in Publication Data

Symposium on Flow Measurement in Open Channels and Closed

Conduits, National Bureau of Standards, 1977. Flow measurement in open channels and closed conduits.

(National Bureau of Standards special publication ; 484)

Includes index. Supt. of Docs, no.: CI 3. 10:484

1. Flow meters—Congresses. I. Irwin, Lafayette K. II. Title. III.

Series: United States. National Bureau of Standards. Special publication ; 484. QC100.U57 no. 484 [TC177] 602'. Is [6a0.r064] 77-14243

National Bureau of Standards Special Publication 484/1

Nat. Bur. Stand. (U.S.), Spec. Publ. 484/1, 479 pages (Oct. 1977)

CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1977

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $12.25 per 2 part set ; sold in sets only

Stock No. 003-003-01845-7
PREFACE

These Proceedings include most of the papers presented at the threeday symposium on fluid flow measurements held at the National Bureau of

Standards, Gaithersburg , Maryland, on February 23, 24, and 25, 1977. The

papers dealt with flow in open channels and closed conduits and a special

related session on international standards.

The impetus to hold the symposium came from several sources. These

included (1) continuing requests for the information presented at the 1974

Flow Measurement Conference held to establish connections between flow

measurement and national needs (conference presentations not published)

,

(2) increasing use of measurement technology by regulators to monitor or

enforce the effects of their rule making, and (3) rising awareness of

the importance of international standards to this nation, particularly,

for quantifying energy-related fluids. The purpose of the symposium was

to bring together a significant group of the producers and users of flow

measurement technology, to provide a forum for presenting and discussing

both the new and the proven ideas for making useful flow measurements,

and to assemble and disseminate the resulting information.

Initially, the theme was to emphasize identification and assessment

of sources of errors, formulation of error analysis techniques and

recommendations for resolving the uncertainties for flow measurement.

The strong interests of investigators in presenting related research,

of instrument developers in describing the performance of flow measurement devices and systems, and of users in discussing applications

resulted in a broader spectrum and larger number of papers than was

envisioned. An optimum ordering of papers in the resulting two volumes

has not evolved.

Lafayette K. Irwin, Editor

iii
Conversions for Customary to International System (SI) Units

Reflecting the transitional stage of the measurement units in this

technological area, both U. S. customary and SI units of measurement are

used in different papers of these Proceedings. The U.S.A. is a member

of the General Conference on Weights and Measures which gave official

status to the SI metric units in 1960. Conversion factors for selected

units in these Proceedings are:

Quantity

Length

Area

Volume

Conversion

customary to SI

inch (in) to meter (m)

foot (ft) to meter (m)

inch^ (in^) to meter^ (ni^)

foot^ (ft^) to meter^ (m^)

inch^ (in^) to meter^ (m^)

foot^ (ft^) to meter^ (m^)

gallon^"^ (gal) to meter^ (m^)

Factor

approximate

0.0254

0.3048

0.0006452

0.09290

0.00001639

0.02832

0.003785

Mass pound, avoirdupois (lb) to kilogram (kg) 0.4536

Force

Pressure, or

Stress

pound (Ibf) to newton (N)

kilogram (kgf) to newton (N)

pound/inch^ (psi) to newton/meter^ (N/m^)

inch of mercury, 60 °F (inHg) to newton/meter^ (N/m^)

inch of water, 60 °F (inH20) to newton/meter^ (N/m^)

millimeter of mercury, 0 °C (mmHg) to

newton/meter^ (N/m^)

inch/second (in/s) to meter/second (m/s)

foot/second (ft/s) to meter/second (m/s)

foot/second^ (ft/s^) to meter/second^ (m/s^)

free fall, standard (g) to meter/second^ (m/s^)

pound/foot^ (Ib/ft^) to kilogram/meter^ (kg/m^)

pound/gallon"^"^ (lb/gal) to kilogram/meter^ (kg/m^)

foot^/second (ft^/s) to meters/second (m^/s)

foot^/minute (ft^/min) to meter ^ /second (m^/s)

gallon"^"^ /minute (gal/min) to meters/second (m^/s)

pound/second (Ib/s) to kilogram/second (kg/s)

Temperature degree Fahrenheit to degree Celsius

Velocity

Acceleration

Density

Flow

4.448

9.807

6895

3377

248.8

133.3

0.0254"^

0.3048t

0.3048"^

9.807

16.02

119.8

0.02832

0.0004719

0.00006309

0.4536

to =(to^-32)l

See NBS SP330, 1977 ed.. The International System of Units (SI) and

ASTM E380-76, Metric Practice Guide for more complete information.

^ Exact

"^"^ U. S. liquid

iv
TABLE OF CONTENTS - Volume 1

PREFACE ill

Conversions for Customary to International (SI) Units iv

ABSTRACT ix

Measuring the Repeatability of Flowmeters 1

A. T. J. Hayward

The Navy Liquid Flow Correlation Program 25

J, H. Tabler and C. G. Kullman

A Laboratory Study of Turbine Meter Uncertainty 33

G. Mattlngly, P. Pontius, H. Alllon, and E. Moore

Evolution of a Modern Petroleum Measurement Manual 55

L. M. Davis

Building Blocks Towards Flowmeter Reliability 61

Dezsoe Halmi

Open Channel Flow Monitoring or Metering, That is the Question ... 83

Kenneth W. Martlg, Jr.

Instrument Errors in Open Channel Flow Measurement Systems 91

D. M. Grant

Marine Dynamics and Its Effects on Current Measuring Transducers . 109

Thomas Mero, Gerald Appell, and Raul S. McQulvey

Some Error Sources in Price and Pygmy Current Meter Traverses .... 123

G. Kulin

Flow Measurements in the Canals of Power Plant Closed Loop Cooling

Systems 145

Joseph F. O'Brien, James Skridulis, John Annett, G. Wayne Slngley

and Harold L. Koenlg

Errors in Flow Measurement and Their Importance in Infiltration/

Inflow Analysis 173

D. L. Guthrie, D. R. Washington and C. Vlncenty

Rating Broad-Crested V-Notch Weirs with Narrow, Sloping Approach

Channels and Sediment Deposits 187

James F. Ruff, Keith Saxton and Clement Dang

Compensating for Construction Errors in Critical-Flow Flumes and

Broad-Crested Weirs 201

John A. Replogle

V
Table of Contents - Volume 1 (Continued)

Numerical Modeling of Two-Dimensional Flumes 219

Ronald W. Davis

The Design of Open Channel Acoustic Flowmeters for Specified Accuracy:

Sources of Error and Calibration Test Results 243

Francis C. Lowell, Jr.

Application Considerations & Performance Capability of the Time

Differential Clamp-On Ultrasonic Flowmeter 267

J. Baumoel

Theoretical and Experimental Assessment of Uncertainties in NonIntrusive, Ultrasonic Flow Measurement 277

Ronald F. Bruner

A New Ultrasonic Flowmeter for the Natural Gas Industry 293

N. E. Pedersen, J. E. Bradshaw, L, C. Lynnworth and P. F. Morel

A New Non- Intrusive Flowmeter 319

R. S. Flemons

Reconstructing Three-Dimensional Fluid Velocity Vector and Temperature Fields from Acoustic Transmission Measurements 335

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro

Validation of Use of Dye-Dilution Method for Flow Measurement in

Large Open and Closed Channel Flows 361

W. H. Morgan, D. Kempf, R. E. Phillips

Dilution Method of Discharge Measurement in Pipes 395

E. R. Holley

Application of a Fluorescence Technique to Dye-Concentration Measurements in a Turbulent Jet 42 3

Hsien-Ta Liu, Jung-Tai Lin, Donald P. Delisi and Frank A. Robben

The Application of Monte Carlo and Bayesian Probability Techniques

to Flow Prediction and Determination 447

Frank J. Berte

Author Index 470-A

TABLE OF CONTENTS - Volume 2

Laser Doppler Anemometry for Flow Measurement 471

William W. Durgin and Lawrence C. Neale

On Improving the Pitot-Tube Determination of Flows in Large Pipes. 479

J. M. Robertson and M. E. Clark

Numerical Modeling and Turbulent Flow Through Thin Orifice Plates. 491

R, W. Davis and G. E. Mattingly

vi
Table of Contents - Volume 2 (Continued)

Studies of Pulsating Incompressible Flow Through Orifice Meters . . 523

R. A. Bajura and M. T. Pellegrin

A Vortex Flowmeter - Calibration Results and Application Experiences 549

R. W. Miller, J. P. DeCarlo, and J. T. Cullen

Temperature Field Measurements in Turbulent Thermals 571

Mohamed Gad-el-Hak

Flow Measurement Up-Date 597

Robert B. Suhoke

Near Wall Velocity Measurements for Wall Shear Inference in Turbulent Flows 621

F. J. Pierce and D. S. Gold

On Reduction of Errors Arising in Hot-Wire Anemoraetry of Thin Turbulent Shear Layers 649

J. Gaviglio and J. P. Dussauge

Probe and Method for Simultaneous Measurements of 'True' Instantaneous Temperature and Three Velocity Components in Turbulent

Flow 659

Gracio Fabris

Measurement of Boundary Layer Transition Using Acoustic Techniques 687

Donald C. Sachs, Thomas F. V. Meagher, and Vernon D. Peckham

Measurement of Recirculating Flow Behind a Cylinder with Spray

Cooling 705

R. S. Rudland

A New Type of Velocity Probe 719

W. B. Brower, Jr. and Alan Servoz

The Influence of Turbulence on Static and Total Pressure Probes . . 737

V. E. Scottron

Determination and Elimination of Errors in Velocity Readings at

Ventilation Tubing Inlets 755

Robert A. Haney

Indirect Two-Phase Flow Measurement: Analysis and Reduction of

Methods Errors 765

N. N. Kondic

vii
Table of Contents - Volume 2 (Continued)

Measurements of Density and Density Gradient With an Oscillating

Conductivity Probe 783

Donald P. Delisi and Robert H. Kirchhoff

New Techniques for Automotive Fuel Flow Measurements 803

Milton Baker

A New Digital Technique for Temperature and Viscosity Compensation

of Turbine and Positive-Displacement Flow Transducers 821

Roger Jennings

Viscosity Effects on the Turbine Flowmeter 847

John M. Ball

Turbine Meters for the Measurement of Gaseous Hydrocarbons 871

Paul J. LaNasa

On a New Method of Gas Flow Measurement Using Cryogenic Techniques 881

D. B. Mann and J. A. Brennan

An Anatomy of the International Standards Producing System - Flow

Measurement 895

L. K. Irwin

OIML and EEC, Their Impact on the International Standards Producing

System 921

D. E. Edgerly

The Interface Between Industrial, National^nd International

Standards 929

Wallace N. Seward

The Adequacy of the Data Bases Available for Promulgating International Standards 935

H. H. Dij stelbergen

Equitable Implementation of National and International Allocation

Programs - Summary of Remarks 945

Jame« A. West

Of Form and Substance for Effective Standards 947

E. A. Spencer

Comments On: Role of International Standards in Measuring and

Allocating Hydrocarbons - A Program for Solution of our Present

Problems 957

F. H. Abernathy

Author Index 960

viii
ABSTRACT

The wide range and complexity of problems and potential solutions that

must be considered for useful flow measurements are emphasized by the

papers contained in these proceedings. Fifty-three presentations cover:

characteristics of new and improved instruments; applications of traditional and new measuring devices in field environments; procedures for

identifying and analyzing errors or uncertainties in data under specific

conditions; uses of physical and numerical models; politico-economic

changes that affect international standards for flow measurement; and

philosophical bases for making measurements. The fluids of most interest

are water and waste water, petroleum and related refined products, air,

natural gas and stack gas.

Experimental and analytical investigations on instrument performance

and interpretation of results include innovative applications of traditional

and new flow measurement techniques to fluid flows in open channels and

closed conduits. The traditional devices or techniques include weirs,

flumes, current meters, orifice plates, turbines, hot-wires, pitot-static

tubes, velocity traverses, dye-dilution, and others. More recent instrumentation developments and procedures such as laser doppler anemometry,

acoustic and thermal imaging, acoustic pulse velocity and doppler anemometry, numerical modeling, vortex shedding and digital computation are

covered for particular measurement purposes.

The most significant trend reflected in these presentations is the

general awareness that uncertainties in measured quantities at the lowest

point in the measurement chain, i.e., in the field or plant, are more

important than accuracy statements derived from controlled laboratory

studies. Other trends in evidence are the rising importance of turbine

meters for use as transfer standards and in-line measurements of liquids

and gases in filled pipes and the increasing number of applications for

acoustics and laser technology for flow measurements in both open channels

and closed conduits.

Key Words: Acoustic flow meters; closed conduit flows; current meters;

dye-dilution methods ; errors in flow measurement ; flow

measurement; fluid flow modeling; fluid velocity; flumes;

gas flow standard; hot-wire anemometry; international flow

standards; laser anemometry; open channel flows; orifice

meters; pitot-static meters; turbine meters; weirs.

ix
DISCLAIMER:

Certain trade names and company products are identified in order

to adequately specify the experimental procedure. In no case does such

identification imply recommendation or endorsement by the National

Bureau of Standards, nor does it imply that the products are necessarily

the best available for the purpose.

X
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

MEASURING THE REPEATABILITY OF FLOWMETERS

A. T. J. Hayward

National Engineering Laboratory

East Kilbride, Glasgow, Scotland

A technique for measuring the repeatability of

flowmeters has been investigated in an extensive series

of tests. The technique involves installing a pair of

nominally identical flowmeters in series and comparing

their readings in a batch of repeat tests; the repeatability is then given by ta//2, where a is the standard

deviation of the ratio of the simultaneous readings of

the two meters and t is Student's t.

The method shows great promise, and has revealed

small but significant drift (sometimes as low as a few

parts per million over a short series of tests) in

meters where drift had not previously been suspected.

Extensions of the method have also been used successfully to determine the repeatability of a flowmeter

calibration system, and to measure the intra-rotational

linearity of a positive displacement meter.

1. Introduction

Although repeatability is a very important property of a flowmeter,

and is glibly referred to very frequently by manufacturers and users alike,

there is as yet no recognized method of measuring it. The reason for this

is fairly obvious. There is no particular difficulty in determining the

repeatability of an instrument that measures some fixed quantity - for

example, the mass, the length, or the elastic modulus of a solid specimen - and standard methods for determining repeatability in the case of

measurements such as these are either in existence or under preparation

by ISO/TC69 and related national standardization committees. It is only

under identical test conditions, and the results can be analysed

statistically to yield a repeatability value.

Unfortunately this method does not work with instruments that measure

transient quantities, since these never remain absolutely constant with

time. Many important industrial instruments fall into this category,

including flowmeters, ammeters, radiation meters, accelerometers , and a

host of others. Any attempt to measure the repeatability of a flowmeter

by the technique mentioned above would yield a result that was adversely

affected by the unsteadiness of the flow being repeatedly measured, since

it is a physical impossibility to produce a perfectly steady flow.

1
There is clearly a need for a standard technique for determining the

repeatability of instruments measuring transient quantities. Indeed,

proposals for such a standard method have been developed at the National

Engineering Laboratory and will be published shortly [1] . This paper

describes the experiments carried out at the Laboratory during the past

year to investigate the validity of the proposed method and to obtain

information about the repeatability of various measuring instruments,

including several flowmeters.

2. Basis of the method

The theoretical basis for the method is described in detail in

Reference 1". It is based upon the fact that the repeatability of a

measuring instrument depends upon the random uncertainties relating to it,

and the assumption that these follow a Gaussian distribution. Probability

theory indicates that where uncertainties are Gaussian they can be added

in quadrature, which means that they can be added by squaring the individual

quantities, adding the squares, and taking the square root of their sum.

If two flowmeters are installed in series in a pipe and a number of

simultaneous readings of both flowmeters, and M2, are made, the variation

of the difference of the two readings (M^ - M2), or of their ratio (M1/M2),

will indicate the combined repeatability of the two flowmeters. Minor

fluctuations in flowrate between measurements will be of no consequence,

since they will presumably affect both flowmeters in exactly the same

manner.

Assuming Gaussian distribution as mentioned above, it follows that if

R22 is the measured repeatability of the combination of the two flowmeters

and Rj and R2 are the individual repeatabilities of the two flowmeters, then

If it is also assumed that the two flowmeters are identical, then

it follows from eq. (1) that

Rl = R2 = (2)

Employing this principle, the following procedure can be used to obtain

the repeatability of a flowmeter.

a Instal two nominally identical flowmeters in the same pipe in such a

way that the wake from the upstream flowmeter cannot significantly affect

the performance of the downstream flowmeter. (This generally necessitates

either having a long length of straight pipe separating the flowmeters, or

using a flow straightener .

)

b Pass flow through the two flowmeters for a period, during which the

flowrate should be as nearly constant and steady as is practicable.

References are listed at the end of this paper on page 13.

2
c Take a number of simultaneous readings

either 10 or 20 readings in a batch) of the

standard deviation, a, of the ratios of the

(at NEL we have generally used

two meters, and evaluate the

readings of the two flowmeters.

d The repeatability of each flowmeter is then given by the expression

R = ta

(3) 72

where t is the value of Student's t for the appropriate batch size and a probability of 95 per cent.

The method requires the flow passing through both meters to be

identical at any instant when a reading is taken. For incompressible flow

this is necessarily so, but it may not prove easy to achieve this with gas

flow. So far experiments have been confined to liquid flows, although it

is hoped to investigate the method with gas flow later. The method can be

applied equally well to flowrate meters and to totalizing meters used for

measuring total volumes passed, although it so happens that most of the

tests described in this paper have been carried out with the latter.

The largest series of tests has been carried out on a vane-type PD

meter, employing three alternative methods of readout and a number of

variations in test method. These are described in detail below. A 5 cSt

oil was used as the test fluid throughout.

3.1 Mechanical readout, standing-start-and-f inish tests

The first tests were carried out with a vane-type PD meter fitted with

a mechanical readout, in the form of a Veeder-Root head and a sandwich

drum enabling volumes to be read to the nearest 10 millilitres. This form

of readout necessitated the use of a standing-start-and-f inish method.

In this, both meters were set to zero, then flow was started by the sudden

opening of a downstream valve and allowed to continue until approximately

1000 litres had been passed by the meters, when flow was stopped by the

sudden closing of the valve, and a final reading of both meters was then

made

.

It is important to ensure, in tests of this kind, that the

discrimination of the meter in relation to the total volume being passed

is small compared with the repeatability values being measured. In this

particular case the requirement was fulfilled, since the discrimination

of 10 millilitres in a run of 1000 litres represents 10 parts per million,

which is very much less than the lowest repeatability value measured in

this experiment.

The results of these tests are given in Table 1. Five complete tests,

each involving at least 10 runs, were made at the maximum flowrate of the

meter, and three tests at each of two lower flowrates. The repeatabilities

found varied widely from test to test, over a range from 60 to 250 parts

per million; the variation in results appears to be random and not related

3. Repeatability tests on positive displacement meters

3
to flowrate. Part of the variation in the test results is probably due to

the variations imposed by the standing-start-and-f inish method of test which

is inherently less consistent then a flying-start-and-finish method. Even

so, however, it was obvious from these first tests that the repeatability

of a flowmeter is a rather variable quantity which is liable to change

significantly from day to day - a conclusion which, as will be seen later,

is borne out by every other test made.

3.2 Photocell pulse-generator readout, flying-start-and-f inish tests

In the next series of experiments the same two meters were fitted

with electrical pulse generators of the photocell type supplied by the

manufacturers and having a discrimination of 10 millilitres per pulse.

The addition of the electrical readout enabled a flying-start-and-f inish

test, again of 1000 litres duration, to be employed. In this method the

flow was allowed to pass at a constant rate throughout each batch of test

runs, and the output from one meter was passed into a preset counter which

automatically gated the counts from the other meter after every hundred

thousand counts. (In anthropomorphic terms, one meter was made to say to

the other meter, 'I have just registered my hundred-thousandth pulse; how

many pulses have you generated in the same period?'). By using two sets

of counters and zeroing one set while the other set is counting, it is

possible to carry out continuous testing without any break between the

runs of one hundred thousand counts. Generally about 20 runs were made in

each batch in this part of the experiment.

The results of this series of tests revealed an interesting and

hitherto unsuspected aspect of the behaviour of this type of meter. The

individual results of three typical batches are shown in figure 1, where

the. difference between the meters (Mj - M2) is shown plotted for each

individual run in a batch. For the sake of uniformity - M2 is expressed

in parts per million - in other words, the difference between the two

pulse counts was multiplied by 10. The type of result which had been

expected, and which in fact was frequently obtained, is that shown in

figure 1(a); here, the results of successive runs are randomly distributed

about a mean value. This value would have been zero if the meter factors

of the two meters had been identical, but since the two meter factors were

slightly different the mean value of - M2 is approximately 1850 parts

per million.

Sometimes, however, the value of - M2 was found to drift steadily

upwards or downwards during the course of a test. A typical result of a test in which steady drift of this kind occurred is shown in figure 1(b).

More rarely, a sudden change in the value of - M2 occurred part

way through a test. An example of a stepwise change of this character is

given in figure 1(c).

It is arguable whether this phenomenon of drift should be considered

separately from repeatability, or whether it should be ignored and its

effect should merely be allowed to increase the repeatability values

obtained from any one run. My own view is that the drift which has been

observed is a distinct phenomenon which deserves recognition in its own

4
right, and that it needs separate study with a view to ascertaining its

cause and preventing its occurrence in future flowmeters.

Consequently, in presenting the results in this part of the experiment

I have as far as possible tried to allow for the effect of drift. In the

case of a stepwise change as in figure 1(c) the only way to do this is to

continue the series of runs until 20 consecutive tests have given a line

without a step in it - and preferably a horizontal line. In the case of

steady drift as exemplified in figure 1(b) it is possible to allow for the

effect of drift by fitting a least-squares line to the experimental points

and calculating the standard deviation about this line instead of the

standard deviation about a mean value; this inevitably gives a smaller value

of standard deviation. The results of the tests are given in figure 2,

where repeatability is plotted against flowrate. Those points in this

figure which are based on batches of runs where steady drift occurred are

denoted by the presence of figures in brackets; these indicate the number

of parts per million by which the ordinate of each point should be increased

if it is desired to ignore drift. Thus the figures in brackets denote the

amount by which the repeatability value has been reduced by basing the

standard deviation on the least squares line through the test results

rather than on a simple mean value.

These results show that, provided corrections for drift are made in

those tests where it appeared, the repeatability of this type of flowmeter

tested in the f lying-start-and-f inish mode is between 30 and 70 parts per

million. Even if corrections for drift are not made, all but two of the

results fall below 70 parts per million and these two fall below 100 parts

per million. This means that there is an approximately three-fold increase

in repeatability over the previous tests with the same basic meters. The

improvement is probably due more to the change from standing to flyingstart-and-finish testing than the change in readout method, since the

electronic readout used this time was superimposed upon the mechanical

head used in the previous tests.

3.3 Shaft-encoder pulse-generator readout, f lying-start-and-f inish

In the next series of tests the same two basic meters were used, but

all the gearing associated with the method of readout used in the two

previous series was removed and an interf erometric shaft encoder was

fastened directly to the main shaft of the PD meter. Since the particular

shaft encoders used generate 2540 pulses per revolution the result was a meter with extraordinarily high discrimination, which was found by

calibration to be approximately 0.9 millilitres per pulse. In this series

the preset counter was set to work on counts of one million pulses, and

consequently the throughput per run was approximately 900 litres. As in

the previous test all the runs were made in immediate succession, but in

some of the tests in this series an additional refinement was introduced.

The data from the counters were fed through a suitable interface into a programmable calculator, which stored the test data and calculated the

repeatabilities obtained from each batch of tests. When this system is

used the testing is carried out almost automatically and the operator has

very little to do except to set the flowrate at the start of each batch of

tests and then press the starting button.

5
Typical results obtained from three tests in this series are shown

in figure 3. When the meters were free from drift, as in the test

illutrated in figure 3(a), exceptionally high repeatabilities were obtained

- usually below 10 parts per million. Sometimes a small amount of drift

was present, as illustrated in the test reported in figure 3(b), but, even

if this drift is not allowed for, the repeatabilities in such tests

usually were still not much more than 10 parts per million. At flowrates

below about 5 litres per second, that is, below about 25 per cent of the

rated maximum for the meter, drift became a very serious problem. Sometimes

it took the form of severe steady drift as shown in figure 3(c), and

sometimes it took the form of very severe stepwise drift. On one occasion

the value of Mj^ - M2 was observed to change by more than 1000 parts per

million between the last batch of runs carried out one afternoon and the

first batch carried out the next morning. In general, the results of tests

at these lower flowrates were so unrepresentative that they have not been

plotted along with the other test results. Since no corresponding

deterioration of performance was obtained at lower flowrates in the previous

experiment with this meter, this erratic behaviour is believed to emanate

from the shaft-encoder employed and is the subject of further investigation

at the time of writing.

The results of this part of the experiment are shown in figure 4, from

which several interesting conclusions immediately appear. First, if the

problem of severe drift at low flowrates can be overcome, this form of

flowmeter has great potential as a tool for standards laboratories, since

its repeatability is better than that of the same flowmeter with a conventional readout: under favourable conditions repeatabilities as good

as 10 parts per million appear to be easily attainable. Secondly, the

repeatability of this instrument appears to be flowrate dependent: between

5 and 11 litres per second, and again between 14 and 19 litres per second,

repeatability is below 10 parts per million, but between 11 and 14 litres

per second the repeatability deteriorates to values around 20 parts per

million. Further investigation is needed to ascertain whether this

deterioration in the middle flowrate range is related to the meter capsule

itself or to the behaviour of the shaft encoder.

4. Repeatability tests on turbine meters

The work at NEL on the repeatability of turbine meters is still in

its early stages. At the time of writing the only tests that have been

completed are a series on a pair of 75 millimetre (3 in) turbine meters

installed in a calibration line supplied with water from a constant-head

tank at flowrates up to 30 litres per second. These meters had been used

extensively before the present experiment took place, and one object of

the work was to assess the improvement in repeatability that could be

obtained by fitting new bearings. The meters were therefore tested over

a range of flowrates in their original conditi(j)n and then re-tested after

the bearings had been replaced with new ones.

This experiment brought to light one practical problem in repeatability

testing of turbine meters. The pulse generation rate of a turbine meter

is rather low, especially at the lower flowrates, and since one repeatability measurement necessitates making at least 10 runs and preferably 20,

6
it becomes prohibitively time-consuming to take more than ten thousand

pulse counts during one run. Unfortunately this introduces a random

rounding-off uncertainty which may be as large as 200 parts per million,

and this is so large that it almost obscures the true repeatability of

the turbine meter.

This point is brought out in figure 5 where the results of this

preliminary experiment are plotted. There does appear to be a distinct

improvement in repeatability as a result of renewing the bearings of

these meters, but the improvement is only large enough to be clearly

visible at the lowest flowrate, where bearing friction is at its most

significant. At the higher flowrates the effect of the new bearings on

repeatability is largely masked by the effect of the rounding-off error

on the repeatability values obtained. Further experiments are now in

hand with 50 millimetre turbine meters where the pulse generation rate

is higher, and where an electronic pulse multiplier is being used, which

should reduce the rounding-off error by a factor of 10.

5. Tests on a proprietary flowmeter calibrator

A proprietary device used for dynamically measuring flow, and hence

for calibrating flowmeters, has also been tested extensively in the author's

laboratory by the methods of this paper, and has thrown additional light

on the problem of drift.

It is a matter of common knowledge that the short-term repeatability

of practically any measuring instrument is substantially better than the

long-term repeatability of the same instrument. Many writers have

commented in qualitative terms on this fact. For example, Ascough [2]

speaks of three classes of errors: Class I errors, which are associated

with the repeatability obtained during a single test period; Class II

errors, which are constant during one test period but vary between batches

of tests, and consequently cause increases in the values of repeatability

measured over a long period of time; and Class III errors, which are fixed

systematic errors. The techniques described in the present paper may

eventually make it possible to assign quantitative values to Ascough'

s

first two classes of error. In particular, the stepwise form of drift

mentioned above is clearly identifiable as one of Ascough' s Class II errors.

The steady form of drift is more difficult to analyse since it

undoubtedly affects both Class I and Class II errors. Some tests on a pair

of ultrasonic flowmeters by a method very similar to that described in this

paper have recently been reported [3] and it is interesting to note that

one of the seven sets of results published there (batch number 118) clearly

reveals steady drift.

Stepwise drift between batches of tests is very evident in figure 6,

which shows the results of tests on the proprietary f lowraeter-calibrating

device, which was specially fitted with dual sets of measuring instruments,

so that two simultaneous measurements of volume passed could be made. Runs

were made in batches of 20 in rapid succession, at each of a number of

velocities. For each batch of 20 repeat readings the mean of the difference

between the two parallel sets of instruments (I^ - I2) was evaluated, as was

7
the standard deviation of (Ii - I2), namely, a. The results of these tests

at different velocities are shown in figure 6, where (I-^ - I2) is plotted

against velocity. Each cross on this graph represents the results of one

batch of 20 runs: the centre of the cross is the mean value of (I^ ~ j and the overall height of the cross represents the effective range (2ta)

of the batch.

It is immediately apparent from this graph that the variations between

batches of runs are often much greater than the variation within any one

batch. In other words, using Ascough's terminology, when this device is

used under these conditions. Class II errors are much greater than Class I

errors; or using my terminology, its long-term repeatability is much worse

than its short-term repeatability. It is worth noting in passing that the

device's repeatability is obviously not a function of velocity.

In the tests reported in figure 6 the runs within one batch were

carried out in continuous succession, using automatic data collection and

a dedicated computer. But it was necessary to shut the apparatus down for

a few minutes after each batch, in order to change the velocity of operation

and to make other necessary adjustments in readiness for the next batch of

runs. In a further experiment, the results of which are given in figure 7,

the apparatus was allowed to run continuously at a velocity of 2.5 m/ s

,

with no interruption whatever between batches of runs except for three

overnight breaks.

The contrast between these results obtained when the apparatus was

running in this continuous fashion (figure 7) and the results already

described when it was being operated intermittently (figure 6) are most

striking. Whenever the equipment was running continuously the between-batch

variations were very small, and were commensurate with the variations within

the batches. The only occasions when a significant between-batch shift

occurred were overnight between 8 and 9 November, and again overnight

between 9 and 10 November. Taken together, figures 6 and 7 provide very

convincing proof that this particular device gives highly repeatable

results as long as it is in continuous operation, but that its performance

is liable to deteriorate markedly whenever it is allowed to rest for even

a brief period.

6. Extensions of the test method

These methods can be adapted to meet a number of other needs in the

field of instrument technology. We have already applied them to two other

flow measurement problems which are discussed below.

6.1 Measuring the repeatability of a flowmeter calibration system

The method of this paper cannot generally be used to measure the

repeatability of a complex flowmeter calibration system directly, because

such systems are generally unique, and even when two identical systems

exist it is usually not practicable to connect them in series and compare

their readings. To measure the repeatability of a flowmeter calibrator it

is therefore usually necessary to connect two identical flowmeters in series

with each other and with the calibrator, as shown schematically in figure 8.

However, by taking simultaneous readings of both instruments and the

8
calibrator it is then possible to use the method to derive three 'combined

repeatabilities ' , namely, the repeatability of instrument 1 against the

calibrator, ^iq, the repeatability of instrument 2 against the calibrator,

R2C> and the repeatability of instrument 1 against instrument 2, ^^2' Then,

provided that the repeatabilities of the flowmeters are not greatly

inferior to that of the calibrator, it can be shown [1] that the repeatability of the calibrator, Rq, is given by

The technique has been used to determine the repeatability at a number of flowrates of a gravimetric standing-start-and-f inish calibrator,

operating with 5 cSt mineral oil. The same positive displacement meters

as were used in the experiment described in Section 3.1, above, were used

in this experiment also.

The results of tests at flowrates of 4, 11 and 16.5 litres per second

are given in Table 2. The repeatability of the calibrator was found to

be between 80 and 220 parts per million during these tests, with a small

but probably significant tendency to better repeatabilities at the lowest

flowrate; this can be attributed to the increased uncertainty imposed by

the periods of acceleration and deceleration in a standing-start-and-f inish

test at the higher flowrates. There has not yet been opportunity to try

the method with a flying-start-and-f inish calibrator, but there seems no

reason why it should not prove equally successful when used in that way.

6.2 Measuring the intra-rotational linearity of a rotary PD meter

The PD meter fitted with a shaft encoder, as described in Section 3.3

above, has been developed for use as a high-precision laboratory tool.

It is intended that it should be used to study the performance, under

flying-start-and-f inish conditions, of other devices used in flow

measurement, such as flowmeter calibrators and meter provers . In such

experiments it will be read to the nearest pulse, that is, with a discrimination of 0.9 millilitres. Since it emits 2540 pulses per

revolution, this means that it will be read to the nearest 1/2540 of a revolution. Calibration has shown that, on average, each pulse represents

0.9 millilitres, but it is necessary to know whether the instrument is

linear within one revolution - that is to say, whether each pulse

represents almost exactly 0.9 millilitres, or whether at some part of a revolution a pulse represents much less than 0.9 millilitres and at other

parts of a revolution a pulse represents much more than this.

This question has been resolved by the use of a technique whose

principle is illustrated in figure 9. A PD meter can be visualised as

being essentially a piston moving in a cylinder, where the distance

travelled by the piston is used as an indicator of the amount of liquid

passing through the meter. In Case 1 of figure 9 a situation is envisaged

in which there are two perfectly regular and identical cylinders connected

in series, with the same volume of liquid passing in succession through

both cylinders. Under these conditions the piston in cylinder A keeps

perfectly in line with the piston in cylinder B: thus at the start of the

9
run Al is immediately over Bl, half way through the run A2 is immediately

over B2, and at the end, A3 is immediately over B3,

In Case 2 of figure 9 the situation is considered where there are

identical bulges in both cylinders, and where these bulges are in

corresponding positions. In this case also the two pistons keep exactly

in line throughout the stroke. At the start, Al is immediately above Bl;

when half the volume of liquid has been passed, A2 is still exactly over

B2, but in this case the distance travelled by both pistons is significantly

less than the distance travelled by the pistons in the first case; and

again, at the end of the stroke A3 is in line with B3.

In Case 3 of figure 9, both cylinders have identical bulges, but the

bulge in A is near to its beginning and the bulge in B is near to its end.

In this case, the fact that the bulges are out of phase means that,

although the pistons are in line in positions 1 and 3, they become out of

line in position 2, where piston A has passed its bulge but piston B has

not yet reached its bulge.

These bulges in figure 9 are an idealized way of representing the

non-linearity within one revolution that might be expected to occur within

a PD meter. If two such meters of nominally identical pattern are connected

in series and their readings compared over less than one revolution, their

readings may or may not be found to be in close agreement. If they are

not in close agreement, it may mean that one or both meters contain some

intra-rotational non-linearity, or it may merely mean that their meter

factors are significantly different. The meter-factor question can be

quickly resolved by means of a calibration, and if the reading of one of

the meters is multiplied by the ratio of the two meter factors, this

procedure (known as 'normalizing') will eliminate the effect of differing

meter factors. If, after normalizing, the two meter readings still

disagree, this proves that one or both meters must have a significant

degree of intra-rotational non-linearity.

On the other hand, if the two meter readings after normalizing are

in good agreement, this does not necessarily mean that both meters have

perfect intra-rotational linearity, as in Case 1 of figure 9; it could

equally well mean that they both have a severe non-linearity, but that the

two linearities are in phase, as shown in Case 2 of figure 9. To ascertain

whether this is so it is therefore necessary to make a number of comparative

measurements between the two meters in series, with the two meters in a

number of different phase positions relative to each other. This can

easily be achieved in practice by inserting a bleed tapping between the

two meters and drawing off a small quantity of the test liquid, so causing

one meter to rotate through the required angle while the other meter

remains stationary.

In an experiment of this kind, where each test run occupies

considerably less than one second, the use of fully automatic data

collection and recording is essential. It was arbitrarily decided to set

the preset counters to collect totals of 2386 pulses, which corresponds to

2386/2540 of a revolution (approximately 338 degrees) and to make 500 runs

in immediate succession in each batch. Figure 10 illustrates

10
diagrammatical ly how, in this method of test, where the end of one run

signifies the beginning of the next run, the initial position of the meter

shaft was caused to precess backwards by about 22 degrees in each test

run. Not only did the starting point change in each test, but, because

the meter factors of the two meters were not identical, there was also a progressive increase in phase angle throughout the duration of the test;

consequently in a test of 500 runs a practically random assortment of

starting positions and phase differences was encountered.

The results from a typical batch of 500 runs are given in histogram

form in figure 11. The normalized discrepancy between the two meters is

remarkably small, being never more than 8 pulses out of 2386 pulses in

any one of these 500 runs. Moreover, the shape of the distribution curve

is not far removed from being Gaussian, which shows that the initial

assumption of randomness was not far from the truth. Consequently it is

reasonable to apply the principle of the method used throughout this paper

to derive a non-linearity constant, N, given by

N = 72

where a is the standard deviation of the normalized discrepancy between

the two meters over the batch of 500 runs.

In physical terms, this constant N has the following meaning. If

the operator uses one of the meters that has been tested to measure the

quantity of liquid passed during part of a revolution, and assumes perfect

intra-rotational linearity for the meter, the error introduced through

this assumption will be less than or equal to N in 95 cases out of 100.

(This non-linearity error, N, is, of course, additional to all the other

errors arising from random uncertainties and systematic uncertainties that

are involved in using any measuring instrument.)

The results of non-linearity measurements at a number of different

flowrates are given in Table 3. As an additional check on the assumption

that the phase difference between the two meters would randomise itself

over a period of 500 runs, the initial phase angle between the meters was

varied in successive tests. That is to say, at each flowrate one batch of

500 runs was carried out with the two flowmeters in a given initial phase

position; in the next batch one of the meters was advanced by 90 degrees

from the initial position of the previous batch before starting the second

batch; in the next batch the initial phase difference was 180 degrees; and

in the final batch 270 degrees. As will be seen from Table 3 this

variation in phase angle at the start of the tests had no detectable

effect - an observation which confirms the assumption of randomness. It

is worthy of note that the values of N determined in this experiment by

successive tests at the same flowrate are in much better agreement than

the values of R obtained in successive repeatability tests on the same

instrument. It is evident that non-linearity is a property that can be

measured more precisely than repeatability, even though similar statistical

methods are used for its determination. This conclusion is reassuring but

not unexpected.

11
It also appears from Table 3 that the intra-rotational non-linearity

of this meter is to some extent dependent upon flowrate. At 17 litres per

second, which is close to the maximum rated capacity of this meter, the

non-linearity approximately doubles. This may mean that some form of

vibration within the meter begins at these high speeds.

Despite this slight increase in non-linearity at the highest flowrate,

the values obtained are remarkably good. They mean that the assumption

of linearity within one revolution of the meter will very rarely introduce

an error of more than 4 millilitres, and at flowrates well below the

maximum capacity of the meter the error is unlikely to exceed 2 millilitres.

7. Conclusions . .

a The technique of coupling two liquid flowmeters in series and comparing

their readings in a large number of repeat tests is an effective method

of obtaining a numerical value for the repeatability of a flowmeter of a given pattern. With meters of sufficiently high discrimination it becomes

an exceedingly sensitive method of assessing the performance of the meter,

and enables effects to be studied that have not previously been detected

by the conventional method of calibrating a flowmeter against a calibration standard.

b There appears to be no reason why the method should be restricted to

flowmeters, since it would appear to be suitable for use with almost any

type of instrument for measuring transient quantities. Depending upon the

nature of the instruments it will sometimes be necessary to couple them

in parallel rather than in series.

c Extensions of the basic method have been successfully applied to the

determination of the repeatability of a flowmeter calibration system, and

to the determination of the intra-rotational linearity of a flowmeter with

an extremely high pulse-generation rate.

d The method can be applied to instruments with manual readout, but it

then becomes rather time-consuming. For instruments with an electrical

readout the tests can be carried out at great speed and with little effort

by the use of fully automatic data collection and processing facilities

incorporating a dedicated computer.

e Using this method has led to the discovery of drift, which sometimes

takes a steady form and sometimes the form of abrupt change, in several

types of instrument. In one case the drift was only a few parts per

million and would have remained undetected but for the present technique.

The existence of unsuspected drift would appear to explain the well-known

fact that nearly all instruments have a much better short-term repeatability than long-term repeatability. The use of this method to detect

and study this phenomenon of drift might enable manufacturers and users

to ascertain its cause in particular cases, and hence to reduce the

severity of drift and thus improve the long-term repeatability of

instruments.

12
f The combination of a positive displacement meter capsule and a shaft

encoder of the interf erometric type yields a promising tool for use in

standards laboratories, since repeatabilities of better than 10 parts per

million seem attainable provided that the problem of drift (which probably

arises from the shaft encoder rather than the meter capsule) can be solved.

g The particular vane-type PD meter used in this experiment appears to

be especially suitable for standards laboratory work of this kind, since

its intra-rotational linearity has been found to be extremely good also.

Acknowledgements

I am indebted to my colleagues J. C. Foot, J. S. Humphreys,

K. I. Jespersen, W. C. Pursley and B. Reed, for this assistance with

the experiments reported in this paper. This paper is presented by

permission of the Director, National Engineering Laboratory, Department

of Industry. It is Crown copyright.

References

[1] Hayward, A. T. J. Repeatability and accuracy . Mechanical

Engineering Publications, London and New York, 1977. (In press)

[2] Ascough, J. C. The accuracy of thrust in flight from engine

calibrations in an altitude test facility. Paper presented at

Tenth International Council of the Aeronautical Sciences Congress,

Ottawa, October 1976.

[3] Westinghouse LEFM May 1976 tests , Westinghouse Electric Corporation,

Annapolis, 1976.

13
TABLE 1

RESULTS OF REPEATABILITY TESTS ON PD METERS WITH MECHANICAL

READOUT (STANDING-START-AND-FINISH TESTS)

"P 1 OXaTT" i5 t" ^

l/s p. p.m.

4 130

4 90

4 170

11 140

11 110

11 180

16.5 200

16.5 250

16.5 120

16.5 60

16.5 110

TABLE 2

RESULTS OF REPEATABILITY TESTS ON A GRAVIMETRIC FLOWMETER

CALIBRATOR OF THE STANDING-START-AND-FINISH TYPE

Flowrate

l/s

Repeatabilities - p. p.m.

he Calibrator

4 350 180 380 80

4 480 190 490 140

4 450 260 490 130

11 220 200 130 190

11 210 130 180 130

11 180 160 130 180

16.5 290 210 220 200

16.5 230 230 150 200

16.5 250 240 170 210

14
TABLE 3

RESULTS OF INTRA-ROTATIONAL NON-LINEARITY TESTS

Flowrate

X// S

Initial

phase angle

uegirees

Non-linearity, N

pulses

4.5 0 1.95

4.5 90 1.98

4.5 180 1.91

4.5 270 1.87

9.4 0 1.45

9.4 90 1.46

9.4 180 1.47

9.4 270 1.57

9.4 0 1.54

9.4 90 1.54

9.4 180 1.46

9.4 270 1.54

14.0 0 1.60

14.0 90 1.49

14.0 180 1.50

14.0 270 1.57

17.1 0 4.00

17.1 90 2.90

17.1 180 3.11

17.1 270 3.22

17.1 0 3.96

17.1 90 3.25

17.1 180 4.02

17.1 270 3.72

Each value is based on 500 flying counts

of Hi - M2 in immediate succession, in

each of which was 2386 pulses, which

is equivalent to approximately 338 degrees

rotation.

15
1700

1800 h . + + +

+ + + , + +++ +

1900 h + + + + + -

2000 J I I -. L

a) EXAMPLE OF TEST FREE FROM DRIFT

-1300

KOO -

b) EXAMPLE OF TEST SHOWING STEADY DRIFT

1200

1300

KOO

1500

1 6 00

1 1 1 1

+

•

+

• i 0 5 10

RUN No.

15 20 25

c) EXAMPLE OF TEST SHOWING STEPWISE DRIFT

FIG 1 TYPICAL TEST RESULTS FOR PD. METERS WITH

PHOTOCELL -TYPE PULSE GENERATORS (BASED

ON 100,000 PULSE COUNTS PER RUN)

16
•uj*d*d 'AiniaViV3d3d

17
10

0 -4.—

- 10

•460 -

-A70 -

-480

-1530

-1540

-1550

% -1560

CM2 -1570

-1580

-1590

-1600

-1610

-1620

-1630

-1640

-1650

-1660

- 1670

1 1 1 1

+ . . + +-+— J. + . + + + + + + + +

+ -If:

I • a) EXAMPLE OF TEST FREE FROM DRIFT

1 1 1

+ ^+-+^+ + + +

-

_ +

1 I 1 • b) EXAMPLE OF TEST SHOWING SLIGHT DRIFT

1 1 1 1

IN

\

h +

V

\

\

+ +

\

\

±

\

0 5 10 15 20 25

RUN No. c)EXAMPLE OF TEST SHOWING SEVERE DRIFT

FIG 3 TYPICAL TEST RESULTS ON P D. METERS FITTED

WITH SHAFT ENCODERS (BASED ON 1,000 000

PULSE COUNTS PER RUN)

18
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19
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nI I LJ

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CASE 3. BULGES OUT OF PHASE

FIG 9 PRINCIPLE OF METHOD FOR MEASURING

INTRA- ROTATIONAL LINEARITY OF PD. METERS

23
(AND SO ON, TO RUN 500)

FIG 10 MODE OF ANGULAR ROTATION OF

ROTARY PD. METER IN EACH OF

500 CONSECUTIVE FLY ING- START

AND-FINISH TEST RUNS

•10 6 -6 -A -2 0 *Z A

DISCREPANCY (IN 2385 PULSES)

FIG 11 DISTRIBUTION OF NORMALIZED

DISCREPANCY BETWEEN METERS

AFTER ROTATING 338o

24
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

THE NAVY LIQUID FLOW CORRELATION PROGRAM

J. H. Tabler and C. G. Kullmann

Navy Type I Standards Laboratory

Naval Air Station, North Island

San Diego, California 92135

. A brief history of the Navy Flow Correlation Program

is described. The present flow correlation program is

outlined in detail. The object of the on-going program

is to assure compatibility of all primary liquid flow

stands in the Navy Metrology System to within ± 0.15%.

This is accomplished by first assuring that the primary

weigh/time dynamic liquid flow stands at the Navy Type I

Laboratory are in agreement with those at the National

Bureau of Standards. The method of maintaining compatibility of the four flow provers at the Type I Laboratory

is discussed. Application of process control charts is

described

.

Approximately 15 primary volumetric and dynamic

weighing flow stands are located at various Navy Metrology

Laboratories and Test Centers throughout the Continental

U. S. The technique used to maintain control is the use

of turbine type flowmeters at discrete frequencies and

the use of a common very stable calibrating fluid. The

excellent short term repeatability of these transfer

devices lends itself to this type of correlation program.

The Type I Laboratory calibrates the turbine meters before

and after a participating laboratory calibration is performed. The report issued to participating laboratories

is described.

1. Introduction

In the mid 1950 's there was a requirement to improve flow measurements

throughout the Navy. This was brought about by the relatively close flow

tolerance on jet engine fuel controls. In 1955, the Navy Bureau of Aeronautics purchased eleven of the Cox Instruments model 311 Primary Fuel Flow

Calibrators. One of these went to the National Bureau of Standards (NBS)

and the others were sent to Navy Overhaul and Repair Activities and Test

Centers located throughout the U. S. In 1960, under the sponsorship of the

Bureau of Aeronautics, the National Bureau of Standards and the Navy Metrology

Engineering Center (MEC) were engaged to conduct a program to determine the

degree of correlation between the calibrator at the NBS and those at the

Navy Activities. The transfer devices used were sharp-edged orifices, a positive displacement meter and turbine meters. Results of this correlation

lead to modifications and improvements in these stands and in standardization

25
of test fluids. The objection to the use of orifice plates was the narrow

range of flow with each orifice plate and the rather stringent data reduction requirements. Most of the fluid measurements being made by personnel

at Naval Aviation activities were with turbine type flowmeters or the

variable area float type meters. Because of the performance characteristics,

high resolution and the small size which makes shipping easy, turbine meters

have been used for all subsequent correlation programs. In 1964, the Navy

Western Standards Laboratory (WSL) was assigned the responsibility of

maintaining compatability of flow measurements in the Navy Metrology System

and assuring traceability to the National Bureau of Standards, A second

correlation program was initiated at this time. Knowledge and experience

gained from these programs and others that followed have resulted in corresponding improvements in the compatability of flow measurements in the

Navy system. Today an on-going correlation program is conducted, and compatability between approximately fifteen of the primary flow calibrators

in the Navy is routinely within + 0.15%.

2. Primary Liquid Flow Provers at the Western Standards Laboratory

The WSL has four primary liquid flow provers. Three of these stands

are Cox 311-AT flow stands containing calibrating fluids of varying viscosities in order to calibrate flowmeters over a large range of viscosities

without the need to change fluids. One stand contains MIL-C-7024B Type II,

another stand contains MIL-H-5606C and the third stand contains a blend of

the two fluids. By using three stands with different fluids, flow calibrations can be performed to 200 gal/min over a viscosity range of 1 to 20

centistokes. The Cox 311-AT flow stands have two complete flow circuits,

a low flow system (0.014 to 2.8 gal/min) and a high flow system (1.4 to

200 gal/min), Each system has its own pumps, valves, filters, heat exchangers, temperature controls, gages and weighing systems. The frequency

meter, electronic counter and timer are common to both systems and are

triggered by proximity type switches which are operated by the weigh beams.

The principle of operation is that of measuring the time interval to collect

a predetermined weight of liquid that has passed through the test flowmeter.

The weigh tanks of each system are supported on weigh beams. The low range

system has a one to one beam ratio and the high range system has a 50 to 1

beam ratio.

The fourth flow prover is a Flow Technology Ballistics Flow Calibrator.

This is also a self-contained instrument with two complete flow systems.

(A high flow section with a 4" ID flow tube and a low flow section with a

1" ID flow tube.) This stainless steel volumetric flow prover is designed

to calibrate turbine flowmeters up to 1000 gal/min using any liquid with

viscosities up to 500 centistokes. Pickoffs are positioned externally on

the flow tubes. A free moving piston actuated by compressed air pushes

fluid down the tubes. As the piston passes each pickoff point an electrical

pulse is generated. The time between pulses is displayed indicating the

time required to displace a known amount of fluid.

26
3. Correlation of the WSL Flow Stands

In order to perform the correlation of the three Cox 311 stands the

following procedure is used. The fluids in the low flow system of all three

stands are drained and replaced with MIL-C-7024B Type II. This fluid has

been accurately measured for density and viscosity beforehand by WSL. Two

turbine flowmeters are then calibrated on the low system of each stand at

80°F and at the same nominal flow rates. Opening and closing measurements

are made on the one stand which normally contains the Type II fluid. Closing

measurements are taken to verify that no shift has occurred in the turbine

meters and to determine the short time repeatability of the calibrator. The

stand which normally contains the Type II fluid is used as the reference

because it is the calibrator used in periodic correlations with the National

Bureau of Standards.

When the low flow systems are in correlation the original fluids are

replaced in each stand and the two turbine meters calibrated to their

maximum flow rate of five gallons per minute. This calibration provides

correlation information between the high and low flow systems of the flow

stands. "Crossovers" between the high and low system are normally within

+ 0.05%. Whenever a customer flowmeter is calibrated using both flow systems,

the compatibility between these systems is determined.

Two higher range flowmeters of 50 GPM capacity are calibrated on the

high range system up to their rated capacity. The pulses per gallon ("K''

factors) obtained for the flowmeters on both flow systems are used as

reference points for control charts maintained for the flow stands.

The ballistics prover is correlated to the 311 stand using these same

four turbine meters. The ballistics prover at the WSL does not presently

have temperature control and the flow rate cannot be set precisely, therefore,

the 311 stand is set up to match the frequency and temperature data obtained

on the ballistics prover.

The turbine meters used for the correlation of the WSL flow stands are

used only for this purpose. These correlations are done periodically and

process control charts are maintained. A complete discussion and test results

of the first in-house correlation at the WSL is given in [l]^. Figure 1 shows

a typical process control chart being maintained at the WSL. The chart includes all values over the range of frequencies investigated and "crossover"

information between the two systems. Since the turbine type meter has its

best performance over the upper 50% of the range, this data has more significance when comparing results. Use of process control charts will not

eliminate the requirement to periodically calibrate individual components

of each flow system. However, confidence in the total measurement system

is greatly improved through the use of these charts.

^Figures in brackets indicate the literature references at the end of this

paper

.

27
^

-0.05/.-

-0.10^

-o.os^ -

-o.;oX

-a.B/.

Wau INi- HOUSE. CQgRELA-riOKl PA.T/v

COMTROU CH/^R.T

PO\MTS FROV*l TO 2.SGAl/wiv;

O.OIa/, 0.000;^ aOAl% O.0£oX^2!^'2i5?5'iSf^ ^TT^btUUni"'

MIME OM.\6e*^-now Poiwre.

MAR SEP MAR SEP

Figure 1. MEASUREMENT PROCESS CONTROL CHART FOR WSL FLOW

STAND CORRELATIONS

4. Correlations with Other Navy Flow Stands

The objectives of the program are:

1. Determine if the primary flow stands in the Navy are performing

within specifications.

2. Determine from the test data such factors as repeatability,

correlation agreement, and determine any corrective adjustment or repairs required.

3. Appraise the knowledge and ability of the operators of

participating laboratories in the field of fluid mechanics,

so training can be provided if required.

At present WSL is using three turbine meters of different ranges for

correlations with other Navy activities. Closing measurements are made at

WSL after the meters are returned from each activity. These closing

measurements are used as opening data for the next activity. See Figure 2.

This "spoked-wheel" scheme has some major advantages for correlation programs

of this type. A shift in the meter coefficient can be detected immediately

and the meter replaced before another customer is sent the correlation. Each

loop is a complete correlation in itself and does not directly depend on all

other participants for statistical data. These meters were selected to

measure flow stand performance over a large portion of each flow system. One

of the meters was selected to be calibrated at the same flow rate on both the

high and low flow systems of the 311 flow stands and provide a measure of the

compatibility between the two systems.

28
ACT = NAVAL CALIBRATION

ACTIVITY

Figure 2 . SPOKED WHEEL CORRELATION SCHEME

The fluid specified for the correlation is MIL-C-7024B Type II at 80°

+ 1°F, In addition, the nominal pulses per second and the weight of the

fluid to collect in the flow stand weigh tank is specified for each flow

rate. A minimum of three runs is made at each setting to determine repeatability.

After closing measurements are made at the WSL, the activity is informed

by telephone of the results. A formal report is then provided comparing the

customer and WSL results. This report includes a graph (see figure 3) which

is divided into two sections. The upper portion of the graph shows possible

systematic errors as a function of flow rate, flowmeter and weigh system.

The lower portion summarizes precision or repeatability. Each plotted point

on the lower portion of the graph is the percent spread of three repeated

observations. This graph suggested to WSL by the Fluid Meters Section of

NBS contains on a single page the following:

a. Flow range investigated

b. Overlap between reference meters

c. Overlap between large and small weigh systems

d. Systematic error as a function of flow rates

e. Repeatability as a function of flow rates and reference meter.

29
i

I002

H»ooo d

^

o.io;i

ao5/o

Ik. aoo/.

1 I

1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

STEM

a

C © ® —

B E B — -. « o

i

B ^

® A.M a-£ UOVW Flo\

©

1

e

1

a

1 1 1 p 1 1 1 1 1 1

iSl A

1 1

H

B

1 1 1 1

1

200 400 GOO IPOO epOO 4poo GPOO ICpOO aqpOO AOpOO loopoo

Figure 3. EXAMPLE OF WSL CORRELATION WITH ANOTHER ACTIVITY,

The formal report gives suggestions for corrective actions if any are

required. Some typical causes of systematic error that have been reported

are

:

a. Errors in measurement of fluid density

b. Beam ratio incorrect

c. Errors in temperature measurements. (Temperature at the stand

can be as much as 5° different from temperature at the turbine

meter.

)

Some of the causes of poor repeatability have been:

a. Low sensitivity of weigh system

b. Leaking dump valve sealing gasket

c. Insufficient tare time

d. Empty or leaking vapor seal on weigh tanks

5. Conclusion

The present flow correlation has been very successful. It has been

possible to correlate the Navy primary flow stands to within + 0.15%. This

uncertainty is near that of the National Bureau of Standards whose possible

systematic uncertainty on dynamic weigh stands is given as + 0.10% with

allowance of + 0.03% for random error based on a standard error of + 0.01%

for a total estimated uncertainty of + 0.13% [2]. No attempt has been made

to discuss the technical aspects of flow measurement in this paper, but

references [3] through [5] discuss flow stand correlation techniques, uncertainties in flow measurements and standardization requirements.

30
Sending three meters of different ranges has caused no problems. The

closing and opening measurements at WSL normally repeat within the turbine

meters random uncertainty. Also, a shift in a meter can be detected immediately and the meter replaced before another customer is sent the correlation

package.

In the past, pairs of turbine meters were calibrated at WSL and then

circulated through a large number of customers before the closing measurements

were made at WSL. A report with a Youden plot (see figure 4) was then sent

to the participants. This method of plotting is given in [6]. Although this

reporting method had advantages, the "round-robin" routing of the standards

resulted in some distinct disadvantages.

a. There is a longer period of time before the customer receives

a report on whether or not his flow stand is within tolerance.

b. There can be long holdups in the cycle if a customer is not

ready for the correlation package.

c. If a turbine meter should shift, WSL might not find out until

other participants have used it for measurements.

We have found the present correlation method successful and in addition

to such factors as repeatability and correlation with WSL, the program appraises

the knowledge and ability of the participating laboratory operators in the

field of fluid mechanics and the measurements of viscosity and density.

COX IMSTRUME^TO MODEL AM\SH

SE^^A^L MUM&EI^e 3Q32 AHP 9053

-03 -CL2 -OJ 0 +0J -»-0,2+03

Figure 4. YOUDEN PLOT, "K" FACTOR DIFFERENCES FROM NBS (%) , TEST

RANGE 5.0 to 50 GPM

31
References

Kullmann, C. G., "'In-House" correlation of the liquid flowmeter

calibrations at the Western Standards Laboratory, Engineering

Report 6-72 of 18 August 1972.

Filban, T. J. and Shafer, M. R. , Flow measurement standardization,

19th annual ISA conference and exhibit, 12-15 October 1964, New York.

Shafer, M. R. , Test stand correlation techniques, 23 Feburary 1956.

Shafer, M. R. and Ruegg, F. W, , Liquid flowmeter calibration

techniques, transactions of the ASMG for October 1958.

Bean, H. S., Comments on how good are good flow measurements, GAS,

November 1961.

Youden, W. J., The sample, the proceedure, and the laboratory.

Analytical Chemistry, December 1960,

32
National Bureau of Standards Special Publication 484, Proceedings of the

Sjmiposium on Flow in Open Channels and Closed Conduits held at NBS

,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

A LABORATORY STUDY OF TURBINE METER UNCERTAINTY

G. E. Mattingly, P. E. Pontius, H. H. Allien

and E. F. Moore

Mechanics Division, IBS

National Bureau of Standards

Department of Commerce

Washington, D.C. 20234

A series of laboratory tests are conducted to

assess, quantitatively, the uncertainties produced

in making fluid flow measurements using turbine

meters. The patterns of uncertainty are analyzed

statistically and remedial modifications are

described and evaluated.

Using a tandemly connected pair of conventional

turbine meters with recommended meter tubes and

straightening vane sections, calibration procedures

are performed with a weigh-time reference for the

liquid flow. The calibration data are analyzed

using a variety of techniques. The results

ind icate that turbine meter performance can, within

the realm of "normal" operating conditions, be

perturbed to exceed specified values. Modifications,

both to hardware and conventional testing procedures,

are described which enhance metering assurance.

Key Words: Accuracy; calibration; flow conditioning;

meter performance; perturbations; precision; turbine

meter; uncertainty.

1. Introduction

A concerted effort at the National Bureau of Standards (NBS) has

been recently directed toward the establishment of Measurement Assurance

Programs (MAPs) for flow. The objective of a flow MAP is to quantitatively characterize the flow measurement process. It is not to force

everyone into state-of-the-art measurements but rather to establish

techniques and procedures which will provide assurance, at all levels

of measurement, that the results are adequate for the intended

purpose [1,2,3,4]^.

Figures in brackets indicate the literature references at the end of

this paper.

33
Throughout the flow metering industry and practice, turbine meters

have earned a considerable reputation for their precision and reliability

The advantages and disadvantages of using turbine meters to measure

fluid flow are widely known and well documented [5, 6, 7 , 8, 9 ,10 , 11 , 12]

The advantages include:

(1) A characteristic performance which exhibits a small

range of variability over wide flow rate ranges. For

normal fluids, this characteristic can be reduced to

a single descriptor using dimensionless parameters.

(2) A pulsed output which is readily digitized. The

resolution is limited only by the least count.

(3) Rotor bearing designs which have evolved to enable

reliable performance over extended periods.

(4) Relative insensitivity to minor variations in

streamwise velocity profile, due to the fact that

the multi-bladed rotor and hub completely fill

the flow cross-sectional area.

(5) Very rapid response characteristics.

In light of these advantages, turbine meters are widely used in aircraft

applications, petroleum metering and other industrial areas where the

user's flow measurement needs are critical. Turbine meters are not

without their disadvantages, however. For example:

(1) The flowing fluid and any materials in it must not

impair the rotor bearing or the rotor and blade

geometry

.

(2) The meter is susceptible to the effects of swirl in

the inlet pipe flow and radical variations in the

streamwise velocity profile have been known to cause

variations in performance characteristics.

In view of these respective advantages and disadvantages, turbine meters

have been selected for the initial phases of the flow MAP program. The

present tests focus on turbine meter performance in water flows in

medium pipe sizes.

2. Experimental Study

The experiments were carried out in the large, water calibration

facilities of the Fluid Meters Section of the National Bureau of

Standards. Shown in Figure 1, these facilities consist of a 60,000 gal.

(%230m3) sump from which water is pumped by means of one or more submerged pumps. For the present tests, the water then flows through

34
filters and straight lengths of PVC pipe to the test sections where the

meters under test are located. The flow is either bypassed to the sump

or diverted into a collection tank positioned on a scale. The correspondence between the meter indication and the collected volume of water is

determined from a static weighing of the water delivered in a measured

time interval. More details on this system and procedure can be found

in [13].

In the present experiments, pairs of turbine meters are tested in

tandem. These turbine meters are commercially available, 4 in. ("^ 10cm)

internal diameter meters, each of which is equipped with a matched meter

tube. These meter tubes have both upstream and downstream sections.

The upstream sections are fitted with flow straighteners consisting of

tube bundles sized and located according to AGA/ASME recommendations [6-7].

This upstream meter tube section is bolted to its respective meter by

means of pinned flanges to assure repeatable, aligned joints. The

assembled meter and matched meter tube sections shall be referred to,

in what follows, as a matched unit.

The present tests were confined to measurements at two flow rates

set to achieve Reynolds numbers of 1.2 x 10^ and 6.0 x 10^ based on

pipe diameter and water temperature. At each flow rate, respectively,

given that a certain meter performance criterion (the ratio of meter

responses) is satisfied, repeated water collections are performed and

the so called "turbine meter constants" are determined in pulses per

gallon corrected to 20 °C. Initially, five such runs were made at each

flow rate. The run-to-run variation obtained indicated that three such

collections were adequate, and this change in the test procedure was

implemented halfway through the test program.

The meter performance criterion used in this test is based upon a particular ratio of the two turbine meter responses. When this ratio

closely approximates an expected value (formulated via averages from

many previous tests) , it is assumed that both meters are performing

"properly" and the test results can be considered credible. Should

the ratio criterion not be satisfied, it is assumed that something has

impaired the meters - such as an anomalous deposit on a turbine blade.

It is assumed that the probability of simultaneous and Identical occurrences in both meters affecting each meter's performance in the same

manner is negligibly small. Given repeatability of the ratio to within

a specified tolerance of the expected value, the test procedure is

continued. Otherwise, the meters are back flushed to remove anomalous

deposits or the flow is stopped, and the pipeline drained and opened so

that the meters can be inspected and cleaned by flushing with alcohol.

To date, it has been possible at NBS to return this ratio to an acceptable

value with one of these remedies,*

*In view of what follows, the actual value of the ratio may be a

characteristic of a given test facility.

35
The specific procedures used to test the meters are given in

Appendix 1. The data produced using this procedure will be referred

to as "conventional." With a conventional data base determined, study

consists of devising and evaluating a variety of schemes that perturb

the meter performance from the conventional values. Once the perturbation effects have been determined, remedial procedures are devised

and evaluated which are intended to re-establish conventional meter

performance in the presence of perturbations.

The perturbation schemes devised include: (1) vibration effects,

(2) radically changing the streamwise velocity profile entering the

stream meter tube, and (3) inserting various levels of "swirl" in the

inflow to the meter tube. It will be shown below that all of these

have some effect on conventional meter performance. Of the three types

of perturbations, swirl exerts the most significant perturbation effect.

The types of remedial procedures devised to re-establish conventional meter performance have included: (1) using a third turbine

meter, installed upstream of the meters under test, (2) a particular

"flow conditioner" installed upstream of the meters under test, and

(3) flow conditioner installed ahead of the upstream section of the

matched units.

To facilitate interpretation of the results of the present

experiment, a list of symbols, shown in Figure 2, has been prepared.

Using this list, particular piping configurations are efficiently

described in the graphs that follow. It should be noted that the meter

tube symbolized denotes both upstream and downstream sections. Thus,

the connection of the meter tube with flow straightener and turbine

meter symbols from Figure 2 denote the matched unit described above.

The flow straightener is of the tube bundle type containing

nineteen tubes having OD = 3/4 in. (1.9 cm), wall thickness 0.049 in.

(1.24 mm), and length 10 in. (25.4 cm). The tubes in this bundle are

arranged in the scallop pattern and the bundle is centered in the tube

by means of spacer lugs. The turbine meters symbolized by the squares

are those under test. The different meters are designated by the

numbers 1 and 2 placed in the proper square symbol.

The flow conditioner used in these tests is a radial flow filter.

As a flow conditioner it is unusual in that it contorts the flow through

four right angle turns. Three such turns produce a flow radially toward

the pipe centerline. This radial flow is then turned again ^o that it

flows axially out of the device. During the radially inward flow, the

fluid passes through a porous element. Although the details of the

conditioning mechanisms in this device are presently unknown, it will be

seen, in what follows, that they are effective.

The third turbine meter, denoted by the propeller s5niibol is of the

same line size as the meters under test. It has approximately the same

angular speed as the test meters.

36
The swirler consists of a semicircular shaped vane. This is

positioned in the pipeline so that its diameter is aligned with the

pipe centerline when the vane is parallel to the pipe. The angular

position of the vane is adjusted from outside the pipe and can impart

swirl to increase or decrease, respectively.

The "out of plane elbows" configuration was specially prepared for

this test. It consists of seven straight 4 in. ('v 10 cm) diameter pipe

lengths that are 6 diameters long and six elbows, and represent a "worst

case" elbow configuration.

The 2:1 contraction, based upon diameters, was used when the meters

were tested in the 8 in. ('^ 20 cm) diameter pipeline. This contraction

is produced by two reducers, one 8 in. ("^ 20 cm) to 6 in. ('^ 15 cm) and

one 6 in. ('^^ 15 cm) to 4 in. ("^ 10 cm) .

3. Results for the Low Flow Rate

3.1 Conventional Configuration

A control chart for the performance characteristics of meter 2 for

the low flow rate is presented in Figure 3. The piping configuration

is symbolically shown as the conventional one having a long, straight

pipe length upstream of the tested meter 2. In this figure the error

bars shown on either side of the round, darkened circles refer to the

± 3 standard deviation spread from the averages of one set of meter

constants. The center of the bars is the average of the set. On the

other side of the large, darkened circle is the repeat set performed

after the pump is turned off and on again. The large darkened circle

is the average of the two set averages. In what follows, the term

"repeatability" is used in several ways. By repeatability is meant the

percentage within which a subsequent determination of meter constant

reproduces a previously determined value. Accordingly, switch off-switch

on repeatability refers to two values which are averages of from three

to five individual runs that are obtained from performing the presently

described test procedure once. As noted from the test procedure, the

meter connections to the pipeline remain untouched between these repeated

tests thus excluding from causes of variation the alignment of the meter

in the pipeline. Analogously, day-to-day repeatability refers, here, to

the percentage within which the meter constant averages of all runs

performed for a single flow rate and meter position on one particular

day are reproduced on another day. Hence, Figure 3 exhibits the degree

of switch off-switch on repeatability via the centers of error brackets

for a single day as well as that from day-to-day via the differences

between the large darkened circles for any two days. The '3 a limits

indicated for all the points are obtained from taking the average and

standard deviation of the nine (9) ordinates given by the large, darkened

circles. These values are given along the ordinate scale with the

standard deviation given in parentheses under the average. It will be

with these "conventional" values that the perturbation and remedial

effects are compared.

37
Figure 4 presents the control chart for the ratio values corresponding to the results presented in Figure 3. This ratio is that of meter

2 divided by meter 1 and the graphical notation is identical to that

used in Figure 2.

3.2 Perturbation and Remedial Effects

To examine the effects of pipeline vibration on turbine meter

performance a controlled vibration source was attached directly to the

pipeline. This source imparted a radial oscillation to the upstream

section of the meter tube near the inlet. Vibration amplitude and

frequency were controlled independently with this source.

The results of the vibration effects are shown in Figure 5. Here,

on the left ordinate are the meter constant results plotted as a function

of vibration frequency for constant amplitude as shown. The conventional

mean results from Figure 3 are shown with 3 a brackets. The nominal

background levels of vibration experienced by the pipeline at these flow

rates is 0.010 to 0.020 in. (0.25 to .5 mm) peak-to-peak. On the right

ordinate are the corresponding ratio results. These results indicate

that, while this imposed vibration causes a decreasing trend in meter

constants at the higher frequencies, the variations are well contained

within the 3 a limits about the conventional mean. Similarly, the ratio

values obtained show neither a systematic variation with frequency nor

any excursion beyond the 3 a limits. It appears that this particular

type of vibration does not significantly influence the performance of

these turbine meters at low flow rates.

The range of perturbations imposed by various elements which alter

the flow into the meters is shown in the modified Youden Plots'^ presented

in Figure 6. Here, the meter constant for upstream meter 2 under test

is compared with downstream meter 1. All points for meter 2 include 3 a bars. Along the right side of the graph is the legend indicating the

corresponding piping configurations. It is found that an out of plane

elbow configuration produces a small decrease in meter constant.

Combinations of elements such as the flow conditioner described earlier

and the third turbine meter or the 2:1 contraction and the third meter

tend to increase the turbine constant slightly. Each of these perturbations, however, does not cause the turbine constant to deviate beyond

the 3 a limit about the conventional mean value. The effect of the third

meter placed upstream of the meters under test is shown to cause an

increase in meter constant of about 0.3 percent. The effect of swirl

induced by the vane is shown by the extreme values in Figure 6, When

the vane induced swirl is in the same direction as turbine rotor rotation

the overspinning effect causes the meter constant to increase 0.4 percent

This demonstrates that a considerable amount of swirl passes through the

straightening vane in the meter tube. Reversing the swirl direction is

shown to decrease the turbine constant about 0.3 percent. The distribution of vorticity across the flow section was not measured.

38
Figure 6 can be used to determine the corresponding data for the

ratio of meter responses. For any value plotted, dividing the ordinate

by the abscissa gives the desired ratio. Therefore, the percentage

changes specified above for the meter constants also pertain, nominally,

to the ratio values. Also evident from Figure 6 is that the downstream

meter unit is relatively unaffected by the perturbations imposed ahead

of the upstream meter unit. In fact, for all these perturbations, the

variation sustained by the downstream meter is about 0.1 percent. This

suggests that the upstream meter unit with its adjacent straightening

tubes forms a rather effective flow conditioner. From the range of

perturbations imposed on the flow into the test meters, the extreme

values, namely those due to vane induced swirl, will now be selected

as the basis for evaluating the various remedial schemes devised to

reduce the perturbation effects.

In Figure 7 are presented the results obtained when the flow

conditioner replaces the upstream section of the matched meter tube.

The results plotted refer to three piping configurations: (1) the

extreme perturbation effects denoted by the square symbols, (2) the

flow conditioner installed with no perturbing effects present, and

(3) the conditioner installed in the presence of perturbations. The

meter constants, as determined with the conditioner and no perturbations

are found to be lowered approximately 0.15 percent from the conventional

mean. This change exceeds the 3 a limits which correspond to about

0.1 percent. Although not presented here, the velocity profiles at the

conditioner exit were measured and found to be more uniform than the

"normal" turbulent profile found in these pipes. It is to this alteration in the velocity profile that the shift in meter constant is

attributed. In the presence of positive swirl, the flow conditioner is

found to dramatically lower the perturbed values which, it is recalled

,

exceeded the conventional mean by 0.4 percent. For the case of swirl

opposite to turbine rotor rotation, the flow conditioned results are

found to be highly scattered. Although the details are, as yet, unclear

these results suggest that the outlet flow from this conditioner which

directly enters the meters produces unstable performance.

Figure 8 presents results for the remedial effects obtained with

the flow conditioner directly upstream of the conventional piping

configuration. With flow conditioning, the averaged meter constants

with and without the extreme perturbation case of positive swirl are

within 0.1 percent of each other. The 3 a limits are nominally the

same with and without perturbations. In view of this set of results,

it is concluded that the flow conditioner installed upstream of the

conventional meter unit produces the most satisfactory arrangement of

the presently devised schemes.

4. Results for the High Flow Rate

The control chart for the No. 2 meter at the high flow rate is

shown in Figure 9 where the notation corresponds to that of Figure 3.

It is noted in Figure 9, that the switch off-switch on repeatability

can be as much as 0.06 percent. The day-to-day repeatability can be

39
as much as 0.12 percent. The ratio of meter responses at the high flow

are given in the control chart presented in Figure 10. It is evident

from this figure that before precise expected values can be chosen for

a flow rate an extensive amount of data must be taken.

The effects of vibration on meter performance at the high flow

rate are shown in Figure 11. Here again, as seen above in the results

for the low flow, no significant variation is produced by this type of

vibration.

The remedial effects produced by the flow conditioner bolted

upstream of the conventional piping configuration are shown in Figure

12. Only the extreme perturbation effects due to vane induced swirl

in the direction of turbine rotation are shown. This swirl increased

meter constants 0.4 percent above conventional values. Vane induced

swirl in the opposite direction produced reductions in meter constants

which plot off this graph.

The meter constants obtained with the flow conditioner in place,

without perturbing effects, indicate that the meter constants are

reduced slightly (0.06 percent) in comparison with conventional values.

In the presence of vane induced swirl, the flow conditioner was found

to eliminate the swirl effects.

5. Conclusions

On the basis of this tightly controlled series of tests at two flow

rates, day-to-day repeatability for turbine meter constants over a

10 month period ranges from ± 0.07 to ± 0.12 percent based upon averages

of at least six runs. The repeatability obtained during a single day's

test when the flow is switched off and then on again ranges from ± 0,03

percent at the low flow to ± 0.06 percent at the high flow rate. The

results for the ratio criterion for "proper" meter performance indicate

that a 3 a tolerance is ± 0.1 percent for the low flow and ± 0,18 percent

for the high flow. The charateristics of these meters thus enable the

investigation of the effects of flow phenomena beyond the "simple"

questions of precision and accuracy of weighing, counting, timing, and

temperature measurements which comprise overall accuracy estimates.

The effects of various levels of swirl, however, were found to

produce the most significant changes in meter constants. These ranged

from -0.3 percent to + 0.4 percent. For selected types of perturbations

on conventional meter performance, those due to vibration and limited

alterations to the streamwise velocity profiles entering the meters

produced no significant changes in the values for the meter constants

Of the various schemes devised to remedy the perturbing effects on

meter constants, the most successful was a radial flow conditioner. In

the absence of perturbations, this device produced a slight decrease in

meter constants which was expected in view of the velocity profile at

the exit of the conditioner. In the presence of the most severe perturbations, the flow conditioner was successful in restoring the meter

40
performance to within the 3 o limits of average conventional values.

Acknowledgment

Several persons in the Fluid Meters Section are to be acknowledged

for their continued patience, attention to detail, and suggestions during

this study. These are Messrs. K. R. Benson, J. D. Melvin, and V. Brame

who conducted the tests and assisted with the preparation of the results.

Mr. L. K. Irwin is acknowledged for his continued support and participation. Messrs. John B. Houser and John Heine are gratefully acknowledged

for their contributions in conducting tests and preparing results. The

secretarial assistance provided by P. Beall, and P. Gurewitz, and K.

Durant is greatly appreciated. Last, but certainly not least, it is

pertinent to acknowledge certain individuals from several of the

laboratories where NBS is conducting the MAP tests. These individuals

who, by agreement, shall remain anonymous have contributed significantly

to the design of the test procedure used in the present tests.

References

[I] Pontius, P. E. , Notes on the Fundamentals of Measurement and

Measurement as a Production Process, f^BSIR 74-545, Sept, 1974.

[2] Pontius, P. E. , Measurement Assurance Programs - A Case Study:

Length Measurements. Part I Long Gage Blocks (5 in. to 20 in.)

NBS Monograph 149, Nov. 1975.

[3] Pontius, P. E. and Cameron, J. M. , Realistic Uncertainties and

Mass Measurement Process: An Illustrated Review, NBS Monograph

103, Aug. 1967.

[4] Youden, W. J. Graphical Diagnosis of Interlaboratory Test Results,

Industrial Quality Control, Vol. XV, No. 11, May 1959.

[5] Shafer, M. R. , Baker, D. W. , and Benson, K. R. , The Calibration

and Performance of Turbine Type Flow Transducers Operating at

Pressure Levels Up to 1400 psig, NBS Report 9088, Apr. 1966.

[6] American Gas Association, Report No. S. Orifice Metering of

Natural Gas, Arlington, Va, 1972.

[7] Bean, H. S., ed. Fluid Meters - Their Theory and Application,

ASME, New York, 1971.

[8] Strohmeier, W. 0., Turbine Flowmeters, Past, Present, and Future,

Symposium Proceedings: Flow - Its Measurement and Control in

Science and Industry, Vol. I, Part II, Flow Measuring Devices,

ISA 1974, pp. 687-693.

[9] Thompson, R. E. and Grey, J., Turbine Flowmeter Performance Model,

Trans ASME, Dec. 1970, pp. 712-723.

[10] Spencer, E. A., private communication; National Engineering

Laboratory, East Kilbride, Glasgow, Great Britain, 1976,

[II] Bellinga, H. and Stronk, H. , The Practical Application of Flow

Straighteners With Turbine Flow Meters for Gas. British Fluids

Laboratory paper No. 242.

41
[12] Rubin, M. Miller, R. W. , and Fox, W. G., Driving Torques in a Theoretical

Model of a Turbine Meter, Trans ASME Journal of Basic Engineering, 1964,

Paper No. 64-WA/FM-2

[13] Shafer, M. R. and Ruegg, F., Liquld-Flowmeter Calibration Techniques,

Trans ASME, Oct. 1958, pp. 1369-1379.

Appendix 1 Test Procedure

1. The meters are bolted to their respective meter tubes with a new gasket

inserted in the upstream flange joint of the meter.

2. The meters, with meter tubes, are bolted in tandem with a specified meter

(No. 2) in the upstream position: the tandem pair of meters are then bolted

into the test section of the pipeline.

3. A "run-in" flow is produced in the pipeline. This flow rate is the higher

of the two test flow rates. It is continued for fifteen (15) minutes,

during which time the ratio of meter responses is monitored and compared

with the expected value.

4. The lower of the two test flow rates is produced in the pipeline. This

flow is valved according to a Reynolds number criterion, i.e., a specified

ratio of upstream turbine meter frequency in cyc/sec divided by the

(temperature dependent) kinematic viscosity of the water in centistokes.

The tolerance on the turbine meter frequency is nominally 2 cyc/sec.

5. The ratio of turbine meter frequencies, i.e., that of No. 2 divided by

that of No. 1 is monitored. When this is within 0.05 percent of the

expected value, the test may proceed. Should this ratio criterion not

be satisfied, the following sequence of remedial procedures is suggested.

The flow in the pipeline can be increased and decreased repeatedly. If

possible a reversed flow in the pipeline, is produced to dislodge particles

from the turbine blades. The meters are removed from the pipeline inspected

for anomalous deposits or adherents to the internal components, cleaned

via a flushing with alcohol and returned to the pipeline.

6. Repeated diversions of the flow into the collection system are performed

and the data is processed to produce turbine constants in pulses per gallon

corrected to 20°C. Five such "runs" are done.

7. The higher flow rate is produced according to a specified upstream turbine

f requency-to-centistoke ratio; the ratio of meter frequencies is monitored.

When satisfactory agreement with the expected value is obtained, repeated

determinations of meter constants are done.

8. After the high flow meter constants are determined the flow is stopped

and the pumps are turned off. After five (5) minutes, the pumps are

switched on and the tests at both flow rates are repeated as pef^ steps

4 through 7.

9. When this has been completed, the flow is again stopped, the pumps switched

off, the line drained, and the positions of the meters with their matched

meter tubes are reversed in the pipelines.

10. Steps 3 through 8 are repeated for this tandem configuration of the meters.

42
1

GO

CQ

1

I—

Q_

Q_

o

CD

GO

•a: o

oo

CO

ooI—

oo

oo

Q_

CD

CM

OO

Q_

SI

OO

a.

Q_

43
LIST OF SYMBOLS

PIPE

METER TUBE WITH FLOW STRAIGHTENER

TURBINE METER

FLOW CONDITIONER

OTHER TURBINE METER

SWIRLER (+)

SWIRLER H

OUT OF PLANE ELBOWS

2 : 1 CONTRACTION

Figure 2. List of Symbols.

44
METER CONSTANT CONTROL CHART

METER No. 2 UPSTPEAN

LOW FLOW-DIAMETRAL REYNOLDS NO = 1.2 X lO'

71.8 -

71.7 -

METER CONSTANT

(CTS/6AL)

71 ,6

71 .5445 (0.0241)

71.5

71 .4

XI* ,1 L I

PIPING: CONVENTIONAL ^ 90 DIAS (J IN PVC PIPE

/ 2 1

71.3

MAY MAY J UN SEP NOV Ptf JAN FEB FEE

3 20 7 28 17 21 f> 2 4

DATE

Figure 3. Control Chart for Meter No. 2. Low Flow.

45
1.007 RATIO CONTROL CHART METER NO 2 UPSTREAM

LOW FLOW-DIAMETRAL REYNOLDS NO U X W

1.006

1.00372

(0.00039)

1.002

1.001

PIPING: CONVENTIONAL

90 DIAS 4 in PVC PIPE <

1 1

MAY MAY JUN SEP NOV DEC JAN FEB FEB

3 20 7 28 17 21 6 2 4

DATE

Figure 4. Control Chart for Ratio of Meter Constants. Low Flow.

46
VIBRATION EFFECTS

METER NO 2 UPSTREAM

LOW FLOW-DIAMETRAL REYNOLDS NO = 1.2 X 10^

7L6

NO 2

METFP

CONSTANT

I' TS/r,AL)

71,5

81° o

D

CONVENTIONAL

MEAN

o

n

o

AMPLITUDES [In. (PK-PKr|

0,030

O 0-095

o

CONVENTIONAL

MEAN

3<r

1,005

RATIO

- 1,003

10 20 30 ^40 50 60

FREQUENCY (Hz)

Figure 5. Vibration Effects on Meter No. 2. Low Flow.
71.9

71.8-

71.7

NO 2 METER CONSTANT

(CTS/GALI

71.6

71.5

71.4

71.3

YOUDEN PLOT

METER NO 2 UPSTREAM

METER NOl DOWNSTREAM

LOW ELOW- DIAMETRAL REYNOLDS NO 1.2 X 10'

±3oCONVENTIONAL MEAN

J I I L

71.2 71.3

NO 1 METER CONSTANT (CTS/GALI

Figure 6. Modified Youden Plot for Perturbation Effects.

Low Flow.

48
71.9

71.8

71.7

NO 2

METER CONSTANT

(CTS/GAL)

71.6

71.5-

71.4-

f ±3o-

- CONVENTIONAL MEAN

71.3

YOUDEN PLOT - REDUCTION OF VARIATION METER NO 2 UPSTREAM

METER NO 1 DOWNSTREAM

LOW FLOW- DIAMETRAL REYNOLDS NO 1.2 X 10*

=\

71.2 71.3

V 2 1

= / 2

NO 1 METER CONSTANT ICTS/GAL)

Figure 7. Modified Youden Plot for Remedial Effects Consisting of

the Flow Conditioner Bolted Directly to the Meter. Low Flow.

49
YOUDEN aOT - REDUCTION OF VARIATION

METER NO 2 UPSTREAM

METER NO 1 DOWNSTREAM

LOW FLOW-DIAMETRAL REYNOLDS NO T2 X 10'

4^

2 1

1

±3c

CONVEf^rriONAL MEAN

1 A

7 1

71.3 ' 1 ' L_

71.2 71.3

NO 1 METER CONSTANT (CTS/GAL)

Figure 8. Modified Youden Plot for Remedial Effects Consisting of

the Flow Conditioner Bolted Upstream of the Meter Tube Assembly.

Low Flow.

50
7?.0 METER CONSTANT CONTROL CHART

METER NO 2 UPSTREAM

HIGH FLOW-DIAMETRAL REYNOLDS NO = 6 X lo'

71 .9

71.8

NO 2

METER

CONSTANT

(CTS/CAL)

71.7

7 1 .6803 (0.0301 )

71,6

71.5

71.

q

PIPING CONVENTIONAL ^ 90 DIAS ^ IN PVC PIPE

HAY MAY JUN SEP NOV DEC JAN FEE FEB

3 20 7 28 17 21 6 2 4

DATE

Figure 9. Control Chart for Meter No. 2. High Flow.

51
i.on;

RATIO CONTROL CHART

METER NO. 7 UPSTREAM

HIGH FLOW-DIAMETRAL REYNOLDS NO = 6 X lO'

1 .006

1 .005

1 .004

RATIO

PIPING: CONVENTIONAL \ 90 niAS 4 IN PVC PIPE

1

1.00318 i

(0.0005)

1 .003

I .002

-I I

1 .001

MAY MAY J UN SEP NOV DEC JAN FEB FEE

3 20 7 17 21 R 7 1

DATE'

Figure 10. Control Chart for Ratio of Meter Constants. High Flow,

52
VIBRATION EFFECTS

^ETER NO 2 UPSTRhAM

HIGH FLOW-niWTRAL REYTOLDS NO = 6 X 10^

AMPLITUDES IN (PK - PK)]

0.030

<0> 0.095

71.8

NO 2

METER

CONSTANI

(CTS/GAL)

71 .7

71.6

T

3cr

o o o

CONVENTIONAL

MEAN

3(r

±

8

ft

O

o

10 20 30 50 40

FREQUENCY (Hz)

Figure 11. Vibration Effects on Meter No. 2.

3cr

<0

CONV.

MEAN

1.004

3ff1.003

RATIO

o 1.002

>

60

High Flow.

53
YOUDEN PLOT-REDUCTION OF VARIATION

METER NO 7 UPSTREAM

METER NO 1 DOWNSTREAM

HIGH FLOW- DIAMETRAL REYNOLDS NO = fi X 10^

72.0

71.9

71.8 -

NO 2 METER CONSTANT

(CTS/GAL)

71.7

71.6803 -

(0.0301)

71.6

71.5

-\ 7 1

71 .4 71.5 71.6

2 1

NO 1 METER CONSTANT (CTS/GAL)

Figure 12. Modified Youden Plot for Remedial Effects

Consisting of the Flow Conditioner Bolted Upstream of

the Meter Tube Assembly. High Flow.

54
National Bureau of Standards Special Publication A84, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

EVOLUTION OF A MODERN PETROLEUM MEASUREMENT MANUAL

L. M. Davis

Oil Measurement Consultant

Peterborough, Ontario

Abstract

Progress from early applications of positive displacement meters to

modern measurement practices are reviewed with particular attention

to liquid petroleum metering. Comments on some advantages of metering

vs tank gaging are made. The realism of the new API Measurement Manual

is discussed and an outline of its contents provided.

1. Background

The petroleum industry has for years measured the liquids it handles

largely by meter; and it measures the liquids several times over; as

it leaves the oil fields, then into and out of pipelines, into the

refineries, then to the marketing terminals, and finally at the retail

level. Each day in North America alone about 100 million barrels are

measured

.

The advantages of meter measurement are considerable, even though metering

has by no means completely superseded tank gaging. The advantages are

spelled out briefly in Chapter 5 Section 1 of the projected American

Petroleum Institute Manual of Petroleum Measurement Standards (API

Measurement Manual) — of which more anon — as follows

:

The principal advantages of metering are:

a) by being a continuous process it can increase the availability

of tankage since no tank need be isolated for measurement alone;

b) it lends itself to the calculation, indication or display of

instantaneous flow rates, and of a volume up to the moment

required;

c) measured volumes can be printed either at the meter or at a distance from the meter, and the meter's coded identity can

be part of the print-out on a measurement ticket;

d) it is labor saving;

e) it can deliver a measured volume taken from several sources

at the same time into a single receiver; or from a single

source into several receivers;

f) buyers, sellers and fiscal authorities, and others, can witness

all the basic steps of dynamic measurement and so be sure that

their interests and those of the public are being safeguarded.

55
There are of course even more advantages to dynamic measurement compared

with static measurement than those above cited, but a short list is enough

to explain why metering is popular. It is also efficient and accurate;

and its requirements have been set forth in a number of API Standards,

the longest and oldest of which is API. 1101. About 30 years ago, the

American Society of Mechanical Engineers (ASME) and API published what

was then called ASME. API Code 1101, and since that time it has been

widely used by all measurement practitioners in the oil business. It

consists of some 100 pages and deals with what was then the only type

of meter in general use, the so-called positive displacement or P.D.

meter. It covers installation, meter provers and their calibration,

meter proving, meter performance, and the operation and maintenance of

metering systems, as well as calculations, tables of correction factors,

diagrams and definitions. It was revised in 1952 and again in 1960.

In this granddaddy of oil measurement documents now called API Standard

1101, a critical observer will detect not only the background but also

what might be called the backdrop. If a backdrop is usually a pretty

scene of trees, flowers and perhaps a fairy castle, installed to set

the mood for a well drilled ballet, and at the same time to conveniently

cover up the stage hands, electricians, and inevitable ropes and pulleys,

then 'backdrop' is the right word. For the picture on the backdrop has

a slightly archaic aspect, namely the time when water meters were the

principal application to large scale measurement, and the hidden ropes

and pulleys were the then warranted assumption that a meter could be

calibrated and adjusted mechanically to read out correctly for a long

period subject only to a tolerable uncertainty.

Archaic expressions which survive in API. 1101 are "accuracy curve" and

"positive displacement meter". They are analogous to the sylvan setting

of flowers and a castle on the backdrop. They are archaic because oil

is not measured on the assumption that a meter can be calibrated once

and for all, and will thenceforward retain its calibration. Nor is any

P.D. meter capable of effecting positive displacement. There is always

some slippage, and slippage is variable according to rate, viscosity,

temperature and so on.

What happened when the oil business started using meters, was that it

decided also to modify the water measurement procedures because a single meter could be used to measure a wide variety of petroleum liquids.

It was not possible to attain the accuracy requ;Lred in oil measurement by

pretending that a meter had an accuracy curve, determined by a set of

calibrations, and that this curve would hold good for a long time. It

was fully expected that its meter factor would change with every change

in liquid, and with rate. So instead of pretending to "calibrate" a meter, with the implied belief that it would hold its calibration, the

expression "to prove" a meter came into common usage. But words die

hard. Some people still talk of 'calibrating' a meter, just as we still

talk of 'shipping' oil by pipeline, with no ships for miles around.

About six years ago API's Committee on Petroleum Measurement (COPM) decided

it had to update and reorganize all its measurement Standards. It soon

56
became apparent that the only possible way to do this was to start all

Over again. It was not practical business to patch up the existing

Standards. There might then be more patches than whole cloth. So the

best way to go about it was to start writing a comprehensive oil measurement Manual divided into Chapters. Chapter 1 is called Vocabulary. In

compiling a vocabulary certain archaic expressions could be dropped, or

at least played down. For example a P.D. meter is now called simply a

displacement meter. There are now just two kinds of provers - tank

provers and pipe provers. "Volumetric prover" has been dropped because

all provers are volumetric , except gravimetric provers and these are

museum pieces; they have no place in the projected API Measurement Manual.

The expression "accuracy curve" survives but in an anaemic condition;

"meter factor curve" is a more suitable term, although of course there

has to be such a curve (as a function of rate) for each and every type of

liquid, or grade of crude oil.

2. A Sense of Realism

What must distinguish the API Measurement Manual is a sense of realism,

and not only in its vocabulary. Its underlying purposes and emphasis

is to get the right answer, not to standardize the hardware. Even to

get the same answer from the same data, however, does require that the

software be standardized, something which the several API Standards had

not really tackled before.

The sense of realism can perhaps be summarized as follows:

(i) A petroleum liquid meter is not and cannot be a sort of

solid gold water meter. Petroleum liquids differ widely in physical

properties, unlike water, and the rates are forever changing.

(ii) Petroleum is much more costly than water, so it is necessary

to re-confirm the industry's established practice of reproving a meter

and obtaining a new meter factor every time operating conditions change

significantly.

(iii) It is not only desirable but necessary to use the same meter,

but not the same meter factor, for each successive batch of petroleum

liquid to be measured.

(iv) It is necessary to have a fixed or portable prover available

as needed for fairly frequent meter proving.

(v) It is desirable to have a standard procedure (i.e., standard

software) for computing the four values that collectively make possible

good measurement. These are:

a) base volume of a prover;

b) meter factor;

c) measurement ticket;

5,7
and, d) changes in line inventory for pipeline measurement.

The absence, so far, of standard procedures for computing the above

values has led to tears, sweat and some loss of hair among measurement

practitioners. A sense of realism must admit that a x b x c does not

necessarily have the identical same value ascxaxborbxcxa,

unless rules for rounding and truncating are spelled out and uniformly

observed. A realist knows that thousands of dollars are spent each

day, in time and long distance calls, in the attempt to find out why

a X b X c does not always equal c x a x b. So Chapter 12 of the API

Manual will attempt to establish standard calculating procedures for

oil quantities. This is especially necessary today when different

persons calculate meter factors, and a computer calculates measurement

tickets, and sometimes no one told " the man who programmed the computer

exactly at what step to round or truncate numbers. The idea is to get

the same answer from the same data, regardless of who or what does the

computing

.

(vi) Sixth and last in the list of items which a realist must keep

in mind is records. Today measurement of oil affects not only the buyer

and seller and the temporary custodian of oil, but many levels of government. Therefore records of measurement should be open and understandable

to all concerned. If something goes wrong, all interested parties should

have access to the records. Happily this is pretty well established, and

no one yet has come up with a wonderful idea to build pipeline-customhouses at every point where a pipeline crosses the US-Canadian border,

for example. If something appears to get out of whack in terms of measurement, a systematic examination of all relevant meter station records,

particularly of the meter factor control charts, is usually sufficient

to find the causes of the trouble. But even here, a sense of realism

must govern so as to prevent looking for trouble that does not exist,

or not looking for trouble that does exist.

3. The API Measurement Manual

We have so far considered the background to existing API measurement

standards and the need for starting all over again, rather than patching

them up. It is hoped that the projected API Manual of Measurement

Standards will be a comprehensive loose-leaf book in which the main

topics are sorted out into Chapters*, and each Chapter of which will

have as many separately printed Sections as it may require. For example

as new or extended compressibility data becomes available, it can be

printed and fitted into its slot without upsetting or intruding into an

established publication. The Chapter subjects are as follows:

CHAPTER 1 - Vocabulary

2 - Tank calibration

3 - Tank gaging

4 - Proving systems

^Manual of Petroleum Measurement Standards, Chapter Introductions and Scopes,

available from American Petroleum Institute, 2101 L Street, NW, Washington,

D. C. 20032

58
CHAPTER 5 Metering

Metering systems

Temperature determination

Sampling

6

7

8

CHAPTER 9 Density determination

Sediment and water determination

Physical property data

Calculation of petroleum quantities

10

11

12

CHAPTER 13 Application

Natural gas

Metrication

Measurement

of statistical methods

14 fluids

15

16 by weighing

Some chapters are already completed; more are now being written by

working groups, which drafts will then be balloted through the committees.

Some should be available this year.

One of the difficulties of the period between the virtual completion of

the new book and the phasing out of the old standards is that there has to

be a sort of minimum critical mass of new publications before they become

useful. To have the metering chapter before the proving chapter, or the

two of them without the calculation chapter, is not very practical.

Chapter 5 deals with metering and covers displacement meters, turbine meters

and an assortment of accessory equipment. Chapter 6 deals with metering

systems; each section of Chapter 6 being devoted to a specialized assembly

for measurement, such as LACT, aviation hydrant fueling, pipeline metering

systems and so on.

3 Primary units will be in SI, that is of course cubic meters, m , for

volumes, Kelvin, K, for temperature, Pascals, Pa, for pressure; with

customary units (bbls., °F, psi) given parenthetically.

The API Manual consists of performance standards, not hardware standards.

The whole idea is to get the right answer, and the publications tell you

how to go about it. For example, central proving of meters, not being a realistic proposition, is not covered. But a central prover-calibrating

facility might well be a very practical proposition. Water drawing a prover in the field is a tricky business at best, and in the Arctic or

on an offshore platform it becomes virtually impossible. So a portable

central-proving facility consisting of a small prover and a transfer

meter is a likely looking candidate for minimizing current problems . In the natural hierarchy of accuracies in oil measurement , provers are

at the top, meter factors are next, and measurement tickets follow.

If the uncertainty at the top of the hierarchy can be squeezed down or

made more consistent, this advantage will have beneficial effects further

down the pyramid.

Of course, NBS calibrated cans and small prover tanks will still be

necessary to the oil industry, and in this regard a passing tribute

4. Hardware

59
to NBS publication 105-3* should be made. It is a hardware document

on field standards and industry has been consulted and a-^ked for comments

which is widely appreciated.

5. Software

In addition to the software aspects of calculation referred to previously

under remarks about Chapter 12, the whole API Measurement Manual has

software aspects because it is primarily a lesson book in oil measurement

API's Committee on Petroleum Measurement has a subcommittee on Personnel

Training which promotes a school held in Bay town, Texas, twice per year;

and also the yearly short course held in Norman, Oklahoma.

API is also financing the important work being carried out at present

by NBS on the thermal coefficients of expansion of crude oils and

petroleum products.

And last but not least, API continues to encourage all measurement

education, in-house training programs and the free flow of ideas such

as is exemplified by its NBS Symposium.

''^NBS Handbook 105-3, Specifications and Tolerance for Metal Volumetric

Field Standards, May 1971 (under revision)

Footnote:

Some of you are old enough to remember Fred Allen's weekly radio program,

in which he always managed to sum up in a bizarre way what he was trying

to get across with what he called "a poem". Taking this cue I too have

written a poem called "The Meter Man", which some of you will realize

I lifted from a work of John Keats. It goes as follows:

0 what can ail thee meter-man,

Alone and palely loitering?

The , rate and temperature have changed.

And no birds sing.

0 what can ail thee, meter-man,

So haggard and so woe-begone?

The floating roof tank's full - And the tender not yet done.

1 met a meter in the ads.

Full beautiful - a faery's child.

It measured everything in sight - It's factors never wild.

It took me to its elfin grot

Showed me its bearings, flanges, seal.

If it was truly all that good —

It weren't for__real!

So this is why I sojourn here

Alone and palely loitering.

The pipeline's full of scraper dirt —

The gol-darn thing!

60
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS

,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

BUILDING BLOCKS TOWARDS FLOWMETER RELIABILITY

Dezsoe Halmi

Senior Principal Engineer

B I F

A Unit of General Signal Corporation

West Warwick, R. I. 02893

The need to analyze the question of flow measurement reliability as a special subject is pointed

out. Attempt is made to give a definition to flow

measurement reliability, review its types, and

uncover the means by which it can be attained, in

terms of "building blocks" which may be found in

the nature of a meter, or in the method of its flow

metering performance investigation.

INTRODUCTION

The first announcement and call for papers for this flow measurement symposium begins its statement "about the symposium"

thus

:

"Reliable and accurate flow measurements are becoming

increasingly important..."

While accuracy is a well-known and extensively analyzed subject

in flow meter literature, author was unable to find any direct

discussion of the subject of flow measurement reliability,

namely, what it is, and how it can be attained.

Due to this lack of information and the importance of the subject,

it is attempted through this paper to place the subject of flow

measurement reliability in the focus of general interest it requires and deserves, by endeavoring to define what is meant by it,

explore its types, and discuss some of the more important building

blocks of which it can be constructed.

DEFINITION OF RELIABILITY

In the lack of anything more directly related to flow metering,

author takes the definition of reliability from the American

College Dictionary:

"The person who or that which is reliable can be relied

upon; from such a one, satisfactory performance may be

expected with complete confidence (it may also have the

suggestion of honesty)..."

61
To see how to apply this definition of reliability for flow

measurement, it helps to realize, that, basically a flow measurement consists of the following elements:

•The flowing fluid and its properties; or fluid properties

like: density, viscosity, temperature, pressure, compressibility, contamination, conductivity, etc.

•The physical phenomenon of fluid flow; or flow properties

like: velocity, Reynolds number, flow pattern (upstream

and downstream piping effect on metering), flow pulsation,

fluctuation, etc.

•The primary part or function of the meter which generates

the flow signal as fluid passes through it by some signal - generating flow mechanism .

•Secondary part or function of the meter that converts the

flow signal into flow indication .

• Accuracy statement that pertains to the flow indication and

Ts provided by the originator of the meter like: manufacturers, codes, standards, etc.

Since this is the case, the subject of reliability shall deal

with forming answers to two basic questions:

•Why should one believe the flow indication to the meter?

•Why should one believe the accuracy statement to the originator of the meter?

Now, since perfect honesty is assumed, the answers to above

questions shall have the task to gage the difference between

the way the series of physical phenomena-- that connects fluid

flow to its i ndi ca ti on--actual ly happen, to the level of their

(possible) understanding from the part of the originator of the

meter. Thus the level of reliability can be defined by qualita^

tive equations, like:

Full Actual

Reliability Happenings

Zero

Rel i abi 1 i ty

Actual

Happenings

Full Level

of Understanding

Ze ro Level

of Understanding

Zero

Lack of

Unders tandi ng

Full

Lack of

Understanding

Since different types of meters may earn different placements

between the full 'and zero reliability categories depending on

how and where they are applied--as the next step toward the

goal; the possible types of measurement reliabilities shall be

expl ored.

62
TYPES OF MEASUREMENT RELIABILITIES

Since in the final analysis the metering performance of any

meter can be only, thus must be verified by flow calibrations

(which may be direct or indirect as becomes evident through the

paper) where the "actual" flow is determined by weight or volume

and time measurement; and since meters are used to measure flow

for other fluids and at different fluid and flow properties than

those encountered at their calibrations; different flow measurements may have different levels of reliabilities typical to the

application; consequently, it becomes necessary to recognize

different types of measurement reliabilities like:

• Reliability of Flow Measurement in the Calibrating Facility

Here the question is how the accuracy can be verified attributed to the flow determination at calibration.

' In Place Flow Measurement Reliability

This case is encountered when a meter is used for flow measurement in the same location, and without a change in

installation, where its flow indication was derived by

calibration (by any means).

Physically Transferred Reliability

This case is encountered when a meter is used at a different

locality, or in different installation at the same locality

than at which its flow indication was derived by calibration

(by any means).

' Meteringly Direct Reliability

This case is encountered when a meter is used to measure

flow for the same fluid and flow properties as applied at

its flow calibration.

• Meteringly Transferred Reliability

This case is encountered when the flow indication is assigned

to a meter--which itself was not flow ca 1 i brated--based on

metering sameness (par. 2.5) as ascertained by "bench inspection" (par 2.3); and the meter is used to measure flow for

flow and fluid properties the same at which those "meteringly

same" devices have been flow calibrated whose flow indication

is transferred to the device in question.

' Meteringly Interpolated Reliability

This case is encountered when f 1 ov/ measurement is performed

so that the device parameters, and/or fluid, and/or flow

properties of the application have not been directly encountered by flow calibration of meteringly similar devices

63
(par. 2.6) but are bounded by them, thus their metering

performance is interpolated to the condition of application

once "metering similitude" of the device in question has

been ascertained by "bench inspection."

Meteringly Extrapolated Reliability

This case is encountered when flow measurement is performed

so that device parameters, and/or fluid, and/or flow properties of the application are beyond the parameters encountered

at the flow calibration of "meteringly same" and/or "similar"

devices whose metering performance is extrapolated to the

conditions of application, once "metering sameness" or "similitude" of the device in question has been ascertained by

"bench inspection."

Time Extrapolated Reliability

Here the question is: How does time affect the metering

performance of a device; where time is considered a possibility that allows: The device to age (aging of components); all surfaces (that of devices and pipes) exposed to

flow to corrode, erode, get coated causing change in diameters, surface finishes which in turn shall cause flow pattern

change

.

Having explored the different types of possible reliability categories, now we shall attempt to find building blocks that may

raise the reliability of flow measurement to the required level.

BUILDING BLOCKS TOWARDS FLOWMETER RELIABILITY

1 . FLOW CALIBRATION

Basically the flow indication of all meters is derived

either by flow calibration of the device itself or by

(metering) transfer, inter-, and/or extrapolation of the

metering performance of calibrated devices by the help of

"metering sameness" and "metering similitude."

While the accuracy of the calibration depends on the ability

of the calibrating facility to measure the actual flow during

calibration and the flow signal (or indication) generated by

the meter as flow passes through it; thus can be known (calculated) and controlled; to assure that this known accuracy

is truly present at all times during calibrations is a question to which the calibrating facility itself is unable to

give continuous answer due to obvious handicaps inherent in

the nature of the work. As a result, the paradoxical statement seems to be valid; that no matter how accurately can a facility flow calibrate devices, the reliability of any cali - hration--in its ultimate sense--can never be assured by the

facility itselTI 64
Flow calibration reliability can be achieved, however, in

the facility and continuously if the metering performance

of the device to be calibrated was already known before

the calibration within some tolerance band. In such a case the calibration may improve the accuracy of the meter,

while the facility and the device shall mutually support

each other's reliability to the extent of the tolerance

band warranted by the background knowledge about the metering performance of the device under calibration. Consequently, as Table 1 indicates, flow calibration helps to

achieve all types of reliabilities except its own.

DEVICE CHARACTERISTICS

2 . 1 Signal is Assigned to the Device by Flow Calibration

Description: A device has this characteristic if its

flow indication is not fixed in its design but can be

adjusted and is intended to be assigned to the device

by flow calibration.

Typical Examples: Magnetic and some ultrasonic flow

meters

.

Main effects on reliability: Devices with this characteristic ought to be called signal generators rather

than true meters due to limits, inherent in their

nature that may hinder them to achieve reliability

in flow measurement. Since background knowledge about

their flow indication cannot be "tied" to them, they

are unable to help to improve the reliability of the

flow calibration; since their set signal may get offset during operation or movement of the device, they

cannot assure direct and time extrapolated reliabilities with in-place or physically transferred applications; also their nature prohibits their use for any

other flow and fluid properties but the ones for which

they have been directly calibrated. (See Table 1)

2.2 Signal Belongs to Device

Description: Such devices belong in this category

whose flow signal is fixed in their "visible" design,

but the flow value indicated by the flow signal is not

known, thus must be assigned to each device by flow

cal i brati on

.

Typical Examples: Differential producers made without

background knowledge about their metering performance.

Main effect on reliability: This characteristic lifts

typical signal generators one step upward on the ladder

leading to reliable flow measurement. Since, by definition, devices possessing this characteristic should

not change their flow signal unless their signalgenerating shape was visibly changed by damage, corrosion, erosion or coating, they can attain direct and

65
time transferred reliabilities but are unable to help

flow calibration reliability and prohibit all types of

metering transfer, inter- and extrapolations. (Table 1)

2 . 3 Bench Predictability of Flow Indication

Description: This characteristic is possessed by a device if its flov^ indication can be assigned to it--

not by its direct flow cal i bra ti on--but based on bench

inspection of its parameters that generate the flow

signal and through a full knowledge about the metering

performance of the device. In its purest sense to

claim this characteristic would require a full, accurate and limitless knowledge and/or understanding of:

•The fluid properties encountered at the measurement.

•The properties of the fluid flow.

•The s i gnal -generati ng flow mechanism utilized by the

device and its interaction with the encountered

fluid and flow properties.

•The operation of the secondary mechanism by which

the flow signal is converted into flow indication

by the device.

•The extent to which the full and exact understanding

is approximated by the actual device operation, because the level of this approximation shall determine

the accuracy of the flow measurement.

Typical Examples: While no device can claim full possession of this characteristic, true Venturi tubes and

nozzles are examples that approximate several of its

aspects

.

Main Effects on Reliability: The most important single

characteristic that can raise a mere signal generator

to the level of a reliable flow meter is the bench predictability of its flow indication. This feature enables the device to check the validity of flow

calibration thus raising its reliability level, also--

and obvi ously--al 1 ows and helps to measure flow reliably under any type of application.

2 . 4 "Distance" Between the Fluid Flow and Its Indication

Description: Here the extent, complexity, sensitivity

to defects and environment and the i nspectabi 1 i ty of

the instrumentation is meant that composes the meter

and by which the device converts flow into flow indication. The more complex, sensitive and the less

inspectable is the instrumentation, the longer is

the "distance."

66
Typical Example: For shorter distance; A differential

producer that measures water flow and where flow is

calculated based on the differential as read on a nearby water manometer.

2. 5 Metering Sameness

Description: Devices are meteringly same if they have

the same metering performance when their physical

parameters generating the flow indication and "visible

on the bench" appear to be the same.

Typical Example: True Venturi tubes, orifice plates,

etc.

Effect on Reliability: As Table 1 shows, this characteristic--if avai 1 abl e--hel ps to attain all types of

reliabilities; if not available, prohibits meteringly

transferred, interpolated and extrapolated application

of a device.

2 . 6 Metering Similitude

Description: Devices are meteringly similar if they

have the same metering performance when their physical

parameters generating the flow indication and "visible

on the bench" differ from each other, but according to

a precisely defined, meteringly meaningful physical

law that assures the same metering performance.

Typical Example: True Venturi tubes.

Effect on Reliability: As Table 1 shows, this characteristic--if avai 1 abl e--hel ps to attain all types of

reliabilities except the time extrapolated ones; if not

available, prohibits the meteringly interpolated and

extrapolated use of a device.

An immensely important fact in connection with metering

sameness and similitude must be emphasized, namely:

To utilize simulated tests for establishing installation effects (flow pattern effects) is impossible for

types of devices that do not possess metering sameness

and similitude. In other words, there is no way to

transfer, inter- and extrapolate up and downstream

piping effects as found with a given unit to other

units of the same or different line sizes; leaving the

question of--how to handle installation effects for all

types of i ns tal 1 ati ons--wi thout a reliable answer.

2 . 7 Metering Work

Description: A device performs metering work if it is

intended to cause some change in the flow and/or fluid

67
properties of the entire body of the flow and utilizes the results of its own effect on the passing

fluid to produce flow indication (on flow signal).

Typical Example: The classical Venturi tube or rather

the Venturi metering principle seems to be the most

suitable example. To explain it in scientific terms

(Ref. 1) here would be a waste of space because it is

widespreadly known; but--for this very reason--it may be

useful to restate it in terms less scientific but

more pointedly enlightening the subject of the paper

to recognize some very important characteristics of

this metering method that has not been properly appreciated so far in flow metering literature. Thus

the author takes the liberty to speak as follows:

The true Venturi tube receives the fluid from the upstream pipe in its own inlet (pipe) section. This

inlet section leads the flow to the cross section

where the inlet tap "looks" at the flow and registers

its static pressure. From here its transition section

accelerates the flow feeding it to the throat section

designed so that the throat tap "sees" and registers

true static pressure. The recovery cone captures now

the high velocity throat jet, tries to slow it down

and expand it to the pipe wall. The accelerationdeceleration process consumes energy, and the greater

the acceleration, the larger is this well-known energy

loss. The little appreciated yet very important fact

about this lost energy is that it is consumed by the

device to do "metering work" that may become beneficial

for the user to improve measurement reliability.

Effect on Reliability: To illustrate the point, the

following chain of thoughts is presented:

Due to its essence, volume flow rate is composed of

area times velocity:

Qa = A x V; (1)

which when applied for a meter:

Qa = Actual Rate of Flow

A = Some area in the meter known from "bench

i nspecti on"

V = Velocity through area "A" estimated from the

flow signal generated by the meter

The mathematical image used to calculate flow rate

for Venturi tubes solves Eq. 1 (Ref. 1):

= Ca'^^^ (2)
where

C

= Coefficient of discharge = Actual Rate of Flow ^ Ideal Rate of Flow

a

AH

g

B

d

D

_ Actual Metering Phenomenon

Ideal Phenomenon as Described

by the Mathematical Image

= Throat Area, measured on bench

= Differential Pressure indicated by the device

as flow passes through it

= Acceleration due to gravity

^ d

D

= Throat diameter, measured on bench

= Inlet diameter, measured on bench

To simplify the discussion, it is assumed that the

actual physical phenomenon is fully described by its

mathematical image, consequently

Qa = a Vrf = a {M^

/1-6'

(3)

where

AH = P

Where

D

2 2

. P. = ^ . l£ = 2g 2g ZgB"* 29

(4)

= Inlet static pressure

= Throat static pressure

Average inlet velocity = inlet local velocities

(because perfectly blunt velocity profile is

ass umed

)

= Average throat velocity = throat local

velocities (perfectly blunt velocity profile

is assumed)

Eqs. 3 and 4 clearly show the drastaic growth in AH,

thus energy loss (which is about 10-20% of the differential) as the Beta ratio of a device is decreased

(at a given flow rate); what it fails to reveal is the

benefit in installed metering accuracy and reliability

the metering work earns for the user at the expense of

this energy loss.
To indicate this simply but with practical effectiveness, we further assume that all metering errors due to

installation effects are caused by "wrong" kinetic

energy content of the flow at the inlet tap cross section of the tube, consequently, the hydraulic shape of

the tube is supposed to normalize the flow before it

reaches the throat tap cross section, so that the throat

flow does not contribute to the error. Now if "Ep" signifies the amount of this "wrong" kinetic energy content

at the inlet tap in part of the ideal or correct one,

then:

\ 2q

!i.

2g 29

)

AHt "

"

Where

AHg = Actual differential

aHj = Ideal or "correct" differential

Eq = Inlet kinetic head error, part of ideal or

"correct" kinetic head

Since

Eq. (5) can be reduced to:

^ = 1 . (6)

(7)

QiND ~ Indicated rate of flow

Consequently the error in % of flow

Eq

70

sequentiy tne error in or tiow

-.(JT - EpB" . l^jx 100 (8)

1-6" /
Table 2 gives a feel of the mechanism Eq. (8) describes.

It also lists actual installation effects as found by

tests (Ref. 2) with classical Venturi tubes, to enable

judgement about the validity of the thoughts presented

in this paragraph. Fig. 1 illustrates some of the data

from Table 2.

Once the meaning of the term "meter i ng work" is fully

understood and the fact is realized that the "same flow"

almost never reaches a meter with "the same flow pattern"

at different installations; or possibly on the same installation from time to time, it becomes a question of

paramount importance from the point of view of measurement

rel iabi 1 i ty--how does the meter estimate "V" for Eq. (1);

namely, just by "looking" at the flow and accepting what

is there, like: Velocity at a point, along a line (diameter), along several lines, in a volume (magnetic meters);

or it performs metering work thus does something to the

full body of the flow and uses its own effect for flow

indication. But why should a difference be suspected in

measurement reliability between devices where one just

looks at the flow and accepts what is there, as opposed

to another that works on the flow, observes its own influence and utilizes this influence to develop flow

indication? The answer lies in the definition of the

ideal meter (from the point of view of installation

sensitivity) which, no doubt, would be the one that indicated the correct flow disregarding the patterns with

which it reached the meter. From this definition follows

that the reliability of devices should improve as in

their flow indication or flow signal the portion generated by the influence of the device on the flow grows,

while the part caused by the influence of the flow pattern on the signal generation diminishes. In view of

these thoughts the data presented in Fig. 1 and Table 2

should obtain new significance, because it shows much

desired benefits which devices doing metering work cannot but assure, while devices that just look at the flow

deny, namely;

•Very large pipe flow pattern defects in terms of

erroneous local velocities, line velocities and/or

total kinetic energy content can be cut down to

comparatively small flow indication errors just by

selecting suitable Beta ratios.

•Installation effects as obtained by simulated tests

can be summarized meaningfully so that the data can

be used to control installation effects at the time

of designing meter installations.

•The laying length of metering sections can be cut in

"tight" situations by selecting smaller Beta ratios.

71.
METHODS OF INVESTIGATION

Since the metering device is only one part of a flow measurement, the other, equally important part, being the knowledge

about its metering performance possessed by the originator;

and since this knowledge can be obtained by different research

methods, it is useful to look at the difference of reliability levels that may belong to the different methods of

the research.

3 . 1 Empirical Determination of the Flow Indication

Description: The flow indication is empirically derived for a device if:

A) The device itself was flow calibrated, or if

B) The flow indication of meteringly same or similar

devices--as found by flow ca 1 i brat i on-- i s transferred, inter-, or extrapolated for the device in

question.

Typical Example: For case A) any meter; for case B)

orifice plates.

3 . 2 Physical Determination of the Flow Indication

Description: If the flow indication is derived for a device based on its bench inspection and through the

full understanding of the fluid and flow properties

encountered at the measurement and of the way the device converts those into flow indication, then the

flow indication is called physically determined.

Typical Example: Have been achieved to a certain

extent for Venturi tubes and true nozzles.

3.3 Combined Determination of the Flow Indication

Description: This case is encountered when the flow

indication of a device was derived by both methods:

empirical and physical.

Typical Example: Venturi tubes and Venturi nozzles

to a certain extent.

Effect on Reliability: The smaller the difference

between the empirically and physically determined flow

indications, the greater is the reliability of the flow

measurement. It is important to recognize the benefit

of this two-pronged method because it can powerfully

raise the level of reliability of the flow measurement

if it arrives to the same flow indication from sources

of derivation located so far apart (flow calibration

vs full understanding of the physical phenomenon involved).
3 . 4 Statistical Determination of the Accuracy of the

Flow Indication

Description: Accepting the definition of accuracy as:

Accuracy = Actual Flow _ I- (g)

Flow Indication of the Device

a statistical estimate of this accuracy can be developed :

A) For flow calibrated devices; by combining the

.tolerances of the equipment used at calibration

with tolerances incurred by the way the calibration work was performed.

B) For devices not directly flow calibrated; by calculating the standard deviation of flow indications

of meteringly same and/or similar sevices which

were flow calibrated and whose mean flow indication was assigned to the device in question. Once

the standard deviation is known, the accuracy can

be estimated with the required confidence level.

Typical Example: For Case A) any device; for case B)

Venturi tubes, orifice plates. (Ref. 3)

3 . 5 Physical Determination of the Accuracy of Flow Indication

Description: If full knowledge of all physical sources

of measurement inaccuracies and their magnitudes applicable to a given flow measurement are available, flow

indication accuracy can be estimated based on the following definition:

Accuracy = Actual Flow (= Full Knowledge) (iq)

The extent to which the full

knowledge was decided to be

represented by the flow indication

Typical Example: A partial effort can be found in Ref. 4«

3 . 6 Combined Determination of the Accuracy of Flow Indication

Description: This case is encountered when the accuracy

of the flow indication of a device was determined by both:

statistical and physical methods.

Typical Example: Partial effort can be found in Ref. 4.

Effect on Reliability: The least recogn i zed- -yet in

author's opinion--the most important single requisite

73
to secure flow measurement accuracy reliably is the

combined, two-pronged method in the accuracy determination. One may say--just by common sense--that due to

the nature of the subject, purely statistically or

purely physically determined accuracies should never

be relied upon. The truth in this statement is well

substandi ated by ASME literature (Ref. 1 and 5) which

drastiaally changed--f rom 1959 to 1971--the accuracies

on the uncalibrated discharge coefficients of orifice

plates and ASME nozzles, and whose present accuracies

are still questioned, for the single reason that the

combined method has never been applied in their determination.

3 . 7 The Value, Behavior, Uncertainty Method

Description: Once a meter is built thus its "nature"

gets fixed, it becomes a task of paramount importance

to find out what this "nature" is; namely, how the

ratio of:

Actual Flow . Indicated Flow

changed as evidenced by flow tests, or expected to

change as determined by physical methods when the

meter gets exposed to different aspects of the total

task of flow measurement; namely, to different flow

and fluid properties or the metering principle is

used to design different line size devices. Since

the v c! 1 u

e

of above ratio may change in a certain,

characteri sti cal fashion as the device is exposed to

the total task, it is helpful to consider this characteristical change as the behavior of the device and

search it as such. Furthermore, since this search canno t

" see" beyond the uncerta i n ti es of the available

information, decision making must be controlled by these

uncertainties. For this reason author recommends the

use of the "value, behavior, uncertainty" method in

flow metering performance analysis. T h i s m e t h o d j^^^

to find the true "naturej^^of a meter, type of meters or

I? eTerrtfcf" me fR'

d

crs"'"'Fy^'~5 aTa n cTri'g"' TH e"" u n ce rt ajWty jd

uTTcTe r s tan ding o f"ThV p Hy sTc a 1 p h e n ome n o n use d for "fTow

measurement wi th the u n c e r t a i n ty. o f t h e d a t a t h a .1 _ir .l£s..

t^,TM'i?-ScXLbe that phenomenon. (Ref. 6)

Typical Example: See Par. 3.8.

Effect on reliability: See Par. 3.8.

3 . 8 Specifically Organized Tests

Description: Since the only and final proof to the

proper understanding of any physical phenomenon is

74
success in tests, but since in case of flow measurement

testing facilities may not be accurate enough to reveal

the true nature of all line size devices, under all

fluid and flow properties, it becomes a powerful tool in

the hands of the i nves t i gator--as well as his obligation--

to organize tests so that their accuracy is sufficient

to uncover the specific aspect of device nature under

investigation. (Ref. 6)

Typical Example for parts 3.7 and 3.8: An interesting

example showing the usefulness of the "value-behavioruncertainty" method as it points at the need and importance of specifically organized tests can be found

in Ref. 7. One of the purposes of this paper is to

substantiate through flow tests a slight slope of the

discharge coefficient (C) in the function of throat

Reynolds number (Behavior of C) as determined by physical methods. The most enlightening data which is part

of Fig. 1 in Ref. 7 is reproduced on Fig. 2 in this

paper. The continuous line of Fig. 2 shows the value

and "shape" (behavior) of the C as determined by the

physical method; the circles are single flow calibration

points; while the dashed line is supposed to be the best

fit of the calibration points. A quick review of this

data clearly shows that while the single point accuracies of the flow calibration were sufficient to support--

within a small tolerance--the correctness of the physical

understanding that led to the calculation of the value of

C^, but it proved to be entirely impotent to verify the

slight slope of C indicated by the physical determination

of C behavior . This is the typical case when by the

value-behavior-uncertainty method limits and possibilities

inherent in the available means to investigate must be

uncovered, and the need of specifically organized tests

recognized, with single point accuracies sufficient to

reveal a slight slope.

Effect on reliability (for Par. 3.7 and 3.8): A beautiful example on how flow measurement reliability can be

improved by the thoughts presented in Par. 3.7 and 3.8

can be composed by reviewing, under the light of these

thoughts, what is stated about transition region in Ref. 8

pertaining to ASME throat tap nozzles and what flow calibration data shows about the same subject in Ref. 8.

Quote from Par. 4.33 in Ref. 8:

"At throat Reynolds numbers below 800,000 the nozzle

boundary layer is laminar; at high throat Reynolds

numbers it is turbulent. In between these two regions

is a zone called the transition region. This transition region should be established during calibration

and should be avoided during test. Flow coefficients

for the nozzles described in this code have been found

to be influenced by transition region at throat Reynolds numbers from 2.0 million to 3.5 million."

75
Now reviewinq Fig. 1 in Ref. 7, one's first impression

would be that above statement about the occurrence of

transition region is substantiated by the flow calibrations, because they reveal the largest single point

scatter between 2 x 10^ to 3.5 10^ throat Reynolds

numbers. Unfortunately, the fact that prohibits the

drawing of such conclusion is somewhat hidden, namely,

that in the "transition region" calibration points taken

with colder water flows at the highest differential pressures (best single point accuracy) and with warmer water

flows at the lowest differentials (poorest single point

accuracy) overlap. Consequently, due to their inherent

i naccuracy--wh i 1 e good enough to verify the value of C -- they are unable to identify the effect of boundary layer

transition on C, i.e., its behavior. Obviously to identify boundary layer transition through C change specifically organized tests are required that assure single

calibration point accuracies with magnitudes a fraction

of the expected C effect of boundary layer transition.

The immense practical importance of what has been said

so far becomes evident if one imagines a coded job

(according to Ref. 8) where a properly built test section goes under flow calibration in a facility where

water at one single sump temperature is used. One may

wonder how boundary layer transition "established" in

this facility would compare to another as "established"

by a different facility that used colder and warmer water

to cover the same Reynolds number range, also what the

revealed single point scatter would have to do with the

sought for transition region.

CONCLUSION

Author hopes that the paper has achieved the following:

•To point out that flow measurement reliability is--in

fact--a new subject that has never been directly and

systematically analyzed in flow measurement literature,

yet should be done.

•To make a first effort toward developing a vocabulary

that helps to discuss this new subject.

•To define what reliability in flow measurement may or

should mean.

•To uncover some of the building blocks of which flow

measurement reliability may be constructed.

•To make it clear that accuracy and reliability are

utterly different characteristics of a meter which

difference becomes obvious when one thinks about the

difference in means that should be and/or can be

utilized to achieve accuracy as opposed to the ones

that assure reliability.

76
To throw light on the fact that different devices that

may yield equal accuracies in flow measurement may be

drastically different in providing it reliably.

To provide help to improve the efficiency and reliability of flow measurement research work by:

•Emphasizing the total task of flow measurement thus

enabling the researcher to measure up realistically

the required effort to cover the total taski or to

address the research effort to the desired or possible portion of the total task and keep conclusions

within the selected boundaries.

•Showing methods by which the accuracy and reliability of data analysis can be improved.

•Showing means by which meter reliability can be

improved

.

To help users to select the device suitable to their

reliability requirements.

To provide a frame within which the subject of reliability can be treated in:

•Scientific papers

•Flow meter standards

•Sales literature

To uncover the fact that certain tyoes of devices due

to their nature cannot provide reliable flow measurement for certain types of applications.

77
REFERENCES

1. ASME. Fluid Meters, Sixth Edition, 1971.

2. Pardee, W. S., "The Effect of Installation on the Coefficient of Venturi Meters," presented at the ASME Annual

Meeting in 1942.

3. Dowdell, R. B. and Che, You-lin, "A Statistical Approach

to the Prediction of Discharge Coefficients for Concentric

Orifice Plates," ASME Paper No. 69-WA/FM-6.

4. Halmi, 0., "Metering Performance Investigation and Substantiation of the Universal Venturi Tube (UVT)," ASME Papers

Nos. 73-WA/FM-3 and 73-WA/FM-4,

5. ASME, Fluid Meters, Fifth Edition, 1959,

6. Halmi, D. "'C Evaluation - Differences Between 'Mathematical' and 'Physical' Possibilities," ASME Paper

No. 74-WA/FM-2.

7. Cotton, Schofield and Carcich, "Experience with Throat Tap

Nozzles for Accurate Flow Measurement," Journal of Engineering for Power, Trans. ASME, Series A, Vol. 94, No. 2,

Apr. 1972, pp 133-141.

8. ASME Power Test Codes, Steam Turbines, PTC 6-1976.

78
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82
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS

,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

OPEN CHANNEL FLOW MONITORING OR METERING. THAT IS THE QUESTION

Kenneth W. Martig, Jr., P.E.

President, The Martig Bub-L-Air

and Senior Engineer Group Leader

in Hydraulic & Sanitary Systems with

Arvid Grant & Associates, Inc.

1600 East Fourth Avenue

Olympia, Washington 98506

For generations we have accurately MEASURED clean fluid

flow conveyed in closed conduits employing precise, reliable

flow meters. Clean fluids, historically, have had "Value".

This "Value" has motivated precise METERING to apportion, buy,

and sell said "Value". In contrast, most open channel conveyed

fluids have by nature been low "value", liability "Value" or

waste liquids containing debris. Therefore, there has been no

incentive to MEASURE open channel flow. We simply have ignored

or only applied loose MONITORING practices to these flows.

The advent of P.L. 92-500 in late 1972, has provided an instant incentive to MEASURE open channel flow more

accurately, using METERING practices instead of less accurate

MONITORING practices.

Instrumentation state-of-the-art does not lend itself to

producing reliable, precise, cost effective METERING equipment

to replace previously employed MONITORING equipment for use in

existing systems. Recognizing this point is a significant

step toward advancing open channel flow MEASURING technology

and reducing the costly mistakes continuously being made by

those who believe as gospel, the output from many new open

channel flow devices being marketed and employed today.

In summary, our instant incentive to accurately MEASURE

open channel flow has placed emphasis on equipment output

means, assuming that technology exists to produce it. Users

have not fully realized that measuring equipment presently

available for use in existing sewers cannot supply METERING

quality data, only MONITORING quality data. OPEN CHANNEL FLOW

MONITORING OR METERING, THAT IS THE QUESTION. The answer is to

provide society with a better understanding of what is

actually possible in terms of open channel wastewater flow

measurement so that "regulations" and required "measuring

equipment" supplied can truly work together to solve our water

pollution problems.

Key Words: Closed conduit flow; instrumentation; measuring

flow; metering flow; monitoring flow; open channel flow;

public law 92-500; sewerage flow, wastewater flow

83
1 Flow Measurement History

Open channel (gravity) conduits and closed (pressure) conduits have

been used for generations to convey fluids from one point to another. Most

clean fluids such as oil (black gold), petroleum products, milk, and water

have economic value and are conveyed in closed conduits to keep them from

evaporating or becoming contaminated. In order to apportion, buy, and sell

these valuable clean fluids, precise reliable flow rate and totalized flow

volume measurements are required. These measurement requirements have

motivated the evolutionary development of precise, reliable, compact,

easily installable, and relatively inexpensive closed conduit flow

measuring apparatus called flow meters for METERING these flows.

Although water is sometimes conveyed in open channel conduits, most

fluids conveyed in them have, by nature, been wastewaters or dirty fluids

with low or negative economic value. Therefore, there has been little or

no incentive for society to develop open channel fluid flow measuring

meters of the same caliber as the meters developed to measure flow in

closed conduits. In fact, society has historically demanded only

occasional loose checking, watching, or MONITORING of these flows.

The advent of Public Law 92-500 in the fall of 1972 has provided an instant incentive to measure open channel flow, particularly wastewater

flow in sewers while paying at least 75% of the cost through Federal Grant

monies. Pursuit of this instant incentive has created problems typical of

those experienced in the past when society takes too large a first step

towards fulfilling a defined objective. The failure of the Teton Dam in

Idaho is a recent example of how haste fosters waste and confusion.

2 Water Pollution Abatement Program Goals

During the late 1960's and early 1970's the impact of applying the

age old philosophy that "Dilution is the Solution to Water Pollution" to

wastewater discharges became quite apparent. Rivers, lakes, water

courses, and even underground aquifers that once could be used for domestic

water supplies, for water contact recreation activities and for supporting

natural fish and aquatic ecosystems had literally become open sewers of

liability value to society in and about heavily populated areas. To

reverse this process and recover these waters dictated that wastewater

discharges be MEASURED in terms of quantity and quality. Once the

knowledge of quantity and quality of wastewater discharges could be

determined, treatment processes could be properly and rationally

prescribed to eliminate the discharge of the wastes that had polluted

society's water resources. Obtaining this knowledge required that

equipment for measuring both flow and water quality parameters be employed

within tens of thousands of old, but still functional, existing open

channel sewer systems. These systems had been designed and installed

without provisions for accomodating measurements of any kind. However,

84
equipment for accomplishing measurements in existing systems had not yet

been developed prior to about 1970, because such equipment had not been in

demand.

The main goal of flow measuring is aimed toward prescribing adequate

treatment of wastewater before discharge. Several other interim goals are:

a) The evaluation of groundwater infiltration and storm water

inflow quantities into existing sanitary sewers,

b) The evaluation of domestic, commercial and industrial waste

discharges for the proper development of user charges and

industrial cost recovery systems, and

c) The evaluation of wastewater flow twenty-four hour peaking

curves

.

Since most pollution sources are identifiable open channel sewer

wastewater discharges from existng systems, flow measurement requirements

at the respective discharge points, have also been incorporated into the

regulations stemming from Public Law 92-500. These measurements satisfy

one requirement of the N.P.D.E.S. (National Pollution Discharge

Elimination System) discharge permit. These permits are required at all

identifiable wastewater discharge points.

Early results of open channel flow measurement efforts indicate that

our flow measuring program in older existing systems as imagined and

requested by society is costing more than it should and is not supplying

adequate data to cost effectively or environmentally evaluate water

pollution control objectives required by law.

The main obstacle, then, in pursuing the achievement of pollution

abatement goals appears to be the employment of equipment in existing

sewers (both public and private) that will adequately measure and record

the flow of wastewater within these open channel conduits and that will

properly extract wastewater samples from these conduits for quality

analysis

.

Measurement, Both Monitoring and Metering

It is obvious today that considerable confusion exists in the field

of flow measurement. Most of this confusion stems from the fact that the

terms MONITORING and METERING have become used interchangeably with the

word MEASURING. However, monitoring and metering represent two completely

different measuring processes and instrumentation abilities. A correct

understanding of these differences will substantially reduce the problems

that have been created out of the misuse of these terms while trying to

accomplish the goals and objectives of flow measurement in water pollution

abatement regulations.

85
The term MONITORING means the process of watching, checking, or

observing. Open channel flow MEASURING employs observed or empirically

derived formulae to convert measured depth of flow to flow rate or

totalized flow volume. Therefore, the association of the term MONITORING

with equipment used to measure and record flow through open channels is

natural, even though it is rarely experienced by most people.

The term METERING means the process of employing a meter. Most

meters employed in flow measurement instrumentation are devices that can

directly output flow rate or totalized flow volume while MEASURING either

the positive displacement of flow during flow through the meter or the

average velocity of flow during flow through a known constant area within

the meter. The only restrictions on the use of these devices is that the

flow must continuously fill meter constant area sections and the flow must

not contain particulate matter or debris that would (i) interfere with

positive displacement gears, buckets, etc., or (ii) interfere with

propellers, pitot tubes, etc.. Meters are commonly used to measure the

volume of domestic water and natural gas flowing into homes, businesses and

industries. Their use is also observed daily at service stations by people

fueling their automobiles. Therefore, the association of the term METERING

with equipment used to measure and record pressure flow of clean fluids

through closed conduits is natural and commonly experienced by most people.

Unlike hydraulic engineers and technicians, most people know very

little about flow measurement. They do not know or appreciate the

difference between pressure flow in closed conduits and gravity flow in

open channels. However, these are the people making decisions on a day-today basis. The decisions they make are based upon what they know or what

they have been directly associated with or exposed to. Therefore, the

common concept that society has, in general, regarding flow measurement is

METERING. They envision a precise, compact, reliable, mechanical device

that can be easily installed directly into the closed conduit pipeline at a relatively low cost. Because they lack knowledge of open channel flow and

have seldom been directly associated with or exposed to open channel flow

measurement, they automatically but erroneously extrapolate their image of

METERING into the field where MONITORING really applies. This erroneous

extrapolation has fostered problems and costly errors by those responsible

for dealing with regulation compliance whether they be the regulator or

discharger (the regulatee).

Open Channel Flow Measurement

The rapid evolution of flow measurement in existing open channel

conduits primarily began with the requirements associated with

infiltration and inflow quantity assessments in the spring of 1973. These

requirements specifically addressed sewers conveying wastewaters requiring

treatment

.

The use of Palmer Bowlus flume pipe inserts and 90° V-Notch weirs,

although used in prior years, was not popular in the 1970's for several

86
reasons: They were difficult to install; they had flow rangeability

problems and reliability problems that when coupled with normal flow

debris, induced signfiicant errors in their measurement output; and our

concern for safety of personnel conducting the measuring, coupled with the

high labor cost of larger crews suggested strongly that the measurements

should be made without entering the manhole. Therefore, initial spot flow

measurements were made using long sticks, rods, or plumb bobs on measuring

tapes without entering the manhole.

The first modern piece of equipment designed to continuously measure

flow in existing sewers was designed to be installed in a manhole, just

under the lid. It produced a continuous recording of flow depth versus

time and was supplied with a table for converting the recorded depths to

flow rates, developed using the Manning Formula. This flow monitor did not

fit the image society had of flow measuring, namely METERING, because it

did not automatically produce an integrated total flow volume. Realizing

this, the device was immediately modified to include an electronic

calculator that would convert the measured depths to flow rates (employing

the Manning Formula) so that the device would output a recording of flow

rate instead of depth and so it could output a totalized flow volume.

This triggered the start of a race by instrumentation manufacturers

to provide equipment designed primarily to provide the output format

requested by regulators (and therefore desired by regulatees) because all

parties envisioned sewer flow MEASURING to be METERING instead of

MONITORING. The race produced high cost equipment called open channel flow

meters that outputted data with signficant errors. These errors were the

direct result of society placing too much emphasis on the output format.

The most important MONITORING/METERING component is the primary sensor,

what it senses, its location in the channel, and its rangeability. The

next most important component is the internal mechanisms dictating the

output. Although the output readability must be accurate, the actual

output format is of least importance as long as it allows the best use of

the chosen emperical relationship for ultimately deriving actual flow. Too

many users concern themselves only with the output format (totalized

volume) without considering the absolute ability of all integral

components of the device to produce that format.

Manufacturers are now looking for ways to produce equipment that will

provide accurate flow output. This can be done, but not on a universal

basis. Too many variations are inherent in the hundreds of thousands of

miles of existing sewers that require flow measuring. Also, the variations

in wastewater quality and flow rangeability logically dictate different

degrees of accuracy of measurements and frequency of measurements when

compared to the goals and objectives of water pollution abatement programs.

Wastewater with critical pollutants such as heavy metals or cyanide must be

more accurately measured than cooling water from ice plants or domestic

sewage discharged from homes.

Also ignored in our haste to meet open channel flow measuring

87
requirements was the fact that two levels of flow measurement exist. The

first level (the level neglected in many cases) is the initial learning

level. It employes the use of relatively inexpensive temporary portable

equipment to determine for the first time what historical flow rates have

actually been. With this information, the sewer system can be properly

evaluated to optimize its use, both economically and environmentally. Then

and only then should larger amounts of money be expended to provide the

second level of measurement. The second level involves the installation of

permanent flow measuring stations to provide system control and to meet the

flow measurement requirement normally contained in most N.P.D.E.S.

discharge permits. Even at this level of measurement, flow MONITORING

equipment may prove adequate instead of the more costly, but more accurate,

METERING equipment.

Recommended Revisions To The Pollution Abatement Process

The purpose of the above four sections is to define the confusion and

lack of knowledge that exists in society today regarding flow measurement

and to show how costly mistakes have been fostered while hastily attempting

to meet required wastewater flow measuring goals without due regard to the

state-of-the-art and understanding in existence at the time.

Even today, February 24, 1977, regulators preparing N.P.D.E.S.

discharge permits, dischargers purchasing and installing equipment in

compliance with their permits, and manufacturers supplying the equipment

that will be used do not completely understand what can and what cannot

practically be accomplished regarding open channel. flow measurement. OPEN

CHANNEL FLOW MONITORING OR METERING, THAT IS THE QUESTION. The answer lies

within the ability of hydraulic and environmental engineers to work

together with congressmen, legislators, regulators, discharges, and

manufacturers to re-evaluate and upgrade the requirements within

individual N.P.D.E.S. discharge permits and to modify the process of

compliance with the permits consistent with existing technology.

In conclusion, the following suggested water pollution abatement

process is offered.

SUGGESTED WATER POLLUTION ABATEMENT PROCESS

Step 1 - The State Sets Preliminary Discharge Standards

The state evaluates the water resource where treated or

untreated wastewater is released by the discharger. This

evaluation sets preliminary standards for acceptable discharge

quantities (as related to quality) based upon the small amount

of initially avai lable data.

Step 2 - The Discharger Conducts Preliminary Measuring Study

The discharger conducts a preliminary study, employing

88
portable temporary flow MONITORING and wastewater sampling

equipment that can provide accurate data when properly

calibrated for each specific monitoring site employed.

Step 3 - The Discharger Evaluates and Optimizes The Use of His System

The discharger evaluates the effective use of his system by

identifying all significant components of the total discharge.

Cost effective evaluations considering wastewater

pretreatment, wastewater flow removal and process

modifications consistent with state preliminary water quality

standards would be conducted in this step to minimize the

impact of the discharge.

Step 4 - The Discharger Obtains Final Measuring Study

After the discharger optimizes the use of his sewerage,

consistent with economic prinicipals but compatible with state

defined environmental quality at the point of discharge, the

discharger again uses portable flow MONITORING and wastewater

sampling equipment to accurately define the impact of his

optimized discharge.

Step 5 - The State Sets Discharge Permit Requirements Using Step 4 Data

The state reviews the data supplied them at the end of Step 4

and uses it to prepare the discharger's initial N.P.D.E.S.

discharge permit. The resulting permit requirements can be

based upon actual discharge characteristics and thus will be

more rational and consistent with pollution abatement goals.

Step 6 - The Discharger Installs Permanent Measuring Equipment

At this point, the discharger can install pretreatment

processes if necessary, and permanent flow MEASURING

equipment (either METERING OR MONITORING) and wastewater

sampling equipment as necessary for compliance with his permit

requirements

.

Step 7 - The State Reviews Its Program Effectiveness

When all dischargers to a given water resource complete Step 6,

the state can evaluate the effectiveness of the implemented

water pollution abatement program and make permit changes as necessary so that the best interest of society can be served.

89

National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

INSTRUMENT ERRORS IN OPEN CHANNEL

FLOW MEASUREMENT SYSTEMS

D. M. Grant

Instrumentation Specialties Co.

P. 0. Box 5437

Lincoln, Nebraska 68505

Certain aspects of open channel flow measurement systems

are briefly discussed, including primary and secondary

measuring devices . The two functions of a secondary

measuring device (open channel flow meter) are discussed:

level measurement and level-to-flow rate conversion.

Some of the common methodologies used to accomplish

each of the two functions of a secondary device are

described, and the possible sources and magnitudes of

errors associated with each of these methodologies are

analyzed. Level measurement methods discussed are:

float, electrical, ultrasonic, bubbler, and dipping

probe. Level-to-flow rate conversion methods discussed are: mechanical cam, electronic analog function

generator, electronic memory device, and opto-electronic

function generator. In the interest of promoting a uniformity of specifications, a standard general format

for open channel flow meters is proposed.

Key Words: Flow measurement, water; flow meters;

instruments, flow measurement; level measurement;

level-to-flow rate conversion; liquid flow; open

channel flow measurement; secondary measuring device.

1. Introduction

The measurement of the quantity of flow in open channels, often in

conjunction with sampling to determine the quality of the flow, is essential to nearly all aspects of water pollution monitoring and control. In

order to fully characterize the nature of the liquid source, it is vital

to know both the quality and the quantity of flow of the source.

This paper will be mainly concerned with one aspect of the flow

measurement process, namely the secondary measuring devices, (commonly

referred to as open channel flow meters) that constitute the "instrumentation" portion of an open channel flow measurement system. The majority

of this paper will deal with the methodologies used in various types of

91
flow meters, and will analyze the possible sources and magnitudes of

errors associated with each of these methodologies, A standard format

for open channel flow meter accuracy specifications will also be proposed.

2. Open Channel Flow Measurement Systems

Open channel flow is defined to be flow in any channel in which

the liquid flows with a free surface. Certain closed channels, such as

sewers and tunnels when flowing partially full, and not under pressure,

are also classified as open channels.

Flow in an open channel is normally measured through the use of a

calibrated restriction inserted into the channel, which raises the level

of the liquid upstream from the restriction. The rate of flow through

or over the restriction is related to the liquid level upstream from the

restriction by a known function. The restricting structures used to

measure flow in open channels are known as primary measuring devices

(see Figure 1) and may be divided into two broad categories: weirs and

flumes. A weir is essentially a dam built across an open channel over

which the liquid flows, while a flume is a specially shaped open channel

flow section providing a restriction in the channel area. Both types of

primary devices result in an increased upstream liquid level which is

related to the flow rate of the channel by either test data (calibration

curves) or by an empirically derived formula. [1] 1

A secondary measuring device (commonly referred to as an open channel

flow meter) is used in conjunction with a primary measuring device to

measure the rate of liquid flow in an open channel. The purpose of a

secondary measuring device is two -fold: 1) to measure the liquid level

in the primary measuring device, and 2) to convert this liquid level into

an appropriate flow rate according to the known liquid level -flow rate

relationship of the primary measuring device. This flow rate may then

be directly displayed, integrated with time to obtain a totalized flow

volume, transmitted to a recording device to be recorded as flow rate or

totalized flow, and/or used to pace an automatic sampler in a flow

proportional mode.

3. Sources Of Errors In Open Channel Flow Measurement Systems

When evaluating the possible sources of errors in an open channel

flow measurement system, both of the integral subsystems of the total

system must be considered: the primary measuring device and the secondary

measuring device. The errors associated with a primary measuring device

may originate with any or all of several factors, including: uncertainties

associated with the basic level -flow rate relationship of the device;

Figures in brackets indicate the literature references at the end of this

paper. ^„
faulty installation and/or construction of the primary device; and

improper maintenance of the primary device. A fairly large body of

literature exists concerning the inaccuracies associated with the various

primary measuring devices, [2,3] and these inaccuracies will not be

further discussed here.

Unlike primary device errors, secondary measuring device errors

have not been widely treated in the literature. The errors associated

with secondary devices may be divided into two classifications: set-up

or "zero" errors and instrument errors. The latter, instrument errors,

originate with inaccuracies related to the two basic functions of a

secondary device: level measurement and level -to -flow rate conversion.

These errors will be discussed at length in the remainder of the paper.

The former, set-up or "zero" errors, originate from improper instrument

installation and/or failure to accurately adjust the flow meter's

indicated liquid level with the actual liquid level in the open channel,

and will be discussed briefly.

It is imperative that the flow meter be properly "zeroed" with the

zero reference level in the primary measuring device. If this is not

accurately done, a systematic level offset error will be introduced,

resulting from the fact that the liquid level indicated by the flow

meter will not correspond to the level actually existing in the primary

measuring device. [4] Due to the non-linear level -flow rate relationship

of most primary measuring devices, this will result in a flow rate error

which becomes increasingly more significant at increased liquid levels,

as shown in Figure 2.

A properly constructed flow meter can greatly aid in minimizing

set-up or "zero" errors. The flow meter should have some type of

accurate visual indication of the instrument liquid level, and the

resolution of this level should be of at least the same order of magnitude as the readibility of the level in the primary device. The flow

meter should also have a mechanical or electronic control which easily

allows the adjustment of the indicated liquid level.

4. Level Measurement MethodologiesDescription And Error Analysis

The first of the two functions which a secondary measuring device

must perform is that of the measurement of level in a primary measuring

device. Some of the more commonly used methods of measuring liquid

level are: float, electrical, ultrasonic, bubbler, and dipping probe.

Float -A float operated system is undoubtedly the oldest method of

level measurement in existence. Basically, it consists of a float which

moves in conjunction with the level of liquid in a primary device. The

float either may be located in a stilling well adjacent to the weir or

flume (float- in -well) , or may ride directly on the actual surface of the

93
flow (float in-flow). See Figure 3. The float is connected by a cable

or arm to a rotating member. As the liquid level in the primary device

changes, the float will rise or fall correspondingly and the cable or arm

will cause the rotating member to be angularly positioned proportional

to the level of the liquid in the primary device.

There are several possible sources of errors in a float operated

level measurement system. By far, the most significant error is that of

"float lag" or hysteresis. In any type of float system, the float is

required to perform a certain amount of mechanical work, arising from

the operation of related mechanical devices by the rotating member and/or

frictional losses in the rotating member. These mechanical losses result

in the float not correctly following ("lagging") a changing liquid level,

in a phenomenon basically analogous to backlash between two gears. The

error caused by these hysteresis losses varies directly with the force

required to move the flow meter rotating mechanism and inversely with

the square of the float diameter. This type of error can never be eliminated from a float system, but can be minimized by reducing the amount

of mechanical work required of the float and/or by making the float

diameter as large as possible. Any additional play or backlash in the

rotating mechanism (e.g., gear reduction systems) will add to errors of

this type.

A cable operated float system can also suffer from temperature and

humidity related errors. A change in ambient temperature will cause the

float cable to expand or contract, resulting in a corresponding shift in

the indicated level. For hygroscopic cable materials, a change in humidity

will have a similar effect. However, proper choice of cable material can

minimize these effects, and except for extremely long cable lengths, the

errors involved are usually so small as to be negligible.

Manufacturing tolerances in a cable pulley or related mechanical

systems can also lead to errors. If the cable pulley is not manufactured

to the correct diameter, the instrument will not track a changing level

correctly. That is, the level indication will be correct at the set

point, but as the actual liquid level in the primary device changes, the

indicated level will become increasingly less reflective of the actual

level. A similar error will also result from inaccuracies in associated

gear reduction or mechanical linkage systems.

Manufacturer's claims of accuracy for float operated level measuring

devices range from ±0.005 ft. to +0.020 ft. For a well constructed float

operated device, accuracies of at least +0.010 ft. should be attainable.

It should be recognized, though, that the accuracy of any float operated

device can be increased by increasing the float diameter; there is, however,

usually a maximum possible float diameter dictated by installation conditions .

Electrical -This type of level measurement system utilizes some sort

of change in an electrical circuit caused by a changing level in order to

94
indicate the liquid level'. In most designs, a probe or some similar sensor

is immersed in the flow stream. The probe functions as an element in an

electrical circuit, and its behavior in the circuit is a function of the

amount of its immersion in the stream. [2] Changes in any electrical

property can be used to sense liquid depth; capacitance appears to be the

property most often used. A capacitive type probe consists of two conductive plates separated by a non- conductive material. The liquid

surrounding the probe functions as a dielectric material, and the capacitance

of the system changes as the degree of immersion of the probe changes.

The capacitance of the probe, then, is a measure of the level in the

primary device.

The first source of error in this type of level measuring device lies

with the instrumentation used to measure the changing electrical property.

Modem electronic techniques should allow the property in question to be

measured with almost any desired degree of accuracy. Therefore, the

size of this type of error can be controlled by the cleverness and/or

design objectives of the circuit designer, and should be stated by the

manufacturer. The probe itself is a second possible source of error in

the system. Appreciable foam or floating oil and grease in the flow stream

may build up on the probe, and could cause serious errors in most designs.

This type of error can be minimized only by frequent maintenance. A

probe in the flow stream can also cause disturbances to the flow which

could affect the basic level -to -flow rate relationship of the primary

device, in addition to being an attractive depository for all types of

debris, which compounds this problem. Errors of this type can be overcome by placing the probe in a stilling well or by using a sensor which

is an integral portion of the primary device.

Also, changes in the electrical properties of the flow stream itself,

which are used in the electrical measurement, can introduce errors. For

example, the capacitive type probe uses the flowing liquid as a dielectric

medium. Numerous factors can cause the dielectric constant of the liquid

to change, including tgmperati^re , presence of impurities, etc. A temperature change from 15 to 25 C will cause the dielectric constant of

water to decrease approximately 51. [5] Thus, a non-uniform dielectric

medium can introduce a significant error into the level measurement system.

Manufacturer's claims of accuracy for electrical type level measuring

devices are all in the range of ±0.5 - 1.0% of full scale level. These

claims would appear to be warranted, provided the probe remains clean and

the electrical property of the flow stream being measured does not vary

with time.

Ultrasonic -This type of level measurement system utilizes the

acoustic instrumentation principles developed for use in SONAR equipment

during World War II. The liquid level is measured by determining the

time required for an acoustic pulse generated by an ultrasonic transmitter

to travel from the transmitter to the air -liquid interface (where it is

reflected) and return to an ultrasonic receiver. The transit time of the

pulse, then, is a measure of the liquid level. The transmitter and

95
receiver may be separate elements, or may be combined into a single transducer. As shown in Figure 4, the transmitter/receiver element may either

be mounted above the flow stream, utilizing air path measurement, or

mounted below the flow stream, utilizing liquid path measurement. [2]

A primary source of error in this type of level measurement system is

with the instrumentation used to generate and receive the ultrasonic pulses,

and to measure their transit time. There are difficult engineering problems

involved with wave propogation, proper detection of the desired portion

of the wave, signal-to-noise ratio problems, etc. The solutions to these

problems are within the bounds of current electronic practice, but all must

be carefully considered in instrument design. The accuracy of the system

will be a fianction of the degree to which the designer has overcome these

problems, and should be stated by the manufacturer. A second instrument

problem results from variation in the velocity of sound. In air, the

velocity of sound is mainly affected by variation in temperature: in liquids

the velocity varies with temperature, pressure, and other factors. [63

Thus, an instrument measuring level using ultrasonic techniques must compensate for changes in the velocity o f sound through the medium in which

the ultrasonic pulses are being sent.

Other errors can result from the use of an ultrasonic measuring device

in narrow channels, round pipes, or other applications with space restrictions. This could result in false echos, with a corresponding loss in

accuracy. The surface of the flow stream itself can also cause problems.

Foam on the surface will absorb the ultrasonic signal, and a choppy or

turbulent surface could diffuse the ultrasonic pulse to the point where

the return signal may not be properly detected. C7j

Manufacturer's claims of accuracy for ultrasonic type level measuring

devices range from ±0.001 ft. (for a 10 Ft. range) to +2.0% of full scale

level. Claims of ±1.0% of full scale are typical, and can probably be

expected of a well designed and properly installed unit of this type.

Bubbler -In this type of level measuring device, a tube is anchored

at a fixed depth in the primary device, and some type of gas is bubbled

out of the tube at a constant rate, as shoivn in Figure 5. The pressure

in the tube necessary to maintain the bubble rate corresponds to the hydraulic head of the liquid above the tube. Thus, the pressure in the tube

is proportional to the liquid level in the primary device. This pressure

can be measured either with a mechanical pressure sensor to provide a

mechanical output or with an electronic pressure transducer to provide an

electrical output, both types of outputs being proportional to the liquid

level.

The primary source of error in a bubbler type level measuring system

is usually connected with the device used to measure the backpressure in

the bubble tube. Mechanical methods, usually consisting of a diaphragm

or bellows type pressure element and a mechanical linkage, can suffer from

hysteresis and non-linearity errors. Electronic methods, using various

types of transducers, suffer from similar errors, with the added problems

96
of temperature related inaccuracies. All of these potential sources of

error can be minimized or held to tolerable levels with competent pressure

measuring system design. However, there will always be a certain degree

of inaccuracy present, and this should be clearly stated by the manufactu:

Another potential problem is related to the density of the liquid in

the flow stream. For any type of bubbler system, the bubble tube pressuris a function of the liquid density. Thus, if the density of the liquid

changes, the indicated level will also change. An instrument system could

be designed which would continuously measure the liquid density and make

corrections for a changing density. However, this correction is rarely

made, as the density of flow streams commonly encountered does not normal

vary an appreciable amount. This potential error source should be recogn

ized, though. LSI

If the exit end base of the bubble tube becomes appreciably reduced

due to build-up of contamiants from the flow, erroneous readings will resul

even though the instrument may appear to be functioning normally. This

problem can be overcome through periodic inspection and maintenance, and

proper selection of bubble tube material.

A final possibility of. error lies with the fact that the transmissio:'

of changes in the liquid level to be measured by the bubbler system may

be delayed due to the effect of the charging time of the air volume in

bubble tube under the liquid surface. This problem will occur if the

liquid level increases at a faster rate than the gas supply can charge

the bubble tube. Thus, the liquid will enter the tube through the outlet

and the pressure sensing element will follow the level increase with a

certain delay. C8] This error can be overcome by one of two stategies.

First, by increasing the rate of gas flow and decreasing the bubble tube

inside diameter the charge time can be decreased; however, this is wasteful of the gas supply. Alternatively, a level increasing faster than the

charge rate can be detected electronically, and the gas flow rate momenta'

increased until the bubble tube is fully charged. If one of these method.,

is not employed, the instrument will not respond as rapidly to rising

levels as it will to falling levels, with a resultant tracking error.

Manufacturer's claims of accuracy for bubble type level measuring

devices range from ±1.0% of full scale level to ±1/4 inch. Accuracies of

±1.0 - 2.0% of full scale can probably be typically expected from a well

designed bubbler level detection system.

Dipping Probe-Of the five methods of level measurement discussed,

the dipping probe type is probably the most recent development. In this

technique, a probe mounted on the end of a cable is lowered, using a motor

driven pulley, until it makes contact with the liquid surface, as shown i

Figure 6. When the tip of the probe touches the surface, a mircoampere

circuit through the liquid to a ground return is completed, and the probe

is raised to a position just above the liquid surface. After a short

period of time, the probe is again lowered and the cycle is repeated. Ir

1 97
a manner similar to a float operated device, the angular position of the

cable pulley when -the probe tip touches the liquid surface is a measure

of the liquid level.

The dipping probe level measurement technique shares cable and pulley

related error sources with float actuated systems. Changes in ambient

temperature and/or humidity may cause the cable length to change, resulting in an inaccurate level indication. [7] Manufacturing inaccuracies

(tolerance problems) in the diameter of the pulley around which the cable

is wound will also lead to level tracking errors.

Collision with surface debris or the action of wind may cause an

oscillation of the pendulum-like dipping probe, which would result in

inaccurate measurement until the swinging motion subsides. It is also

possible that the probe might become coated after extended periods of use,

which would result in a level zero offset. L21

Manufacturer's claims of accuracy for dipping probe type level

measuring devices range from ±0.005 ft. to ±1.01 of full scale level. It

appears as though accuracies of ±1.0% of full scale can probably be

expected from this type of device.

5. Level -To-Flow Rate Conversion Methodologies

Description And Error Analysis

The second of the two functions which a secondary measuring device

must perform is that of converting the liquid level measured in the primary

measuring device into a corresponding flow rate, according to the known

liquid level-flow rate relationship of the primary device. Some of the

more commonly used methods of converting level to flow rate are: mechanical

cam, electronic analog function generator, electronic memory device, and

opto-electronic function generator.

Mechanical Cam- The mechanical cam is probably the oldest method

of accomplishing the level -to -flow rate conversion. In this method, a

mechanical cam, whose profile follows the level -flow rate relationship

of the primary measuring device in question, is rotated by the level

measuring device to an angular position proportional to the liquid level

in the primary device. The radial distance from the cam centerline to the

cam follower centerline is then proportional to the flow rate, as shown

in Figure 7.

The major error problem in a mechanical cam level -to -flow rate conversion system is the degree of accuracy with which the cam can be manufactured to match the level -flow rate relationship of the primary device.

In any manufacturing process, there are limits to the precision with

which any part can be economically made in quantity. These accuracy

limits, of course, apply to the profile of a typical level-to-flow rate

conversion cam, compounded by two additional factors. First, the usually

non-linear, and often complex, level -flow rate relationship of most

98
primary measuring devices makes the manufacturing process difficult. And

second, because of space limitations in the flow meter package, the cam

must of necessity be relatively small, placing further limitations on

its accuracy. The manufacturer should state the precision with which his

cam matches the level -flow rate relationship of the primary device.

No published manufacturer's claims of accuracy for a mechanical cam

level-to-flow rate conversion system could be found. However, normal

accuracies of the level-to-flow rate conversion which can be expected

from this type of device are probably in the ±3-5% range.

Electronic Analog Function Generator -In this type of level -to -flow

rate conversion system, a solid state integrated circuit device is used

to convert an electrical analog of the level into a flow rate analog,

according to a presettable power formula. The heart of this type of system

is an electronic algebraic module which essentially performs a transfer

function defined by the following expression:

Q = KL"*

where: Q = flow rate

K = constant multiplier

L = liquid level

m = power

An electrical analog of the liquid level in the primary device, L, is

provided by the level measuring system as an input to the algebraic module.

The values of the multipling constant K and the power m are established

as reference analog signals by associated circuitry, and are usually selectable (within limits) by flow meter front panel controls. These values are

selected so as to match the level -flow rate relationship of the primary

device in question. The output of the module is then an analog signal

proportional to the flow rate for the level input, according to the

multiplier and Dower preset.

The major problem with electronic analog function generator type

level -to- flow rate conversion systems is their inability to accurately

reproduce the level -flow rate relationships of many of the commonly used

types of primary measuring devices. The majority of primary devices have

a level -flow rate relationship which can be expressed as a simple power

function, similar to the above expression, and thus theoretically may

be accurately accomodated by this type of conversion system. However,

instruments using this system usually have only two or three discrete

values of the power m available, for example 1.5 and 2.5. A number of

primary devices have~simple power formula type level -flow rate relationships

whose power is close to, but definitely different from 1.5 or, 2. 5. Thus,

for these devices the instrument can only approximate the level -flow rate

relationship of the device, with a resultant error in the conversion.

A fairly large minority of commonly used primary devices have level

flow rate relationships which do not fit a simple power function. Their

level-flow rate relationships are expressed by more complex equations or

by data only, and thus are not directly compatible with an analog function

generator tvne conversion system. These types of primary devices caii

99
be accomodated only by approximating their level -flow rate relationship

with a simple power function. This, of course, necessarily leads to

errors in the level-flow rate conversion.

Systems of this type also have difficult engineering problems regarding the application of the function generator module and the associated

electronics. First, there is a certain amount of inherent inaccuracy

in the accomplishment of the transfer function, an inaccuracy which may

be compounded by improper electronic system design. [9] There are also

problems associated with temperature sensitivity and long term stability

of the module and of the analog references which establish the values of

K and m. Proper design can minimize these errors, but they will always

Be present to a certain extent, and should be so stated by the manufacturer.

Again, no published manufacturer's claims of accuracy for an electronic

analog function generator type level-to-flow rate conversion system could

be found. However, for primary measuring devices having a simple power

formula type level -flow rate relationship with a power exactly matching

one found on the instrument, conversion accuracies of ±1-2% can probably

be attained. For other primary devices whose level -flow rate relationships

have to be approximated, conversion accuracies in the ±3-6% range would

seem reasonable.

Electronic Memory Device -In this type of level-to-flow rate conversion

system, a large scale integrated circuit memory device is programmed such

that for a given level input, the output of the device is the flow rate

corresponding to that level. The type of memory device most often used

is a programmable read only memory (PROM) . The PRCM is programmed by

the manufacturer with the level -flow rate relationship of the primary

measuring device in question. The level measuring device supplies to the

PROM a digital code representing the level in the primary device. The

PROM is essentially a look-up, table, in which each digital liquid level

value has a corresponding flow rate value. Thus, for a given digital

level input, the output of the PRCM is a digital code for the flow rate

corresponding to that level.

A PROM type level -to -flow rate conversion system does not have any

actual "error" associated with the memory device; that is, for a correctly

programmed PRCM, a given level input will always result in the pre-programmed

flow rate as an output. However, there are flow rate resolution problems.

The inputs to the PROM are normally based on equal increments of liquid

level, the outputs usually being the flow rate value corresponding to the

level at the midpoint of the level increment. Because of the exponential

nature of most primary measuring device level-flow rate relationships,

equal level increments will result in nonequal flow rate increments, the

flow rate increments for each level increment becoming increasingly larger

at higher level values. Thus, the majority of the flow rate range will

be contained within a minority of the level range, and vice versa. This

results in a decreased resolution of the flow rate for higher level values.

The PROM'S normally used in this type of instrumentation have 256

level -flow rate pairs (words), meaning that the level will be resolved

100
into 256 equal increments. Manufacturers commonly state for PROM's of

this size that ". . .the level -flow rate function is accurate to better

than 0.4V. What they actually mean is that the level resolution is better

than 0.41 (1 part in 256). At low levels, the flow rate resolution is

better than 0.4%; at high levels it is worse than 0.4%. Thus, the

resolution of the flow rate (which is, after all, the desired quantity)

varies over the level range and is much worse than the stated 0.41 at

higher level ranges. In general, of course, the larger the PROM, the

better the flow rate resolution.

Opto -Electronic Function Generator -In this final type of level -to

flow rate conversion system, a disk which has been optically coded with

the level -flow rate relationship of the primary measuring device in

question is used to accomplish the level -flow rate conversion. The coded

disk (see Figure 8) is rotated to a level proportional angular position

by the level measurement system. The disk is divided into a number of

angular increments, the angles representing equal increments of flow rate

(not equal increments of level) . Each angular flow rate increment on the

disk has its own numeric coded value corresponding to a flow rate value

for that particular angular increment. An. optical sensor electronically

reads the coded flow rate value corresponding to the measured level, and

transmits this value to further decoding and processing circuitry. Thus,

liquid level is converted into a corresponding coded flow rate value by

the interaction of the disk and the optical sensor.

The major problems of an opto -electronic function generator type

level-to-flow rate conversion system are associated with the degree of

accuracy with which the information can be encoded on the disk, and the

degree of accuracy with which the instrument can read the information

from the disk. The angular bands of flow rate information must be

optically encoded on the disk by the manufacturer. There is a certain

amount of inaccuracy inherent in the encoding process, which will be

reflected by inaccuracies in the level -flow rate relationship recorded on

the disk. There will also be errors resulting from the detection of

the coded bands by the optical sensor. Both of these errors will affect

the accuracy of the level -flow rate conversion.

The manufacturer of this type of device claims a level -to -flow rate

conversion precision of ±0.05% of full scale flow rate for a 0.2% flow

rate resolution device (1 part in 512) . These claims would appear to

be justified.

6. Flow Meter Accuracy Specifications

Various methods of accomplishing each of the two basic functions of

an open channel flow meter have been discussed separately, along with the

accuracies which might be expected from each. Any of the level measurement

methods may be combined with any of the level-to-flow rate conversion

methods to form a complete, functional flow meter. The temptation here

is to combine level measurement accuracies with level -to -flow rate conversion accuracies, in an attempt to obtain a comprehensive flow meter

101
accuracy. Such a combined accuracy would not, however, be truly reflective

of the situation, due to the non-linear relationship of liquid level and

flow rate for most primary measuring devices. Errors in the liquid level

are usually not reflected by equal errors in the flow rate, and a linear

error in the level-to-flow conversion process often leads to non-linear

(with respect to level) flow rate errors. Thus, it is usually better

practice not to combine the two flow meter functions, but rather to specify

the accuracy of each separately.

With this in mind, the following standard general format is proposed

for open channel flow meter accuracy specifications. The format is

proposed in the interest of promoting a uniformity of specifications to

aid prospective users. This is, of course, only a general format, and

certain categories may have to be added or deleted, dependent upon the

design of the particular flow meter in question.

A. LEVEL MEASURgvENT

1. Accuracy : The accuracy of the level measurement system should

be stated. May be combined total error or may be broken down

to indicate linearity, hysteresis, repeatability, sensitivity,

etc., errors separately. May be stated as an absolute level

error, or error as a percentage of full scale level.

2. Temperature Error : Both the magnitude and applicable temperature range of any temperature related errors should be

stated as error per unit of temperature.

3. Resolution : Any level resolution limitations inherent in

the system should be stated, along with the resolution of

the liquid level indication device (if applicable)

.

B. LEVEL -TO -FLOW RATE CONVERSION

1. Accuracy: The accuracy of the level-to-flow rate conversion

system should be stated. Usually to be stated as a percentage of full scale flow rate.

2. Temperature Error : As Above.

3. Resolution : Any resolution limitations inherent in the system

should be stated. Also to be stated is whether the limitation

deals with input (level resolution) or output (flow rate resolution.)

A review of product literature revealed that many manufacturers

currently state flow meter accuracies in a manner quite similar to the

above proposed format. However, many manufacturers utilize a different

format in quoting the accuracies of their respective instruments. In

any case, typically a manufacturer's accuracy specifications are written

in such a way as to emphasize the strong points of their instrument and

de-emphasize any weak points. There is nothing untruthful or dishonest

in this practice; it is simply an outgrowth of a competitive marketplace.

However, a prospective purchaser and user of flow measuring instrumentation

has the right to know the accuracy specifications of the instrument, in

terms he can understand and which relate to his particular application.

Most flow meter manufacturers are more than willing to discuss accuracy

specifications, and a prospective customer should not be hesitant to

ask the manufacturer exactly what his specifications mean in relation to

102
the prospective application. A flow meter is usually a fairly significant

investment, and the user should know what he is paying for in terms of

accuracy. While a user should not have to pay for more accuracy than the

application demands, the flow meter should provide the accuracy required

to supply meaningful results.

7. Conclusion

Instrumentation errors in open channel flow measurement systems have

been discussed. While the errors associated with secondary (and primary)

measuring devices are, of course, of vital concern in controlling the

accuracy of an open channel flow measurement system, probably of equal

concern should be the sizing of the system to minimize the effect of the

errors associated with the measuring instrumentation. The accuracy

tolerance of a flow meter is usually specified as a percentage of a full

scale quantity, this being the position where the factors contributing

to the error are most controllable. As the quantity being measured

decreases, the possible error as a percentage of the actual reading

increases. Consider, for example, a level measurement system which

has a specified accuracy of +1% of full scale for a 0 to 1.0 ft. level

range. The potential maximum error at any position will then be +0.01

ft . At a 1 ft . level , the maximum error as a percentage of the reading

will be 1 percent (0.01 ft./l.O ft.). However, if the level decreases

to 0.2 ft., the maximum error as a percentage of the reading will increase

to 5 percent (0.01 ft./0.2 ft.). Thus, as is typical with many instrument

systems, the user is advised to keep the magnitude of the quantity

being measured as high up on the full scale range as possible.

This can be accomplished by proper selection and sizing of primary

and secondary measuring devices. The primary device should be sized such

that it generates as large a level as is practical for the flow rate

expected. A nontraditional primary device, such as a combination of a

V-notch and rectangular weir, may be employed to assure an acceptable

level over a wide range of flow rates. The secondary device should be

selected such that the anticipated levels in the primary device are

well up in the liquid level range of the secondary device. In this

manner, the effect of possible errors in the flow rate being measured

can be minimized. This runs contrary to the popular practice of oversizing

primary and secondary devices to allow for future expansion, a practice,

however, which should be avoided if errors are to be minimized.

103
References

1. United States Department of Interior, Bureau of Reclamation, Water

Measurement Manual , Second Edition, Superintendent of Documents, U.S.

Government Printing Office, Washington, D.C. (1967).

2. Shelley, Philip E., and Kirkpatrick, George A., Sewer Flow Measurement - A State -of -the -Art Assessment , Environmental Protection Technology

Series No. EPA-600/2-75-027, National Technical Information Service,

Springfield, Va. (1975).

3. Kulin, Gershon, and Compton, Philip R. , A Guide to Methods and

Standards for the Measurement of Water Flow , National Technical

Information Service, SpringfielH^, Va. (1975)

.

4. Mougenot, G., "Measuring Sewage Flow Using Weirs and Flumes",

Water § Sewage Works , July 1974, pp 78-81.

5. Hodgman, Charles D. , ed. , Handbook of Chemistry and Physics, 43rd

Edition, The Chemical Rubber PublisHTng Co., Cleveland, Ohio (1961).

6. Baumeister, Theodore, ed. , Mark's Standard Handbook for Mechanical

Engineers , 7th Edition, McGraw-Hill Book Company (1967)

.

7. Schontzler, Gordon J., "Fact Sheet on the Use of Ultrasonic and

Dipping Probe Flowmeter Application", Manning Environment Corp.,

Santa Cruz, Calif.

8. Buzay, K. , "Pneumatic Level and Open Channel Flow Measuring Systems",

Water Resources Instrumentation , Vol. 1, Krizek, Raymond J., ed.

,

Ann Arbor Science Publishers, Inc. Ann Arbor, Mich. (1974).

9. Sheingold, Daniel H. , ed. , Nonlinear Circuits Handbook , Analog Devices,

Norwood, Mass. (1974).

104
WEIR

Figure 1

FLUME

- Primary Measuring Devices.

DISCHARGE OF 90° V-NOTCH WEIR

.4 .5 .6 .7 LIQUID LEVEL, ft. 1— .05 ft. error

Liquid Level

Indicated on Flow Meter

Actual liquid

level In weir

.05 ft. error

Figure 2 - Flow Meter "Zero" Errors.

105
FLOAT-IN-WELL FLOAT-IN-FLOW

Angulor

Position /-"^^ ^Pulley

Proportional /

To Level / \\

Figure 3 - Float Operated Level Measurement Device.

Seporote Tronsmitter Combined Transducer and Receiver

AIR PATH MEASUREMENT

LIQUID PATH MEASUREMENT

Figure 4 - Ultrasonic Level Measurement System.

106
Figure 5 - Bubbler Level Measurement System.

Figure 6 - Dipping Probe Level Measurement System.

107
Figure 7 - Mechanical Cam Level-to-Flow Rate Conversion Systenn.

Figure 8 - Opto-Electronic Function Generator Level-to-Flow

Rate Conversion System.

108
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS

,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

MARINE DYNAMICS AND ITS EFFECTS ON CURRENT MEASURING TRANSDUCERS

Thomas Merc and Gerald Appall

Test and Evaluation Laboratory (T^EL)

National Oceanic and Atmospheric Administration

and

Raul S . McQuivey

Sutron Corporation

1925 North Lynn Street

Arlington, Va.

A growing interest in measuring current velocities

in coastal zone and estuaries prompted the creation of

a Dynamic Analysis Program at T^EL (formerly NOIC) . The

dynamics of near-shore current flows are more severe

than that of the deep ocean. Therefore, it is important

to qualify the performance of current measuring transducers in a dynamic environment. A study was contracted

to determine what scales and intensities of turbulence

exist in the near-shore environment. The major effort

of the study is concerned with turbulence from 1 to

20 Hz and scales up to 30 cm.

Tests are being conducted to determine the response

characteristics of several current measuring transducers

including both rotor and electromagnetic instruments.

These tests include grid-produced turbulence measurements as well as simulated mooring-line dynamics' tests.

Measurements have revealed several transducers that

have 10 to 20% sensitivity increases when turbulence

intensities of 6 to 12% are present.

This paper contains a description of turbulence that

exists in the marine environment along with dynamic

response characteristics of several current measuring

transducers

.

INTRODUCTION

A growing concern about the preservation and management of our

marine resources has precipitated the need for a more complete understanding of the physical, chemical, and biological processes in the

marine environment. The estuarine and coastal zones are of particular

interest because they are the receiving bodies for disposal of large

109
amounts of solid and liquid wastes, excess heat and many other chemical

and radioactive materials. Understanding how pollutants are transported

and how they affect the marine environment requires accurate, reliable

instrumentation for measurement of mean and fluctuating flow characteristics.

The Test and Evaluation Laboratory (TJ^HL) formerly the National

Oceanographic Instrumentation Center (NOIC) is involved in the calibration and testing of current measuring systems. Recognizing the lack

of information available on the dynamic response characteristics of

current measuring transducers and systems, TfjHL established a laboratory

testing program in 1975 to evaluate the effects of both large scale

dynamics, i.e., wave generated and small scale (high frequency) turbulence.

Preliminary results indicated some electromagnetic type flow sensors were

biased by nearly 15 percent of actual flow velocity when certain scales

and intensities of turbulence were present. Large scale dynamics created

errors of over 50% of actual velocity on rotor-vane type instrumentation.

T^LL expanded its dynamic analysis program in 1976 in an effort to better

understand the range of dynamics present in tlie marine environment and

to quantify their effect on current measuring transducers.

Sutron Corporation was awarded a contract to study marine environmental dynamics. A major objective of the study was to conduct a literature search to identify the mean and fluctuating flow measurements

which liave been made in describing the marine environment and to determine

a realistic range of natural environmental dynamics. This information

would then be utilized in establishing tlie relevancy of the dynamic test

program and in the development of realistic laboratory simulation methods.

The following is a description of marine environmental dynamics as indicated from the study and laboratory simulation tests which have been

conducted at T^EL to describe current measuring transducers performance

in the dynamic environment.

THE ^1ARINE ENVIRONMENT

The marine environment can be divided into three regions or zones:

the estuarine zone, the coastal zone, and tlie deep ocean. It was

attempted in the study to quantify a realistic range of environmental

dynamics in each of these regions by summarizing the mean and fluctuating velocity measurements tliat have been reported in the literature.

Each region experiences particular scales of dynamic behavior wliicli

are primarily dependent on environmental factors. A brief description

follows of the flow structure in each region along with a range of

environmental dynamics obtained from the literature review.

Estuarine

Estuarine flow structure is generated by many natural forces

creating complex patterns of motion. In describing this flow structure.

110
tlie important characteristics to be considered are:

1. The cyclical rise and fall of the water surface which results

from ocean tides and the periods of these tidal waves;

2. The geometry of tidal estuaries and the rise and fall of the

tide at the mouth, which results in an exchange of water masses through

the entrance with large amounts of sea water temporarily stored in

the estuary during high tide;

3. The state of diffusion of fresh water to salt water;

4. The variation of salinity over the depth and the resulting

internal density currents;

5. Surface waves and currents.

listuarine flow dynamics are largely two-dimensional in nature with

each estuary having distinct characteristics depending on location and

geometry. Table 1 summarizes the range of environmental dynamics

obtained from the literature review.

Coastal Zone

Coastal zone processes are important in designing coastal structures,

understanding the stability of a coastline, and evaluating the effects of

pollution on the coastal environment. The currents in the coastal zone

are extremely complex, with wind-induced currents a primary source of

energy.

An attempt at briefly describing the coastal circulation process

follows. Coastal currents flow roughly parallel to the shore constituting a relatively uniform flow in the deeper water adjacent to the

surf zone. A near-shore current system may be superimposed on the

inner portion of the coastal current or, in the absence of a coastal

current may exist independently. The near-shore system is associated

with wave action in and near the breaker zone and consists of shoreward

mass transportation of water caused by wave motion, movement of the water

parallel to the coast as longshore currents, or seaward return flows

(rip currents) . In some cases local currents result in the near-shore

area where water from rivers and nearly enclosed bays is discharged into

tlie ocean with considerable momentum.

By comjiarison with the otiier zones, tlie coastal zone probably has

more energy in the small-scale structure (turbulence) because of its

three-dimensional dynamics and high intensities. Literature indicates

that investigators are taking measurements of currents in the coastal

zone with instrumentation that is not capable of responding to the

three-dimensional dynamics. Many investigators are not aware of the

problems associated with taking reliable/accurate measurements in the

coastal zones. A summary of a range of dynamics obtained from a literature review is presented in Table 2.

Ill
Table 1.

Cliaracteristic

Currents (cm/sec)

Intensity of u Turbulence _

(percent

J

Scales : Tides

(m)

Wind

Macro

Micro

Listuarine Environmental

Fresh Water

0 to 500 cm/sec

5 to greater than

50

10^ to 10^

1 to 10

1 to 10

0.01 to 0.1

Dyn ami cs

Saline Water

0 to 500 cm/sec

5 to greater than

50

103 to IQS

1 to 30

1 to 10

0.01 to 0.1

112
Table 2. Coastal Environmental Dynamics

Characteristi c Range

Currents (cm/sec) : Coastal

Longshore

Rip

Wave Induced

Up to 60

Up to 50

Up to 75

Up to 25

Intensity of Turbulence | ^ (percent) ^ u

^

U

5 to greater than

Scales (m)

:

Tide

Wind

Macro

Micro

10-^ to 10^

1 to 25

1 to 10

0.01 to 0.1

113
Deep Ocean

Deep ocean flow structure is generated by a variety of natural

forces acting to keep the ocean in a continual, complex pattern of

motion. Ocean currents can be categorized into several groups,

one of which is currents related to the distribution of density in

the sea. This class of currents include large-scale ocean current

such as the Gulf Stream, the Kuroshio, and the equatorial currents.

Wind stress on the sea surface directly causes currents, which in

turn tend to alter the density distribution; tidal currents and those

associated with internal waves are also present. In describing general

oceanic flow structure, the ocean can be divided into: (a) the mixed

surface layer, (b) the thcrmocline and interior layer, and (c) the

bottom layer. The dynamics associated with the mixed layer is threedimensional in nature wliile the thermocline and interior layer, and

the bottom layer contain largely two dimensional dynamics. Tlie

range of environmental dynamics associated with these layers as indicated in the literature is sunmiarized in Table 3. Considerable

variation in these dynamics should be expected depending on location.

DYNAMIC PERFORMANCE TESTING OF CURRENT-MEASURING TRANSDUCERS

Tree's Dynamic Analysis program was established to describe the

dynamic performance of current-measuring transducers by first developing

laboratory simulation metliods. Dynamics in the marine environment has

been categorized in the program into two groups: (a) turbulence being

that from 1 to 10 Hz with lengtli scales up to 10 cm, and (b) macroscale

dynamics which describe motions from 0.05 Hz to 1 llz with length scales

greater than 10 cm.

Turbulence

TfjEL's submerged jet facility at the Washington, D.C., Navy Yard

is being used extensively in turbulence investigations. The mean flow

turbulence intensity of the facility is less than 1 percent over its

operating range of 0.5 to 200 cm/sec. Screens of various sizes are

inserted at the outlet of the jet to produce turbulence with intensities

up to 10 percent of mean flow. A Delft miniature propeller meter and

a Disa hot- film system are being used as flow measurement standards.

Tests conducted on an Aandcraa rotor-type current measuring transducer found jt to be largely unaffected by the turbulence generated in

tlie facility. The rotor acted as a low-jiass filter, averaging the

small-scale velocity fluctuation. Mean flow measurement errors were

generally less than 2 percent.

Several electromagnetic current meter (EMCM) sensors have been

studied. A cylindrical EMCM sensor was tested having a 2.5 cm diameter

with extended electrodes. Two spherical EMCM sensors having extended

electrodes with a 1-cm diameter and a 3.8-cm diameter, respectively

were evaluated.

114
Table 3. Deep Ocean Environmental Dynamics

Characteristic

Mixed Surface

Layer

Tliermocline 5 Bottom

Interior Layer Layer

Current Velocity

(cm/ sec) Up to 250 Up to 250 Up to 25

Intensity of ^ Turbulence u 5 to greater

(percent) — than 5

1 to 5 (variable) 1 to 5

Scales: Tide

(m)

Wind

Macro

Micro

10^ to 10^

1 to 25

1 to 10

0.01 to 0.1

10^ to 10^

5 to 50

1 to 25

0.01 to 0.1

Depth

(m)

Up to 100 0 to 600 Below 100

115
A streamlined elliptical LMCM sensor of French design and an "open"

design EMCM sensor developed by J. R. Olson of the Naval Undersea

Center (NUC) were included in the testing. The "open" sensor utilized

a Helmlioltz coil design with its electrodes mounted in the center of

the coil.

The operation of HMCMs is based on Faraday's law of electromagnetic

induction. An electromagnetic field is generated, and as water (an

electrical conductor) moves perpendicular to the magnetic flux lines an

electromotive force (potential) is generated. Orthogonal electrode

pairs sense two voltage components that define the magnitude and

direction of the water velocity in a plane perpendicular to the magnetic

field. The magnetic field strength diminishes as the square of the

distance from the magnet; therefore, the largest electrical potential

is produced closest to the surface of the transducer.

Mean flow measurement errors detected wlien turbulence scales of

approximately 2 cm with intensities of 4 percent were introduced are

shown in Figure 1. All the EMCM sensors tested but the small spherical

and the "open" EMCM sensor indicated a positive averaging error of

greater than 2-6. These errors are believed the result of a disturbance

in the boundary layer flow around the probes introduced by the presence

of turbulence in the flow. The small spherical probe was relatively

unaffected. It is conjectured that the probe size is small with respect

to the scales of turbulence; therefore, the boundary layer flow was not

significantly affected. The "open" design EMCM sensor was also unaffect

because it measures free stream velocity and produces, by design, no

boundary effect.

Investigations are continuing in an effort to more fully understand

the dynamics of boundary layer flow and its resulting effects on EMCM

sensors, and at improving laboratory simulation methods.

Macros cale Dynamics

Current measuring systems in the marine environment encounter macro

scale dynamics from water partical motions and from system platform

motions. In an effort to define the large-scale dynamic response of

current-measuring systems in the laboratory, T^EL contracted tlie design

and fabrication of a dynamic test apparatus to the Naval Ship Research

and Development Center (NSRDC) . The apparatus mounts on the No. 1 tow

carriage at NSRDC and can produce nearly sinusoidal motions in the

horizontal and vertical plane and circular motion in the vertical plane.

These motions are generated while the tow carriage is undenvay, tliercby

simulating wave generated water partical motion or mooring line and

platform-induced motions on current meter systems.

A 10-cm diameter spherical EMCM Marsh McBirney MMI-555, Aanderaa

RCM-4 rotor type meter and a General Oceanics (CO.) Model 2010 drag

force type meter were selected as representative instruments for testing

116
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Test data from the Aaiuleraa and (1.0. current meters were of

limited value since only speed was sampled from these instruments

precluding a vector analysis. However, test conditions in which

the mean flow velocity exceeds the angular velocity of the sensor

are valid measurements of vector magnitude and indicate significant

over register from both meters. Analysis of motion picture films

show unresponsiveness of the Aanderaa to liigh frequency and reversing

type flows. Tl\e G.O. meter was more responsive to dynamics and yet,

it became unstable under high-frequency oscillations.

The MMl-555 measures two components of the flow; therefore, a vector analysis was performed comparing the system input vector with

the instrument output vector. Test results from averaging 100 data

samples acquired at 2 per second are summarized in Table 4. Real-time

vector analysis was also performed and sensor response to 32 cm/sec

and 77 cm/sec orbital velocities are shown in Figures 2 and 3, respectively. A 1-second pliase shift and attenuation at high orbital

velocities results from tlie instrument's internal time constant. The

MMl-555 was able to measure mean flow in the presence of the simulated

large scale dynamics with errors generally below 4 cm/sec.

SUMN1ARY

The literature search conducted by Sutron Corporation confirmed

that both micro and macroscale dynamics are present in tlie marine

environment

.

Laboratory tests conducted to define dynamic response characteristics of various current measuring transducers indicated that the

electromagnetic sensors tested, accurately measured mean flow in the

presence of macroscale dynamics with errors generally less than 4 cm/sec.

However, microscale turbulence cause significant positive averaging

errors in certain EMCM sensors. The rotor tested was relatively

immune to microscale turbulence errors and yet, it over registered by

over 18 cm/sec when subjected to macroscale dynamics.

Additional investigation is required in the form of improved

laboratory simulation of marine dynamics as well as field measurements

to verify the significance of laboratory test techniques.

Understanding both tlie dynamics of the environment and the response

of various current sensors to tliose dynamics will enable investigators

to select sensors and platforms wliich are best suited for eacli particular

application.

118
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120
RliFERBNCOS

1. Unpublished Report, "Mean and Fluctuating Measurements in the

Marine Environment" by Raul S. McQuivey, Phd. , Sutron Corporation, 1925 North Lynn Street, Arlington, Virginia.

2. Appell, Gerald P., "Evaluating the Performance of Flow Measuring

Instrumentation", Plessey STD/Ocean Systems Conference, San

Diego, California, January 19-21, 1977.

3. Bennett, George and Graliam Wells, "Flow Improvement Modifications

and Flow Quality Evaluation of the NOIC Current Meter Calibration

Facility", Prepared by Engineering and Industrial Research

Station, Mississippi State University, for National Oceanographic Instrumentation Center, NOAA, Department of Commerce,

Report Number EIRS-ASE-75-2 , April 1975.

121
'I'al MM I Ml DATA SUMMAl^Y Ul- Tllli STROKli AMPLITUDl: = 122 CM NSRDC

FLOIV PERIOD ORBITAL

VELOCITY

CM/SEC SEC CM/SEC

ORBITAL MO'

10.52 12 32.00

10.52 8 47.56

10.38 5 78.86

36.12 12 32.01

36.10 8 47.60

36.55 5 77.32

72.23 12 32,00

72.35 8 47.64

72.20 5 77.40

ORBITAL MOTION =

10.67 12 32.04

10.62 8 47.65

10.39 5 76.98

36.60 12 32.07

36.26 8 47.76

36.16 5 77.26

72.23 12 32.08

72.23 8 47.85

72.25 5 77.08

ORBITAL MOTION =

0.00 12 32.10

0.00 8 47.74

0.00 5 76.94

10.87 12 32.12

10.24 8 47.81

10.49 5 77.09

36.29 12 32.12

36.42 8 48.78

36.45 5 77.12

72.04 12 32.11

72.76 8 48.48

72.84 5 77.20

le 4

niil, 555

nVNyUllC TEST I^iSULTS

7/21/76

MMI 555 SPEED % SPEED

OUTPUT ERROR ERROR

CM/SEC CM/SEC CM/SEC

= INTO THE FLOW

13.51 2.99 28.42

15.51 4.99 47.43

19.16 8.78 84.59

37.40 1.28 3.54

40.06 3.96 10.97

41.05 4.50 12.31

73.56 1.33 1.84

70.91 -1.44 -1.99

71.90 -0.30 -0.42

45 DEGREES TO THE FLOW

11.86 1.19 11.15

13.19 2.57 24.20

14.85 4.46 42.93

35.75 -0.85 -2.32

39.06 2.80 7.72

38.40 2.24 6.19

72.90 0.67 0.93

69.91 -2.32 -3.21

70.90 -1.35 -1.87

90 DEGREES TO THE FLOW

5.89 5.89 0.00

8.87 8.87 0.00

13.85 13.85 0.00

13.85 2.98 27.41

13.52 3.28 32.03

14.18 3.69 35.18

37.74 1.45 4.00

38.40 1.98 5.44

38.40 1.95 5.35

72,24 0.20 0.28

71.90 -0.86 -1.18

71.57 -1,27 -1.74

122
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

SOME ERROR SOURCES IN PRICE AND PYGMY

CURRENT METER TRAVERSES

Gershon Kulin

Mechanics Division

National Bureau of Standards

Some performance characteristics and effects of

methods of calibration and use of Price and Pygmy water

current meters are examined with emphasis on their

application to velocity-area traverses for flowrate

determination. The limits of applicability of group

rating equations relative to individual ratings are

discussed using available calibration data. The

repeatability of meter performance at low velocities,

0.1 < V < 0.5 ft/s (3.0 < V < 15.2 cm/s),l is investigated. The effect of cable-and-weight suspension

relative to rod suspension of Price meters is examined

experimentally and analytically for Columbus and Elliptic

weights. Errors due to lateral velocity gradients and

bottom and sidewall proximity effects on both types of

meters are also investigated.

Key words: Current meter calibrations; current meters,

water; flow measurement; Price meter suspensions; Pygmy

meter performance; velocity-area method.

1. Introduction

Price and Pygmy water current meters, shown in figure 1, are vertical axis meters with rotating wheels composed of six conical buckets.

The outside diameters of the Price and Pygmy bucket wheels are 5 inches

(12.7 cm) and 2 inches (5.1 cm) respectively and the meter geometries

are approximately similar. The Pygmy meter was developed specifically

to replace the larger Price meter in very shallow flows [1],^ but its

actual use in the field is not restricted to such cases. These meters

are among the most widely used devices in this country for open channel

flowrate determination by the velocity-area method.

"^ Non-metric units are given precedence in this paper because they are

still customarily used in the field.

^Figures in brackets indicate the literature references at the end of

this paper.

123
The accuracy of such flowrate determinations depends on some combination of current meter accuracy and the accuracy of the spatial and temporal

sampling of the velocities. This note deals only with the first element,

i.e., the current meter accuracy, and its purpose is to provide users with

an estimate of the accurary that can be expected from Price and Pygmy

current meters calibrated and used in various ways. Accuracy requirements

differ among various users, some needing perhaps only a few percent while

others are anxious to shave every possible percentage point off of the

uncertainty. It is reasonable to expect the latter category to increase

as needs increase for accurate water and wastewater flow measurements.

In this work, more than routine attention has been given to low

velocities, i.e., velocities less than one foot per second (30.5 cm/s).

Although guidebooks generally caution users to make traverses in sections

of moderate velocity, sometimes lower velocities are unavoidable, e.g.,

when traversing upstream to calibrate a weir or flume. Meters in good

condition are assumed. Smoot and Carter [2] pointed out the sensitivity

of Price meter performance to dented buckets, and because the low velocity

performance of the meters can depend upon the internal friction, it is

also assumed that the user continuously monitors the spin tests of the

instruments

.

The data included here were obtained from: low velocity, i.e., less

than 0.5 ft/s (15 cm/s), experiments in a short towing tank; limited

experiments in a 20- inch (50.8 cm) diameter gravity-flow water tunnel;

and from the records of numerous NBS calibrations of submitted meters

during the 1960's, when NBS was still operating a large towing-tank

calibration station that was later transferred to the U.S. Geological

Survey

.

Price meters submitted to NBS were routinely calibrated at eight

velocities between 0,25 ft/s (7.6 cm/s) and 8 ft/s (244 cm/s). Pygmy

meters were calibrated at six velocities between 0.25 ft/s (7.6 cm/s)

and 3 ft/s (91.4 cm/s). At each velocity two passes were made in opposite

directions along the towing tank. The resulting 12 sets of velocityrotation rate data for the Pygmy meters were fitted with a rating equation

of the form

where V is the velocity in feet per second, N is the rotation rate in revolutions per second, and M and A are constants. The 16 sets of velocityrotation rate data for the Price meter were fitted with two straight line

equations which matched at N = 1, so that

2. Calibrations in the Large Towing Tank

V = MN + A (1)

Vi = MiN + Ai, N < 1 (2)

V2 = M2N + A2 , N > 1

Vi = V2, N = 1

124
where Mi, M2, Ai and A2 are constants.

3. Repeatability of Rating Equations

Price meters . We consider first the dependability of the rating

equations themselves. One rod-mounted Price meter was repetitively

calibrated a total of 18 times. Five of these runs were made in 1968,

shortly after the meter was purchased. Eight runs were made in 1972

and five more runs were made shortly thereafter with a new lower bearing

and a sharper pivot pin replacing the older one which had been dulled.

Between the 1968 and 1972 runs the towing carriage was completely rebuilt

by the Geological Survey, so that effectively two different towing systems

were involved.

The results are shown in terms of observed revolutions per foot of

advance, N/V, in figure 2 for each of the three groups of calibrations.

The lower curve of figure 3 shows the standard deviation of the rating

equations from the average of the eighteen equations computed for selected

velocities over the calibration range. (In these figures and in many of

the following a logarithmic scale is used for the velocity abscissa for

clarity in plotting the lower velocities.) The deviations are very small

at moderate and high velocities and exceed 0.5 percent only at velocities

below about 0.3 ft/s (9.1 cm/s). It is inferred from this result that a

single carefully made calibration of a Price meter comes very close to

describing the correct, i.e., average, V-N relation, at least for nonturbulent conditions.

Pygmy meters . The corresponding results for a Pygmy meter are shown

in the lower curve of figure 4. In this case one meter was repetitively

calibrated seven times over a period of a few days. Although the deviations

at the higher velocities are slightly larger than for the Price meter, they

are still small and one again can infer the utility of a single rating

equation as a description of the "true" or average towing-tank rating for

that meter in specified condition.

A. Calibrations of Groups of Meters

Price meters . Often, Instead of having a meter individually calibrated,

users employ a "standard" or representative equation developed by a manufacturer or by other means to describe the V-N relation for a group of

meters. We examined the rating equations of Price meters, representing

at least three different manufacturers, which were submitted in groups for

calibration. It is assumed that the meters in each group were manufactured

at the same time or under the same conditions. In figure 3 are shown the

standard deviations of the individual rating equations from the average

rating equation for each group. The dashed curve typifies the deviations,

which are of course larger than the deviations for an individual calibration

but are still quite small and could well be adequate for many applications.

This result agrees with the conclusions of Smoot and Carter [2]

.

If this course is taken it is necessary to consider how closely a

"standard" equation describes a group average. The U.S. Geological Survey

handles this problem by supplying meter dies and forms, for which standard

ratings have been developed, to the manufacturer which is awarded the meter

125
contract. However, these comments are intended mainly for individual

purchasers of individual meters, and there is little information available

on which to base a judgment. One manufacturer's standard rating that is

known to us shows deviations from the average performance of its meters

which exceed the meter-to-meter deviations shown in figure 3 and which

exceed 2 percent for some batches of meters. Based on this limited evidence

it is concluded that an individual calibration is necessary if a meter

rating within 1 percent is desired.

Pygmy meters . Corresponding results for groups of Pygmy meters are

shown in figure 4. The standard deviations are larger than those obtained

for the Price meter groups, but might still be satisfactory for many applications .

Concerning standard or group ratings for Pygmy meters, many users apply

the approximate rating equation

V = N (3)

The results of this action are shown in figure 5. For velocities higher

than about 1 ft/s (30.5 cm/s), standard deviations are within about 2

percent, again a value whose acceptability depends upon the application.

As the velocity decreases, however, the deviations become unacceptably

high for virtually all purposes. Actually the deviations at the low

velocities are not entirely random; equation (3) always underestimates the

velocity in that range. A simple albeit crude adjustment can be made by using

the weighted average calibration equation for all of the meter groups of

figure 4.

V = 0.969N + 0.043 (4)

The standard deviation of the groups of calibration equations referenced

to eq. (4) is typified by the dashed line in figure 5 and indicates a

substantial improvement over the use of equation (3) at low velocities.

Equation (3) is not presented here as a valid average equation for all

Pygmy meters, except perhaps for those manufactured during the 1960 's from

which these data were derived. However, it is noted that when eq. (4) is

compared with a more recent group average cited by Schneider and Smoot [3]

,

agreement is within about 2 percent over the entire velocity range and

within about 1 percent for velocities higher than 1 ft/s (30.5 cm/s).

5. Precision of Calibrations

The repetitive runs on the single Price and Pygmy meters were made

differently than the routine calibration runs, in that between each velocity

run a measured time interval was allowed which was longer than that used

during the routine calibrations. Also, a stilling period of two hours was

allowed after the highest velocity runs were completed before starting the

0.25 ft/s (7.6 cm/s) runs for the next calibration. The standard deviations

of the measured values of N/V from the values computed at the same velocities

using the calibration equations are shown in figure 6 for the Pygmy meters.

The deviations are slightly smaller for the repetitive runs on the single

126
meters. However, this is not expected to affect the deviations of the

calibration equations appreciably, and the conclusions drawn from figures

3 and 4 regarding the advantages of individual calibrations are still

valid

.

6. Deviation of Velocities from Rating Equations

The deviations shown in figure 6 can be due to: approximating the

rating with straight-line equations, a factor which does not appear to be

important for velocities greater than 0.25 ft/s (7.6 cm/s); factors inherent

in the calibration method; and random differences in meter response.

Usually it is not possible to separate the deviations due to meter response

from those due to calibration method. By tacitly assigning all deviations

to meter response we are being conservative. In any event, the deviations

shown in figure 6 are small so long as the velocities are higher than 1 ft/s

(30.5 cm/s). Also, if these deviations are random their probable contribution to the total velocity-area error can be estimated by dividing the

deviation for each point velocity by the square root of the number of

sampling points, making the resultant effect still smaller.

To further investigate meter responses at low velocities, where deviations are likely to be higher, a series of runs on two Price and two Pygmy

meters was undertaken in a small towing tank 3 ft (91. A cm) wide by 2 ft

(61.0 cm) deep by 40 ft (12.2 m) long towing tank using velocities between

about 0.1 ft/s (3.0 cm/s) and 0.5 ft/s (15,2 cm/s). This range overlaps

the velocities from 0.25 ft/s (7.6 cm/s) to 0.5 ft/s (15.2 cm/s) covered

in the usual calibrations in the large towing tank, thus serving in part

as a check on the equipment. The limiting high velocity was governed by

the short tank length. The limiting low velocity was essentially the lower

limit of consistent meter behavior.

The deviations of N/V can be computed using each pass down the tank,

as was done by Smoot and Carter [2] for deviations of V. Alternatively they

can be computed using the "round trip" average N/V for each velocity,

recalling that the rationale for using round trips in the calibrations is

the cancellation of the effects of steady thermally induced currents in the

tank [4] . Computing standard deviations based on each pass in effect charges

the meter for such currents in the tank. On the other hand, using round

trips could mask some deviations which should be blamed on the meter.

Extreme examples are shown in Table 1, where run 6-11-74 exhibits obvious

drift while run 6-12-74, made one day later, shows no such quasi-steady

effect. For practical estimating purposes, a figure somewhere between the

round trip and single-pass values is considered to representative, a conclusion which is supported by data of figure 7 discussed later.

Results for two Price and two Pygmy meters repetitively run at constant

velocities in the small towing tank are given in figure 7, which also

includes available data from the full calibrations in the large towing tank.

These results represent the deviations of measured N/V values from the

average N/V at a given velocity. It should be noted also that the percentage

standard deviation of N/V, which is equivalent to the percentage change in

N for a fixed velocity, is not the same as the percentage deviation in V for

the same N. But for velocities over 1 ft/s the two are the same for

127
practical purposes.

Pygmy meters . For meter no. 1 the spin test time was about one minute,

a value which is below the desired time of approximately 1-1/2 minutes but

above the practical minimum of 1/2 minute as recommended by the U.S.

Geological Survey [1]. The reason for the larger internal friction was

not apparent since the bearings appeared to be in good condition. Meter

no. 2 had a spin test time longer than 2 minutes. The minimum speeds for

reliable and consistent rotation were about 0.16 ft/s (4.9 cm/s) and

0.11 ft/s (3.4 cm/s) for meters no. 1 and no. 2 respectively.

Price meters . The spin tests on the two Price meters varied but in

figure 7 only runs with similar spin tests are compared. The lowest

velocity for consistent performance was about 0.1 ft/s (3.0 cm/s) for both

meters. When the penta gear was removed from meter no. 2, the minimum

velocity was reduced to about 0.07 ft/s (2.1 cm/s). However, at these

velocities (not shown in figure 7) the deviations of N/V were high; for

example, at V = 0.075 ft/s (2.3 cm/s), the standard deviation from the

average N/V was 11 percent for each pass and 6 percent for a round trip

computation.

Although the data of figure 7 were treated as though all deviations

were random, deviations were noted in the day-to-day averages when runs

were repeated over a period of several days. These are noted on the curves.

It is possible that, as far as a day's field work is concerned, these have

to be considered as systematic errors, so that at 0.25 ft/s (7.6 cm/s), for

example, we should add about 1 percent uncertainty to the uncertainty in

the calibration equations themselves.

The towing tank results of figure 7 are in reasonable agreement with

the deviations obtained in a 20- inch (50.8 cm) diameter gravity flow water

tunnel, where the Price and Pygmy no. 1 meters were subjected to velocities

from 0.25 ft/s (7.6 cm/s) to 8 ft/s (2.44 cm/s) with runs repeated on several

different days. In spite of the fact that only two meters of each type were

used, the agreement between two such disparate testing methods suggests that

figure 7 presents a usable and conservative picture of current meter

reliability.

7. Calibration of Price Meters With Cable Suspension

The two most commonly used weight suspensions are shown in figure 8

using the 15-pound (6.8 kg) weight as an illustration, but heavier weights

are available in both the Columbus and Elliptic weight series. Recommended

spacing between meter and various sizes of Columbus weights can be found

in references [5] and [6].

The deviations of the rating equations of groups of cable-rated Price

meters from the group average are notably larger than the corresponding

deviations for rod-mounted meters, as shown by the upper dashed curve in

figure 3. The same figure also shows that the deviations for repetitive

ratings of a single cable-suspended meter are larger than for the rod-mounted

case. Probably part of the increased deviation is due to additional

currents generated in the towing tank by the submerged weight.

128
8. Comparison of Rod and Cable Ratings

Columbus weights . One Price meter was repetitively calibrated, from

8 to 10 times, with rod suspension and with 15, 30 and 50-lb (6.8, 13.6

and 22.7 kg) Columbus weights. The rating equations are compared in

figure 9. Also shown are results for a group of 11 Price meters which

were calibrated both with rod and 15-pound (6.8 kg) Columbus weight

suspensions. In general, the results support the contention of the U.S.

Geological Survey that properly used Columbus weights cause no change

from rod mounted ratings [5] . However, consistent deviations in the

Columbus weight curves are noted, particularly at the lowest velocities.

Placing the results in terms of N/V in figure 10, it is seen that

there are consistent trends toward slower rotation for cable suspensions

at the lowest velocities and toward slightly faster rotation at the moderate

velocities. Possible hydrodynamic explanations for this behavior will be

noted later.

Elliptic weights . No single-meter repetitive runs were made with

Elliptic weights, but the results of comparative rod-cable calibrations

on two groups of meters are shown in figure 9. The meter-to-weight spacings

are different for the two groups, 5.3 inches (13.5 cm) and 6.7 inches

(17.0 cm). It is seen that both groups show a higher velocity for the

cable rating throughout the range, with the difference increasing markedly

at low velocities. The same effect in terms of N/V is shown in figure 11,

where N/V is always smaller for cable mounting.

The slower rotation with decreasing velocity of cable-mounted relative

to rod-mounted meters is common to both types of weights, and possible

non-hydrodynamic causes were considered. First was considered the possibility that the rod-mounted results were too high at low velocities. There

was a certain amount of "lurching" of the carriage which was apparent to

the observer at low velocities. Conceivably this could have given a sawtooth velocity vs. time curve of such shape that meter inertia rounded off

the "teeth" and gave an erroneously high rotation rate. However, available

rod-mounted meter data at low velocities for three different carriages do

not account for differences of this magnitude. Second it was considered

that the meters were not properly balanced on their pivots and therefore

operated at a tilt when the velocities were too low for the fins- to develop

enough lift for balance. Such a tilt could increase the internal friction

and make the meters rotate more slowly. To check this, a meter was repetitively run at low velocities with a deliberate tilt of 1.5 degrees but no

consistent effect was found.

9. Hydrodynamic Effects of Weights

The most obvious hydrodynamic effect of the weight is the distortion

of the velocity field around it. Figure 12 shows the velocity distribution

for potential flow around a shape closely approximating that of the 15-pound

(6.8 kg) Elliptic weight as developed by Lofquist [7]. A typical position

for the bucket wheel is shown by the dashed rectangle, and the center of

the wheel is seen to be in a zone of little or no velocity distortion. The

rear of the wheel is subject to an increased ambient velocity of more than

one percent. Because most of this part of the periphery exerts a retarding

129^
torque, this could be translated into a fractional percentage reduction

in N, but not enough to account for the differences in figure 11.

Figure 13 shows the potential flow velocity distribution around a

15-pound (6.8 kg) Columbus weight [7]. Here the meter is situated in

a zone of at least + 1 percent in velocity. This can account for part

of the effect seen at moderate velocities in figure 10, but cannot account

for the observed change in the apparent suspension effect with decreasing

velocity. We note in passing that this change cannot be accounted for

by a larger boundary layer thickness at the low Reynolds numbers because

such an increase, even if significant, would provide an effect in the

opposite direction.

A remaining potential source of the increased cable-weight effect at

the low velocities is the interaction between the rotary motion induced

by the bucket wheel and the supporting strap and fin. The strap is about

2 inches (5.1 cm) closer to the wheel than the rod (see figure 8), and

the fin also is a little closer than when the rod is used. Figure 12

suggests that data available for a 6.7-inch (17.0 cm) spacing between meter

and Elliptic weight would be essentially free of velocity-distortion effects.

These data for a group of 12 meters, plotted in the manner of figure 11,

suggest a strap interference effect of about 1 percent at 8 ft/s (244 cm/s)

gradually increasing to about 4 percent at 0.25 ft/s (7.6 cm/s). These

values are supported by limited data available for a 50-pound (22.7 kg)

weight at 9.8-inch (24.9 cm) spacing. Also, in configurations where the

meter is particularly close to the top of the weight, there can be an

additional meter-weight interference as is suggested by the 30-pound

(13.6 kg) Columbus results of figure 10. However, there is insufficient

data to evaluate this effect for all situations. Because the various

interference effects are at least as large as the theoretically obtainable

velocity-distortion effect, it remains necessary to experimentally determine

any needed adjustments to rod-mounted calibrations, and the curves of

figure 9 can be used for that purpose.

10. Measurements With Wading Rod

Recommended practices for wader position relative to the current meter

are cited in the literature, e.g., [5], but the data base or analytical

reasons for these recommendations are not given. As a. result, users have

no guidelines as to the errors involved in failure to adhere to recommended

practice.

If the wader can be considered as a vertical circular cylinder immersed

in a low Froude number potential flow, the solution is well known and is

shown in figure 14 in terms of local-to-undisturbed velocity ratio [7]

.

Recommended practice places the bucket wheel somewhere within the dashed

rectangle in the figure, depending upon the particular geometry and upon

whether a Price or Pygmy meter is used. In that area the increases in

velocity average about 3 percent. Slenderizing the water to an elliptical

cylinder of 3:4 breadth-length ratio reduces the velocity increase to about

2 percent. Experimental verification is needed here with shapes more closely

simulating a body. Existing field data on wading-cable suspension comparison

are very limited. Schoof and Crow [8] actually found differences in the

opposite direction but some of their field work was done under adverse

130
conditions. In the meantime this idealized analysis has been presented

to show that a potential for systematic error exists in wading measurements.

11. Effect of Lateral Velocity Gradient

Figure 15 shows the elements of a theoretical model of bucket wheel

current meters developed by Lofquist [7] . The torque is an integral around

the periphery of the product of a drag coefficient, the square of the

tangential component of the relative velocities felt by the buckets, and

obvious geometric factors. It is assumed that the drag coefficient has

only two discrete values depending upon whether the open or closed side

of the bucket faces the relative tangential velocity. One result of this

analysis is that the change in rotation rate, u, in rad/s, can be expressed

as

Aw/o) = (2+k2) (afi/V)/3k (5)

where k is the value of (aco/V) at large V, and a and Q. are the wheel radius

and velocity gradient as defined in figure 15. For example, if the velocity

increases (in the direction shown in figure 15) by 10 percent over a lateral

distance of 1 foot (30.5 cm), and if we use experimentally determined values

of k of 0.34 and 0.30 for the Price and Pygmy meters respectively, the

rotation rate is increased by 2.9 percent for the Price meter and 1.2 percent

for the Pygmy. These numerical results have not been confirmed experimentally. But if they are even approximately correct their magnitude is

sufficient to produce errors in traversing a badly skewed profile if the

sum of the gradients at the sampling points is not zero. The possibility

of lateral gradient effects should also be considered when using figure 16

to position meters for wading measurements.

12. Wall and Bottom Proximity Effects

Wall effect . We assume that the total "wall effect" in flowing water

is due to the velocity gradient created by the wall, superposed upon another

effect due only to wall proximity. The velocity gradient effect has already

been considered and the proximity effect can be treated separately by making

near-wall experiments in a towing tank. This was done by repetitive towing

of meters in the small MBS tank at various positions near the wall and along

the centerline, with the runs for each position repeated over a period of

several days. Because of the short length of the towing tank, experiments

were restricted to velocities of 0.5 ft/s (15.2 cm/s). The results obtained

are expected to be conservative for velocities higher than 0.5 ft/s

(15.2 cm/s), since the cable-suspension data analysis indicated that interference effects decrease as velocities increase. Unfortunately, this low

velocity introduces considerable scatter into the results,

A dimensional analysis shows that the relative change in rotation rate,

Aw/(jj, where o) is the rotation rate unaffected by boundary proximity, is a

function of a Reynolds number and of s/a', where s is the distance from

the meter vertical axis to the wall and a' is the outside radius of the

bucket wheel. Assuming a negligible effect of Reynolds number [7], Aoj/w

becomes a function only of s/a' for each meter. The experimental data

131
for this relationship are shown in figure 16 for both meters. Each point

of Pygmy meter data represents the average of one day's runs, which

typically consisted of about 6 near-wall and 6 centerline runs. For the

Price meter only the overall averages are plotted.

The response of the current meters to wall proximity clearly depends

upon the meter orientation, i.e., whether the "up stream- facing" or

"downstream facing" cups are next to the wall. The results in figure 16

suggest that wall proximity effect does not exceed about 0.5 percent if

the meter is kept a dimensionless distance of s/a' = 2 from the wall,

corresponding to bucket tip clearances of 1 inch (2.5 cm) and 1-1/2 inches

(3.8 cm) for the Pygmy and Price meters respectively.

Bottom effect . Neglecting any possible Reynolds number dependence,

Aw/o) due to bottom proximity is a function of h/a, where h is the distance

from the horizontal axis of the meter to the bottom and "a" is the radius

of the bucket face. Experimental results are shown in figure 17, with the

individual points again representing daily averages. The interpretation

of these data is somewhat more uncertain than for the wall effects, but

it appears that bottom proximity effects can be kept under 1 percent if

h/a is larger than about 8 for the Pygmy meter and about 6 for the Price,

corresponding to axis distances of about 3 inches (7.6 cm) and 6 inches

(15.2 cm) respectively. These figures are in reasonable agreement with

existing U.S. Geological Survey recommendations [6]. Surface proximity

errors were not investigated because free-surface effects cannot be covered

with such a limited velocity range.

13. Summary

Some performance characteristics and effects of methods of calibration

and use of Price and Pygmy water current meters have been examined with a

view toward their application to velocity-area traverses for flowrate

determination. The improved accuracy obtained by calibrating meters

individually as opposed to the use of group or type calibrations is shown,

based on NBS calibration experience. The repeatability of meter performance at low velocity, 0.1 < V < 0.5 ft/s (3.0 < V < 15.2 cm/s) , was

investigated with the results shown in figure 7. Differences between

rating equations for rod-mounted and cable-suspended Price meters were

determined experimentally and explained analytically where possible for

Columbus and Elliptic weights. These differences were found to be generally

small for Columbus weights, except at low velocities. For both types of

weights no single constant correction factor could adjust for the cable-rod

differences over the entire 0.25 to 8.0 ft/s (7.6 to 244 cm/s) velocity

range. The potential for systematic errors of 2 to 3 percent in wading

measurements and a consequent need for further research was pointed out

.

A theoretical model for vertical-axis bucket-wheel current meters was

outlined which permitted an estimate of a substantial effect of lateral

velocity gradients. The effects of wall proximity in the absence of

velocity gradient and of bottom proximity were investigated experimentally

at a velocity of 0.5 ft/s (15.2 cm/s) with the results shown in figures 18

and 19. The potential hydrodjmamic effect of turbulence in the flowing

stream on Price and Pygmy meters has not been considered here and will be

the subject of future NBS research.

132
14. Acknowledgment

The repetitive runs for comparison of rod-mounted and Columbus-weight

suspended Price meter ratings described in figure 11 were made by T. Robusto

of NBS using the U.S. Geological Survey towing tank. We gratefully

acknowledge this U.S.G.S. cooperation.

15. References

[1] Smoot, G. F. and Novak, C, E., Calibration and Maintenance of VerticalAxis Type Current Meters, Teohni-ques of Water-ResonToes Investigati-ons

of the United States Geological Survey^ Book 8, Ch. B2, 1968.

[2] Smoot, G. F. and Carter, R. W., Are Individual Current Meter Ratings

Necessary?, Pvoo. Amev. Soo. Civ. Eng., 94, HY2, March 1968, pp. 391-397.

[3] Schneider, V. R. and Smoot, G. F. , Development of a Standard Rating

for the Price Pygmy Current Meter, Jour. Researahj U.S. Geol. Survey

,

4, 3, May-June, 1976, pp. 293-297.

[4] Dickman, R. H., The Rating of Water Current Meters, Water Power,

Sept. 1951, p. 330.

[5] Buchanan, T. J. and Somers, W. P., Discharge Measurements at Gaging

Stations, Techniques of Water Resources Investigations of the United

States Geological Survey, Book 3, Ch. A8, 1969.

[6] Bureau of Reclamation, Water Measurement Manual, U.S. Dept. Interior,

Second Ed., Revised Reprint, 1974.

[7] Lofquist, K. E. B., Notes on Current Meter Thoery, unpublished

manuscript. Fluid Mechanics Section, National Bureau of Standards,

1975.

[8] Schoof , R. R. and Crow, F. R. , Discussion of Reference (3) , Proc. Amer.

Soc. Civ. Eng., 9h_, HY6, Nov. 1968, pp. 1596-1601.

133
Table 1

Evidence of Effect of Drift Current

at V = 0.24 ft/s. Pygmy Meter

Run of 6-11-74 Run of 6-12-74

N/V N/V

Round Trip Northbound Southbound Northbound Southbound

1 0.837 0.896 0,862 0.862

2 0.826 0.913 0.857 0.840

3 0»861 0.880 0.876 0.845

4 0,843 0.900 0.876 0.890

5 0.841 0.885 0.874 0.882

6 0.847 0.894 0.871 0.884

7 0.835 0.905 0.885 0.878

8 0.849 0.900 0.880 0.870

9 0.846 0.895 0.877 0.878

10 0,842 0.915 0.880 0.883

11 0.846 0.902 0.874 0.871

12 0.855 0.904 0.870 0.878

13 0.837 0.920

134
^i*^^^)^ iiii i M

FIGURE 1. PRICE AND PYGMY WATER CURRENT METERS.

0.46

^ 0.44

\

S 0.42

0.40

0.1

1972, new bearings--^... '

j". j^^^'^'^^C^^ "

^1972

— /

y

/

'~~~l968j meter new

- std deviation

>> spread, based on

' round trips

f1 1 0.3 1.0

Velocity, V, ft/s

3.0 10.0

FIGURE 2. n/v for THREE SETS OF REPETITIVE CALIBRATIONS

OF A SINGLE ROD-MOUNTED PRICE METER.

135

I
3.5

"T

«: .0

:30- 2.5

ki

rat

c:

1.0

0.5

0^

one meter^ \ repetitive ratings \

(catile)

^ a

'

\\

\ o \ A \

coble (groups)

-rod

one meter

repetitive ratings (rod)'

Price teeters

O Group of if maters

A+

V

X

-oi2159

722

^9

30

40

0.2 0.5 1.0 2.0

Velocity , fps

5.0 8.0

FIGURE 3. STANDARD DEVIATIONS OF CALIBRATION EQUATIONS

FOR PRICE METERS.

3.5

<0

.0

to

3.0

2.5—

2.0

^ l.t

«0

1.0

0.5

X

1'^

Pygmy Meters

-o- V

one meter, repetitive

calibrations -. ^

o Croup of 25 meters

m II 22

+

a II 6

V

H « 9

X

It 7

•

0 II 49

n

0 M 25

A

M H 25 H II 7

-•-

m II 24

0

m M 47

-0 a a 30

A a a 20

f

0.2 0 5 1.0 2.0

Velocity , fps

5.0 8.0

FIGURE 4. STANDARD DEVIATIONS OF CALIBRATION EQUATIONS

FOR PYGMI METERS.

136
FIGURE 5. STANDARD DEVIATIONS OF PYGMY METER RATING

EQUATIONS FROM GROUF EQUATIONS.

137
^ o

2

O

1

o

a\

1 1 1 1 1

1

(a) Each Pass

1

1 1 1 1 1 1 pygmy Meters

o Group of 25

„ "IS

A " " g

o- 2 NB5(repetitive)

-o ^-^^^

o-

^

(b) Round Trips

1 1 1 1 l"^ f — V r <.-rrgrr^

1 1 1 1 1 1

0.1 0.3 1.0

\/(^lnrifv V ff .^s

3.0 lO.C

FIGURE 6. DEVIATIONS OF MEASURED VELOCITIES FROM INDIVIDUAL CALIBRATION

EQUATIONS.

138
8

C:1)

>^

\

I

1

~ A

1 1 1 1 1

1

1

1 1 ! 1 1 1

A

\ ^

o Price meter no. 1 A Price meter no. 2

(solid symbols are

for round trips)

-k. \

A \

1

A A

^ tow tank

1

~~°L — J C£— 0

water tunne!

• •

-

1

•*

1 r 1 1 1 1

2

o

0.1 O.J 1.0

Velocity, V, ft/5

3.0 10.0

FIGURE 7(a). DEVIATIONS OF n/v FOR TWO PRICE METERS.

1

A

1 1 1 1 1 1 1

0 Pygmy meter not A Pygmy meter no. 2

(solid symbols are for round trips.)

1 1 1 1 1 1

A a

•\ o

\d,

A

daily ai^gs --..^•

1

/water tunnel rz o s

1 ^1 1 M 1 • 1 1 1 1 1 1 1 1

^0

0.1 03 1.0

Velocity, V, ft/s

3.0 10.0

FIGURE 7(b). DEVIATIONS OF n/v FOR TWO PYGMY METERS.

139
15 -pound Ellipfic weight 15- pound Columbus weight

FIGURE S. PRICE METERS SUSPENDED WITH ELLIPTIC AND

COLUMBUS WEIGHTS.

8.0

6.0

4.0

2.0

O

Si -2.0

-4.0

1—I—r—

r

I I I I

Key. 15 05.3

"1 1—I—I—r—r

distance, ins. , between

meter and wgt mounting

holes. —Columbus,C; Elliptic, E

-Weight, pounds.

Columbus weights- -I Price (repetitive)

I5C5.3

—-30C4.9

5006.7

I I

I I I I

0.1 03 1.0

Velocity, V, ft/s

3.0 8.0

FIGURE 9. COMPARISON OF RATING EQUATIONS FOR CABLE AND ROD SUSPENSIONS.

140
0.46 \ I

I

1 1—I—I

I

I

I

1 1

1 1—I—r—T—

r

O./ 0.3 1.0 3.0 10.0

Velocity, ft/s

FIGURE 10. REPETITIVE CALIBRATION OF A PRICE METER

WITH ROD AND COLUMBUS WEIGHT SUSPENSIONS.

0.1 02 0.5 1.0 2.0 5.0 8.0 Velocity , fps

FIGURE 11. COMPARISON OF N/V FOR ROD AND ELLIPTIC

WEIGHT CALIBRATIONS.

141
X /a

FIGURE 12. VELOCITIES AROUND AN ELLIPTIC WEIGHT.

x/a

FIGURE 13. VELOCITIES AROUND A COLUMBUS WEIGHT.

142
FIGURE lif. EFFECT OF "CYLINDRICAL WADER" ON VELOCITIES.

1 - pa^J C [v -0)0) de

C = C + AC, for (v -a)Q)> 0

e e

Cq=-(C-AC), for (Ve-cjQ)<0

Vq = V cos 9

FIGURE 15. ELEMENTS OF ANALYTICAL MODEL.

143
1.0 2.0 3 0 4.0

s/a

FIGURE 16. WALL EFFECT

5

•V. k

\

c:

1.030

1.0 20

I OlO

I.OOO

.990

.980

.970^

\

^ 1.030

1 1 1 1 1 1 1 - , (a) PYGMY

/I

^/r^ r''

X

o

///.

X

o

' 74 data

,

'75 "

• Average

1.0 20

1.0 10

I.OOO

.990

.980

.9 70

O

(b) PRICE

o

as. .•

77777777777777777777777777 o

o

o

8

Xo

J.

1 2 3 4 5 6 7 8

h/a

FIGURE 17. BOTTOM EFFECT

144
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS

,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

FLOW MEASUREMENTS IN THE CANALS OF POWER PLANT

CLOSED LOOP COOLING SYSTEMS

Joseph F. O'Brien, James Skridulis, John Annett, and G. Wayne Singley

NUS Corporation

Rockville, Maryland 20850

Harold L. Koenig

Commonwealth Edison Company

Chicago, Illinois 60690

Measurements were made of the flows in the

intake and discharge canals of three large power plants

as part of a comprehensive study on the thermal performance of various closed loop cooling systems. The

types of cooling systems studied were spray canal,

cooling lake, and spray canal/cooling lake combination.

The flows were measured using velocity-area and dye

dilution techniques. Velocity-area flow measurements

were made at two of the plants under constant plant

operational conditions. Analysis showed that the

experimental error in these measurements was generally

less than the estimated total standard error. This result

was expected because of the relatively uniform nature

of the canal cross-sections. At one of the plants, dye

dilution and velocity-area flow measurements were used

to develop a stage-discharge equation for a wing-tipped

weir. Subsequent validation measurements indicated

that the variation of the measured flow rates from the

flow rates predicted by the weir equation was less than

the probable experimental error. Discharges calculated

by the velocity distribution method were compared to

those found by the one-point and two-point methods for

various numbers of verticals per transect. It was found

that the calculated discharge was not particularly sensitive to the number of points per vertical, but was affected

by the number of verticals per transect,

145
Key Words: closed-loop cooling system; cooling lake;

dye dilution method; error analysis; flow rate measurement; open channel flow measurement; power plant;

spray canal; stage-discharge equation; velocity-area

method; weir equation,

1 . Introduction

Flow measurements were made in the intake and discharge canals

of three large power plants with different types of closed loop cooling

systems: spray canal, cooling lake, and spray canal/cooling lake combination. These measurements were part of a comprehensive study on the

thermal performance of the cooling systems carried out for Commonwealth

Edison Company. Data on canal flows were required to obtain mass

balances for the plant steam electric system. In addition, plant discharge flows were used in conjunction with condenser heat rejection

models to determine variable heat loading on the cooling system. The

flows were measured by the velocity-area and dye dilution methods.

Approximately 50 transects of the various canals were made using

an Ott propeller current meter. The data generated in these transects

were used to compute canal flow rates by velocity-area techniques. For

two of the transects, sufficient vertical resolution was maintained so as

to allow a comparison of flow rates calculated by the velocity-distribution

method with those found by the one-point or two-point methods. Another

area of investigation involved transects made while plant operational conditions were constant. The flow rates calculated for these transects

permitted an assessment of potential errors in measurement procedures.

At one of the plants, data from dye dilution and velocity-area flow

measurements were used to derive a stage-discharge equation for a wingtipped weir. Subsequent velocity-area measurements were made to

validate the derived weir equation.

2. Description of Plants

Powerton Station (Figure 1) consists of two 840 MWe(net) coalfired fossil units. The station is designed for closed-cycle operation

using a cooling lake. Circulating water is discharged from the condensers

into a discharge canal which flows to a 1 ,426 acre cooling lake. Water

withdrawn from the lake passes through an intake canal and is returned

to the condensers by six circulating water pumps (each rated at 256.6 cfs)

.

146
Dresden Nuclear Station (Figure 2) consists of three boiling water

reactors: one 200 MWe unit and two 809 MWe(net) units. The smaller

unit is designed for once-through operation. The larger units are designed

for closed-cycle operation using a spray canal/cooling lake combination.

Circulating water is discharged from the condensers of the larger units

into a discharge/spray canal where the water passes through an array of

spray modules. Six lift pumps (each rated at 372,4 cfs) raise the water

approximately 22 feet to a 1,275 acre cooling lake. Water from the lake

flows over a gated- spillway structure to the intake canal and is delivered

to the condensers by six circulating water pumps (each rated at 350.1 cfs).

Quad Cities Nuclear Station (Figure 3) consists of two 809 MWe (net)

boiling water reactors. The station is designed for closed-cycle operation

using a spray canal. Circulating water is discharged from the condensers

to the discharge canal where six lift pumps (each rated at 372.4 cfs) raise

the water approximately 17 feet to the spray canal. The water passes

through an array of spray modules and over a wing-tipped weir to an intake bay where six circulating water pumps (each rated at 350.1 cfs)

deliver the water back to the condensers.

3. Velocity-Area Method

J Measurement of flow rate by the velocity-area method [l,2,3,4,

5,6] is explained by reference to Figure 4. The depth of each sub-area

is measured at prescribed width stations. The velocity is measured with

a current meter at one or more points in the vertical. The measured

widths, depths, and velocities are used to calculate the flows for each

sub-area. These flows are then summed to give the total flow.

2

An Ott current meter with a type 1 propeller was used for the

velocity measurements (Figure 5). The current meter propeller is designed to be accurate when set to within + 5° of the flow direction. The

meter is mounted on a relocating device that slips over a 20 mm diameter

support rod. The support rod is marked at 0.1 foot intervals. Propeller

revolutions are counted over a pre-set time by an Ott counter. The

meter was calibrated by the manufacturer.

Figures in brackets indicate the literature references at the end of

this paper.

Model C31 Ott universal current meter and Model ZlOO counter. Epic,

Inc., 150 Nassau St., New York, N.Y.

147
Steel cables stretched across the width of each canal were marked

at 5 foot intervals. Depth and velocity measurements were made at these

5 foot intervals from a flat-bottomed boat attached to the cable. Measure

ment procedure involved lowering the support rod, with current meter

attached, until it touched the canal bottom. After careful positioning to

insure that the rod was vertical, the canal depth was measured to the

nearest 0.1 foot. Then, after checking that the propeller was directed

into the flow, the velocity at each desired point in the vertical was

measured by counting propeller revolutions for 60 seconds.

The average velocity for each width station was determined by the

two-point and/or the velocity-distribution method. For the two-point

method, the velocity was measured at 0.2 and 0.8 of the water depth.

For the velocity-distribution method, the velocity was measured at 0.2,

Oo4, 0.6 and 0.8 of the water depth for depths between 2 and 5 feet, and

at every tenth of the water depth for depths greater than 5 feet. For both

methods, the velocity was measured at 0.6 of the water depth for depths

less than 2 feet. The average velocity at each station was the arithmetic

mean of the measurements.

The total flow rate for the cross-section was calculated by the midsection method as follows.

+ (1)

where (refering to Figure 4)

Q flow rate (cfs)

water depths at Station 1 and 2,

respectively (ft)

V , V

1' 2

velocities at Stations 1 and 2,

respectively (fps)

^l'^2'^3 distances along water surface from

shore to Station 1 , Station 1 to Station 2

,

and Station 2 to Station 3, respectively (ft)

148
4. Dye Dilution Method

Measurement of flow rate by the dye dilution method [ 3,7,8,9,10]

involves injecting a known amount of dye into the flow and determining

the dye concentration downstream. Two methods are commonly used for

dye injection: slug injection and continuous injection. The latter method,

which was the one used in this study, involves the injection of a dye

solution of known concentration at a known, constant rate. The point of

injection should be sufficiently upstream to insure uniform dye concentration throughout the downstream measurement cross-section. As shown

in Figure 6, a plateau, or level of concentration, develops at the measurement cross-section. Under these conditions, and, assuming that the

injection rate is much less than the stream flow, the rate of flow is given

by

Q=q-| (2)

where

Q = flow rate (cfs)

q = dye injection rate (cfs)

c = dye concentration in injection

solution

C = dye concentration at measurement

cross-section

The dye concentrations can be determined by measuring fluorescence.

However, since the ratio c/C is the dilution factor, absolute values of the

concentrations are not required. A set of standards is usually prepared

by serial dilution of the injection solution. Thus, if F^^ is the fluorescence

of the injection solution diluted by a factor K, and F^ is the fluorescence

of flow sample, then the flow rate is given by

^1 Q=qK~ (3)

2

3 Rhodamine WT dye was diluted with water to give an approximate

injection solution concentration of 10%. A serial dilution process was

Rhodamine WT (20% aqueous solution), E.I. Du Pont, P.O. Box 96240,

Chicago, 111. 60693

149
used to prepare a set of dilution standards. The dilutions were made using

5 ml volumetric pipets and Class A volumetric flasks. The tolerance for

the pipets was + 0.01 ml, while the tolerances for the volumetric flasks

were + 0.2 ml, + 0.3 ml, and + 0.5 ml for 500 ml, 1000 ml, and 2000 ml

flasks, respectively.

4

The dye solution was injected by a positive displacement pump.

The pump was driven by an A.C, frequency locked motor to assure constant pumping rates. The stroke length was variable to allow pumping

over a range of from near 0 to 108 ml/min. The pumping rate was measured before and after injection by pumping into a 500 ml graduated

cylinder for a time interval measured with a stopwatch.

Samples at the measurement cross-section were collected in

polyethylene bottles at five minute intervals. Samples were checked for

fluorescence as collected to determine the arrival of the dye cloud.

With the arrival of the dye cloud, three samples across the width of the

cross- section were taken in each time interval. Sampling continued at

this schedule until the dye concentration decreased to the pre-injection

level.

Fluorescence of Rhodamine WT is affected by changes in temperature

as follows

where

= F^ exp|o.02 6 (T^-T^)} (4)

F = calculated fluorescence at reference

r temperature T

r F = measured fluorescence at sample

s temperature T^

T = sample temperature (°C)

s T = reference temperature ( C)

4Model RP-l-G-400 positive displacement pump. Fluid Metering, Inc.,

48 Summit St., Oyster Bay, N.Y. 11771

All fluorescence measurements were made using a Turner Model 111

fluorometer. (G.K. Turner Associates, 2524 Pulgas Ave. , Palo Alto,

Calif. 94303)

150
A 1 C rise in sample temperature will result in a decrease in the

fluorescence of about 2.6%, Thus it is essential that the fluorescence

of samples and dilution standards be measured at the same temperature.

The samples and the dilution standards were allowed to reach room

temperature before fluorescence levels were determined. The arithmetic

mean of all the fluorescence values in the plateau was used in calculating the flow rate,

5, Powerton Station Measurements

Flow rate measurements at the Powerton Station were made using

the two-point velocity-area method. The locations of the measurement

cross-sections are shown in Figure 1, The cross-sections are located

in straight reaches of the canals and are remote from bends and artificial

obstructions. Both the intake and discharge canals are trapezoidal in

shape with a bottom and a top width of about 35 feet and 95 feet,

respectively, and a depth of about 10 feet (Figure 7), The bottoms of

both canals are fine dirt with an occasional 12 inch stone. The sides

are covered with 1-2 foot stone rip-rap.

Seven measurements were made with one unit (3 circulating water

pumps) operating. Plant operational conditions were essentially constant for these measurements. The results are shown in Tables 1 and 2

for the intake and discharge canals, respectively. The mean flow in

the intake canal was 786.5 cfs, with a maximum deviation from the mean

of 15,2 cfs (1,9%) and a standard deviation of 8.8 cfs. The mean flow

in the discharge canal was 853.8 cfs, with a maximum deviation from the

mean of 38.5 cfs (4.5%) and a standard deviation of 20.9 cfs.

6. Dresden Nuclear Station Measurements

Flow rate measurements at the Dresden Nuclear Station were made

using the two-point velocity-area method. The locations of the measurement cross-sections are shown in Figure 2. The cross-sections are

located in straight reaches of the canals and are remote from bends and

artificial obstructions. Both the intake and discharge canals are

trapezoidal in shape (Figure 8), The intake canal has a bottom and top

width of about 25 feet and 85 feet, respectively, and a depth of about

9 feet. The discharge canal has a bottom and top width of about 45 feet

and 95 feet, respectively, and a depth of about 15 feet. The bottoms

of both canals are fine dirt with an occasional 12 inch stone. The sides

are covered with 1-2 foot stone rip-rap.

151
Five measurements were made with one unit (3 circulating water

and 3 lift pumps) operating. Plant operational conditions were essentially

constant for these measurements. The results are shown in Tables 3 and

4 for the Intake and discharge canals, respectively. It should be pointed

out that the relatively large variations in canal top width and flow crosssectional area noted in the discharge canal data (Table 4) are due to

imbalances between lift and circulating water pumps. These variations

should not materially effect the magnitudes of the flows being measured.

The mean flow in the intake canal was 1110.4 cfs, with a maximum

deviation from the mean of 24.2 cfs (2.2%) and a standard deviation of

16.8 cfs. The mean flow in the discharge canal was 1079.8 cfs, with a

maximum deviation from the mean of 14.1 cfs (1.3%) and a standard

deviation of 8.7 cfs.

7. Quad Cities Nuclear Station Measurements

Flow rate measurements at the Quad Cities Nuclear Station were

made using the velocity distribution and two-point velocity-area methods

and the continuous injection dye dilution method.

The location of the measurement cross-section for the velocity-area

measurements is shown in Figure 3. The cross-section is about 50 feet

upstream from a wing-tipped weir and about 50 feet downstream from an

array of spray modules. The spray canal is trapezoidal in shape with a

bottom and top width of about 120 feet and 190 feet, respectively, and a

depth of about 10 feet (Figure 9). The canal bottom is sand over a PVC

liner. The sides are covered by 3-6 inch stone rip-rap.

The locations of the injection and measurement cross-sections for

the dye measurements are shown in Figure 3. The measurement crosssection is about 1 ,000 feet downstream from the weir. Before reaching

the cross -section the flow traverses a concrete- lined flume and an

energy dissipater. Dye concentrations at the measurement cross-section

were not well mixed when the dye was injected at the weir or from a

bridge about 1 ,100 feet upstream from the weir. Injection from a bridge

about 11 ,000 feet upstream from the weir did give good mixing for 2 and 3

lift pump operation, but not for 5 lift pump operation. For the 5 lift pump

case, injection at the lift pumps, about 16,000 feet upstream from the

weir, resulted in good mixing. The time histories of dye concentrations

at the measurement cross-section for 2,3, and 5 lift pumps are shown

in Figure 10.

152
An initial set of dye and velocity-area measurements was made at

Quad Cities to determine a stage-discharge equation for the wing-tipped

weir in the spray canal. The form of the equation was

Q

where

Q

C

e

L

e

H

The data from these measurements are presented in Table 5. A subsequent

set of flow measurements was made to confirm the validity of the derived

relationship. These data are given in Table 6.

8. Analysis and Discussion

Analyses were carried out in three areas with regard to the flow

measurement data collected for the present study. These included: (1)

the estimation of the total standard error expected for the velocity-area

method measurements; (2) an assessment of the calibration and validation

of the stage-discharge equation developed for the weir in the Quad Cities

spray canal; and (3) a comparison of the discharges calculated by the

one-point, two-point, and velocity distribution methods. Each area of

investigation is discussed below.

Velocity-Area Measurements: Error Analysis

The overall error associated with a physical measurement consists

of a random component and a systematic component. A simplified estimate

of the random standard error for flow meter measurements of the type performed for this study is given by [5] as

= C L H

' e e

= flow rate (cfs)

= effective discharge coefficient

= effective crest length (ft)

= depth of water above weir crest

upstream from the weir (ft)

153
where X' is the percentage random standard error in the discharge 0.

The othe? quantities in equation (6) are defined as follows.

X = m

error associated with the number of

verticals per cross-section. Typical

values of X are + 5,3 and 1% for

8,15, and ?b verticals/cross-section,

respectively [5] .

X^ = error associated with depth measurement.

A value for X^ of + 0.7% has been estimated

for alluvial channels of the depths encountered

in this program [5] .

X = error associated with the determination of

the mean velocity for each vertical.

m = number of verticals per cross-section.

The error associated with the determination of the mean velocity for each

vertical can be expressed by combining the local point velocity error

with that for the number of points at which velocity measurements are

made [5] . That is,

X = +

V —

^f— 2

+ X

P o

1/2

(7)

where

X^ = error associated with the local point velocity

measurement

X = error associated with the number of points

per vertical

p = number of points per vertical

A value for X^ of + 6% has been suggested where the period of each

measurement is 40 seconds or approximately 100 propeller revolutions

[5] . These conditions are similar to those in the present study. Guides

as to the value of X^ have been given as + 3.5, 3.0, and 0.5% for the

one-point, two-point, and velocity-distribution methods, respectively [s]

15A
The major source of systematic error for velocity area measurements

will arise from errors in current meter calibration. This error is on the

order of + 0.5 to + 1% for an individually rated meter [5] . A second

potential systematic error may be caused by differences in water temperatures at calibration and under field conditions. The manufacturer of the

Ott current meter used in this program states that special oil used in the

hub of the propeller allows measurements to be made at various temperatures without any decrease in accuracy [ll] , However, an Ott propeller

current meter of a different type exhibited a change in velocity of about

1% for a change in temperature of from 70 to 40 f[6] . Since water

temperatures in the canals were frequently more than 90 F, the 1% value

for this constituent of the systematic error was adopted. As a consequence, by using an equation similar to (6), and by assuming the calibration error to be + 1%, the systematic standard error (X" ) can be

estimated as being + 1 .4%,

The overall estimate of the standard error in the discharge is given

by [5]

1/2

(8)

was calculated for various numbers of verticals per cross-section for

the one-point, two-point, and velocity distribution methods. The estimated values for the individual error components presented in the foregoing were used in the calculations. The calculation results are presented in Table 7.

The total percentage standard error for the two-point velocity-area

measurements (15-20 verticals) at Dresden and Powerton were 1.5 and

0.8% for the Dresden intake and discharge canals, respectively, and 1,1

and 4.5% for the Powerton intake and discharge canals, respectively.

From Table 7 it can be seen that these experimental errors are generally

less than the estimated total standard errors. This result would be

anticipated because of the relative longitudinal and cross-sectional

uniformity of the canals and the generally well-behaved nature of the

measured flows. The reason for the higher than expected experimental

errors in the Powerton discharge canal is not known at this time.

Weir Equation Calibration and Validation

As discussed in Section 7, an initial set of dye and flow meter

measurements were made at the Quad Cities Nuclear Station to determine

a stage-discharge equation for the wing-tipped weir in the spray canal.

155
These data are shown in Table 5. The form of the proposed weir equation

was

Q = C L H^"^^ (9)

e e

where the quantities in equation (9) are the same as those defined for

equation (5) . The terms and Le in equation (9) are functions of weir

head (H) , velocity of approach (v) , and weir geometry. It was assumed

that the dependency of Cg and Lg on these parameters could be expressed

through a representation similar to the Francis formula. That is.

e e \ H / \ H

2

V /2g

(10)

To complete the discharge equation development, the velocity of approach

was correlated to weir head by making use of the Ott meter data, and the

constant M was determined from a statistical analysis of both the Ott

meter and dye dilution data. The resulting weir equation is given below.

Q = 772 {(H.h)3/2-h^/2} (11)

where

h = (0.78H -0.31)^/2 g

2

g = gravitational acceleration (ft /sec)

A plot of the weir equation, along with the initial set of data is presented

in the upper graph of Figure 11.

A subsequent set of flow measurements were made to validate the

weir equation. These data are shown in Table 6 and the lower graph of

Figure li. From Table 6 it can be seen that the validation flow measurements agreed with the weir equation predictions to within 2%. From Table

7 it can be seen that this variation is less than the probable experimental

error for the measurement procedures used.

Velocity-Area Measurements: Comparison of Methods

The velocity-distribution method was used for a mid-canal transect

made in conjunction with the weir equation validation tests at Quad Cities.

The data from this transect were used to compute discharges by the

156
one-point and two-point methods as well as by the velocity-distribution

method. These calculations were carried out for mid-section intervals of

5,10, and 20 feet. The mid-section intervals correspond to about 40,20,

and 10 verticals per transect, respectively, or about 2.5, 5, and 10% of

canal width per interval, respectively. The results of the calculations

are given in Table 8.

These results show that the calculated discharge was not particularly sensitive to the number of velocity measurements per vertical.

For example, for the 5 foot mid-section interval, the discharges calculated

by the three methods differed from the mean of the three discharges by less

than 0.5%. This finding, which is consistent with the standard error

estimates given in Table 7, was an important one for the present study.

This was because of a more than four-fold decrease in measurement time

for the two-point method as compared to the velocity distribution procedure.

Another finding, also consistent with the standard error estimates, is that

experimental error increases relatively quickly with decreasing number of

transects per cross-section. For example, the means of the flows computed for the 10 and 20 foot intervals differed from the 5 foot interval

mean by 0.4% and 6.9%, respectively. This error, however, appears to

be more systematic than random in that the increased flows calculated for

the larger intervals were primarily due to over estimates of mean velocity

in near shore areas.

9. Conclusions

The following conclusions are reached regarding the flow measurements and analyses reported herein.

(1) The experimental error in the flow rates measured at Powerton

Station and Dresden Nuclear Station were, as expected, generally less

than the estimated total standard error.

(2) For measurements of flow rate by dye-dilution methods,

preliminary experiments should be carried out to insure complete mixing

at the measurement cross-section under all expected flow conditions.

(3) The stage-discharge equation for the weir in the Quad Cities

Nuclear Station spray canal predicts flow rates to within the probable

experimental error of validation measurements.

(4) Flow rate calculations for these canals are not particularly

sensitive to the number of points per vertical, but are affected by the

number of verticals per transect.

157
10. References

Buchanan, T.J. and Somers, W.P., "Discharge Measurements at

Gaging Stations," Book 3, Chapter A8 of Techniques of WaterResources Investigations of the U.S. Geological Survey , Geological Survey, U.S. Department of the Interior, 1969,

"Current Meters," Chapter 5 of Water Measurement Manual , Bureau of Reclamation, U.S. Department of the Interior, Second

Edition, Revised Reprint, 1974.

"Instruments and Methods of Observation, " Chapter 2 of Guide

to Hydrometeorological Practices , WMO-No. 168. TP. 82, World

Meteorological Organization, 1970.

Kulin, G. and Compton, P.R., "A Guide to Methods for the

Measurement of Water Flow," NBS Special Publication 421,

National Bureau of Standards, U.S. Department of Commerce,

May 1975.

"Liquid Flow Measurement in Open Channels — Velocity-Area

Method," International Standard I. S.O. 748, 1973. (Obtainable

from America 1 National Standards Institute, 1430 Broadway,

New York, N.Y. 10018).

Johnson, R.L. , "Laboratory Determination of Current Meter

Performance," Technical Report No. 843-1, Division Hydraulic

Laboratory, North Pacific Division, U.S. Army Corps of Engineers,

November 1966

.

Wilson, J.F., "Fluorometric Procedures for Dye Tracing,"

Book 3, Chapter Al 2 of Techniques of Water-Resources

Investigations of the U.S. Geological Survey , Geological

Survey, U.S. Department of the Interior, 1968.

Replogle, J. A., Myers, L.E., Brust, K.J., "Flow Measurements

with Fluorescent Tracers," Journal of the Hydraulics Division,

ASCE, Vol. 92, HY5, pp. 1-15, September 1 966

.

Kirkpatrick, F.A. , Sayre, W.W. and Richardson, E.V., "Flow

Measurements with Fluorescent Tracers" Journal of the Hydraulics

Division, ASCE, Vol. 93, HY4, pp. 298-308, July 1967.

158
[lO] Kirkpatrick, F.A. , "Flow Calibration by Dye-Dilution Measurement," Civil Engineering, February 1968, pp. 74-76.

[ll] "Instruction Manual, " Ott Universal Current Meter C31 (10.002)

Table 1. Flow Rate Measurements for the Powerton

Station Intake Canal

Date Top Width Cross-Section Flow Rate Average Velocity

Area

(ft) (ft^) (cfs) (fps)

21 Aug 75 95„3 658o3 801 .7 1 .22

30 Aug 75 92.8 641,2 778.0 1.21

30 Aug 75 92.7 647.9 790.2 1.22

31 Aug 75 94,0 646.8 786,1 1 ,22

31 Aug 75 94.1 647.9 787.7 1 ,22

1 Sept 75 94,0 653.8 774,4 1.18

1 Sept 75 94,2 654.5 787.3 1.20

Table 2. Flow Rate Measurements for the Powerton

Station Discharge Canal

Date Top Width Cross-Section Flow Rate Average Velocity

Area

(ft) (ft^) (cfs) (fps)

21 Aug 75 91.2 714,4 866.1 1.21

29 Aug 75 90,5 705,5 857.1 1.21

30 Aug 75 90.0 705,0 883.2 1.25

31 Aug 75 91.2 699,1 857.5 1,23

31 Aug 75 91,5 696,7 852,8 1,22

1 Sept 75 91.5 702.7 815.3 1.16

1 Sept 75 91.5 701.8 844.5 1.20

159
Table 3. Flow Rate Measurements for the Dresden

Nuclear Station Intake Canal

Date Top Width Cross-Section Flow Rate Average Velocity

Area

(ft) (ft^) (cfs) (fps)

7 Aug 75 85.0 483o3 1111.0 2.30

7 Aug 75 85,0 479.3 1126.1 2o35

8 Aug 75 85c3 480<,4 1125.9 2.34

9 Aug 75 85.0 475c4 1086.2 2.28

10 Aug 75 84„7 472o2 1102.6 2.34

Table 4. Flew Rate Measurements for the Dresden Nuclear

Station Discharge Canal

Date Top Width Cross-Section Flow Rate Average VelocityArea

Ift) (ft^) (cfs) (fps)

7 Aug 75 91.5 916.3 1081.5 1.18

7 Aug 75 93.0 930,4 1077.9 1.16

8 Aug 75 86.5 832.2 1093.9 1.31

9 Aug 75 93.8 928.6 1072„9 1.16

10 Aug 75 94e8 948,0 1073.0 1.13

Table 5. Quad Cities Nuclear Station Flow

^feasurements (weir Calibration)

Date

10 July 75

8 July 75

22 May 75

19 July 75

3 Aug 75

Number of

Lift Pumps

2

3

3

5

5

Weir Head

(ft)

1.07

1.33

1.38

1.90

lo92

Flow Rate Flow Rate

(dye dilution) (Ott meter)

(cfs) (cfs)

833

1252

1269

2072

804

1216

2043

160
Table 6. Quad Cities Nuclear Station Flow Measurements

(Weir Validation)

Date Number Weir Flow Rate (A) Flow Rate (B) Percent

of Lift Head (Ott meter) (weir equation) Difference

Pumps

(ft) (cfs) (cfs)

(A-B/B)

(%)

28 July 76 3 1.34 1221 1208 1.1

13 Aug 76 3 1.36 1241 1236 0.4

14 Aug 76 3 1.35 1202 1222 -1.6

Table 7. Estimated Total Standard Error

in the Discharge (%)

Method Number of Verticals

10 15 20 40

One-Point 5.1 3.8 3.4 2,4

Two-Point 4.9 3.6 3.3 2.3

Velocity Distribution 4.7 3.4 3.1 2.1

Table 8. Comparison of Calculation Procedures

Mid-Section

Interval

(ft)

5

10

20

Q

(one-point

method

(cfs)

1295

1302

1382

Q

(two-point

method

(cfs)

1289

1289

1377

Q

(velocitydistribution

method)

(cfs)

1296

1304

1389

Mean

(cfs)

1293

1298

1383

161

163

FIGURE 4

CROSS-SECTION SHOWING

LOCATION OF POINTS OF

VELOCITY MEASUREMENTS

165

I II III

DISTANCE

(a) SPATIAL DISTRIBUTION

(b) TIME - CONCENTRATION CURVES

FIGURE 6

DYE CLOUD AS IT DISPERSES DUE

TO SINGLE POINT, CONSTANT RATE

OF DYE INJECTION UPSTREAM, AFTER

KIRKPATRICK, ET AL., [9]

167
DISTANCE (FEET)

FIGURE 7

CROSS-SECTIONS OF POWERTON STATION CANALS

ON 1 SEPTEMBER 1975

168
DISTANCE (FEET)

FIGURE 8

CROSS-SECTIONS OF DRESDEN NUCLEAR STATION

CANALS ON 9 AUGUST 1975

169
DISTANCE (FEET)

FIGURE 9

CROSS-SECTION OF QUAD CITIES NUCLEAR

STATION SPRAY CANAL ON 13 AUGUST 1976

170
-1 \

^

\

^ \

^

\ r 10 JULY 1975

Q = 833 CFS

L

8 JULY 1975 , Q= 1252 CFS

3 AUG 1975 , Q=2072 CFS •

> > I

\ \ \ I I '

I * 50 100 150 200 250

TIME ( MINUTES)

FIGURE 10

SAMPLE FLUORESENCE AT MEASUREMENT

CROSS-SECTION

171
T—I—I—

r

2.2

2.0

1.8

1.6

1.4

1.2

1.0 —

0.8 ' I I I I

1—I—I—

r

1—I—I—

r

T—I—I—

r

0.8

500

A DYE METHOD

• VELOCITY-AREA METHOD

2.2 1—I—I—I—

r

2.0

1.8

1.6

1.4

1.2

1.0

J I I L

CALIBRATION IN 1975 —

J I I L J I I L J L

T—I—I—

r

1—I—I—

r

T—I\—

r

VALIDATION IN 1976 —

»

I I I J I I L J I I L

1000 1500 2000

FLOW (CPS)

FIGURE 11

WEIR FLOW CALIBRATION AND VALIDATION

IN THE QUAD CITIES NUCLEAR

STATION SPRAY CANAL

2500

172
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, M), February 23-25, 1977. (Issued October 1977)

ERRORS IN FLOW MEASUREMENT AND THEIR IMPORTANCE IN

INFILTRATION/INFLOW ANALYSIS

D. L. Guthrie

Environmental Quality Systems, Inc.

Rockville, Maryland

D. R. Washington and C. Vincenty

Donald R, Washington and Associates

Residencial Tintillo, Bayamon, Puerto Rico

For inf iltration/inflow analysis, flow measurement problems

encountered in warm weather and humid climates include rapid

build-up of floe particles and solids caused by high bacterial

growth rates. Environmental in situ factors not only have a

tendency to shorten equipment life but also to reduce the

accuracy of flow determinations and to lengthen the time

required for field study of infiltration/inflow problems.

Several common flow-measurement devices including the

weir, Parshall flume, and scow were used in field studies of

inf iltration/inflow in Puerto Rico. Mathematical determinations of inf iltration/inf low were made by calculating the

hydraulic gradient, the height of water in the sewer, and the

velocity head loss. Both of these instrumental and physicalmathematical methods for flow determination are reviewed,

stressing the advantages and disadvantages of each, their

sources of error, and methods used to resolve that error,

wherever possible.

Results of infiltration/inflow studies are presented,

emphasizing flow determination errors and their effect upon the

inherent errors in infiltration/inflow assessment. Sewer system

evaluations are not discussed because they are concerned with

the evaluation of the physical problems of the existing sewer

system which would increase infiltration/inflow.

Errors resulting from operational problems with physical

flow-measurement devices are presented, along with a strategy

to reduce these errors. The impact of operational problems

in flow determination on infiltration/inflow assessment are

also presented, along with field-tested resolutions for

those problems.

1. Introduction

Flow determination is one of the most important and critical steps in

infiltration/inflow analysis. There are many ways to measure flow, depending

upon the application of the chosen flow-measurement device.

173
The infiltration/inflow (I/I) studies providing the data base for this

paper were conducted in Puerto Rico. Isabela [l]^and Bayamon [2], Puerto Rico,

including ten other urbanizations were studied by the authors under contract

to the Puerto Rico Aqueduct and Sewer Authority (PRASA)

.

Errors in flow measurement can result from many sources in the field.

The errors, of course, are different for different flow measurement devices.

The most commonly used devices for measuring flow in semi-closed conduits or

sewers are:

(1) Water level recorder

(2) Manning "dipper"

(3) Parshall flume

(4) Pitot tube

(5) Weir

(6) Scow

These devices are either universal in concept and design or are manufactured

commercially.

Sewer flows can also be measured physically. There are two methods:

(1) Hydraulic grade line measurements

(2) Hand measurement of depth of flow

Both instrumental and physical methods will now be reviewed in respect

to their (1) installation, where applicable; and (2) sources of error. Some

field-tested approaches to minimize the error in flow results will be detailed.

Water Level Recorder

Several water level recorders are commercially available. The recorders

used by the authors were manufactured by the Stevens Company.^ Other variations of this instrument have been modified for use as tide gauges and test

well recorders for depth-to-water table measurements. The authors used the

Stevens recorders to measure the head (1) above weirs, (2) in Parshall flumes,

and attached to a scow.

The recorder is mounted on either wooden, metal, or plastic platform,

attached to the wall of the manhole (or flume, etc.) by brackets and nail studs.

Care must be taken to align the float carefully over the exact center of the

usually semi-circular channel in the middle of the manhole. A stilling basin

must then be constructed using PVC pipe, for example, so that the float will

maintain a relatively stable position and will not be subjected to the force

of the current. Only through experience and judgment on a case-by-case basis

can the installer select the correct length of PVC pipe so periodic high flows

will not cause wastewater to spill over on top of the float, perhaps dislodging

it from the stilling basin.

Figures in brackets indicate the literature references at the end of this

paper.

Leupold and Stevens Company, Beaverton, Oregon.

174
Debris accumulation is a problem in using the water level recorder

method. A scum layer of 1/2 to 3 inches invariably forms. It must be

frequently removed (every one to two days) to insure accurate measurements

and to minimize errors.

Another source of error is the dislodging of the stilling basin. The

stilling basin was usually positioned by the authors by wire to nail studs

shot into the walls of the manhole in a triangular fashion radiating from the

stilling basin. During periods of surcharging, not only are the chances of

the stilling basin to be dislodged great, but the recorder itself may be

inundated. Either of these events would cause a loss of data for a significant

period of time.

Calibration of the chart may also cause problems. The water level must

be measured from the bottom of the invert. With an inexperienced field crew,

more training must be initiated by the field engineer to ensure that this is

done properly. The water level must be measured at the same place in the

manhole every time. Slight construction irregularities in the sewer can

introduce a significant source of error to an infiltration/inflow study.

The pen ink must be regularly changed to minimize ink dry-ups and ink blotting

the chart.

Manning "Dipper"

A very common device used in infiltration/inflow studies is the "dipper,"

developed by the Manning Corporation.^ The instrument consists of a cylindrical

housing and a probe. Dippers are available in several lengths, the most common

of which are 15 and 18 feet. Upon contact with the water surface completing

the circuit by grounding, the probe retracts slightly to descend again.

Electronic circuitry transforms the change in the water or wastewater elevation

to a circular chart recorder. The unit is powered externally by a battery.

This device was found to be reliable for only short periods of time. Over

extended time periods, the chance for problems to occur is increased. The units

had to be regularly serviced every one to two weeks. Field problems are as

follows

:

(1) Dead batteries (batteries should be charged every 2 days at a minimum)

(2) Acid spills

(3) Failure to establish a proper ground

(4) Cable corrosion

(5) High temperature and humidity causing arcing and short-circuiting

(6) Fouling of the probe with scum, trash and debris

(7) Inking problems on the chart

Each of these seven field problems, if not rectified promptly and

efficiently, could result in significant data losses. During an I/I study in

Bayamon, Puerto Rico, an 80 percent data loss was sometimes realized at certain

'Manning Environmental Corporation, Santa Cruz, California.

175
stations. Again, the dipper (and probe) must be centralized in the middle of

the channel so that the water level can be measured as accurately as possible.

Parshall Flume

As a temporary field measurement technique, the use of the Parshall flume

is ludicrous from the standpoint of capital expenditure, construction

practicality, and cost-effectiveness. Therefore, their use is largely confined

to permanent facilities, such as sewage treatment plants.

Parshall flumes were found to work well under normal conditions. The

deposition of solids and debris is not extensive, reducing errors in flow

measurement caused by that deposition. The largest problems encountered by

the authors, especially at the Levittown and Sierra Bayamon sewage treatment

plants, were that the flow meters operated by the Puerto Rico Aqueduct and

Sewer Authority were inoperative a large percentage of the time. They had to

be replaced by Stevens meters. Unfortunately, the flow in Parshall flumes was

never calibrated with another flow measurement device such as a weir so that

the discrepancy in flow resulting from using the two methods was never calculated.

Pitot Tube

A usually reliable and accurate flow measuring device, the pitot tube, has

been used for measurement of closed conduit liquid flow for many years. However,

due to the accumulation of solids and debris in open channel sewers, a pitot tube

would become clogged within several minutes. This tendency for solids build-up

increases in small diameter sewers because, in warmer climates, higher evaporation

rates cause the percent solids in the wastewater to be substantially higher than

larger diameter sewers, where dilution tends to reduce the solids concentration.

Thus, the pitot tube is completely impracticable for wastewater flow measurement.

Weir

Because Manning "dippers" are impracticable for flow measurement in shallow,

small diameter sewers (their probes do not retract far enough) , Parshall flumes

are impracticable as temporary flow measurement devices, and pitot tubes have

too many fouling difficulties, the weir becomes a reasonable alternative for

wastewater flow measurement. It is used in conjunction with a Stevens or other

equivalent water level recorder. I

Many problems result in searching for a complete weir formula due to the I

inconsistency of the nomenclature and the specificity of the formulas presented. 1

The foremost of these is the problem of a submerged weir. The authors have found

that V-notch weirs tend to be the most reliable. The formulae most commonly given

for the low flows caused by infiltration are very limited in their accuracy as

their flow approaches the average sewage flow. For normal and peak flow, a

suppressed rectangular weir is preferred. In an effort to obtain the benefits

afforded by both types of weirs, a combination weir was developed which, in the

authors' experience, is a preferable method of metering sewage flow to either

component part. The derivation of a weir formula will be discussed subsequently.

176
Although the weir is a relatively good and maintenance-free flow measurement device, its installation is rather difficult. First, a manhole in good

structural condition, having no sediments on the apron, and preferably 4-7 feet

in depth should be selected. After selection, the manhole surfaces should be

cleaned by scrubbing or water spraying.

Measurements for weir construction appear in Figure 1. These measurements

are optimal, but they can be easily modified depending upon field conditions.

The weir is pre-cut to reduce labor and ensure standardization. Then, the

sluice gate must be cut. Obtaining a proper fit of the sluice gate is very

important to reduce leakage and therefore increase weir accuracy. Several

refittings are necessary to obtain a tight fit, done while the wastewater

is flowing. Finally, a rubber gasket is placed around the edge of the sluice

gate to maintain a waterproof seal.

FIGURE 1. WEIR PLAN VIEW

Before actual weir installation, the sluice gate is fitted to the pre-cut

weir, marking screw hole locations. Four screws or bolts are usually sufficient.

The weir is then installed, placing the aluminum strip on the upstream side. Two

wood braces attached to the weir hold it in position against the manhole wall

using nails shot into the concrete.

177
Packing material must then be placed around the weir sides. It should

be somewhat flexible, allowing for the seal to hold when the wood bends due

to water pressure. The following materials are recommended:

(1) hydraulic cement

(2) tar

(3) plastic.

After the weir is installed, the sluice gate must be positioned. In

order to do this, the flow must be stopped with buckets, sandbags, inflatable stoppers, or whatever else is available.

The sluice gate is lowered and forced into its final position. This

is a real* "boots and gloves" job. Once installed, more insulation is placed

around the weir and attached sluice gate, if needed. Finally, the water

level recorder platform and stilling basin are installed on the upstream

side of the weir, and the water level recorder (Stevens or equivalent) is

placed in operation.

Two major problems are created by weir installation which can result in

significant flow measurement problems. These are:

(1) Solids accumulation behind the weir can become rather extensive.

At a weir installed in the Sierra Bayamon system, a scum mat

six inches thick accumulated [1].

(2) Leaks often create critical flow measurement problems. There is

no actual method for determining the percent volume of wastewater

leaking from the weir, but either regular maintenance to add

packing should be done or a percent volume leak could be estimated.

A derivation of a combination flow equation for the combination weir was

presented in a recent Water and Sewage Works Journal article [3]. The final

expression is a result from a combination of Bernouli's equation and the

standard weir formula for a 90° V-notch weir, and integrating to yield the

formula for the combination weir. Taking into account the expression for C

as revealed by King [4] , the final equation for the combination weir including

velocity of approach is:

This equation says that A is the area of (L) (P + H) , or more correctly,

in a weir with the dimensions shown in Figure 1, A =

= 2.52 (a)

(b)

178
Tables for the solution of the second term in equation (a) are given

in reference 4 starting on page 4-54. A graphical solution for a 90 degree

V-notch weir with L = 44 in. shows that there is little difference for teninch and 16-inch diameter sewers. These appear in the aforementioned Journal

article by the authors.

Despite the small differences between sewers in the 10 to 16-inch range,

it is very easy to have errors. The authors noted that it is most difficult

to measure flow closer than 1/4 inch in the field to set the zero points on

graphs. For example, in an eight-inch sewer at Bajura, Isabela, Puerto Rico,

this amounts to a range of 12,000 ± 5,000 gal/day [1] - a 143 percent maximum

error! For low flow over the weir, using equation (a), the flow at = 1/4 in

is 115 gal/day. At Hj^ = ± 1/4 in., the average relative error, or the average

of the maximum and minimum errors divided by the accepted value, is 176 percent

If weirs are to be used for making infiltration/inflow analyses, these

corrections should be noted. Despite their occasional difficulties of

installation, they have proven to be one of the most reliable and efficient

methods for determination flow in small diameter sewers.

Scow

A scow is a flotable block attached to the side of a manhole by a

vertically-pivoting rod. A wire from the top center of the scow is

connected to a water level recorder located at the top of the manhole.

The scow is particularly useful in estimating flows in extremely deep

manholes or in manholes subject to wide flow fluctuations. The total

change in head allowed for by a scow using a 5 foot radius is approximately

8-9 feet. This type of scow was installed in the Calle Marina section of

Bayamon, where manhole depths approximated 15-20 feet.*^

Errors can occur when frequent and violent surcharging are evident.

In the Calle Marina system, the manhole covers would blow off during periods

of high- intensity , short-duration rainfall, which were frequent. Obviously,

this does not tend to prolong equipment life!

Other problems were created during surcharging conditions. The most

notable of these were:

(1) The balled cable sometimes fell off the cylinder pulley. A man

had to be sent down to retrieve the cable.

(2) During one particularly violent surcharging (exact rainfall data

could not be easily obtained because of convective precipitation.

A standard USGS rain gauge several kilometers away was not representative of areawide precipitation) , the scow was completely

ripped off the wall of the manhole. It was never recovered.

^ It should be noted that in the installation of the scow, the field worke

must wear gas masks when working in such a hazardous environment.

179
(3) When the water level drastically and quickly changed, the drum

recorder charts were almost impossible to read. Sometimes the

ink smeared, blending several lines into one very thick one.

Hydraulic Grade Line Measurements

Flows were occasionally calculated by measuring the height of water

and in order to compute the hydraulic grade line. The technique is useful

for measuring flows in sewers located directly upstream from pumping

stations. This method's reliability, however, is jeopardized by only

periodically measuring the flow. A constant flow record is impracticable

to obtain by this method.

Hand Measurement of Depth of Flow

When rapid measurements of wastewater flow are needed to cross-check

previous measurements and/or calculations, the water level in the sewer

can be measured by an extremely heavy plumb-bob and wire. The Manning

equation [4] could be used to calculate the flow, if the slope is known.

However, this is only a rough estimate for one point in time and should

not be relied upon to contribute toward the study data base.

Measuring Flow in the Field

Field problems are sometimes created by circumstances independent of

the measuring device used. These problems usually require an innovative

approach in their solution and have the potential of causing a great loss

of data.

Major considerations in attempting to measure sewer flow in the field

are:

(1) Traffic diversion problems

(2) Sewer gas buildup and the risk of explosion

(3) Theft of equipment

(4) Deep manholes

(5) Vibrations caused by heavy traffic flow.

These will be further explored at this point. Traffic diversion problems

are minor, but they create a nuisance. Standard orange colors must be worn by

all field workers, and flashing yellow lights may be necessary. An additional

worker may have to be given the job of flagman. This creates the need for more

f ieldworkers , reduces the speed with which the survey is conducted and thereby

reduces the number of stations which can be monitored daily, and lengthens the

duration of the sampling program.

Sewer gas buildup and the risk of explosion tends to increase during

periods of warm or hot weather. In the installation of weirs, the monitoring of flow patterns, and the other miscellaneous tasks necessary to conduct

an I/I study, the manhole must be removed and several minutes be allowed to

pass before entry. Even so, gas masks protecting the eyes, nose, and throat

must be worn upon entry. Common sense prevails here.

180
Not necessarily a cause of error but a cause of data loss, is the

theft of equipment. In an I/l study for Levittown, Puerto Rico [2], a

theft rate of approximately 30 percent was noticed. In other words,

something was stolen one out of every three days. Heavy chains and padlocks

were insufficient. The study had to be extended an extra two months in this

area.

Deep manholes present problems previously mentioned. Again, these

are

:

(1) Difficulty in measuring water elevation

(2) Difficulty in assessing groundwater elevation

(3) Difficulty in centralizing "dipper" probe in the channel

At first thought, a discussion of vibrations from heavy traffic may

appear to be irrelevant to this paper. Several years ago, the authors

thought that also. However, not so. In a manhole in Sierra Bayamon, a Manning dipper was securely installed. Several days later when it was

checked, the dipper, battery, cables and rack were gone. It showed up

on the bar rack of the Puerto Nuevo sewage treatment plant approximately

two months later! Needless to say, a week's data was lost, and the

company would not repair the dipper!

Examples of Flow Measurement

Three examples of flow measurement are given, from the best-case to

the worst-case. These flow measurements were made during August-November

1974 [2]. The sampling locations are shown in Figures 2 and 3.

The first case is shown in Figure 4 and represents wastewater flow

and rainfall vs. time for the Bayamon Gardens Sewage Treatment Plant

effluent. A continuous wastewater flow measurement was obtained, which

correlated very nicely with precipitation in the same area.

Figure 5 shows a worsening field condition, where only a partial

flow measurement was able to be obtained. The sampling station was

located in the Calle Zinia area in Bayamon. The "S" indicates a surcharging condition. However, for more than 50 percent of the time period,

rainfall could be correlated to wastewater flow and/or surcharging conditions.

The worst case of infiltration/inflow causing a loss of data appears in

Figure 6 and represents data from the Calle Columbia Station in Bayamon.

Rainfall could not be correlated to wastewater flow, as the rainfall gauge was

more than two miles away from the sampling station. Additionally, the sewer

surcharged more than 80 percent of the time. The peak flow was more than

300 percent greater than the highest measurable wastewater flow by conventional

methods

.

181
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182
183
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These problems cover the range of those expected in the field. Not

every I/I study will encounter such difficult environmental conditions.

However, the techniques presented herein can serve as a guide to those

attempting to minimize errors while conducting l/l studies, especially

in tropical or semi-tropical environments.

References

1. Donald R. Washington, Inc. and Ing. P. Tejada, Inc. Infiltration/

Inflow Study, Isabela, Puerto Rico for PRASA (in print). 1975.

2. Basora and Rodriguez, A. Heres Gonzalez and Donald R. Washington,

Inc. Facilities Planning for the Bayamon Wastewater Region . Vol. IV: Infiltration/Inflow Analysis for PRASA. April, 1976.

3. Guthrie, D. L. and Washington, D. R. "Using the Weir in Infiltration/

Inflow Analysis." Journal of Water and Sewage Works . 123, 10, 26 (1976)

4. King, H. W. and Brater, E. F. Handbook of Hydraulics . 4th ed.

,

New York: McGraw-Hill. 1954.
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

RATING BROAD-CRESTED V-NOTCH WEIRS

WITH NARROW, SLOPING APPROACH CHANNELS AND SEDIMENT DEPOSITS^

James F. Ruff, Assistant Professor

Civil Engineering, Colorado State University

Keith Saxton, USDA-ARS , Washington State University

Clement Dang, Engineer, Grinnel Fire Protection Systems Inc.

An accurate and reliable precalibrated streamflow measuring device is required to determine the discharge from small

watersheds because rapidly rising and falling streamflow stages

prevent accurate field determination of flow rates by conventional stream gaging techniques. Broad-crested, V-notch weirs

were developed and calibrated by the U. S. Soil Conservation

Service about 1940 to meet these requirements and they have

been used extensively. Because field calibration of these

weirs is difficult and inaccurate, the calibration curves

often have not been verified. Recent evidence has shown that

different approach channels cause significant deviations from

the original calibrations.

Model studies were conducted to investigate the effects

of the different approach channels slopes, cross-sections,

and sediment deposits on the weir calibrations. The fixedboundary channels were tested at slopes of 0.0, 0.5, 1.0, and

1.5 percent. Rectangular, trapezoidal, and circular crosssections were tested with sediment deposits of 0.0, 0.5, and

1.0 feet depth. The calibration curves developed from the

broad-crested V-notch weirs, will provide more accurate discharge measurements for small watersheds with narrow sloping

channels

.

Key Words: Broad-crested V-notch Weirs; small watersheds;

streamflow measurements; calibration; approach channels;

stream gages; weirs.

1 . Introduction

Streamflow measurements are an important requirement of many research

and design activities. These measurements have greater significance with

the increased emphasis on environmental water quantity and quality. The

economic, social, and scientific interpretations based on these streamflow

measurements require that the measurements be accurate and reliable.

Numerous measurement techniques and instruments have been developed for a

wide range of flow conditions and criteria, and each system has unique

characteristics. Watersheds with areas of less than 1 mi^ (2.59 km^) often

have streamflow with stages that rise and fall rapidly. Measurement of

streamflow from small watersheds where maximum peak rates are expected to

be 300 to 1,000 cfs (8.49 to 28.3 cms) presents gaging requirements that

^This study supported in part by USDA-ARS Specific Agreement

No. 12-14-3001-556.

187

^
are best met by precalibrated flumes or weirs because these watersheds

often have rapidly varying, short-duration streamflow, where field calibration is difficult or impossible. Additional measurement problems exist

because good accuracy is required throughout a wide range of flow rates,

and the flow often contains significant amounts of trash and sediment which

may interfere with the gaging device.

A broad-crested, V-notch weir was developed and tested by researchers

of the U. S. Soil Conservation Service (SCS) for use as a precalibrated

streamflow measuring device for small watersheds [3, 4, 5, 6] The calibrations were developed using laboratory models with fixed-bed approach

channels with geometric cross sections. These weirs have been used extensively, often with no further laboratory or field calibration verification.

However, the weir calibrations are dependent on the approach channel conditions; thus, only when the laboratory condition is closely deplicated can

the laboratory rating be reliably applied to field installations.

Although most hydraulic handbooks for flow measurements describe the

required approach channel conditions for the weir ratings listed, few

suggest the consequences if these conditions are not met. In many situations,

appropriate approach channels cannot be constructed and, often, properly

constructed approach channels are later significantly modified by sediment

deposition. In this article, we review the effect of approach channels on

weir ratings, and present preliminary research results that indicate the

potential effect of approach channels on the calibration of broad - crested,

V-notch weirs.

2. Determining Weir Calibrations

The theoretical discharge of a sharp-crested, triangular weir is

Q = A (2g)l/2 Tan (|) H^^^ (1)

3-1 7 where Q is the discharge, L T , g is the gravity constant, L T , 9 is the

internal weir angle and H is the head above notch, L.

The assumptions are: (1) no approach velocity, (2) parallel flow, (3)

no energy loss within the approach reach, and (4) an energy coefficient of

unity. Since these ideal conditions do not exist. Equation (1) is modified

to account for the approach velocity by the addition to the static head, H,

of the velocity head, aV_ ; where a is the kinetic energy correction factor

and V is the average approach velocity at the gaging section. Also, a coefficient is incorporated with the constants of Equation (1) to form a

discharge coefficient, Cq , so that the discharge is expressed by

Q = Cj^ Tan |

(H + ai ^li) ^''^ (2a)

or Q = '^^^ I "t ^^^^

'^Figures in brackets indicate the literature references at the end of

this paper.

188
where Cq is the discharge coefficient, L T , is the kinetic energycorrection factor, Vi is the average approach velocity, L T-1, and Hj is

H + aj Vl^ is total head L. The discharge coefficient is assumed to

2g

account for losses, the gravitational constant, and for other changes

without having to alter the other variables, i.e. H, V. and 9.

To precalibrate V-notch weirs, then, requires that the discharge coefficient, Cq , be determined for the entire flow range by laboratory models

or other means. This coefficient is primarily dependent on the weir angle,

weir cross section, and approach channel geometry. Usually, the coefficient

is not constant through the full range of flow. The weir ratings are commonly

expressed as graphs of versus Q or as Q versus H.

Field installations must closely simulate the laboratory setting to

allow application of the laboratory-developed discharge coefficients, or

some method must be used to modify these coefficients. For rectangular,

thin-plate weirs with rectangular, level approach channels, Kindsvater and

Carter [7] developed modification criteria to estimate discharge coefficients

for a variety of weir heights and approach channel widths. The calibrations

for the SCS borad-crested V-notch weir, were made dependent on the

upstream cross section (which can be expressed as weir coefficients); but,

again, only level channel slopes were considered [2]

.

3. The Broad-Crested, V-Notch Weir

The broad-crested, V-notch weir developed by SCS researchers at Cornell

University, the National Bureau of Standards, and the University of Minnesota

was economical to construct, durable, and accurate throughout

the flow range for small channel slopes. The weir thickness was 16 in.

(40.6 cm). The crest cross section had 6-in. (15.2 cm) wide sections sloping

3 horizontal to 1 vertical on the upstream and downstream side and a 4-in.

(10.2 cm) wide horizontal section at the center. Weir section details

are presented by Holtan [2], page 35.

Weirs with side slopes of 2, 3, 5, and 10 horizontal to 1 vertical were

calibrated using the head 10 ft. (3.0 m) upstream of the weir. The approach

channels for the weir calibrations had several sizes of rectangular or

trapezoidal cross sections, a 0.0-percent grade, and bottom elevations 6 in.

(15.2 cm) or more below that of the weir notch. These calibrations were

first reported by Huff [3, 4, 5, 6]; again by Harrold and Krimgold [1]; and

later by Holtan [2].

Early field installations soon disclosed that sediment transported by

the streamflow often deposited in the approach channel, which resulted in

a channel gradient and an effective channel bottom at or above the weir

notch elevation. Further testing was done on several modified approach

channels for alleviating the deposition problem and head-discharge relationships were determined [6]. Many of these tests showed significant deviations

from the published ratings, irregular rating curves due to standing waves.

189
and approach velocities near critical- -but the summary reports were not

widely disseminated.

The broad-crested, V-notch weir remains a useful stream-measuring device

for numerous applications, both in the U. S. and in other nations. Usually,

the field sites closely approximate those represented by the laboratory

conditions; thus, the laboratory ratings are applicable. For cases of

obvious approach channel deviations from the calibrated weir geometry or weir

modifications, field calibration is necessary and accomplished; in other

situations, field calibration is necessary, but is impossible or impractical.

The only recourses here are either to accept the inaccuracy (usually of

unknown magnitude) or to use a different measurement device, but suitable

devices are often neither available nor practical. Thus, the broad-crested,

V-notch weir continues to be used despite the difficulties of accurate

ratings.

Weirs cause hydraulic characteristics conducive to upstream sediment

deposition. Streams carrying even moderate suspended sediment loads or

those with mobile beds naturally assume some channel gradient and equilibrium

cross section. If this results in a weir approach channel different from

that represented in the laboratory, there may be significant rating shifts

and inaccurate streamflow measurements. Streams carrying heavy bedloads,

such as sandbed streams, or those with suspended loads greater than 50,000

to 100,000 ppm are particularly susceptible.

The sediment deposition in the approach channel may be dynamic during a

flow event. These deposition changes may cause a variable rating shift

that is difficult to document and assess. The deposit may become apparent

only as the flow recedes, and only then if observers are tending the

measurement station. Even with reasonable estimates for deposition depths,

cross sections, and slopes during an event, no reliable method is available

to adjust the weir rating.

4. Recent Weir Tests

Model tests were conducted to further define approach channel effects

on the calibrations of broad- crested, V-notch weirs [9] . Fixed-bed models

with a model-to-prototype ratio of 1 to 5 were installed in a 4 x 8 x 200 ft.

(1.2 X 2.4 X 60.8 m) tilting flume. The weirs tested had side slopes of

2 and 3 horizontal to 1 vertical (127° and 143° internal weir angle,

respectively). Head measurements were made 10 ft. (3.0 m) (prototype)

upstream of the weir center line.

The approach channel cross sections tested are shown in Figure 1. Cross

sections 1 and 6 were the entire rectangular flume width. Cross sections 2

and 9 were similar to the original weir calibration tests [3, 4, 5, 6] and

were used to verify similarity of results between the two test series. Cross

sections 3, 4, 5,7, and 8 represent alluvial channels being gaged on research

watersheds in western Iowa, near Treynor [8, 10]

.

Channel slopes of 0.0, 0.5, 1.0 and 1.5 percent were tested for all

cross sections by tilting the test flume. The rating curves in Figure 2

indicate the effects caused by channel slope. All curves are for the number

190
3"1 Weir

(1) Rectangular Channel

2:1 Weir

(6) Rectangular Channel

-I 4.25' U- o.5'

(2) 4:1 Trapezoidal Bed Channel R = I7'

(7) 2:1 Circular Bed Channel

^Bed

R = 24.7

(3) 3:1 Circular Bed Channel (8) 2:1 Circular Bed Channel

with Severe Deposition

(4) 3:1 Circular Bed Channel

with Severe Deposition (9) 3:1 Trapezoidal Bed Channel

(5) 3:1 Circular Bed Channel

with Moderate Deposition

(10) 3:1 Trapezoidal Bed Channel

with Severe Deposition

Figure 1 - Approach channel cross sections used in tests.

All dimensions in prototype units (1 ft = 0.30 m)

191
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192
3 channel cross section (Figure 1) . As expected, the channels with steeper

slopes cause higher flow rates for a given head as a result of increased

approach velocity. Some small irregularities in the rating curves for 1.0-

and 1.5-percent slopes caused by standing waves at the measuring section

were smoothed. The slope effects were not as large nor as irregular for

the 4:1 trapezoid channel cross section (No. 2, Figure 1).

The rating curve for 0.0-percent channel slope developed from the

1938 to 1941 tests and reported by Holtan [2] is also presented on Figure 2.

The rating curves for the other slopes based upon Holtan 's rating table do

not vary significantly from that for the 0.0-percent slope. The discrepancies

between the rating curves developed in the recent tests and those based on

the rating table in Agricultural Handbook No. 224 [2] are evident in Figure

2. The rating table accounts for slope and channel geometry only indirectly

by a change in area. This indirect approach is not satisfactory and the

ratings may be in error by as much as 100-percent when comparing the Handbook

No. 224 results with those at 1-percent slope. For example, representative

discharge values are given in Table 1 for discharges read from the rating

curves at different channel slopes and for discharges based upon the Handbook

rating table.

Table 1

Representative discharge values for effects of channel slope, 3:1 Weir;

Circular Bed Channel; No Deposition (1 ft = 0.03 m, 1 cfs = 0.028 cms)

Head

ft

Discharge, cfs

From Figure 2

Slope,

0.0 0.5 1.0

2 52. 0 58 .0 100.0

3 170. 0 175 .0 300

4 375 420 700

5 710 810 1360

From Ag. Handbook 224

Slope, ' 0.0 0.5

51.4

155.5

344

667

51 .7

156.5

360

695

1.0

52.5

156.5

370

-775 1-

Value extrapolated beyond table,

The effect of approach velocity is demonstrated in Figure 3. The curves

of Figure 3 are derived from the same data as those shown in Figure 2. The

velocity head is added to the static head to obtain the total head. A value

of a = 1.116 was used for the calculations and is the average value determined

from velocity profiles for all channel slopes and cross sections tested. By

using the total head, the rating curves tend to converge toward a single curve.

Determination of the discharge is easier since the differences are smaller

for a given total head.

193
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194
The effect of sediment deposition was tested by filling the channel

bottom as shown by the dashed lines on cross sections 4, 5, 8, and 10 of

Figure 1. Two deposit depths were tested to represent 0,5 and 1.0 ft.

(15.2 and 30.5 cm) depths (prototype) at the channel section 10 ft. (3.0 m)

(prototype) upstream from the weir. The deposit depths were linearly

reduced to zero at the weir notch and 25 ft. (7.6 m) (prototype) upstream.

The effect of deposition in the approach channel at 0.5% channel slope

is demonstrated in Figure 4 for the 0.5 and 1.0 ft. (15.2 and 30.5 cm)

(prototype) deposits in the number 4 and 5 cross sections. For the 0.5-percent

channel slope, the discharge for specified heads was significantly reduced

below about 50 cfs by the 1.0 ft deposit. This general trend was observed

at all channel slopes with more deviation from the no deposit case evident

at greater slopes. The use of total head gave more consistant results and

less deviation in the rating curves. However, most rating curves used in

the field still relate to static head. Therefore, representative discharge

values relative to static head are given in Table 2 to aid in assessing the

effects of channel deposition on such rating curves. If the discharge rating

for the 1.0-percent channel slope is representative of field site situations,

discharge measurements based on ratings of 0.0-percent slope may be more

than 100-percent in error throughout the flow range. However, the laboratory

model is probably smoother than most field sites; thus the expected error

would be somewhat less, but further comparisons with field data are needed

for verification.

Table 2

Representative discharge values for the effects of channel deposition, 3:1

Weir; Circular Bed Channel (1 ft = 0.30 m, 1 cfs = 0.028 cms).

Head Discharge, cfs

ft 0.0 % Slope 0 .5% Slope 1 .0% Slope

Deposition, ft Deposition, ft Deposition, ft

0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0

2 46.5 45.0 52.5 58.0 49.5 60.8 109.0 78.3 73.2

3 165.4 160.3 166.5 184.6 180.0 176.4 316.9 270.6 303.8

4 374.2 360.0 365.8 407.4 412.9 403.3 700.0 648.9 597.4

5 670.1 661.0 676.3 806.3 789.1 758.6 1169.6 1044.0 1017.6

Values taken from head versus discharge data used to develop Figure 4.

Observations in the flume showed that the deeper deposit caused high

velocities between the weir and 10 ft. (3.0 m) (prototype) upstream. An

alluvial stream bed would probably not maintain this situation for any length

of time.

The effects of approach channel cross section at 0.5-percent channel

slope are demonstrated in Figure 5. As expected the smaller channel cross

sections caused increased flow rates due to increased approach velocities.

Similar results were observed for all other slopes and for the 2:1 weir

195
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Figure 4 - The Effect of Deposition in the Approach Channels on the

3:1 V-notch Weir Rating (1 ft = 0.30 m: 1 cfs = 0.028 cms)

S 10 20 50 100 200 500 1000 2000 5000

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Figure 5 - The Effect of Approach Channel Cross Section on the

3:1 V-notch Weir Rating (1 ft. = 0.30 m; 1 cfs - 0.028 cms)

196
with a few exceptions below total heads of 1.5 ft. For those cases below

1.5 ft, the trapezoid or circular channel would indicate less discharge for

a specified head. This occurred at the lower discharges tested, generallyless than 20 cfs (0.56 cms), and the differences when compared to the

rectangular channel were on the order of 20-percent or less.

The results obtained for the 2:1 weir and different approach channels

were similar to those described for the 3:1 weir. For example, the effect

of channel slope on the 2:1 weir rating curves for the number 7 channel

cross section (Figure 1) is demonstrated in Figure 6. The 0.0 and 0.5-percent

slopes are essentially coincident over the full range of discharges. The

1.0 and 1.5-percent rating curves deviate at the lower discharges, but tend

to converge toward the 0,0-percent curve at the higher discharges. Errors

on the order of 100-percent can also be obtained if the 0.0-percent channel

slope rating curve is used for a channel with a slope of 1.0-percent.

The discharge coefficient, Cq , was defined in Equation 2b. The

relationship of Cq with the total head provides another means of determining the rating curve for a weir-channel system.

The discharge coefficient curves obtained for the data used to develop

Figure 6 are shown in Figure 7. The curves for 1.0 and 1.5-percent slopes

also tend to deviate from the curves for 0.0 and 0.5-percent slopes as was

evident in Figure 6. In some cases, it may be necessary to develop a discharge rating curve. The discharge rating curve can be developed for the

total head versus discharge coefficient relationship in a more satisfactory

manner than using the rating curves. This is because the values of , although subject to some scatter, tend to group much closer together, and

interpolation is easier since the graphs are plotted on rectilinear scales

rather than logarithmic scales.

5 . Summary and Conclusions

Broad-crested, V-notch weirs have many characteristics that make them

desirable for streamflow measurements where expected peak flow rates range

from 300 to 1,000 cfs (8.4 to 28.0 cms). A weir developed by the U. S.

Soil Conservation Service is often used. This weir was originally calibrated

with varying approach channel cross sections but with the channel bottom

level and at least 0.5 ft. (15.2 cm) below the weir notch. For alluvial

streams carrying sediment, the approach channels establish a slope, and the

channel bottom at the weir is usually near the notch elevation. Because the

weir head-discharge relationships are dependent on approach channel conditions, any deviation from the original calibration conditions causes a

rating change.

Recent model tests have been conducted to determine the effect of

approach channel slope and cross section and of deposition on the ratings

of these broad- crested, V-notch weirs. These laboratory results provide

improved predictions for the head-discharge relationships of these weirs

for many common field sites. Further, improvements between the model and

field installations can be obtained if one or two field data points are

available to adjust for differences in roughness between model and prototype.

197
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Figure 7 - Discharge Coefficient Curves

199
6. References

[1] Harrold, L. L., and Krimgold, D. B., 1943, Devices for Measuring Rates

and Amounts of Runoff, USDA, Soil Conservation Service, SCS-TP-51.

[2] Holtan, H. N., Minshall, N. E., and Harrold, L. L., 1962, Field Manual

for Research in Agricultural Hydrology, Agriculture Handbook 224, USDA,

Agricultural Research Service.

[3] Huff, A. N., 1938, Tests of 16-inch Weir and Comparison with 30-inch

Weir, Report No. 1, Cornell University, Section of Watershed and

Hydrology Studies.

[4] Huff, A. N., 1941a, Calibration of 2:1 V-notch Measuring Weirs, St.

Anthony Falls Hydraulic Laboratory, Minneapolis, Minnesota, Report

No. MN-R-3-6.

[5] Huff, A. N., 1941b, Calibration of 3:1 V-notch Measuring Weirs, St.

Anthony Falls Hydraulic Laboratory, Minneapolis, Minnesota, Report

No. MN-R-3-8.

[6] Huff, A. N. , 1942, Calibration of 3:1 and 5:1 V-notch Measuring Weirs,

Supplement to Calibration of 2:1 V-notch Measuring Weirs, St. Anthony

Falls Hydraulic Laboratory, Minneapolis, Minnesota, Report No. MN-R-3-7.

[7] Kindsvater, C. E., and Carter, R. W., 1959, Discharge Characteristics

of Rectangular Thin-plate Weirs, Transactions of the American Society

of Civil Engineers, 124, 772-822.

[8] Minshall, N. E., and Spomer, R, G., 1965, New Type Weir Construction

for Small Watersheds, Proceedings of the American Society of Civil

Engineers, 91 (IRl) , 11-15.

[9] Ruff, J. F., and Dang, C. , 1977, Rating Broad-Crested V-notch Weir,

Interim Report prepared for USDA, Agricultural Research Service.

Colorado State University, Engineering Research Center, CER77JFR-CD42

.

[10] Saxton, K. E., Spomer, R. G., and Kramer, L. A., 1971, Hydrology and

Erosion of Loessial Watersheds, Proceedings of the American Society

of Civil Engineers, 97 (HYll), 1835-1851.

200
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

COMPENSATING FOR CONSTRUCTION ERRORS IN

CRITICAL-FLOW FLUMES AND BROAD-CRESTED WEIRS

John A. Replogle

Research Hydraulic Engineer

U. S. Water Conservation Laboratory

Agricultural Research Service, USDA

4331 East Broadway, Phoenix, Arizona 85040

Critical-flow flumes and broad-crested weirs can now

be routinely and accurately calibrated (+ 2%) by computer

techniques over a wide range of flow rates and flume cross

sections, including trapezoidal and complex shapes. Tnis

ability permits detailed compensation for errors introduced

by construction anomalies . Procedures were developed and

used on a series of primary devices in irrigation canals,

including trapezoidal flumes and broad-crested weirs, which

identified construction errors and accurately related a

readout mecnanism to the primary device, as constructed,

so that the original intended accuracy could be restored.

Templates were used to accurately define field dimensions

of the flume throat section, its most sensitive portion.

Errors in defining the throat cross-sectional area can be

shown to be nearly equal to the error in discharge. Errors

in the approach and converging sections are about one-tenth

as important. The accurate field measurements of flume

dimensions were then used to compute a tailored calibration

table for the particular construction. The template can

also be used to accurately relate the readout device to the

flume. Direct reading discharge gages that were prestamped

could be mounted, using a modification of the technique, to

reserve the highest accuracy for any selected flow rate on

the gage. Discharge errors, if any, could thus be relegated

to the least used flow range.

Key Words: Broad-crested weirs; errors; flow measurement;

flumes; open-channel flow.

1. Introduction

Calibrations of flumes and broad-crested weirs have traditionally been

based on laboratory ratings which, if done carefully, can produce a discharge calibration for a particular structure that is repeatable and accurate to witnin + 1% of reading over a 20:1 range in flow rate. This same

Contribution from the Agricultural Research Service, U. S. Department of

Agriculture

.

201
device when translated to field construction and read by field installed

instruments, too frequently produces errors of more than + 5%. Part

of the problem may be that no general understanding of the error sources

or guidelines for correcting construction anomalies and registration errors

are available at the field level. The main objective of this paper is to

discuss calibration, construction, and registration errors, and to suggest

techniques for accurately compensating for their presence or reducing their

influence. A realistic goal would be to reduce errors in the primary

device to less than half the total system error, including the readout

method, i.e., for a 5% desired total accuracy, a flume, itself, should be

within + 2.5%.

2. Errors — Types and Sources

Errors associated with flow measurements using flumes or broad-crested

weirs can usually be traced to primary calibration errors, construction

anomalies, zero registration problems, and readout-device shortcomings.

These errors can be classified as systematic, random or spurious — the

latter linked to mistakes and malfunctions. Many errors are random when

applied to a population of flumes but are systematic when applied to a

single flume. For example, several flumes constructed by the same method

would be expected to have random errors associated with the dimensional

tolerances of the variables listed in Table 1, Appendix 1. These dimensional errors would cause a random spread of indicated discharge from flume

to flume, but each flume would show a systematic, not random, deviation

from calibration. Since it is possible and practical to measure the flume

dimensions "as constructed" these will, therefore, be treated as systematic

errors. Likewise, registration errors (zero determination) will be treated

as systematic errors. Both construction and registration errors, with

proper technique, can be reduced to less than +1%, at least for the most

frequently used flow range on a particular irrigation canal.

Other error sources are associated with the secondary or readout

device. The methods available to detect flow depth are many and varied,

ranging from direct-indicating staff gages, point gages, and float operated

indicators, to pressure transducers, purge-bubbler systems, sonic-level

indicators and capacitance-level detectors. They may be used with or

without separate stilling wells. The types of errors associated with these

secondary devices have been adequately described in measurement handbooks,

and the effects of errors in detecting flow depth of weirs and flumes has

been well documented (reference [5]).^

Yet another source of error results from the method for obtaining the

discharge rating for a particular flume. As mentioned earlier, a laboratory rating on an individual device may be within + 1%, especially if a

Numbers in brackets refer to appended references.

202
good primary standard like a weighing-tank system, is used. Inferring

the calibration from scale models and from similar devices of a type is

often and successfully used, but at some sacrifice of accuracy in exchange

for more flexibility to mass produce similar structures. More recently,

computer modeling has been applied to produce individual ratings for massproduced structures.

3. Computer Model for Flumes

Critical-flow flumes and broad-crested weirs can now be routinely

calibrated by computer modeling, accurate to about + 2% for a wide

range of flume cross-sectional shapes, including trapezoidal and complex

shapes [6, 8]. Typically, laboratory data fit to a third-degree polynomial, coincides with the computer-predicted calibration curve, within

+ 2% in the upper two-thirds of the maximum design flow depth, and be

within + 5% at one-tenth of the maximum flow stage. Figure 1 illustrates

a typical comparison of computed discharge coefficient and laboratorymeasured discharge coefficient for a trapezoidal flume.

Flumes must have nearly parallel flow in the contracted throat section

(Section 3, Figure 2) and in the approach channel (Section 1, Figure 2, for

the stage discharge relation to be computer predictable. Also, the converging section (Section 2, Figure 3) should not produce flow separation.

Tnis requires converging the flow into the flume throat with rounded

entrances, or with flat-surface construction, at less than 1 unit laterally to 3 units longitudinally. x\lso, the throat section and the approach

sections should each be at least twice as long as the depth of flow to

insure nearly parallel flow. However, when the throat is twenty times

larger than the flow depth, frictional effects, whose prediction at deeper

flows could be rather casual, now dominate and can cause rating errors over

5%. Thus, the practical measuring limits for a critical-depth flume, or a

broad-crested weir with rounded upstream edge is

:

^ < Yl < f3 (1)

where Yl is the water depth in Section 1 referenced to the elevation of

the floor in Section 3, and L3 is tne length of Section 3.

Long-throated flumes with proper diverging sections can withstand

up to 0.95 submergence, while causing no more than a 1% change in discharge.

This is the definition of "modular limit" (2). Shorter throated devices

have modular limits that are 0.65 or less, which causes submerged flow

operation to be frequently encountered or else forces the design of extra

overfall height into a canal system.

Other advantages of these computer- rateable flumes include the ability

to tailor the flume for a particular canal measuring problem, most often

eliminating the requirement for measuring with submerged flume flow, which

is inconvenient and inaccurate, as is discussed in more detail in

reference [8].

203
Computer Model Programs; A simple version of a Basic computer program

that will compute the depth-dis charge relation for trapezoidal flumes is

given in Appendix 1. This version will not print out Froude i<lumber information or channel velocity information. More complex programs have been

presented elsewhere [6, 7].

4. Model Verification

Laboratory Verification: Many shapes of measuring flumes — trapezoidal,

rectangular, triangular, and complex — have been successfully calibrated with

the computer model and verified by laboratory tests [6, 8]. These shapes

have included structures with bottom sill only (broad-crested weirs), side

contractions only, contractions from one side only, and even side contractions with a depressed floor (negative sill). Discharge ratings for all can

be predicted, if the basic length criteria for parallel flow, (mentioned

above) are observed. Historical data, as published, usually fit the model.

When it did not fit, recalibration , when possible, nas confirmed the computer

prediction. Usually, the differences were on the order of 5 to 7% and were

reduced by recalibration to about + 2%, indicating that the original

calibration had a systematic or registration error.

Computational Verification; Computational verification of the computer

model can be established by comparison to hydraulic modeling. Suppose that

a scale model were made of a flume. If the model is made of plexiglass and

the prototype made of concrete, an appropriate length ratio would be 1:7.

Now, if we use the computer technique to directly compute the rating curves

for both the model and the prototype, then appropriately expand the rating

for the model by Froude Modeling Techniques, the Froude Model expansion and

the direct prototype rating by computer will differ by only a fraction of a

percent. This small difference probably represents the slight calibration

change caused oy Reynolds I'lumber effects, which Froude modeling assumes is

negligible.

Field Verification: Field checks by others have also confirmed the

model. Four separate examples include the following:

I. A complex-flume [6] having discharges ranging from 30 to 220 cfs,

was field-checked with a series of 64 current meter traversings over a

2-year period. The current-meter ratings substantiated the computer

predicted ratings to within + 5%.

II. Flume Number 2, ASAE Standard S 35 9.1, was originally calibrated

using an outdoor flow facility. At flow depths below about half the maximum

flow depth, the published flume calibration deviates from computer prediction by 5 to 7%. Recent recalibration^ against a suppressed rectangular

Recalibration made by Allan Humpherys and James Bondurant, Snake River

Conservation Research Center, Agricultural Research Service, USDA,

Kimberly, Idaho, fall of 1976.

204
weir confirmed the computer prediction within a standard deviation of + 1%.

The original calibration, although systematically overpredicting discharge

by only 5 to 7% in a seldom-used flow range for this size flume, will be

updated in future standards and publications, since better information is

now available.

III. A trapezoidal flume, designed for use in concrete canals, was

field-checked against a constant-head orifice that was carefully adjusted

to 20 cfs. The flume, installed downstream and fitted with a staff gage

attached to the canal wall, confirmed this discharge.

IV. A sill placed in a trapezoidal canal was fitted with a sloping

approach. Thus, it was a trapezoidal broad-crested weir with the equivalent of a rounded upstream edge. A staff gage was fastened to the sloping

canal wall. Irrigation District personnel, using a propeller meter in the

delivery pipe to the canal lateral, reported the meter read slightly over

15-1/2 cfs when the meter was in place. About 1000 ft downstream, 15.7 cfs

was read, indicating excellent agreement.

The nearly universal capability to rate flumes permits detailed examination of the magnitude of discharge errors introduced by construction

anomalies, registration problems, and aging effects of erosion or

deposition on the flume surface.

5. Error Evaluation and Compensation

Deriving an error equation for the flumes that involves so many variables is impractical. The most straightforward approach is to examine the

influence of introducing an incremental change of + 1% in each variable,

in turn, and noting the corresponding change in indicated discharge rate.

A problem arises when applying this percentage formula to a quantity like

sideslope, Zl and Z3. These quantities, for example, are zero for a rectangular flume, so a percentage change is meaningless. However, for both of

these examples, the sideslope is 1, and changing these values by 1% is

consistent with changing the flow areas by 1%, if a corresponding 1% change

is simultaneously made in bottom widths . A specified rotation applied to

the sideslopes is a viable alternative.

The effects of construction and registration error on discharge prediction were examined using the model for two specific types of field installations, a broad-crested weir and a trapezoidal flume. The specific variables

examined were those listed in Table 1, Appendix 1.

Broad-crested Weir Example: A broad-crested weir was designed to

measure irrigation canal discharges primarily at 15 cfs with an expected

range of 12 to 16 cfs, and an occasional need to measure as low as 2 cfs.

A one-time measurement, i;ising a portable, adjustable, trapezoidal flume [7]

indicated that 16.1 cfs was flowing in the canal at a depth of 1.8 ft. This

left a freeboard of 0.7 ft, more than adequate for installing a criticalflow device to obtain the desired measurements. The weir was designed to

205
cause 0.25 ft of water-depth increase upstream at 15 cfs to avoid submergence. This was a liberal allowance since this configuration is expected

to operate with a head difference of only about 0.15 ft at 15 cfs. (Equal

to 85% submergence (modular limit) with water depths referenced to the

floor or weir crest elevation, not the channel floor). Since the tailwater

control in this case was primarily due to channel friction, this liberal

allowance was needed to handle seasonal changes in channel roughness due to

algae growth and Bermudagrass infringements over the canal edges. For construction economy, diverging Section 4 was omitted. This could be added

later if submerged flow were threatening to increase the modular limit to

nearly 95%. Tne sill could also be easily raised by pouring a concrete cap

over the original sill and recomputing the calibration, but at sacrifice of

some upstream freeboard.

Construction Procedure: The broad-crested weir was poured directly

into an existing trapezoidal concrete canal witti nominal 2-ft bottom,

1:1 sideslopes, and 2.5-ft depth. Calculations with the computer model

indicated the need for a 1.1-ft sill height to achieve the proper submergence limits discussed previously. Pljwood end forms (1.1 ft high) for the

sill were pre-cut to fit the channel shape. These were held upright in the

canal with 2-ft-long timber spacers and wire ties. The top edges of the

forms were carefully leveled across the channel and with respect to each

other before pouring the concrete. The form tops served as finishing guides

for the top surface of the sill. After the initial set of the concrete in

the sill, the upstream form was removed and a concrete ramp was formed.

This was essentially hand plastered into place with little requirement for

accuracy, except that there be no sharp discontinuities between this section

and the previously poured sill. The sill was 2 ft long and the ramp

approximately 3 ft long.

A carefully measured sheet-metal template, precut to matca the intended

finished cross -sectional shape, and with a height corresponding to the design

flow depth at 15 cfs, was inserted to check the finished dimensions. The

fit was not very close because the nominal existing canal had a sideslope

which, measured with the template, was 0.931 horizontal to 1-unit vertical

instead of 1:1. When the two end forms were finally level, the top sill

surface ended up about 0.05 ft higher than planned, making Y8 = 1.15 instead

of 1.1. The width of the sill was 4.25 ft instead of the intended 4.2 ft.

The only measurements remaining unchanged were the cross leveling and longitudinal leveling of the sill surface, as read with a 4-ft carpenter's level.

No special effort was exerted to correct or maintain the dimensions, except

for leveling, since the other dimensions could be readily handled in the

final calibration.

The template was used for two purposes — to determine the true constructed dimensions of the contracted throat section (Section 3) , and to

minimize errors due to mounting a gage on a rough canal wall that may not be

exactly on a 1:1 slope. After recalibrating by coii:5)uter modeling, a sidewall gage scale was calculated and marked directly in cubic feet per second.

This pre-marked gage was mounted about 1 ft upstream from the end of the

sloping transition section (Section 2) using the 15-cfs mark for registration

at the appropriate elevation above the throat section floor. The actual

206
registration was transferred with a surveying level from the top of the

template, placed near the center of Section 3, to the appropriate mark on

the gage in Section 1. Tnus, small depressions or high spots on the sill

that a surveying rod might encounter became quickly apparent and were somewhat averaged when the template was appropriately placed. Ponded water

over the sill can sometimes be used in place of the surveying technique,

but the template is needed, using either method. By registering on the

most frequently expected flow, errors in gage mounting slope or even in

gage marking are relegated to seldom-used ends of the gage.

Detailed discussions on location of depth sensing and construction

precautions are found in References [1], [2], [3], [5].

Error Detection and Compensation: The systematic errors tnat field

measurements revealed were evaluated with the computer model. Recapitulating, the noted differences in dimensions and slopes. A, from intended

were

:

A B3 = 0.05 ft = + 1.19%

A Z3 = 0.069 ft = - 6.90%

A Y8 = 0.05 ft = + 4.55%

From calibration tables computed for the uncorrected weir, 15 cfs

occurs at a Yl reading of 0.95 ft. Figure 3 shows the computed change in

discharge corresponding to a 1% change in the respective variable. At

0.954-ft flow depth or a Y1/L3 ratio of 0.477, these tnree variables

caused the following indicated cnanges in discharge:

B3: A Q = + 1.19% x 1.05 = + 1.25%

Z3: A Q = - 6.90% x 0.17 = - 1.17%

Y8: A Q = + 4.55% x 0.38 = + 1.73%

The absolute value of the combined detected error is 4.15%. In this

case, the errors cancel somewhat to yield a net systematic error in A Q

of only +1.81% from the intended original calibration for a flow of 15 cfs.

Recalibration using the actual dimensions placed the 15 cfs mark at

0.954 ft above the throat bottom elevation. The gage scale which had been

premarked based on the expected dimensions was still useable by registering

on the most losed flow range. Applying the calculated corrections and the

registration-with-template procedure restored the flume to intended accuracy.

Sensitivity: The relation between discharge, Q, and changes in upstream

depth, Yl, can be readily studied with the model. A familiar approach is to

obtain a power function for the weir by plotting the computed head and discharge values on log-log paper. The calibration equation for this particular

weir can be approximated to within + 1% by

Q = 16.1(Y1)

'•^^

207
for between 2 and 16 cfs. Thus, a 1% error in Yl will cause 1,65% error in

Q. A more complete representation of variation, dQ , with Yl is shown in

Figure 3 (plotted as Yl -r 10 to accommodate the figure). To tiold the error

contribution from the readout device to less than 2.5%, the weir gage must

be capable of less than (2.5/1.65) = + 1.52% random error of detection. At

15 cfs, this translates into detecting + .0144 ft vertically or + 0.02 ft

on the sloping wall gage. In the field, even with gusting winds~~of 20 to

30 MPH, the canal water surface was stable enough to be easily read to

within half of 0.02 ft on the gage. Thus, sensitivity in the most used

flow range was no problem. For 2 cfs, however, similar reasoning requires

detecting + .006 ft on the sloping gage, a much more difficult task but

certainly not unusable.

This particular broad-crested weir was not field rated with any

planned field test program. However, an opportunity presented itself to

check at least one flow rate using a propeller * meter in the 30-in supply

line from the main canal. Tlie check is that mentioned earlier as the

example IV of field verification of flumes and weirs.

All other dimensional errors associated with the remainder of the

variables caused 0.2% or less change in discharge for each percent change

in that variable, except for the variable A3, which is really a combination of errors in B3 and Z3 (Figure 3). Discharge errors are almost

directly proportional to errors in the determined area of flow, A3,

related to a particular flow depth. The template, when fitted to the

throat of a flume or broad-crested weir, visually shows whether area determinations will be suitable, assuming that proper zero registration is made.

Even some cross-slope error, if properly accounted for by the registrationwith-template procedure, is not critical because the correct flow area will

be closely approximated.

Trapezoidal Flume Exanple: A trapezoidal flume (Figure 2) was designed

with a constant throat (Section 3) cross-sectional shape, but could be constructed using variable sill heights, Y8, to adjust for submergence situations. This type flume was built using an inside mold or form for Section 3

that was reused for many flumes of the same style. It was not subject to

inaccuracies due to canal shape and could be used in unlined canals with

proper channel protection. This has advantages for contractors because

standard tables can be established for a range of sill heights, and individual calibration by computer modeling is not required. Also, they have

greater sensitivity than the broad-crested weir, and will drain down after

use better than the weir without having to use special drain pipes.

Provided the original form is accurate, and the flume throat is level

in the direction of flow, registration becomes the major source of error.

Using prestamped gages, the template method should be applied to align the

most used flow range to the correct registration, and relegate any error

of gage or canal slope to the little-used ranges.

For this style of flume, plotting the rating curve on log-log paper

produces

:

2 1

1

Q = 5.34(Y1) ^'-^^

208
for the region of 10 to 20 cfs. Thus, a 1% error in Yl causes 2.11% error

in discharge. As was computed for the broad-crested weir, this requires

that the readout device be readable to no more than + 1.17% random error to

retain less than 2.5% error in discharge. At 15 cfs, this means detecting

+ 0.019 ft vertically or + 0.028 ft on the sloping gage. Thus, accuracy of

readout should be no problem. At 2 cfs, the equation changes to Q

= 5.31(Yl)l-36 and the gage must be readable to + 0.011 ft which is still

detectable. At 1 cfs, this flume would require detecting + 0.008 ft, which

is getting near the limit of accurately reading a wall-mounted staff gage.

However, if 5% random error in Q is acceptable, the gage may still be

useable

.

Since this flume has a throat length of 4 ft it will be somewhat more

sensitive to roughness changes than the broad-crested weir, so it may be

instructive to examine the expected calibration shifts due to deposits, etc.

Irrigation waters are sometimes used to apply anhydrous ammonia fertilizer to the crops. In hard waters this leaves a wall deposit on the canal

and flume that may be 1/8 in. thick, or more, but will dissolve during a

clear-water flow period. Other temporary changes are caused by sediment

deposits in the approach section (Section 1). The changes in discharge due

to a 1/4-in. throat deposit, a roughness change from glass smooth to broomed

concrete, and sedimentation (Section 1) equal to the sill height, are shown

in Figures 4 and 5. These coating effects are almost the same as a registration error of the same magnitude. Very subtle differences result from

the coatings in the approach channel and the way that the uniform coating

woiJ-d geometrically change the flow areas

.

Figures 3 and 4 show how changes in B3 and Z3 influence the discharge

difference, but the combined influences plotted as changes in A3 for both

structures are similar.

More than 50 of these trapezoidal flumes, designed for the Soil Conservation Service (SCS), have been installed by contractors, and checked by the

SCS for conformance to specifications. Upstream from one flume an existing

constant-head orifice structure was used to obtain a flow of 20 cfs. This

is one of the field checks discussed previously as example III of field verification of flumes and weirs. The example output of the computer model in

Appendix I is for this style flume with a 0.2-ft sill height. The flume

calibration curve of Yl vs Q can be plotted directly from this table.

6. Summary and Conclusions

Critical-flow flumes and broad-crested weirs can now be routinely

calibrated by computer techniques accurate to about +2% over a wide range

of flow rates and flume cross-sections, including trapezoidal and complex

shapes. This ability permits detailed compensation for errors introduced

by construction anomalies. Procedures were developed and used on a series

of primary devices in irrigation canals, including trapezoidal flumes- and

broad-crested weirs that identified construction errors and accurately

209
related a readout mechanism to the primary device, as constructed, so that

the original intended accuracy could be restored. Templates can accurately

define field dimensions of the flume throat section, which is the most

sensitive portion of a flume. Errors in defining the cross-sectional area

of the throat can be shown to be nearly equal to the error in discharge.

Errors in the approach section and converging section are about one-tenth

as important.

The accurate field measurements of flume dimensions were then used to

compute a tailored calibration table for the particular construction. A

registration-by-template procedure accurately relates the readout device

to the flume.

Direct reading discharge gages that were prestamped could be mounted,

with the registration tecnnique, to reserve the highest accuracy for any

selected flow rate on the gage. Discharge errors, if any, are thus

relegated to the least-used flow range.

References

[1] American Society of Agricultural Engineers, ASAE Standard: ASAE 359.1,

Trapezoidal flumes for irrigation flow measurement, pp 565-7 (1976).

[2] Bos, M. G. (Ed.), Discharge measurement structures. Publication No.

161, Delft Hydraulics Laboratory, Delft, Netherlands (1976).

[3] British Standards Institution, Standard No. 3680-4B, Methods of

measurement of liquid flow in open channels, 4B: Long-base weirs.

Available from American National Standards Institute, 1430 Broadway,

New York, NY (1969).

[4] King, Horace Williams, Handbook of hydraulics, Fourth Edition, McGrawHill, New York, Wi (1954).

[5] Kulin, Gershon, and Philip R. Conpton, A guide to methods and standards

for the measurement of water flow, NBS Special Publication 421, U. S.

Department of Commerce, National Bureau of Standards, 89 pp (1975).

[6] Replogle, John A., Critical-flow flumes with complex cross section,

Proc. of ASCE Irrigation and Drainage Division Specialty Conference

held at Logan, Utah, August 13-15 (1975).

[7] Replogle, John A., Portable, adjustable flow-measuring flume for small

canals, Paper No. 75-2558, presented at the 1975 Winter Meeting,

American Society of Agricultural Engineers, December 15-18, Chicago,

Illinois (1975).

[8] Replogle, John A., Tailoring critical-depth measuring flumes, In Flow;

Its measurement and control in science and industry, Rodger B. Dowdell,

Editor, Vol. 1, Instrument Society of America, pp 123-32 (1974).

210
I.I

1.0

T3

o

0.9

0.8

LABORATORY DATA

THIRD DEGREE POLYNOMIAL^

THROUGH LAB DATA

COMPUTER MODEL

1 1

0.1 0.2 0.3

YI/L3

0.4 0.5

Figure 1: Typical laboratory results fitted with a third-degree

polynomial and computer model prediction of discharge

coefficient, C,, for a trapezoidal flume,

d

211
APPROACH

SECTION 1

CONVERGING

SECTION 2

THROAT

SECT 3

rm7777T7777777777m "jT

^Y8 = l.lft.

*-XU- L2 = 3ft. —4*

1 f t. *- '

L3=2ft.-^

SIDE VIEW

t 2

END VIEW

= 1ft.

TRAPEZOIDAL BROAD- CRESTE D WEIR

(a)

APPROACH

SECTIONl

CONVERGING

SECTION 2

THROAT

SECTION 3

Y8 = 0.2tt

L3= 4ft.

SIDE VIEW

TRAPEZOIDAL FLUME

(b)

Figure 2: (a) Broad-crested weir in concrete canal.

(b) Trapezoidal flume in concrete canal.

212
UJ

+ 1.4

QQ +1.3

<

E +1.2

<

>

+

ct: o

u.

UJ

<

X

o

41.1

+ 1.0

+ .9

+ .8

+ .7

+ .6

+.5

+.4

+.3

UJ

CD

O

+1

DISCHARGE, FT 7S

2 3 4 5 10 15

1 1—i—I

I I I I

I

> I I I

I I I

^8f +

^83 ;

( AYI -MO )

-.3 J L J L J L 1

0.1 0.2 0.3 0.4 0.5

YI/L3

Figure 3: Effects on discharge through a broad-crested weir caused

by changes in weir dimensions.

213
LxJ + 1.4

00+1.3

<

q: 41.2

<

> 41.1

LJ +1.0

<

X

O +.8 - 4.7

(T

O +.6

UUJ +.5

5 +.4

o +.3

CO

Q 4.2

CD

<

X

o

0

-.1

? -.2

-fl -.3

DISCHARGE, FT VS

2 3 5 7 10 15

~l—I—I I

' I > T T

20 25

—I r-

( ^YI-I-IO)

ALI a A L2

AL3 AY8

3"

ABi a zi

1

0.1 0.2 0.3

YI/L3

0.4 0.5

Figure 4: Effects on discharge through trapezoidal flume caused by

changes in flume dimensions.

214
UJ

_l

GQ

<

cr<

>

Q

Ul

I-

<

o

o

>-

CD

o

UJ

CO

<

o

UJ

CD

<

X

o

CO

UJ

o

<

X

o

+ 4

+ 3

+ 2

+ 1

0

- I

-2

-3

-4

-5

-6

-7

-8

-9

-10

1 \—\—I—I—

r

SiZT" — OF SliT . SECT i .

0.2 FT OF SiLT IN

_ SECT. 1 , FLUME1/4-INCH COATING 0

TRAPEZOIDAL

FLUME

•12 h

13

1/4-INCH COATING ON

BROAD-CRESTED WEIR

0 0.1 0.2 0.3

Y3/L3

0.4 0.5

Figure 5: Effects on discharge through trapezoidal flume and broadcrested weir caused by fertilizer coatings and canal

siltation.

215
Appendix 1

The attached BASIC computer program prints out the depth, Yl , in feet or

meters, and discharge, Q, in ft^/s or m^/s, depending on the units of the input

data. More extended versions provide critical depth, Froude Number, velocities

in Sections 1 and 3, discharge coefficient, and the estimated value for the

velocity distribution coefficient (listed as A3 in the program, not to be confused with area, A3, of the text and Table 1, which the program uses in the

form of FNC(X)). Program versions of this type can be found in references [6]

and [7]. The differences are only in the information format printed out, not

in the values computed.

The remarks statements refer to an equation number. These are the equations as described and used in reference [6]

.

This version is suitable for trapezoidal flumes, and the limiting cases,

triangular and rectangular. The approach channel and the throat need not be

similar. A rectangular channel approach section may have a trapezoidal throat

section, but the converging section will have to be warped or multiplaned.

For parabolic, circular, and non-symmetrical channels, the five functions

at the beginning of the program need to be modified to give area, top width and

wetted perimeter, for each shape required. For more complex shapes, refer to

reference [ 6]

.

Table 1

DATA INPUTS

TO PROGRAM: Comments

Bl, B3 Bottom widths Sections 1 and 3. (Feet or meters).

Zl, Z3 Sideslopes, Z (horizontal): 1 (vertical). Sect. 1 and 3.

S, 1, LO Starting depth, increment and limiting depth for Yl. (Feet or meters

K Absolute roughness height. (Feet or meters).

L3 Length of Section 3. (Feet or meters).

Y8 Sill height. (Feet or meters).

Ll LI = XI; distance, depth-sensing location from downstream end of

Section 1. (Feet or meters).

G Gravitational constant (32.16 ft/s^ or 9.803 m/s^)

.

T Water temperature (°C).

Yl Flow depth in Section 1 referenced to elevation of flume floor in

Section 3.

Al Flow area of Section 1 (not input directly but computed from Bl,

Zl as FNE(X))

A3 Flow area of Section 3 (not input directly but computed from B3,

Z3 as FNC(X)).

216
0001 REM COMPUTER MODEL FOR TRAPE20IDAL FLUMES

0005 REM FUNCTIONS FOR A 1 , T3* A3> P 1 , P3 RESPECTIVELY

0016 DEF FNE(X) =(Bl+2 1*X)*X

0028 DEF FNN(X) =B3+2*23*X

0038 DEF FNC(X) =(B3+Z3*X)*X

0048 DEF FNP(X)=Bl+2*X + ( 1+21+Zl ) t.5

0058 DEF FN0(X)=B3 + 2*X*( 1+23*23) t .5

0109 READ B1,B3,21*23>S,I>L0*K>L3,Y8*LUL2

01 10 DATA 2, 1 , 1 > 1 0120 DATA .

1

» .2,2.5

0140 DATA . 0005> 4> .2, 1 ,

4

0189 READ G>V1

0190 DATA 32. 1 6, 1 .228E-5

0200 PRINT ••YrS"Q"

0210 FOR Y1=S TO L0 STEP I 0220 LET Y=Y8+Y1

0236 LET M=0

0237 LET G5=0

0238 LET H6=0

0239 LET Q=0

0240 LET Al=l

0250 LET A3=l

0260 LET Y3=.7*Y1

0261 LET Q3=Q

0279 REM COMPUTE Q (EG. 1)

0280 LET Q = (G*FNC( Y3) t3/( A3*FNN( Y3) ) ) t . 5

0290 IF ABS(Q-Q3)<.0001*Q THEN GOTO 0360

0300 LET Y4=Y3

0309 REM COMPUTE Y3 (EQ. 4)

0310 LET Y0=Q*G/(2*G*FNE( Y) t2)

0312 LET Y0=Y0*A1+Y1-H6

0314 LET Y3=Y0-FNC( Y3) /(2*FNN(Y3)

)

0330 IF ABSC Y3-Y4) < .0001+Y4 THEN GOTO 0261

0350 GOTO 0300

0360 IF M>0 THEN GOTO 0390

0380 LET M=l

0390 IF ABSCQ5-GX. 001*0 THEN GOTO 0510

0400 LET Q5=Q

0410 GOSUB 0570

0429 REM COMPUTE A3 (EG. 17-21)

0430 LET E=1.77*C6t.5

0440 LET A2=l . 5*FN0( Y3) /FNNCY3) -.

5

0450 IF A2<2 THEN GOTO 0460

0455 LET A2=2

0460 LET A4=.025*L3/(FNC(Y3) /FNO( Y3) ) -.05

0465 IF A4>0 THEN GOTO 0470

0468 LET A4=0

0470 IF A4<1 THEN GOTO 0480

0475 LET A4=l

0480 LET A3=l+C3*E*E-2*Et3)*A2*A4

0489 REM ASSIGN Al A VALUE FOR OPEN CHANNELS

0490 LET Al=l .04

0500 GOTO 0310

0510 PRINT Y1,Q

0540 NEXT Yl

0550 END

21?
0560

0569

0570

0579

058 0

0589

0590

0609

0610

0620

0630

0640

0645

0650

0660

0670

068 0

0690

07 00

07 10

07 19

0720

0729

07 30

07 39

07 40

07 42

07 44

07 49

07 50

07 59

07 60

0770

07 79

0780

0790

0800

08 0 5

08 10

0820

Yl

• 1 .3

. 5

.7

.9

1 . 1 1.3

1 . 5

1.7

1 .9

2.1

2.3

2.5

REM

REM

LET

REM

LET

REM

LET

REM

LET

LET

SUBROUTINE FOR FRICTION LOSS

COMPUTE A LENGTH REYNOLDS NO. BASED ON L3

R = Q/FNC( Y3)*L3/\/l

COMPUTE A

R5=350000+L3/K

COMPUTE

CI =1 .328/Rt .

5

COMPUTE

CRITICAL LENGTH REYNOLDS NUMBER (EG. 8)

A LAMINAR DRAG COEF., CI ( EQ . 10)

THE TOTAL DRAG COEF . ( CORRECTED) , C7

X 1 =R

X2=L3

GOSUB 0780

LET C6=C0

IF R<R5 THEN GOTO 07 10

LET X1=R5

LET X2=R5*V1 /(Q/FNC( Y3)

)

GOSUB 0780

LET C8=C0

LET C7=C6-(R5/R)*(C8-C1

)

GOTO 0720

LET C7=C1

COMPUTE FRICTION IN THROAT*

C7*FN0( Y3)*L3*Q*Q/(2*G*FNCCY3) t3)

ESTIMATE Y2 (EG. 15)

Y3+< 5/8)*( Y

1 -Y3)

COMPUTE FRICTION, SECT.

FNP(Y)*L2*Q*Q/FNE(Y) t3

H2+FN0C Y2) *L2*Q*Q/FNC( Y2) t3

H2* .00235/C 4*G)

COMPUTE FRICTION, SECT.

.0023 5*FNP( Y )*L1 *Q*Q/C2*G*FNE( Y ) t3)

COMPUTE TOTAL FRICTION, H6 (EQ.

(EG. 7)

REM

LET

REM

LET

REM

LET

LET

LET

REM

LET

REM

LET

RETURN

REM

LET

LET

LET

LET

IF

H3 (EQ. 13)

H3 = (

Y2=^

H2:

H2:

H2:

HI =

2, H2 (EQ. 16)

1 HI (EQ. 14)

5)

H6=H1+H2+H3

COMPUTE

C0=.005

C9=C0

j=( . 544*C0t .5)

C0=J/( 5. 61*C0t .

5

ABS(C0-C9)>. 00001

C0 (EQ. 9)

.638 -LOG ( 1 /(X1*C0)+1 /( 4.8 4*X2/K*C0t .5)))

THEN GOTO 0790

RETURN

0

9 .26352E-2

. 58 59 46

1 .4546

2.7227

4.42 69

1

6. 60579

9 .29756

12.5397

1 6.3688

20.8202

25.9286

31 .7275

38 .2499

218
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

NUMERICAL MODELING OF TWO-DIMENSIONAL FLUMES

Ronald W. Davis

Mechanics Division

National Bureau of Standards

Flumes are commonly used devices for the measurement

of open-channel flow rates. Their range of operation has

been limited, however, because, outside of experimentally

calibrating each configuration used, one has had to resort

to one-dimensional theory for flow rate determination. This

theory is restricted to fairly low values of upstream-head

to crest-length ratio, as well as to almost level flumes with

a reasonably uniform incoming velocity profile and nonconverging sidewalls. The present study is concerned with

extending the operational range of a particular type of

flume (the two-dimensional version of the Palmer-Bowlus

flume) into areas where one-dimensional theory loses its

validity. The determination of the head-discharge relationship for this flume is done numerically by use of the SOLA

finite difference routine for two-dimensional free-surface

flows. Effects of changes in flume geometry, channel slope,

and upstream velocity profile are investigated. The numerical results are verified experimentally.

Key Words: Flumes; hydraulic models; hydraulics; numerical

models; open-channel flow; weirs.

1 . Introduction

Flumes are commonly used devices for the measurement of open-channel flow

rates. The fluid depth at a particular location in a flume is measured, this

depth then being related to the discharge by means of a head-discharge calibration curve. This paper will present a new means of determining this calibration curve. Previously it has been determined either experimentally or by

means of one-dimensional flow theory. It will be shown here that, under certain

conditions, the use of a two-dimensional numerical flow field simulation has

significant advantages over these other two methods in determining the calibration curve. As will be seen, it is clearly far cheaper and less timeconsuming to calibrate by computer program than to experimentally rate each

type of flume prior to field installation. Also effects of errors in field

installation of the flume can often be more readily assessed on the computer

than in the laboratory. The use of one-dimensional flow theory [1]-^ can lead

to errors in discharge determination for flumes with such features as converging sidewalls, short crests, steeply sloping channel beds, and severely

1 Figures in brackets indicate the literature references at the end of this

paper

.

219
distorted incoming flow profiles. It is in situations such as these that the

numerical simulation to be discussed here will be shown advantageous.

As noted previously, the present investigation is concerned with the

application of two-dimensional theory to flow through measuring flumes. The

particular flume chosen for study is the two-dimensional version of the PalmerBowlus flume (i.e., a broadcrested weir), which is often used in sewers [2].

This flume serves as a convenient test case for the present study and provides

the first data on this device not derived from either experiment or onedimensional theory [2,3,4]. The computation of flow through this flume is

performed numerically by use of the SOLA finite difference routine recently

developed at the Los Alamos Scientific Laboratory [5]. This code solves the

complete two-dimensional, time-dependent Navier-Stokes and continuity equations

for incompressible flows with free surfaces. It is one of a line of Marker-andCell codes for the computation of free-surface flows which have appeared during

the past decade [6, p. 196] . SOLA is used here to determine the head-discharge

relationship of the given flume for situations in which one-dimensional theory

is generally inadequate. These include large upstream-head to crest-length

ratios, sloping channel beds, and highly nonuniform incoming velocity profiles.

In addition, effects of changes in flume geometry are investigated. In order

to verify the numerical results, an experimental investigation of the baseline

flume is made. The experimental apparatus consists of a flume set at the exit

from a constant-width plexiglass channel. The head-discharge relationship and

the effects of channel slope on this relationship are obtained experimentally

and compared with the numerical results. Free-surface profiles traced off

the plexiglass sidewall are compared with those generated numerically. Results

of all these comparisons are seen to be satisfactory.

2. Numerical Scheme

The SOLA finite difference routine is a simplified version of the Markerand-Cell (MAC) method first proposed by Harlow and Welch [7] for incompressible

free-surface flow problems. Since a complete description of SOLA can be found

in ref. [5], only a brief review of the numerical scheme will be presented

here.

The Navier-Stokes and continuity equations for incompressible flow are

written in primitive variables as

It- + (q • \7)q = - Vp + $ + vV^q (1)

0 t

V • q = 0 (2)

in which q = (u,v), where u and v are the velocity components in a two-dimen

sional cartesian reference frame (x,y); p is the ratio of pressure to constant

density; $ is the body acceleration (gravitational here) ; v is kinematic viscosity (zero for the present study); and t is time. The two components of

eq. (1) are put in conservation form [8, p. 28] and finite differenced using

an explicit marching scheme in time. The convective terms (q • V)q are differenced using a mixture of central and upwind differencing [6, p. 19 and 73].

The precise mix can be varied by means of a parameter. Thus, the velocity

field computed from eq. (1) at time step (N + 1) is solely a function of

the velocity and pressure fields at time step (N) . However, this velocity

220
field at time step (N + 1) is determined only from eq. (1) and will probably

not satisfy the continuity equation, eq. (2). Incompressibility is imposed

by adjusting the pressure in each computational cell until the velocity divergence

for that cell approaches to within a certain tolerance, e, of zero, where e is

usually of order 10"^ or less. Thus, if there is a net inflow into a cell, p

is increased to eliminate this; for a net outflow, p is decreased. Since the

pressure adjustment to a given cell affects the velocity divergence of neighboring cells, this process must be done iteratively. The velocity changes in

each cell which occur as a result of these pressure changes are computed from

a finite difference form of

||=-Vp (3)

which is the appropriate form of eq. (1) for impulsively applied pressure gradients

[8, p. 471]. Once the divergence condition for each cell is satisfied, the iteration stops. Since eq. (3) has not changed the vorticity field determined from

eq. (1), both vorticity and divergence are correct at time step (N + 1) . Thus,

the correct velocity and pressure fields at time step (N + 1) are now known, and

the calculation can proceed to time step (N + 2) once the free-surface location

has been updated.

In the original MAC code, massless marker particles moved with the fluid

and determined the location of the free surface. These particles have been

eliminated from SOLA. Instead, the location of the free surface is determined

from the kinematic equation

3h

3^ - q • Vh^ = V (4)

in which h is free-surface height. The pressure along the free surface is

maintained constant (zero) by appropriately adjusting the pressures in the

computational cells through which the free surface passes. In addition to

computing free surfaces, SOLA can also handle rigid curved boundaries by

appropriate modifications to the pressure iteration along these boundaries.

As with most explicit numerical schemes [6], there are time step restrictions required to maintain numerical stability. For the present investigation

At = i min { t\ , } (5)

3 |u| I v|

in which At is the time step; and Ax and Ay are the cell dimensions. After

each time step, the mesh was swept to find the value of the right hand side

of eq. (5). Thus At changed as the computation proceeded. Also for numerical

stability reasons, the differencing of the convective terms in eq. (1) was

split equally between central and upwind differencing.

3. Numerical Modeling of Flume

The model of the Palmer-Bowlus flume that is used in this study is shown

in figure 1. The values of 9 for both bottom slopes are the same. The flow

enters parallel to the channel bed on the left and leaves almost parallel to

the bottom slope on the right. The exit condition on the right was chosen so

as to simulate free flow over the brink and also to avoid the computing dif221
iculties associated with any hydraulic jump that might occur downstream. The

length of the bottom slope downstream of the brink was adjusted until further

increases had negligible effect on the flow upstream of the brink. Typical

distances in the x-direction traversed by this slope are between L and 2L,

depending on flume geometry. The components of gravitational acceleration, g,

are denoted by g and g . The value of g is always negative, while g is nonzero only if the'^entire^f lume is sloped. -^A positive g denotes positive flume

slope, i.e., flume tilted downward to the right.

The problem being considered here is time-dependent, as the location of

the free surface, ^iS hg(x,t) = hg.(x,o°), is not known a priori. Thus, h^(x,0)

must be guessed; the better the guess, the less computer time required. The

incoming velocity profile, U(y,t), is arbitrarily specified at each time step

at a distance approximately L/2 upstream of the base of the crest. The drawdown of the free surface begins downstream of this point. The outgoing velocity

on the right is closely approximated from Bernoulli's equation for flow parallel

to the bottom slope [9, p. 40]. It is specified at each time step, but only

prior to the pressure iteration required to satisfy the continuity equation,

eq. (2). Thus, the outgoing velocity at a given time step changes as pressure

changes, as do all other velocities except U(y,t). This makes for minimum

interference with the upstream flow [5], which is insensitive to disturbances

at this location anyway since the flow downstream of the brink is supercritical

[1, p. 192]. The upstream depth, h, is allowed to vary with time. Because of

this, the discharge through the flume can also vary with time. The computation

proceeds until the greatest change in [h (x,t) + P + y ] from one time step to

the next is less than 0.01 percent. The final value o? h is then h (0,°°),

which is the primary quantity sought from the calculation. This alfows the

head- discharge relationship to be determined for the specified geometry, channel

slope, and inlet velocity profile.

As the fluid being modeled is nearly inviscid, the kinematic viscosity

in eq. (1) was set equal to zero. Thus, viscous boundary -layer effects in

the flume were ignored. This is justified by the relatively large upstreamhead to crest-length ratios being considered here. Also, boundary layer

effects upstream of the flume can be accounted for by an appropriately

specified upstream velocity profile. A typical mesh consisted of 29 cells

horizontally and 9 fluid-filled cells vertically at the inlet location. All

computations were done in double precision on the NBS UNIVAC 1108. Computation

times ranged from one to five minutes,

4. Numerical Results

A summary of the cases to be presented is given in table 1. Flume 1 is

the baseline configuration. Effects of changes in slope and inlet velocity

profile on the head-discharge relationship are evaluated using this flume.

Flumes 2 and 3 show the effect on this relationship of geometry changes from

the baseline configuration.

The definitions of discharge, Q, and head, H, to be used here are as

follows: „ , •y +P+h

•^o Q

= U(y)dy (6)

o

222
(7)

in which y and P are defined in figure 1; g = /g^ + g'^; h = h (0,°°); and U(y)

= U(y,o°) , as time is now irrelevant. All quantities c^n be nondimensionalized

according to the following:

d' = ^ , 5' = ^ , Q' = -2- (8)

/IL L/iL

in which d is an arbitrary length. Primes will now be dropped; all quantities

referred to hereafter are nondimensionalized according to eq. (8) and are

steady-state values.

The main purpose of this study is to extend determination of the headdischarge relationship into areas where one-dimensional theory is inadequate.

For values of H less than about 0.33, one-dimensional theory should generally

provide satisfactory results, at least for level flumes with reasonably uniform

incoming flow profiles [1, p. 212]. Thus, this investigation is concerned

solely with values of H between the approximate limits of 0.33 and 1.30. The

upper limit of about 1.30 is rather arbitrary. It was chosen partially on the

basis of computer costs and partially because h is approximately 1.0 near the

upper limit of H. It will be seen that the results presented here do in fact

approach the one-dimensional results as H tends to its lower limit.

Figure 2 shows discharge versus head for flume 1 with zero slope and

constant inlet velocity profile, U(y) = U. The head-discharge relationship,

Q = .581 H-^'^^^, is obtained by least-squares curve-fit. The RMS error of

the fit is about 0.6 percent. This is based on percentage error in Q for a given H. Figure 3 shows the same least-squares curve-fit along with those

for flumes 2 and 3. Also shown is the one-dimensional relationship [1, p. 211].

The RMS errors of the fits for flumes 2 and 3 are about 1.0 percent and 0.6

percent, respectively. All three curve-fits are based on from 21 to 28 data

points, which are equally weighted. It can be seen that the effect of the

change in 6 (flume 2) is much greater than the effect of the change in P

(flume 3). In fact, the increase in P has almost no effect. Figure 4 presents

the difference between the two-dimensional and the one-dimensional values for

Q taken from the curves in figure 3. As expected, these differences decrease

as H decreases.

Figure 5 illustrates the effect on the head-discharge relationship of

sloping flume 1. The effect is, of course, greatest at low values of H

where gravitational cons iderations are most important. In fact, the inlet

Froude number, U//h + P, is about 2/3 near the top of the curve and 1/3 near

the bottom.

Figure 6 is a comparison between the linear inlet velocity profile and

the uniform inlet profile for flume 1 with no slope. For the linear profile,

the surface velocity is 2.33 times greater than the bottom velocity. This

profile is chosen so as to give typical values of energy and momentum coefficients, a and 6, of 1.16 and 1.05, respectively [9, p. 28]. The effect

of inlet velocity profile on the velocities at two points along the crest of

the flume is depicted for two values of H. The two locations chosen to

depict the profiles are at distances of 0.56 and 0.19 upstream from the brink.

At the first of these locations, the Froude number, u/A\^, is equal to one

223
(critical) for the solid line with H = 0.37. If there were a free overfall at

the brink, the critical depth would be about 1.40 times the brink depth [1, p. 195]

For the present case, the corresponding value is 1.39. Figure 7 shows the negligible effect that the distorted inlet profile has on the head-discharge relationship. The maximum change in Q for a given H is about 2 percent.

5. Experimental Flume

In order to try and assess the accuracy of the numerical results, it was

decided to set up an experimental model of flume 1 . Figure 8 shows the experimental apparatus. It consisted of an 8-foot (2.44 meter) -long clear plexiglass

channel with a constant width of 6 in. (15.2 cm). The flume was set at the exit

from the channel and terminated in a free overfall. The crest length, L, was

6 in. (15.2 cm). Flow entered the channel through a 1-in. (2.5 cm)-wide slot in

the vertical 2-in. (5.1 cm) -diameter pipe at the upstream end. An orifice meter

in the pipe upstream of the slot was used to measure flow rate. In order to

smooth out the high level of turbulence produced by the flow into the settling

basin just downstream of the pipe, two screens and a series of spaced honeycomb

sections were employed as shown in figure 8, To assess the effectiveness of

these elements in smoothing the flow, pitot tube and propeller meter traverses

were made at the depth measuring station 3 in. (7.6 cm) (L/2) upstream of the

base of the flume. The traverses came to within 0.37 in. (0.9 cm) of the sidewall and 1.30 in. (3.3 cm) of the bottom. Water depth varied from 3 in. (7.6 cm)

to 7 in. (17.8 cm). As observed from the numerical results (figure 7), vertical

variations in velocity were not deemed as important as transverse variations.

Two-dimensionality of the flow required a reasonably uniform velocity profile

across the channel. The traverses showed that the greatest change in velocity

in any given transverse or vertical plane within the region covered was about

15 percent. This was felt to be about as good as could be expected in a channel

of this size, and the experiment proceeded.

The experimental procedure consisted of reading the pressure drop across

the orifice meter with a mercury manometer and obtaining the upstream height

at the depth measuring station using a point gage. Free-surface profiles

were traced off the channel sidewall at the flume location with the aid of

a floodlight. Flume slopes of 1 percent, 2 percent, and -1 percent were

obtained by simply sloping the entire channel. At the conclusion of the

experiment, the orifice meter was calibrated in place using a weighing tank.

6. Experimental - Numerical Comparison

Figure 9 shows the least-squares fit to the experimental head-discharge

data at zero slope. The fit is based on 20 data points and has an RMS error

of approximately 0.7 percent. Head, H, and discharge, Q, are defined here

as previously. Discharge, Q, is obtained by dividing the total flow rate

measured with the orifice meter by the channel width. Figure 10 is a comparison

between the experimental and the numerical head-discharge relationships. Also

shown for comparison is the one-dimensional relationship. For H = 0.33 (log

H = -0.48), the difference between the numerical and experimental values of

Q is about 3.3 percent. For H = 1.30 (log H = 0.11), the difference is about

1.6 percent. The experimental curve does not approach the one-dimensional

curve with decreasing H as it should for an ideal fluid, such a fluid being

assumed in the numerical computations. In fact, it is suspected that viscous

boundary- layer effects along the channel sidewalls, which grow in importance

224
as H decreases, are largely responsible for the difference between the experimental and numerical results. In the present study, the thickness of these

boundary- layers was not measured. Further, the exact nature of the boundary

layers in the channel is difficult to estimate because of the honeycomb and

turbulence upstream. However, if one can assume that a turbulent boundary

layer begins to grow just downstream of the final honeycomb, then the following

formula for boundary- layer displacement thickness on a flat plate is appropriate

[10, p. 599]:

6^ = .04625 D {Re^) , Re^ = ^

^ ^

(9)

in which 6^ is the nondimensional (with L, as usual) displacement thickness;

D is the nondimensional distance from the final honeycomb to the depth measuring

station (about 4) ; v is the nondimensional kinematic viscosity; and Re^ is the

Reynolds number based on D. The boundary layers along the channel sidewalls

will reduce the effective channel width and thus effect the experimentally

calculated values of Q and H. Note that the boundary layer along the channel

bottom does not effect the experimental calculation of Q and has only a negligible effect on the experimental calculation of H. Also, the numerical results

indicate that the effect of vertical variations in velocity is negligible. If

the experimental data is corrected for the assumed sidewall boundary layers,

then the curve-fit indicated by the dashed line in figure 10 results. The RMS

error for this fit is about 0.6 percent. The values of Re^ that occur in the

fitted region are roughly between 2.0 x 10^ and 5.0 x 10^. This is in the

laminar-turbulent transition region for flow over a flat plate [10, p. 600],

but because of the high levels of turbulence in the channel, it is sufficient

for this approximation to assume that the boundary layer is always turbulent.

For H = 0.33, the difference between the numerical and boundary-layer-corrected

experimental values of Q is about 1.0 percent, while for H = 1.30, it is about

0.2 percent. Table 2, which summarizes the curve-fits obtained in this study,

clearly shows that the boundary- layer-corrected curve-fit more closely approximates the numerical curve-fit than does the uncorrected experimental fit.

Also, it is seen from figure 10 that the boundary- layer-corrected experimental

curve now approaches the one-dimensional curve for small H. Thus, the suspicion

that sidewall boundary- layer effects in the channel are important appears to

be justified, at least based on this approximate analysis. It is unfortunate

that the experiment could not have been performed in a wider channel, thus

reducing boundary-layer blockage as percentage of width.

Figure 11 compares numerical and experimental free-surface profiles over

the unsloped flume for two values of H. The energy (h^ + |qp/2) along the

computer-calculated profiles is constant to within five percent within the

region shown. It is noted that one-dimensional theory gives no free-surface

profile for the case H = 0.96, while the one-dimensional flow for the case

H = 0,37 is subcritical everywhere. Figure 12 shows the difference between

the zero-slope curve-fit value of Q, denoted by Qq , and the value of Q at the

same H for a sloped flume. This difference is plotted as a function of head

for the experimental, the boundary- layer corrected experimental, and the

numerical cases. The computed slope effect is seen to agree quite well with

the experimental effect for each of the three slopes studied.

7. Conclusions

It has been shown that a relatively simple, easily obtained numerical

225
routine, SOLA, can provide significant advantages over current methods of

calibrating two-dimensional flumes. The routine can easily accommodate

upstream velocity profile and channel slope effects, as well as changes in

flume geometry. The accuracy of the method has been demonstrated by experimental means. The head-discharge relationships presented in table 2 show

both the high quality (small RMS error) of the curve-fits and the very good

agreement between computer and experiment. It is recalled that this agreement was obtained under the simplifying assumption of inviscid flow for the

numerical computations. Thus, the convenience and low cost of using this

code make the excellent numerical-experimental comparison quite impressive.

The main conclusion of this investigation is that the computer in

general and SOLA in particular have an important part to play in the study

of two-dimensional measuring flumes. Such things as the effect of sediment

build-up (i.e., geometry change) on the flume calibration curve can now be

readily determined. Errors produced by incorrect flume installation are

easily assessed. The limits of validity of one-dimensional theory can now

be defined for a given flume and use of this theory suitably restricted.

Outside of these limits of validity, the errors associated with the currently

common practice of using one-dimensional theory are now amenable to calculation.

It is hoped that other open-channel flow researchers will make use of

SOLA. In particular, it can readily be adapted to compute fully threedimensional flows [5]. Thus, the study of three-dimensional flumes with

converging sidewalls should be possible with this code. The main limitation

in this regard would seem to be computer time.

8. Acknowledgments

The author gratefully acknowledges the helpful advice of Drs . Gershon

Kulin and George Mattingly. He is also grateful for the assistance of Mr.

Louis Lembeck in performing the experiment.

226
9. References

[1] Henderson, F. M., Open Channel Flow^ (The Macmillan Company, New York,

1966)

.

[2] Kulin, G. and Compton, P. R., A guide to methods and standards for the

measurement of water flow. National Bureau of Standards Special Puhlioation 421, (U. S. Government Printing Office, Washington, D. C., 1975).

[3] Ludwig, R. G. and Parkhurst, J. D., Simplified application of PalmerBowlus flow meters, J. Water Pollution Control Federation 46 , 2764 (1974)

.

[4] Wells, E. A. and Gotaas, H. B., Design of venturi flumes in circular

conduits, Froo. Amer. Soo. Civ. Engr. 8£, Proc . Paper 928 (1956).

[5] Hirt, C. W. , Nichols, B. D., and Romero, N. C., SOLA - A numerical

solution algorithm for transient fluid flows, Los Alamos Scientific

Laboratory Report LA-5852j (U. S. Government Printing Office, Washington,

D. C., 1975).

[6] Roache, P. J., Computational Fluid Dynamics, (Hermosa Publishers,

Albuquerque, New Mexico, 1972)

.

[7] Harlow, F. H. and Welch, J. E., Numerical calculation of time-dependent

viscous incompressible flow. Physics of Fluids 8^, 2182 (1965) .

[8] Batchelor, G. K. , An Introduction to Fluid Dynamics, (Cambridge UniversityPress, London, 1967)

.

[9] Chow, V. T., Open-Channel Hydraulics, (McGraw-Hill Book Company, New

York, 1959).

[10] Schlichting, H,, Boundary-Layer Theory, (McGraw-Hill Book Company,

New York, 1968)

.

227
10. Notation

The following symbols are used in this paper:

D = distance from final honeycomb to depth measuring station in experimental

channel

;

d = arbitrary length;

g = gravitational acceleration;

g^ = component of gravitational acceleration parallel to channel bed;

g^ = component of gravitational acceleration normal to channel bed;

H = upstream head;

h = height above crest of upstream flow;

h^ = height above crest of free surface;

L = crest length;

N = N^^ time step;

P = crest height;

p = ratio of pressure to constant fluid density;

Q = discharge through flume;

Qq = baseline flume discharge from zero-slope curve-fit;

q = velocity vector; . Q .

Re^^ = Reynolds number based on D =

;

t = time;

U = inlet velocity profile;

u = component of velocity parallel to channel bed;

V = component of velocity normal to channel bed;

X = coordinate parallel to channel bed;

y = coordinate normal to channel

y = height of channel bed in xy-coordinate system;

ry +P+h

^o

.y

[U(y)]3dy

a = energy coefficient = —TrrFrBTuVi > QV(P+h)

[U(y)]2dy

y +P+h ^ o

3 = momentum coefficient = pr^- Q^/P+h

At = computational time step;

Ax, Ay = computational cell dimensions;

228
6^ = boundary- layer displacement thickness;

e = maximum bound on velocity divergence for a computational cell;

e = flume bottom slope angle;

V = kinematic viscosity;

$ = body acceleration; and

V = gradient operator.

229
TABLE 1

NUMERICALLY MODELED FLUMES

Flume No. P/L Tan e Flume Slope, %

Inlet Velocity

Profile, U(y)

1 0.19 0.25

0, 1, 2, -1 constant

0 linear

2 0.19 0.33 0 constant

3 0.33 0.25 0 constant

TABLE 2

LEAST-SQUARES CURVE-FITS OF HEAD-DISCHARGE RELATIONSHIP

FOR UNSLOPED FLUMES

FLUME HEAD-DISCHARGE RELATIONSHIP

(RMS Error, %)

NUMERICAL EXPERIMENTAL

EXPERIMENTAL

(Boundary-Layer

Corrected)

1

Q = .581 H^'^SO

(0.6)

Q = .570 h1-5S3

(0.7)

Q = .579 h1'556

(0.6)

2

Q = .596 h1*5^2

(1.0)

3

Q = .584 H^'^55

(0.6)

230
231
232
233
234


-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

LOG H

FIGURE 7. EFFECT OF INLET VELOCITY PROFILE ON HEAD-DISCHARGE

RELATIONSHIP

237
I 1 1 1 1 1

ISo

iH

•H6OOc

bO

•H

c

03

•ua

(1)

Soao

H<

<:

Cm<d

<;HZWM

PiW

PhXw

00w

CDM

ft,

238
239
-0.1

-0.2

-0.3

-0.4 —

-0.5

o

o

-0.6

-0.7

-0.8

-0.9

-1.0

-1.1

Computer

Computer

1.550

Q = .581 H

and

Exp. (B. L. Corr.)

Q = .579H^-^^^

Experimental

Q = .570

1-D Theory

Q = .544 H

1.500

Experimental (Boundary-Layer Corrected)

Q = .579 H^-^^^

Experimental

1-D Theory and

Exp. (B.L. Corr.)

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

LOG H

FIGURE 10. EXPERIMENTAL AND NUMERICAL HE AD- D I S CHARGE

RELATIONSHIPS

0.1

240
O 00 <1- CNI o Ho• • • • O O O

s

241
u

oo

6-S

4-1 4-1 T-l Q) IS) m

II II II e s o

•H -H -H

d) 0) Q) S-l !-l t-l (X ft P< 0) 01 OJ o o O a< Pu B

iH

CO to w w g; o < O X) •

>6

o

1^

4tl

I kl

o

O

ino

o

O

O

CNl

242
l^ational Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

'the design of open channel acoustic flowmeters for

specified accuracy: sources of error and calibration

'test RESULTS

Francis C„ Lowell, Jr.

Ocean Research Equipment, Inc.

Accusonic Division

Falmouth, Massachusetts 02541

The design of open channel acoustic flowmeters is more

difficult than the design of pipeline flowmeters because of

errors introduced by the free surface, as well as because

the open channel environment is usually more susceptible to

attenuation and multipath conditions which can inject large

errors unless special signal recognition and filtering

techniques are employed. The sources of error in open

channel acoustic flowmeters are tabulated and discussed.

These include 1) Line velocity errors 2) Level measurement

errors 3) Integration errors. Assuming that the required

sophisticated techniques are used to prevent the large

errors (20-50%) which can occur due to signal attenuation

and multipath, the total flow rate error contribution of

Items 1 and 2 above can be kept below 1% for liquid velocities greater than .5 FPS . Any error in excess of 1% of

flow is contributed by Item 3, Integration error. For an

accurately known cross section, this error is a function of

the number and spacing of acoustic paths and the integration

method used, and can be predicted in advance. Therefore,

the desired overall system accuracy should be taken into

account during the design phase in order to establish the

minimum number of paths required. Two design examples are

given for 2% specified accuracy with maximum flow rates of

1000 and 6000 CFS . In both cases the meters were calibrated

using standard stream gauging techniques and demonstrated

accuracies of better than 2%.

Key Words: Acoustic velocity, acoustic flow rate, flow rate,

liquid flow rate, liquid velocity, open channel flow rate,

open channel velocity.

243
1, Introduction

Acoustic methods have been used for velocity and

flow rate measurement for several years {1} ^ {2}

,

Their chief advantages and limitations are summarized in Figure 1, and typical applications are listed in

Figure 2. In general, they are economically practical for

use where one or more of the following apply: (l)high

accuracy is required, (2) channel widths are large, (3)

head loss must be avoided, (4) calibration or rating costs

are prohibitively high.

This report discusses the use of acoustic flowmeters

to determine volume flow rate in open channels with particular emphasis on sources of error and the design approach

required to produce a meter of specified accuracy for a

given application.

The design problem is more difficult for open channels

than for pipelines because of errors introduced by the free

surface, as well as because the open channel environment is

usually more susceptible to attenuation and multipath conditions which can inject large errors unless special signal

recognition and filtering techniques are employed. Nevertheless, relatively high accuracies have been achieved in

a wide variety of applications.

2. Theory of Operation

Acoustic flowmeters operate on the established

principle that the propagation velocity of acoustic signals

in liquids is changed when a component of the liquid velocity is parallel to the direction of acoustic propagation.

The average liquid velocity on each path is determined by measurement of the acoustic traveltime in each

direction. These paths have one acoustic transducer at

each end, and are located at various elevations in the

flowing cross section. The liquid velocity measured at

each path is integrated across the flowing area to determine the total volume flow rate. Figure 3 summarizes a

mathematical derivation for the acoustic line velocity.

Figures in brackets indicate literature references

at the end of this paper.

244
An acoustic flowmeter (see Figures 4, 5, and 6)

consists of an electronics/processing unit and several

pairs of acoustic transducers and associated cabling. In

addition, it must include one of several possible means

to measure water level (stage). Normally, the measurement

section is part of the channel, and a separately fabricated

"meter section" is not required.

When several acoustic paths are to be located in a

given cross section, there are limits to the proximity

of the acoustic path to the surface and bottom. This

limitation results from acoustic multipath reflections at

the boundary which interfere with the direct signal arrival

from the opposite transducer (see Figure 7), The time required to detect the presence of a signal (say one carrier

frequency period), establishes the interval required before the first interfering signals can arrive. Thus the

acoustic path must be located a certain minimum distance

(clearance height) from the surface or bottom, a distance

which depends on operating frequency and path length.

The operating frequency chosen depends on the application, but the following general rules apply.

1. - Higher operating frequencies increase system

accuracy by providing higher timing accuracies

and hence smaller line velocity errors. Higher

frequencies also allow path locations closer to

reflecting surfaces; e.g. surface and bottom,

hence velocity estimation errors can be reduced.

2. - Lower operating frequencies yield reduced signal

attenuation, particularly where silt or entrained air may be a problem. This increases system

reliability

.

3. - For a given liquid velocity and acoustic path

angle, AT becomes smaller as the channel becomes

narrower, thus requiring a higher operating frequency to maintain system accuracy. Thus,

"small" systems tend to be high frequency systems

Typical values of frequency and path length are

shown in Figure 8.

2A5
3. System Design

Acoustic flowmeter design is, like many other things,

an iterative process.

Usual specifications include range of volume flow

rate, type of channel, type of access available for transducer installation, maintainability requirements, accuracy,

type of outputs, and others.

Additional information required includes a knowledge

of the general hydraulic conditions around the meter, and

its location relative to sources of entrained air such as

pumps, turbines, and spillways upstream. Furthermore, in

river systems the expected silt or sediment load should be

considered, as well as shipping traffic.

There are several possible system trade-off's. The

most usual one is to trade-off the cost of increasing

system accuracy by incorporating additional acoustic paths,

vs the cost to obtain the same increase in accuracy by performing more extensive post installation tests and calibrat ions

.

A related problem with open channel flowmeter installations is that the velocity distribution in the channel

is not well known, either because it changes with different

flow/stage conditions or the channel is new and of a

sufficiently unusual design that the velocity profile cannot be predicted accurately. In either of the above cases,

additional acoustic paths can reduce system errors.

The economics of open channel flowmeters strongly

favor the addition of acoustic paths because the marginal

cost of additional paths is low given that the site work,

electronics, stage measuring system, cable runs/conduits

and transducer mounting means have already been paid for.

Therefore increased system accuracy can often be achieved

more economically with an increased number of acoustic

paths than by field calibration after installation.

Furthermore, as the channel wall and bottom roughness

change between maintenance and cleaning cycles the velocity

distribution will change, and even if a calibration is

performed after installation, the meter coefficient may

change with time, unless a sufficient number of acoustic

paths are installed.

246
A second trade-off is the relation of system cost to

site location and site development work required. An

example of this is whether the system should be installed

in a location requiring extensive site work for a favorable

measurement section vs. the possible cost savings of reduced site work on the channel and more complex acoustic

path placement to compensate for unknown cross flow components and velocity distributions.

Needless to say, the best system designs occur when

the flowmeter is considered in the planning process so that

an optimum meter location can be picked at minimum cost.

4. Sources of Error

There are several important sources of error which

must be considered when acoustic flowmeters are being

designed o These are summarized below:

A) Line Velocity Errors

(1) Installation or Survey Errors

a) Referring to Figure 3, it can be seen that the

measured line velocity is proportional to L/T^, and since

T is proportional to L, then V is proportional to 1/L.

Therefore any error in L will introduce the same error in

V. In general, this is not a problem because L can be

measured to a high degree of accuracy.

b) Path angle error. Since v^l/cosO, any error in

9 will produce an error in V. For example, at a 45° path

angle, a 1 degree error in 0 will produce a 1.7% error in

the line velocity.

c) Non-liquid and/or stationary liquid delays:

Any delays in the system which add to T without changing

AT will introduce errors in the meter and must be compensated for. These errors include:

i) Cable/logic/detector delays

ii) Acoustic delays associated with transducer

windows

iii) Liquid delays which occur when transducers are

separated from the moving fluid, for example,

recessed in the walls of an aqueduct

.

247
(2) Errors introduced by transducer installation

These can be caused by transducer projection, if

any, into the streamflow, and any unsampled flowing area

behind the transducers. Correction for these errors

requires analysis of the exact installation configuration.

In general the errors produced by the projection effect

are less than the ratio of transducer diameter to channel

width, which is generally less than 0.1 ot 0.5 percent,

and can be compensated for.

(3) Errors due to variable acoustic signal strength,

receiver circuits, and timing accuracy.

The signals traveling along any acoustic path are

subject to variable attenuation in addition to normal

spreading loss. This attenuation can be caused by silt,

entrained air, marine life, shipping or fouling to name

a few. Furthermore, acoustic signals can be distorted

by raultipath reflections from shipping or debris.

The most serious manifestation of this problem in

its effect on flowmeter accuracy occurs when the acoustic

signals are sufficiently weaked or distorted so that the

receiver detector misses the desired detection point on

the waveform by one acoustic wavelength (one period) the

effect on accuracy can be very severe. Typical errors

resulting from this problem are shown in Figure 8.

If the meter is to be installed in areas where

weak of distorted signals are a possibility, then it is

essential that signal recognition circuits be incorporated

into the system receiver in order to prevent this type

of error.

248
(4) Crossflow Errors

One of the most frequent problems in acoustic flowmeter design efforts is the prediction of streamline direction components which are not parallel to the measurement section axis. The errors produced by this effect are

exactly analagous to angular errors in the assumed path

position (survey errors) covered in Section 4A(l)b except

that the crossflow component is, in general, a function of

flow rate and stage.

These errors can occur, for example, when the measurement section is too close to an upstream bend, transition

or obstruction which causes the flow to become detached

from one or both walls, thus producing asymetical and/or diverging streamlines in the meter section. The problem may

be aggravated in bi-directional flow installations because

any correction factor used for one direction may not be

correct when the flow is in the other direction.

A third case occurs in unlined channels where the

sidewalls are so irregular that it is impossible to determine the "axis" of the channel.

In any of the above cases, errors can be reduced by

the addition of one or more acoustic paths, at the same

elevations as the original ones but installed at the

opposite angle (see Figure 9). Exact cancellation of errors

can be accomplished on the crossed paths, and an estimate

of the cross flow component can be used to adjust the

readings on the non-crossed paths.

(5) Other Sources of Error

a) Errors due to time base accuracy are generally

negligible because crystal oscillators are used which

have very high stability and accuracy (+.005%).

b) Differential detector traveltime measurement

errors are eliminated by using the same detector for time

measurement in both directions, or by careful balancing of

two detectors if simultaneous measurements are made in both

directions. The use of one detector, while eliminating one

source of error, adds one, at least theoretically in that

the two traveltimes must be measured sequentially rather

249
than simultaneously. However, in practice, this does not

introduce significant error because the time required to

make a complete pair of measurements is short compared to

the rate of change of velocity in the measurement section,

even under very turbulent conditions, and the possibility

of long term differential delays developing between two

separate receivers is eliminated,

c) Errors associated with the dicrrete traveltime

counting periods or quanta, are reduced by using a relatively high counting frequency (10-40 mHz, depending on

channel width) and the averaging of many individual line

velocity readings. This is possible because while an error

of +1 count is, in general, made with every traveltime

measurement, this is a true random error and can be reduced

by averaging over a suitable period (usually 10 sec-10 min,

depending on the rate of change of flow rate),

B ) Level Measurement Error

The effect of a given uncertainty can be easily calculated given a specific application, as it has a direct

effect on the flowing area between the highest acoustic

path and the surface.

C) Integration Error

Integration error is defined as the difference between the flow rate calculated by an exact integration of

the velocity profile and the flow rate calculated using

exact line velocities measured on the several acoustic

paths in the system.

The error associated with flow rate integration is

conveniently divided into three parts. They are: (1)

Highest layer error: the volume flow rate between the

highest acoustic path and the surface is uncertain to the

extent that the velocity distribution is uncertain. The

highest acoustic path cannot be placed arbitrarily close

to the surface because of multipath effects. Therefore an

estimate of the velocity distribution must be made, usually

on the basis of the velocity gradient between the two

highest paths extrapolated to the surface, (2) Lowest

layer error: the region bounded by the lowest acoustic

path and the channel bottom has a flow rate uncertainty

resulting from the unknown velocity distribution near

250
the bottom. The error contribution here is generally

less, however, because the velocities are lower, and the

boundary condition that the velocity is zero at the bottom

is known. (3) Layers bounded by acoustic paths: the

errors associated with layer flows which are bounded by

acoustic paths are completely under control of the designer

5. Designing For Specified Accuracy:

Worst Case Analysis For Acoustic

Path Placement

With proper system design, it is possible to keep the

total error contributions due to line velocity error and

uncertainty in stage to within +1% of flow for average

velocities in excess of .5 FPS (.15.m/sec), (see Figure 16)

Any error in excess of 1.0% of flow results from

integration error. For an accurately known cross section,

this error is a function of the number and spacing of

acoustic paths and the integration method used, and can be

predicted in advance. Therefore, the desired overall

system accuracy should be taken into account during the

design phase in order to establish the number of paths

required

.

It is not possible to produce a standard "cookbook"

design for open channel flowmeter acoustic path placement

because variable stage, and high Reynolds numbers and

thermal gradients can produce velocity distributions which

bear no resemblance to theoretical curves which are derived assuming fully developed turbulent flow. Thus

numerical integration techniques which require a specific

distribution of acoustic paths are of limited usefulness,

and every system should be designed with its particular

characteristics in mind.

In this and the following section, two designs are

discussed. In each case the design required an accuracy

of 2% of flow rate. In both cases, the aqueduct was mortar

lined and its cross section was accurately known. The

location, angle and elevations of the acoustic paths were

accurately determined by standard surveying techniques,

and the water level was determined by a stilling well and

float connected to a digitizer yielding an overall resolution of +.005 ft (+. 0015m) of stage.

251
A) In the first example, the channel was rectangular in

cross section, with a 30' width and 10' depth (9.14 m by

2„74 m deep). The water level varied from 2 feet to 10

feet (o61 m to 2.74 m). An overall accuracy of +2% was

specified.

The design sequence involved first, the determination

of the number of acoustic paths required to meet an integration accuracy of approximately 0.5% using a theoretical

logarithmic velocity distribution (4 paths).

A "worst case" velocity distribution or gradient was

then constructed which produced additional integration

error o The "worst case" profile, which consisted of a

linear velocity gradient between the highest acoustic path

and water surface, was chosen to represent the uncertainty

in flow rate which could possibly occur, but which could

not be corrected for, since the gradient occurred above the

highest acoustic path and therefore could not be detected.

The magnitude of this gradient was chosen to be +5% of the

average surface layer velocity. This average velocity was

determined by a weighted average of the velocity measured

on the highest acoustic path and the surface velocity

estimated by linear extrapolation from the two highest

acoustic paths.

Using this "worst case" velocity gradient, the integration error was calculated as the stage was increased.

At the stage where the integration error approached 1%, a

5th acoustic path was added at the minimum distance below

the water surface (.5 ft in this case, or .15m). This procedure was repeated, adding acoustic paths as required

up to maximum stage.

Finally, the positions of the acoustic paths were

adjusted to equalize the estimated error peaks, since the

highest acoustic path, in general, did not fall at the

maximum distance from the highest water elevation. The

final path elevations are shown in Figure 10.

B) This system was built and installed at the inlet canal

to Lake Skinner, near Hemet , California, and was subsequently calibrated by Metropolitan Water District of

Southern California and U. S. Geological Survey personnel,

252
using standard stream gaging techniques {3}. On the first

calibration, agreement was better than +2% on all readings except for two at low stage, where flow separation

and excessive turbulence made accurate conventional stream

gaging impossible. A subsequent low stage calibration

with the stream gaging equipment installed upstream of

the turbulence produced readings within the 2% tolerance,

(see Figure 12).

The turbulence observed at low stage was caused by

an S bend and transition from trapezoidal to rectangular

cross section upstream of the meter section (see Figure 5).

This caused separation of flow, turbulence and non-axial

flow streamlines through the meter section. The effect of

this can be seen on the calibration curve, since the meter

error varies from - 1.5-2% at stages over 7 ft (2 m) to

+ lo5-2% at a stage of approximately 3.5 ft (1 m). From

Figure 9, the average cross flow component at this stage

is therefore about 2°, and is a function of stage. The

statements above are supported by streamflow direction data

taken by Metropolitan Water District personnel {3}.

This calibration illustrates a problem common to high

accuracy open channel meters of any type: the available

standards to which the meters can be compared are of comparable accuracy to the meter being calibrated. Therefore,

it is difficult to determine an error or accuracy for the

acoustic meter on this basis.

C) Recently, computer simulations have been run on this

flowmeter design at O.R.E. The purpose of the simulation

was to obtain a more accurate estimate of integration

errors produced by different vertical velocity profiles.

Two types of tests were performed. The first consisted of

several runs using undistorted or "typical" velocity profiles produced by several possible logarithmic curves

simulating different degrees of channel surface roughness.

Velocity profiles used during these runs are shown on

Figure 10, and the resulting errors are shown on Figure 11a.

The important thing to notice here is that the integration

errors are small, and that the several assumed flow profiles do not effect the error significantly.

In the second set of runs, see Figure 10, one of the

assumed logarithmic distributions was distorted with two

253
different profiles. These profiles were scaled so that

the maximum distortion velocity occurred at maximum stage,

with a linear decrease with lower stages so as to maintain

a constant maximum near surface velocity gradient. These

velocity profiles were chosen to represent the possible

results of a strong upstream or downstream wind or other

effects such as ice or debris which could cause unpredictable velocity gradients near the surface.

The two velocity distortion profiles used differ in

that one is mostly confined to areas relatively near the

surface, and hence cannot be "seen" by the two upper

acoustic paths at all stages. The other, while more gradual,

has a larger total flow rate effect but it can be "seen"

by the upper paths. The flow rate change caused by the

distorted profile can thus be partially corrected for.

The plots in Figure lib show the effect of these

assumed profiles. In order to make the worst case integration error approximately 1%, the velocity distortion

was increased to +20% of the undistorted surface velocity

at maximum stage. (The 20% distortion figure was regarded

as unrealistically high and hence conservative). The

maximum integration errors occur at water depths ,5 ft

(„15 m) above each transducer elevation, which corresponds

to a maximum velocity extrapolation distance.

Since, in our opinion, a 20% surface velocity distortion is unrealistically large, our conclusion is that

we were conservative in our acoustic path spacings and we

could have made the meter meet specification with one or

two fewer acoustic paths. Also, given that all nine paths

were to be used, we could have distributed them more

favorably, so as to reduce the peak error slightly.

However, in the real world of fixed price contracts,

penalty clauses and performance bonds, one or two extra

acoustic paths is a small price to pay for a conservative

design, customer confidence, and a shorter acceptance

periodo

D) Two additional flowmeters were designed and built for

the State of California Department of Water Resources, and

install in the California Aqueduct near check structures

12 and 21. Both meters operate in trapezoidal cross

254
sections of very similar dimensions, and have overall

design accuracies of +2% of flow. The pertinent dimensions and acoustic path locations for Check 12 are shown

in Figure 13.

The main difference between these trapezoidal meters

and the previous example, other than size and shape, is

the relatively small variation in stage (27 to 31 ft or 8

to 9 m at Check 12).

This design problem differed from the proceeding

example because the potential surface layer flow rate

estimation error was relatively small due to the small

variation in stage. However, because of the anticipated

daily variation in flow rate (pumping of aqueduct water

is normally done during off-peak hours) and the strong

density gradients which could be produced by near zero

flow during the daylight hours, irregular velocity profiles could develop which would not resemble the classical

logarithmic velocity profiles.

This problem was illustrated by several profiles,

taken by The Department of Water Resources at the flowmeter

locations, which were examined during the design stage. {4}.

Some of these have subsequently been published {5}. An

error analysis was performed using "worst case" criteria

for this application, based on the measured velocity profiles. The resulting design is shown on Figure 13. The

analysis showed that the total flow rate error contribution of upper and lower paths was constant with equal path

spacing because the upper layers had larger areas and

lower velocity gradients while the opposite was true of

the lower layers.

E) Computer simulations have been run on this design, in

this case using several actual velocity profiles. The

curves selected for computer analysis did not represent

a random sample of expected velocity profiles, but were

selected to include the "worst" ones from the point of

view of possible integration error, together with one

considered more nearly typical (Check 21 Run 144). These

curves are shown on Figure 15, together with the integration errors.

255
Four of the six velocity profiles show integration

errors of less than 1%, and demonstrate the ability of the

flowmeter to provide accurate results even when the velocity

profile does not resemble theoretical curves. Even the two

worst profiles demonstrated errors of less than 2% and the

velocity distribution producing this error could only be

expected to persist for a small fraction of the time since

they were probably produced by system flow transients.

After installation, the flowmeter was calibrated by

Department of Water Resources personnel {6}, The results

of the calibrations are shown in Figure 14A, together with

the velocity distributions which were obtained (Figure 14B),

The calibration results demonstrate the high accuracy which

is achieved when near-typical velocity profiles occur in

the meter section.

F) A summary of error sources for a typical open channel

application is shown in Figure 16. From this illustration

it is clear that for high accuracy flow rate measurement,

the potentially large errors associated with incorrect

determination of traveltime (Item 5b) must be positively

eliminated by suitable design.

Summary and Conclusions

Acoustic flowmeters can be used to measure volume

flow rate in a wide variety of applications.

Using the design approach discussed above, system

flow rate accuracies of two percent or better are

achievable over a broad range of flow rates for mortar

lined channels. Similar design techniques are applicable

to any open channel acoustic flowmeter requirement.

In all cases, total installed system cost can be

minimized during the design stage by a suitable trade-off

between equipment, site work and calibration costs on

the one hand, and desired accuracy on the other.

256
Acknowledgements

The author wishes to thank the Metropolitan Water

District of Southern California and the State of California

Department of Water Resources for making calibration and

flow profile data available for this paper.

In addition, valuable ideas, particularly concerning

acoustic design and error analysis have been provided from

discussions with personnel of the Hydro-Instrumentation

Department of Ri jkswaterstaat , the Netherlands.

Within our own organization I would like to thank

Mr. Tom Campbell for his help and suggestions used in the

computer analysis portion of the work.

References

{1} Smith, Wo, Experience in the United States of America

with acoustic flowmeters, Symposium on river gauging

by ultrasonic and electromagnetic methods. University

of Reading, England, Dec 1974

»

{2} Schuster, J.C„, Measuring water velocity by Ultrasonic

Flowmeter, Proc ASCE„ 5 of Hydraulics DiVo , 101:HY12,

Dec 1975

{3} Whitsett, AoM., Ramsay, W^B., Angularity flow tests at

acoustic flowmeter installation. Lake Skinner, Unpublished Report, Metropolitan Water District of

Southern California, 29 Jan 1975

«

{4} Ohriti, Ro , L.Eo Flowmeter flow profiles near Check 12

and 21, Unpublished Data, State of California Department of Water Resources, 20 April 1970 thru 10 Feb 1971o

{5} Lanning, C„ , Ehrhart, RoWo, Application of Acoustic

Velocity Flowmeters, J. Am. Water Works, 69 Feb 1977

o

{6} Innes, Go, Trefethen, Go, Gage, Do, Current meter

measurement, California Aqueduct bridge at Acoustical

flow meter site near Check 12, Unpublished Data, State

of California Department of Water Resources, Dec 10,

11, 1975; Jan 4, 1976o

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INTEGRATION ERROR VS STAGE FOR DIFFERENT

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NORMALIZED DEPTH

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WHERE C=NORMALIZING CONSTANT, Z=STAGE, X=N0RMALIZED DEPTH, B=0.2

FIG.IIB INTEGRATION ERROR VS STAGE FOR DISTORTED

PROFILES OF FIG.IO
WATER LEVEL, FT

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SPEC

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FIG.I2 FLOWMETER CALIBRATION RESULTS

INLET CANAL TO LAKE SKINNER (REP 3)

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FIG.I4A FLOWMETER CALIBRATION RESULTS

AT CHECK 12 (REF 4)

FIG.I4B VELOCITY PROFILES MEASURED

DURING CALIBRATION AT CHECK 12

265
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266
National Bureau of Standards Special Publication 484, Proceedings of the Symposium on Flow in Open Channels and Closed Conduits held at NBS

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

APPLICATION CONSIDERATIONS & PERFORMANCE CAPABILITY

OF THE TIME DIFFERENTIAL

CLAMP-ON ULTRASONIC FLOWMETER

J. Baumoel

Controlotron Corporation

111 Bell Street

West Babylon, New York 11704

ABSTRACT

Substantial information has been developed on the application history

and performance capability of the time differential clamp-on ultrasonic

flowmeter during approximately four years of in-service experience.

These application considerations and performance capabilities are reviewed together with a review of the principles of operation and listings

of the types of uses to which this instrument has been applied.

1 . INTRODUCTION

The impact of a practical clamp-on flowmetering device would be significant

enough for many users who are faced with the need to measure flow rate of

liquids whose physical or chemical properties make conventional flowmeters

difficult or impossible to apply. But the availability of a clamp-on

flowmeter whose basic performance characteristics challenge and, in many

cases, exceed the performance capabilities of conventional flowmeters,

permits application of the time differential clamp-on ultrasonic flowmeter

to ordinary applications with significant economic and performance benefits

not previously available.

Such a device is the time differential clamp-on ultrasonic flowmeter.

This device works on the principle of transmitting an ultrasonic beam

between two transducers merely clamped to opposite sides of an existing

defined pipe in such a way as to transmit the beam at a definable angle

relative to flow. A flow display computer obtaining data from these transducers is charged with the task of extracting the relatively minute effect

of flow rate on the upstream versus downstream transmission time of the

sonic beam through the liquid. (See Fig.l) This flow display computer

converts the data which it has obtained to those forms which are standard

in industry, i.e., a digital display in engineering units of choice, a

scalable analog display with settable high and low flow alarms, plus analog

outputs 0-lOV and 4-20ma, a pulse rate output proportional to flow (100 Hz

per foot per second), and a totalizer register in units of liquid volume.

The time differential clamp-on ultrasonnc flowmeter (hereinafter referred

to as the At clamp-on ultrasonic flowmeter) should not be confused with

the Doppler ultrasonic flowmeter which uses only a single transducer mounted

267
on one side of the pipe which obtains data from the change in apparent

reception frequency of a continuous ultrasonic beam being reflected from

bubbles or solid particles which may be suspended in the liquid. Nor

should it be confused with insertion time differential ultrasonic flowmeters which employ a spool section having sonic transducers embedded

within fixed holders welded into the side of the spool section. The

former device has the convenience of the /\T clamp-on flowmeter, but is

highly dependent on the presence of free solids or gases in the stream

for operation, and has a calibration factor dependent upon the sonic

velocity of the liquid - a factor very frequently unknown or variable.

The spool insert time differential ultrasonic flowmeter has the accuracy

capability of the clamp-on ^T flowmeter, but of course does not have its

convenience of installation or maintenance.

2. DESCRIPTION OF /\t CLAMP-ON ULTRASONIC FLOWMETER

The clamp-on T ultrasonic flowmeter consists of a set of ultrasonic

transducers mounted on opposite sides of the pipe at a preferred displacement ordinarily fixed by a spacing bar interconnecting the two

elements. Installation on a pipe whose diameter, wall thickness and

Material has been previously designated is accomplished by merely applying a sonic coupling compound between each transducer element and pipe

exterior, and tightening a self-contained clamp mechanism. This simple

mounting procedure makes practical the use of this device as a portable

flowmeter

The flow display computer operates the transducers and measures the

time differential between transmission and reception of sonic energy.

It then converts this time differential, which is proportional to flow,

to conventional data format. Models are available which provide digital

display in preferred flow units, analog display, analog outputs (0-lOV &

4-20ma) , totalizer display, and pulse rate proportional to flow. Thus

the T clamp-on ultrasonic flowmeter provides any of the functions ordinarily provided by any conventional flowmeter of any type, and in many

applications can be used interchangeably.

In general, it is necessary to specify only the pipe size, schedule and

material as the flow display computer will automatically correct for the

effects of changes in sonic velocity of the liquid regardless of the cause

of such change (i.e., temperature changes or changes in liquid chemistry).

This flowmeter has been used in applications on pipes as small as h'*

and is available for pipes larger than 60". In principle, there is no

upper limit so long as the liquid is sonically conductive - a property of

most homogeneous liquids free of excessive undissolved gases or solids.

3. APPLICATION CONSIDERATIONS

The prime requirement for operation of the clamp-on flowmeter is that

the sonic beam which is injected into the exterior wall of the pipe be

268
successfully transmitted through that wall, through the liquid, and then

through the far wall of the pipe so as to be received with sufficient

amplitude at the receive transducer to permit signal detection. For this

to be achieved only two conditions are required —

1. The pipe wall must be sonically conductive (including the

effects of any internal coatings or linings)

.

2. The liquid must be sufficiently sonically conductive.

These two conditions are almost always achieved in real applications.

All metallic pipe has been found to be an excellent conductor of sonic

energy. Many lined pipes have also been found to be sufficiently sonically

conductive, provided that the lining is well bonded to the interior of the

pipe wall. Most linings themselves such as plastics are excellent sonic

conductors. However, some linings such as cement may be absorptive of the

sound beam if during the mixing of the cement air bubbles have been retained

during the curing of the cement.

Most plastic pipes are excellent sonic conductors. Included in this

class are PVC and filament wound FRP pipe. However, FRP pipe made of

woven mat generally includes sufficient entrapped air within the mat to

seriously interfere with the sonic conductivity of the pipe wall.

Nature has been very kind to us in that all homogeneous, non-aerated

liquids have been found to be excellent sonic conductors. The only

property of the liquid other than its sonic conductivity which is of

any material importance is the sonic velocity of the liquid itself, which

for most liquids lies in the range of 1000-2000 meters per second. The

/\T clamp-on flow display computer computes the sonic velocity from the

data which is received from the ultrasonic beam and uses this data to

correct the scaling of the system so that the system calibration is essentially independent of the sonic velocity.

However, there are conditions under which a liquid will lose part or

even all of its sonic conductivity. A primary cause of this condition is

what is commonly called aeration. Aeration is a condition in which free

bubbles of either air or of the vaporized liquid pass through the sonic

beam. These bubbles tend to disperse and diffuse the sound beam if

present. However, this is a condition which can generally be avoided

since its causes are rather well defined. For example, aeration can be

caused by adding liquid to a holding tank with the inlet above the liquid

level, causing air to be mixed as the stream passes through the surface

of the liquid. A low inlet (below the surface) has always avoided this

condition. Other causes of aeration are leaky fittings at the suction

end of a pump, or cavitation caused by a defective pump or an upstream

throttling valve. Since these causes are generally destructive in themselves, their effect on the flowmeter which is announced by the flowmeter's "fault monitoring system", enables the user to be aware of the

condition and take the necessary corrective action.

269
Undissolved solids in the liquid can also serve to reduce the liquid's

sonic conductivity. In general, this condition is considerably more tolerable

than aeration. Examples of systems providing excellent operation with free

solids content of up to 50% have been found on many occasions. While no

hard and fast rule has been established, it has been found that solids whose

acoustic impedance is reasonably similar to the liquid itself (such as

paper pulp) , are more tolerable than solids whose acoustic impedance is

considerably higher (sand particles)

.

Since the principle of operation of the /\T flowmeter does not in itself

depend upon the physical properties of the liquid, operation is generally

not affected adversely by reasonable variations of liquid temperature. The

only effect of viscosity will be on the flow profile - an effect which is

among those for which means of correction are available within the computer.

Means also exist for taking into account non fully developed flow profile

or flow profile skew, which may exist if the pipe section on which the

clamp-on transducers are placed is immediately after an elbow. Correction

is effected by placing the transducer in a plane which is coincident with

the plane of the elbow itself, so that the sonic beam passes through the

peak of the flow profile itself.

Transducers have been supplied for applications ranging from cryogenic

temperatures up to 500°F. Since introduction of the sonic beam from the

transducer to the pipe wall requires use of a coupling compound to fill

any gaps which may otherwise exist between the pipe surface and the transducer surface, it is necessary to use a coupling medium which is suitable

for the temperature at which the process will operate. A variety of

liquids such as oil, greases, and settable rubbers and epoxies have been

found which cover all of these conditions very satisfactorily.

4. PERFORMANCE CAPABILITY

The performance parameters of interest to the user of the ^T clamp-on

ultrasonic flowmeter are the same as those parameters which are currently

used to rate conventional flowmeters. These parameters include:

1. Linearity

2. Stability (freedom from drift)

3. Range of operation

4. Resolution

5. Calibration accuracy

6. Response time

Linearity

The independent linearity of the clamp-on ultrasonic flowmeter approximates 0.25-0.5% for liquids and pipes of normal sonic conductivity. Evidence

of this capability is presented in Fig. 2 representing data obtained at

Alden Research Laboratories on a typical production unit manufactured for

Atomic Energy of Canada Ltd.

270
Stability

Zero drift stability of the Z\t clamp-on flowmeter has been found not

to exceed variations of .05 ft/sec independent of pipe size. In many cases

where flow can be shut down at will, any long term zero drift can be eliminated

even though .05 ft/sec represents an exceedingly small maximum drift.

Span drift has not been found to exceed 0.5% of reading over long periods

of time approximating one year. This high stability is due to the digital

nature of the computation not ordinarily subject to variation. Portable

calibration equipment is available to permit testing span calibration without

removing the flow display computers from their installations.

Range of Operation

The operating range of the Z\T clamp-on flowmeter is guaranteed from

-30 ft/sec to +30 ft/sec including zero flow. This is evidenced by incorporation of a 3h digit display which has the capacity to display flow

in any pipe size for velocities of up to at least 30 ft/sec. An automatic

flow direction indicator is also provided.

Resolution

The resolution is defined as the smallest change in flow which can be

detected and displayed by the flow display computer. The resolution of the

^T clamp-on ultrasonic flowmeter is of the order of magnitude of 1/1000

ft, per second independent of pipe size. The reason for this extremely high

resolution is because friction is not a factor in the measurement technique.

A particular benefit of this high resolution is that very low flow velocities

can be displayed with very high resolution. For example, a flow velocity of

1 ft/sec in a 2V' pipe can be displayed with a resolution of 1/100 of a gallon per minute. Thus, even at this nominally low velocity, 1% resolution

can be obtained.

Calibration Accuracy

Calibration accuracy refers to the degree to which the data displayed by

the /\T flowmeter truly represents actual flow. Since the linearity specification defined above applies primarily to the degree to which a change

in flow causes a correspondingly proportional change in indication, the

calibration accuracy can be viewed as a statement as to how closely the

proportionality constant corresponds to unity.

In general, the clamp-on flowmeter can be calibrated either intrinsically,

which is basically an electronic calibration method, or by setting its

calibration controls while the flowmeter is operating on a flow calibration

stand. As would be expected, calibration performed under actual flow

conditions will be more accurate in general than that performed intrinsically.

This does not mean to imply that an intrinsic calibration cannot be perfectly accurate, but rather that the confidence factor associated with an

actual flow calibration is greater. It has been found by many tests performed

271
at recognized flow laboratories that the calibration of the clamp-on flowmeter

can achieve accuracies of 0.5% or better. Intrinsic calibrations have generally

fallen within the range of 1-4% when performed electronically and then checked

on an actual flow calibration stand. Continued experience in intrinsic calibration, which is of course less expensive than actual flow calibration, has

shown a general improvement as experience has been gained. There is the

expectation that intrinsic calibrations should lie in the 0.5%-2% region in

the relatively near future.

It should be noted that the clamp-on flowmeter is somewhat responsive to

the flow profile of the liquid which it is measuring. This, of course, is

similar to the flow profile sensitivity of many conventional flowmeter

techniques

.

The clamp-on flowmeter has the ability to compensate for the difference

between the flow profile shape in actual service as compared to the flow

profile shape achieved during calibration which is ordinarily performed with

a liquid such as water. This correction is performed by the flow display

computer by means of an internal program into which the user of the machine

merely provides an instruction as to the Reynolds number of the process

being monitored. The flow display computer then automatically corrects

for this new flow profile shape on the presumption that it is fully developed.

Non -fully developed flow requires a special program which in general can be

provided.

Response Time

Since the clamp-on flowmeter operates on the basis of a sonic beam sent

through the liquid, there is no inertia involved in extracting flow data

from the flowing stream. For that reason, the response time of the clampon flowmeter can be extremely fast. Raw data is available with each transmission passing through the pipe. In general, for liquids similar to water,

that transmission travels through the pipe in a time equal to approximately

sixteen millionths of a second per inch of pipe diameter. For most practical

models of the clamp-on flowmeter, the data processing uses approximately

1/100 of a second so that actual output data is available at a frequency

bandwidth of approximately 100 Hz. This is considerably faster than most

flowmeters and, in fact, is considerably faster than that required for most

applications. For that reason, the flow display computer is generally

equipped with a user selected control of response time which can be made

as fast as .01 second or as slow as 10 seconds for applications in which a

smoothing of a normally fluctuating flow is desired.

5. APPLICATION LIST

Successful applications of the ^T clamp-on ultrasonic instrument are

found in essentially every industry. A representative list of such applications follows:

272
INDUSTRY APPLICATION

Paper Black liquor reactor two 10 channel

multiplex operation for 3/4" carbon

steel schedule 80 lines

Nuclear power Reactor cooling balance, pipe sizes

sizes from 2-4" and multipipe surveys

to 36'

Petrochemical

Pharmaceutical

Chemical processing

Crude oil and other petrochemical flow

measurements and control functions.

1 through 24" pipes

Measurement and control of products for

the preparation of antibiotics

Numerous and diverse chemicals in pipes

from 1" to 12"

Government

Municipal

Measurement of cryognic liquids, liquid

TNT, phosphorous and other liquids

Waste water and sewage with pipes up

to 42" in diameter

Aerospace

Food

Jet engine fuel flow (aircraft tubing)

Quality and batch control in food

processing. 1" to 4" pipes

Marine Submarine ballast measurement and control

(3" copper nickel pipe)

6. CONCLUSION

The clamp-on ultrasonic flowmeter is a new tool available to industry,

enabling it to accomplish flow measurements with a convenience, economy and

performance capability not previously available in one instrument. Substantial

field experience and calibration test results have established its capability

in a wide diversity of practical applications. The availability of performance data, limits of acceptable application, and the body of knowledge

governing successful installation and use, has been established and is available to industrial users as a base for the increase in the utilization of

this instrument.

273
SKETCH

d

Ts = d

Vs cos -e

ATg = Vp sin -e- Ts

Let d=l", Vs= 5000 ft. /sec, Vp = .001 ft. /sec ^=50^

= 1 =19.25 ^isec

(60X10-^) (.866)

ATg = .001 (0.5) (19.25X10-^)

ATg = 1.925 X 10-12 seconds

Figure 1. AT clamp-on flowmeter differential time computation,

274

PERFORMANCE

Zero Stability

Linearity

Accuracy

Intrinsic Calibration

Actual flow Calibration

Repeatability

Flow Range

Digital Display Resolution

Digital Display Update

Interval

Analog Update Interval

Totalizer Update Rate

Liquid Characteristics

Standard:

Approva 1 Required

:

,05 1/sec

1% of reading above 1 fps

1-4% of range (typical)

O.S-1.5% of range (typical)

+^3 digits of s.d.

0 to 30 ft/sec (minimum)

20,000 digits (two ranges)

4 seconds

0 . 1 sec (nom)

Faster update on special order

Continuous (Real Time)

Homogeneous

No Undissolved solids or

gases, Vl from 900 to 2100 m/sec.

Non-homogeneous

liquids or undissolved solids in liquid

Note: For nominal pipe size, wall thickness, in specified

pipe material and for homogeneous non-aerated liquids

.

Figure 3. 241 flow display computer specification.

276
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

THEORETICAL AND EXPERIMENTAL

ASSESSMENT OF UNCERTAINTIES IN

NON- INTRUSIVE, ULTRASONIC FLOW MEASUREMENT^

Ronald F. Bruner

Naval Ship Engineering Ce^nter

Philadelphia Division

Philadelphia, PA 19112

In recent years the development of commercially avail-^

able, externally mounted, ultrasonic flow measuring systems

has made it possible to measure liquid flow rate without disturbing the flow or modifying the ducting. Such systems rely

on the highly accurate measurement of both the time for sound

pulses to travel from an upstream transducer to an oppositely

mounted downstream transducer, and the time for sound pulses

to travel from downstream to the upstream transducer. (This

technique does not employ the Doppler shift, and scattered

sound is undesirable.) This paper treats only the steady

flow of homogeneous liquid in full ducts, although the theory

is easily extended to open channel and gas flow.

In making such measurements the duct itself becomes part

of the measuring system, as the path and intensity of the

transmitted sound depend on the material composition, wall

thickness, and surface condition of the duct. Furthermore,

the outside dimension of the duct (i.e. diameter), velocity

of sound in the liquid, and flow velocity field also figure

in the theoretical determination of the flow rate. Add to

the uncertainties in the above parameters the positional (i.e.

mounting) uncertainties plus uncertainties in the electronic

equipment, and the feasibility of accurate flow measurement

outside the laboratory becomes questionable.

A theoretical analysis is developed which aims to quantify the uncertainties associated with these parameters, and

to determine their effects on overall system accuracy and repeatability. Experimental data have been obtained for estimating the uncertainties which might normally be encountered.

These are presented for comparison with the theory. The result is a clearer picture of what we can realistically expect

for the overall accuracy of non-intrusive, ultrasonic flow

measurement both in the field, and under controlled conditions.

Key Words: Ultrasonic flowmeter; -acoustic rays; flow measurement.

^The opinions expressed herein are those of the author and

not necessarily those of the Department of Defense nor the

Navy Department.

^The flow measuring system referred to for the experimental

data in this paper was manufacturered by Controlotron Corp.,

Ill Bell Street, West Babylon, NY 11704.

277
1. Introduction

The development of commercially available, externally mounted, ultrasonic flow measuring systems may well be the most significant advance in

flow measurement technology in recent years. In principle, this type of

system leaves little to be desired for the measurement of homogeneous liq

uid flow. It is portable, and does not require any disturbance of the

flow or the piping, other than removal of lagging and/or scale from the

pipe surface. The essential idea behind this type of flow measurement is

that it takes longer for ultrasound to traverse the flow when directed

against the flow (upstream) than when it is directed with the flow (downstream) (see Figure 1). The greater the flow, the greater is the difference between these two traverse times.

GABLES

UPSTREAM

TRANSDUCER

-FLOW NW. WWWWW c 4^ XXX

DOWNSTREAM

TRANSDUCER

PIPE WALL

FIGURE 1 SKETCH OF FLOW MEASURING SYSTEM

The major elements of the system are seen in Figure 1. The flow computer is the originator and receiver of electrical pulses. For example,

a pulse is sent to the downstream transducer. Natural frequency vibrations are thus induced in the transducer crystal (usually in the lOOkHz5MHz range) and an ultrasonic pulse travels across the pipe, exciting

the upstream transducer crystal. This results in an electrical signal

which is received and analyzed by the flow computer to give a measure of

the upstream traverse time. The direction of the signal is reversed (i.e

sent to upstream transducer and received by downstream transducer) to obtain a measure of the downstream traverse time. These two measurements

must be the bases for the subsequent calculation of flow. Approximately

13,000 measurements per second were made by the unit tested, so that averaged readings, displayed about every four seconds, were relatively sta278
ble, with most fluctuations less than 1/2% of reading from the average

(except near zero flow)

.

2. Laboratory Experience

The simplest setup for testing the flow measuring system is a long

straight pipe with a "standard" flowmeter of proven accuracy and reliability installed in series, downstream. This was done for the specific

case of water flow in an 8 foot (2.4m) section of 4 inch (10cm) (IPS),

schedule 40, carbon steel pipe. A 4 inch (10cm) turbine flowmeter was

used" as the standard. Periodic in situ calibration of the turbine meter,

using gravimetric (weight-time) methods, verified its accuracy within

+1/2% of reading throughout. The transducers were mounted with approximately 20 diameters of straight piping upstream and a silicone base sonic

couplant applied between them and the pipe surface to enhance acoustic

transmission.

The flow display computer offered two readouts, one indicating flow

in gallons per minute (gpm) , the other indicating total gallons of flow

since the last scale resetting (1 gallon = 3.785 liters). At least 25

displayed readings were averaged for each flow measurement. Each of these

averages was accomplished in less than 30 minutes, a sufficiently short

time to consider the data randomly distributed about a constant mean {1}.

The totalizer was calibrated versus the standard six times over a period

of twelve days, and the flow display was calibrated five times over the

last seven of those days. After the initial zero setting and calibration

scale setting were made, nothing was changed in the setup during the

twelve day period except to disconnect and secure the flow computer each

night. To estimate the magnitude of random changes in calibration over

several days in this permanent type installation, the average meter correction factor (true flow divided by measured flow) and its standard deviation were calculated at each flow rate for the number, n, of reliable

calibration points obtained. The results are given in Tables 1 and 2.

Table 1 Summary of Permanent Installation Data from Flow Display

Nominal Flow Rate (gpm) Average Correction Factor Standard Deviation n.

580

480

380

280

180

80

1.0100

1.0169

1.0193

1.0359

1.0342

1.0452

.0032

.0032

.0086

.0019

.0075

.0233

4

5

5

4

5

5

3Figures in brackets indicate the literature references at the end of this

paper.

279
Table 2 Summary of Permanent Installation Data from Totalizer

Nominal Flow Rate (gpm) Average Correction Factor Standard Deviation n

580 .9991 .0048 6

480 1.0003 .0062 6

380 1.0022 .0085 6

280 1.0095 .0121 6

180 1.0111 .0223 6

80 1.0063 .0502 6

It is apparent from these data that the standard deviation tends to

increase with decreasing flow rate. It is surmised that this is due to

drifts in the zero setting. To check the consistency of this assertion

with the data, all readings were corrected for zero drift by taking the

average of true zero (no flow) readings before and after each calibration

and subtracting this from the flow readings. The resulting, zero-corrected

data are presented as Tables 3 and 4. The decrease in standard deviation

is quite apparent at the lower flow rates, supporting the contention that

zero drift is an additive correction to the flow readouts.

Table 3 Summary of Zero-Corrected Permanent Installation

Data from Flow Display

Nominal Flow Rate (gpm) Average Correction Factor Standard Deviation n.

580 1.0002 .0048 4

480 1.0072 .0051 5

380 1.0048 .0067 5

280 1.0163 .0042 4

180 1.0018 .0030 5

80 0.9717 .0077 5

Standard Deviation of Average Correction Factors 0.0135

Table 4 Summary of Zero-Corrected Permanent Installation

Data from Totalizer

Nominal Flow Rate (gpm) Average Correction Factor Standard Deviation n

580 .9924 .0034 6

480 .9920 .0031 6

380 .9918 .0030 6

280 .9974 .0051 6

180 .9885 .0066 6

80 .9554 .0134 6

Standard Deviation of Average Correction Factors 0.0174

280
There are two other prominent tendencies in the data in Tables 3 and

4. First, the meter correction factor is different (lower) for the totalizer than for the flow display by about 1.4% on average, with a slight

trend towards larger discrepancies at lower flow rates. Furthermore, the

correction factor is dependent on flow rate. This is borne out by the

fact that the standard deviation of the average correction factors (center

column) is greater than the standard deviations computed at each flow rate

(right column). Also, there is an obvious decrease in correction factor

with decreasing flow for the three lowest flow rates.

To summarize further the data obtained in the permanent installation,

it is assumed that the variations in meter correction factor can be divided into two independent components: one originating from a simple, additive zero drift (which may be minimized by setting true zero before reading), the other originating in electronic and mechanical drifts in calibration factor and expressible as a percentage of flow rate. This latter

source of variation is estimated from the root-mean-square (rms) of the

standard deviations listed in Tables 3 and 4. For the flow display the

rms deviation is 0.00552, and for the totalizer, 0.00683. If the uncertainty in a flow reading, R, is taken to be 2.576 standard deviations

(which would include 99% of normally distributed data) , then we have

+0.0142R uncertainty in the flow display, and +0.0176R uncertainty in the

totalizer. In addition, the zero drift standard deviation was estimated

at 1.30 gpm for the display calibrations and 2.47 gpm for the totalizer

calibrations. However, the flow display zero drift decreased monotonically

over the twelve days, starting at true zero, and decreasing to -7.1 gpm.

Based on this observation it seems reasonable to assign an uncertainty of

at least +7 gpm to the zero setting if true zero is not set for several

days. It should be recalled that the overall uncertainty in this circumstance is the square root of the sum of the individual squared uncertainties. Also, all of the above data refer to a system which is accurately

calibrated in place, and therefore does not include uncertainties due to

piping differences, installation procedures, or any other observed parameters.

Next the removal and reinstallation of the transducers (on the same

pipe) was isolated as a source of uncertainty. This variable is present

in the common case whereby the ultrasonic flowmeter is calibrated at one

location for use at another location with similar piping. The zero-corrected data for this temporary type installation is summarized in Tables

5 and 6.

Nominal Flow Rate (gpm) Average Correction Factor Standard Deviation n

Table 5 Summary of Zero-Corrected Temporary

Installation Data from Flow Display

580

480

380

280

180

80

.9289

.9617

.9741

.9670

.9677

.8758

.0146

.0264

.0155

.0234

.0248

.0566

10

10

10

10

8

8

281
Table 6 Summary of Zero-Corrected Temporary

Installation Data from Totalizer

Nominal Flow Rate (gpm) Average Correction Factor Standard Deviation n

.0141 10

.0141 10

.0146 10

.0179 10

.0186 10

.0268 8

The rms standard deviation from Table 5 (excluding data at 80 gpm where

unavoidable zero drift during the calibration may make a significant contribution) is 0.0214, and, from Table 6, 0.0160. With the same interpretation as for the permanent installation data, this implies an estimated uncertainty of +0.0551R in flow display readings, temporary installation,

and an estimated uncertainty of +P'0^12R in totalizer readings, temporary

installation. The increase over the corresponding permanent installation

uncertainties (+0«0142R and +0.0176R respectively) must be due to uncertainties originating from removal and reinstallation, without recalibration, as this was the only additional variable.

To summarize Tables 1 through 6, lumping flow display and totalizer

data together, a repeatability of +1.67o of reading is estimated for permanently installed, undisturbed, transducers if true zero setting is maintained. If in situ calibration is performed, this is -also equal to the

estimated uncertainty. If transducers are calibrated in one place and

used in another, the estimated uncertainty is +4.9% of reading, assuming

true zero setting, and not counting the uncertainties due to piping differences .

Finally, in discussing experimental data, it should be recalled that

the above data are from only one flow measuring system which is not a large

enough sample to infer reliably the parameters which characterize the performance of this type of flowmeter. However, calibration and shipboard

data have been obtained with several different flow measuring systems of

this type {1, 2, 3} and they indicate comparable linearity, data scatter,

and repeatability to the estimates reported herein.

3. Theoretical Analysis

The evaluation of uncertainties associated with additional variables,

such as pipe diameter and wall thickness, acoustic velocity in the pipe

wall, and flow profile and acoustic velocity in the liquid, would require

a fairly long and costly program if done experimentally. A theoretical

analysis was undertaken instead, which permits direct calculation of the

uncertainties in flow measurement due to each of these variables except

flow profile. Furthermore, this analysis provides considerable insight

into the acoustic principles underlying the operation of externally mounted

ultrasonic flowmeters in general.

282

580 .9580

480 .9601

380 .9620

280 .9617

180 .9545

80 .9481

I
The idealized model used to simplify the analysis incorporates the

following: (1) The sound is emitted and received at a geometric point

in each transducer. (2) The sound wavelength is sufficiently short

that geometric acoustics pertains. (3) The sound path lies entirely in

a plane containing the point transducers and the pipe axis. (4) The

pipe is exactly symmetric about its axis. (5) The flow velocity is constant everywhere inside the pipe, and in time. Figure 2 depicts this model.

OOVHSTREAH TRANSDUCfR.

UPSTREAM TRANSDUCER

FIGURE 2 DIAGRAM FOR ANALYSIS OF ACOUSTIC

RAY TRANSMISSION BETWEEN ULTRASONIC TRANSDUCERS

A simple argument based on sjrmmetry and the identity of the transducers

leads to the conclusion that any sound ray traveling between the two transducers must pass through the midpoint of the line joining them. To see

this consider the symmetry operation of time reversal. Inasmuch as the

transducers are identical, this operation must result in no real physical

change to the system. However, if it is conjectured that the sound ray

path does not go through the midpoint of the line joining the transducers,

then time reversal (which amounts to a reversal in the directions of sound

propagation and flow) would not leave the path unchanged (i.e. A path

which crossed the x-axis upstream of the midpoint initially, would cross

the x-axis downstream of the midpoint if time were reversed) . Since time

reversal is a fundamental syiranetry of classical physics, it follows that

the upstream and downstream ray paths must intersect each other, and the

pipe center line, at a point midway between the transducers.

The space and time dependence of a sound wave in a uniformly moving

liquid has the form {4} :

exp{i(k-x - (kc + k.u)t)} (1)

283
This result shows that the sound will travel along a straight line in the

idealized flow considered. The central problem now is to determine the

angle of refraction of the sound as it enters the liquid. To do this, we

notice that the boundary conditions on the solution to the wave equation

lie only in the plane normal to the y-axis. The correct solution must

therefore be invariant under translation along the x direction {4}. This

requires that the x component of the wave vector be continuous across the

solid-liquid interface:

^2x = ^3x (2)

where the notation of Figure 2 applies, and k is the magnitude of the

wave vector. Geometric analysis gives the expression for 'k.^x (see Appendix")

to/c^ sine3

291

(1 - -^2 cos 63)^ + ^ sin^e - ^

^3x

(3)

2 2 1 1 - -o cos^e„ + - sine„ (1 - " cos^e„)^

C3 J C3 J (.^ J

Since k2 is simply a)/c2, and since the angular frequency, w, is the same in

both media, equations (2) and (3) combine to give the desired generalization

of Snell's law relating the refraction angles at the interface of the pipe

wall and the moving liquid:

Slll{ 2 _ (4)

C3(l-^2cos2e3)^

while, at the interface of the transducers with the pipe wall we have the

usual form of Snell's law:

sine2^ sin62

Equations (4) and (5) together with the equation constraining the sound

ray path to begin and end at the transducer locations, namely,

L/2 = h tan 6^ + t tan 62 + D/2 tan 63 (6)

284
uniquely determine the ray path. The resulting equation is developed in

the Appendix, and it determines 9^ only implicitly. A program was developed to calculate sin92 iteratively on a hand-held programmable calculator

The calculated value of sin92 was used then to calculate the time of traversal of a sound pulse. (The equations, as listed herein and in the

Appendix are for downstream traversal, and the same relations pertain for

upstream traversal except the sign of u is changed.) Since the ultrasonic

flow measuring system under discussion must determine flow based on measurements of upstream and downstream traverse times, the ability to calculate these as a function of the various parameters permits a theoretical

determination of the effects which changes in these parameters will have

on actual flow measurements.

Before listing the results of the numerical investigation, it is interesting to note that the upstream and downstream paths are not coincident (This is somewhat exaggerated in Figure 2). However, the fact that

both paths intersect the midpoint of the line joining the transducers

requires that each path be symmetric about that point. This means that

the analysis may be restricted to the portion of each path which is between a transducer and the midpoint since the remainder of the path is

just a reflection of this. As may be expected, the upstream pulse takes

a more direct path across the liquid in an "effort" to avoid the retarding effects of the flow, while the downstream ray is lengthened in the

liquid to "take advantage" of the free boost in velocity. This is in

accord with Fermat's principle, which states that sound takes the path of

least time.

The calculations were made using an idealized version of the system

actually used to obtain the experimental data of section 2. The nominal

parameters were: pipe inside diameter, D = 4.026 in. (10.23cm); pipe wall

thickness, t = 0.237 in. (0.602cm); acoustic velocity in carbon steel pipe

0-2 - 5940 m/s; axial separation of transducers, L = 5.57 in. (14.1cm);

distance between transducer crystal and pipe surface, h = 1.0 in. (2.54cm)

acoustic velocity in liquid (fresh water), C2 = 1496.7 m/s; and, acoustic

velocity in transducer housing, c^ = 2620 m/s. With these nominal parameters, it was calculated that sin03 = 0.2497474683 for u=o, and the traverse time, for a sound pulse was t = 107.4664 ys. With u = 5 m/s, for example, the difference between upstream and downstream traverse times was

At = 117.740 ns.

It was found that the difference between upstream and downstream

traverse times. At, is linearly proportional to the flow velocity, u, to

an accuracy of better than 1.4 x 10~3% for velocities up to 50 m/s. The

difference in the reciprocal times, A(1/t), is similarly a linear function

of u over all commonly experienced flow velocities. More importantly,

though, A(1/t) was calculated, with u = 10 m/s, for six values of liquid

acoustic velocity ranging from c^ = 800 m/s to 2000 m/s and found to be

independent of the liquid acoustic velocity to better than 1.8 x 10'

This means that the flometer calibration may indeed be considered independent of the metered liquid provided an acceptable signal/noise ratio

can be maintained. Furthermore, the product of the liquid acoustic veloc285
ity with the average of the upstream and downstream traverse times is

essentially constant (to better than 6.2 x 10~^%) over the same range of

acoustic velocities, so that the flow computer may actually "know" what

liquid it is metering (i.e. the acoustic velocity of the liquid). This

is important in the case where the transducers must be repositioned to

improve the signal to noise ratio, since a correction factor for the

change in position can be calculated only if the liquid acoustic velocity

is known.

The uncertainties associated with certain parameters were calculated

using the nominal values listed previously, and are given in Table 7. The

uncertainties assigned to pipe wall thickness and pipe Inside diameter

correspond to tolerances given in military specifications for carbon steel

pipe. The uncertainties in pipe acoustic velocity, liquid acoustic velocity, and transducer axial separation were chosen arbitrarily to illustrate

their effects on At and A(1/t).

Table 7 Uncertainties Arising in

Measured Quantities Due to Uncertainties

in Certain Parameters

Parameter

Resulting Resulting

Uncertainty in Uncertainty in Uncertainity in

Parameter (%) At (%) A(1/t) (%)

Pipe wall thickness +12.5

Pipe inside diameter +0.65

Pipe acoustic velocity +1.0

Transducer axial separation +2.24

Liquid acoustic velocity +50

-0.24, +0.22

+0.62

-1.0, +1.1

+0.06, -0.07

-56, +400

-0.31, +0.28

+0.15

-0.76, +0.78

+0.92

<2 X 10

-4

It is apparent that the values in Table 7 are considerably smaller

than the uncertainties determined experimentally in Section 2. Apparently,

the electronic stability and signal processing techniques are the principle sources of uncertainty. For example, implicit in the idea of

averaging out the noise in the signal by taking a large number of measurements is the assumption that noise is entirely random. In fact, signals from the transmitting transducer might reach the receiving transducer

via certain other "favored" paths in addition to the idealized path used

for the analysis. This could skew the distribution of the received noise,

resulting in a biased flow reading. (It has been pointed out by the inventor that a higher signal/noise ratio generally leads to improved

readings)

.

In closing this section, it should be emphasized that all calculations

were made assuming a constant velocity profile. Although the constancy

in time is not expected to alter the results significantly (based on calculations done by the author describing the interaction of sound with

turbulence {5}, and on laboratory observations which showed no significant

increase in readout fluctuations as the flow was increased from laminar

286
to turbulent) , the assumed constancy of velocity throughout the pipe

overlooks the essentially non-linear changes in the real velocity profile

as the Reynolds number and pipe roughness change. In reality, therefore,

At and A(1/t) are not linearly related to the flow rate, a fact borne out

by the data of Tables 1 through 6. Also, it is hoped that this analysis

will help permit the eventual use of a single flow measuring system for

use in various piping situations, for, so long as an acceptable signal can

be transmitted across the pipe, it is possible, in principle, to correct

for the various paramter changes and infer the flow from the measured upstream and downstream traverse times.

The development of non- intrusive, ultrasonic flow measurement systems

represents a significant advance in our technology. They enable flow

measurement, by means of portable instruments, of homogeneous liquid flow

in systems which may not be opened, a capability previously unavailable.

Experimental and theoretical results indicate that, while commercially

available models may be adequate for many industrial and military applications, there remains significant room for improvement in overall accuracy and repeatability.

There is a need for additional work in this area of flow measurement

to provide improved statistics on a representative sample of instruments,

and to extend and deepen our understanding. For example, scant information is available on the effects of upstream piping, a problem commonly

encountered in flow measurement applications. Also, an accurate analysis

of sound propagation in real, viscous and turbulent, shear flows could

help to improve linearity. These and other problems are under investigation presently by the author.

The derivation of the equations determining the sound ray paths and

traverse times is most easily visualized using Figure 2 of the text. The

central problem is to find a means of calculating the upstream and downstream traverse times for sound pulses given D, t, h, c-j^, C2,c^, L and u. As discussed in the text, the analysis may be confined to half of the

ray path, due to symmetry, and the traverse time calculated will be half

the complete traverse time.

The most difficult of the necessary relationships to establish is

the one which relates the refraction angle, 63, in the moving liquid to

the angle 62 in the pipe wall. If there were no flow, Snell's law would

be adequate, as it is in the case of refraction at the transducer-pipe

wall interface:

4. Conclusions

APPENDIX

sin0

1

sin6

2 (lA)

287
To determine the necessary generalization of Snell's law when one

medium is moving we consider the boundary condition that the x component

of the wave vector be continuous across the interface:

-v In the pipe wall, the magnitude of ^2 is simply aj/c2, and the direction of k2 is along the direction of the ray path. In the moving medium,

however, neither the magnitude nor direction of k-^ is so simple. As seen

from the expression in the text giving the time dependence of the wave,

the angular frequency is {4}:

0) = Cjkj + u • ^3 (3A)

By a general theorem of physics, the velocity of this wave is {4}:

If (4A)

Transposing u, squaring both sides, and solving for c^' gives:

^3 = ^3

u 2 u

(1 -

—J- cos 9 ) H sin9. (5A)

Now, solving for k^ in (4A) and substituting the result in (3A) , we obtain

the relationship between o) and k^;

1 + 2 C3

sine

,

u

(6A)

Substituting for c ', from (5A) and solving for k yields:

3

w/c. (7A)

1+1 sine

3 U^ 2 !< U

(1 - -2 cos e,) +7" sine.

u

- T2

In order to determine k3^, we take the scalar product of k^ with u

which, by definition is k^^u. Using (4A) as well as (7A) , we obtain:

ukq

k u = —- (c.' sine- - u)

3x 3 3 (8A)

288
Substituting the previously determined expressions for c^' and k from

(5A) and (7A) respectively gives: ^

w/c. u sin9o (1 -

3 ^ c

'3x

cos 6^)'^ + ^ sin^e

3 C3 3

_uc1 - ^2 cos2e3 + I

sine. (1 - --^^

(9A)

4

cos 0„)

C*! J

u^ 2

1- Factoring sinG^ from the numerator and (1 - '^'^cos 9 3) from the denominator, and equating with k-^ gives the explicit expression of (2A) :

sin93

u/c^ cscG^

1 - —

sine

2 _

1^ u

(1 - ^2cos^e3)''^ + c3 5±nQ^

(lOA)

u 2

' 03 (1 - -2 cos 93)

Equation (lOA) is the generalization of Snell's law to the case where

medium 3 is moving with velocity u relative to medium 2. In order to determine the ray path completely, we must also require that it begin at one

transducer and end at the other. This condition is expressed mathematically from a direct analysis of the geometry in Figure 2:

L/2 = h tan 9^ + t tan 02 + D/2 tan 93 (llA)

Equations (lA) , (lOA) , and (llA) uniquely determine the three unknowns,

9-|^, 92 and 93. Unfortunately, these three equations cannot be solved

explicitly for the angles. If we can determine 93, however, 92 and 0^^ may

be calculated from (lOA) and (lA) directly.

Before developing an exact expression for the implicit determination

of 93, we shorten the notation in (lOA) by letting:

X = sin93 (12A)

e = U/C3 (13A)

y = cos 93 (14A)

289
F =

(1 - (15A)

G =

(1 - £ y

)

- F (16A)

n = c /c

13 r 3

(17A)

and

23 2' 3

(18A)

Now, (lOA) becomes;

(19A)

and (lA) becomes:

sin9, = n, „ G '1 " "13 (20A)

Substitution of (19A) and (20A) into (llA) leads to an implicit expression which determines x:

L/2 = h i^jL3 G t n G

+ — +

Dx

(1 - 2 G 2)^ (1 - n,, 2 G 2^H 2(1 - x")"^

(21A)

23

Although the actual numerical solution of (21A) for x is a fairly

lengthly and tedious undertaking, it can be done iteratively on a programmable calculator of 224 step capacity. Once the value for x is

determined for given conditions, the next problem is to calculate the

traverse time for the path corresponding to x. This is accomplished by

dividing the length of the ray path in each medium by the acoustic

velocity in that medium and summing (see Figure 2). The traverse time,

in terms of the new variables, is:

2h

T =

^1 (1 - ^3

2^h

+

2t +

D

^2 " ''23

^ G h'"^ c^yia - eV)^' + ex) (22A)

290
This expression applies to a pulse traveling from the upstream to

the downstream transducer when e is positive, and vice versa when e is negative.

References

{1} "Evaluation of Controlotron Corporation Ultrasonic Clampitron

Flowmeter", by Fania Wanenchak, Naval Ship Engineering Center, Philadelphia

Division report of Project A-1811 (August 6, 1976).

{2} "Report on Performance Testing of Main Propulsion and Auxiliary

Machinery Using Presently Installed and Portable Instrumentation", by Paul

Drnjevic, Code 1861, PERA (CV), Puget Sound Naval Shipyard, Bremerton,

Washington 98314 (August 1976).

{3} "Test Equipment Application Project Report: Controlotron Model 240

(Clampitron) Ultrasonic Flowmeter (3" and 4" Pipe Size)", by John Flude,

SMMSO, Naval Ship Engineering Center, Washington, DC, (June 1975).

{4} L.P. Landau and E.M. Lifshitz, Fluid Mechanics , Pergamon Press,

New York (1959), pp. 254-259.

{5} Unpublished paper, "Sound Propagation in a Turbulent Medium", by

Ronald F. Bruner, Naval Ship Engineering Center, Philadelphia Division.

{6} A.R. Wenzel, "Propagation Speed and Attenuation Coefficient for

Plane Coherent Acoustic Waves in a Turbulent Medium", J. Acoust. Soc.

Am., v51, 1683-1687 (1972).

{7} S.F. Clifford and E.H. Brown, "Acoustic Scattering from a Moving

Turbulent Medium", J. Acoust. Soc. Am., v55, 929-933 (1974).

{8} Lynnworth, L.C., "Clamp-on Ultrasonic Flowmeters: Limitations and

and Remedies", Instrumentation Technology, v22, n9, Sep 1975 pp. 37-44.

{9} Segall, J., "Application Considerations For Controlotron Ultrasonic

Clampitron Flowmeter", ISA Industry-Oriented Conference and Exhibit, paper

822, Milwaukee, WI, Oct. 6-9, 1975.

291

National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

A NEW" ULTRASONIC FLOWMETER

FOR THE NATURAL GAS INDUSTRY

N. E. Pedersen, J. E. Bradshaw,

L. C. Lynnworth and P. R. Morel

Panametric s, Inc.

221 Crescent St.

Waltham, MA 02154

An ultrasonic flow metering system is being

developed for the measurement of the flow of natural gas in pipelines having diameters from less than

1 ft to 3 ft or more. The system is designed to measure flow over a line pressure range from less than

100 psig to 1500 psig. Problems relating to high frequency dispersion and attenuation by the gas are

avoided by the use of a patented ultrasonic flow metering technique which utilizes the accurate measurement

of the absolute phases as well as the phase difference

of various discrete low frequency (25 Hz and 5 kHz)

components in the identical but sequential upstream

and downstream transmitted waveforms. The measurement system utilizes a dedicated microcomputer

which has been programmed to provide outputs of flow

rate (ft/ sec), supercompresibility, flow rate (std cu

ft/hr), totalized flow (std cuft), Reynolds number,

energy flovv^ rate (btu/hr) and totalized energy flow

(btu), as well as a real time statistical analysis of

intermediate data. A prototype system has been

tested on an in-service gas line of 2 ft diameter.

Good tracking was observed between the ultrasonic

meter and a downstream orifice station on a pointby-point basis at 30 second intervals, although the

flow rate varied rather substantially over such intervals. Representative preliminary data are presented.

293
1. Introduction

The standard of measurement of the flow of natural gas in pipelines

has traditionally been the multiple tube orifice station. Such systems

represent a substantial capital investment, and require a significant

amount of space for their location. By contrast, the utilization of ultrasonic techniques makes possible the measurement of natural gas flow at

a relatively low cost and with essentially zero pressure drop. The

space requirements are minimal. Although it is not the primary objective of the present development to replace orifice metering stations as

custody transfer devices, it is believed that a substantial number of

measurement requirements exist in which an ultrasonic system of proven reliability and reasonable accuracy is of benefit to the natural gas

industry. An example of such an application is the metering of flow

from new gas sources,

1 2

In a previous ultrasonic gas flowmeter development, ' a mechanical device produced sonic shock waves which were alternately propagated upstream and downstream along a diagonal path within the pipeline. Measurement of the individual transit times permitted calculation

of the gas flow velocity within the pipeline. The device was capable of

providing one gas flow measurement in a period of several minutes, and

utilized a rather expensive and complex electromechanical shock generator.

The present system is being developed for gas flovv measurement

in pipelines having diameters ranging from less than one foot to three

feet or more, and over a gas pressure range of from less than 100 psig

to 1500 psig. Although the macroscopic parameters (pipe diameter,

flow velocity, sound speed, etc.) affecting the system point to the use of

ultrasonic pulses having frequency components on the order of 1 MHz or

higher, the dispersion and attenuation by the gas at such frequencies are

very high. The resulting need for very high energy transmission is

avoided by the use of a patented ultrasonic flow metering technique which

accurately measures the absolute phases as well as the phase differences of various discrete low frequency (25 Hz and 5 kHz) components

in identical but sequential upstream and downstream transmitted waveforms. These waveforms are efficiently transmitted by means of modulating a carrier wave, whose frequency is high enough (100 kHz) to provide good directionality, but low enough to be below the high attenuation

region of the predominantly methane gas mixture. The rate of data acquisition of the present system is approximately 4500 times higher than

that of the aforementioned ultrasonic system. Upon reception and after

demodulation of the carrier wave, the above mentioned 25 Hz and 5 kHz

signal components are passed through high Q bandpass filters. These

filters serve the dual purpose of providing a high signal-to-noise ratio.

294
and of isolating the particular frequency components, of the received

waveform, whose phases are to be measured. The utilization of these

narrow-band and highly directional techniques permits the use of piezoelectric transducers of relatively conventional design, operated at very

modest voltage and power levels.

The measurement system utilizes a dedicated microcomputer to

provide the various outputs such as flow rate (ft/sec), supercompressibility, volumetric flow rate (std cu ft/hr), totalized volumetric flow

(std cu ft), Reynolds number, flow profile correction factor, flow velocity statistics, profile corrected flow rates, energy flow rate (btu/hr)

and totalized energy flow (btu). The computations relating to energy

flow rate are dependent upon a priori knowledge of the mole fractions of

the various gas constituents. The supercompressibility computation

also utilizes the above data on gas constituents, as well as continuously

measured values of flowing gas temperature and pressure. The various

site parameters, such as pipe diameter, gas constituents, etc. , are

entered via front panel thumbwheel switches. The microcomputer also

provides a real time statistical analysis of intermediate data. Individual flow measurements and computations are made at the rate of 25 per

second. Upon accumulation of a statistically meaningful (e. g, 100) number of preliminary flow computations, the mean and the standard deviation among the measurements are computed. Then a statistical criterion is invoked to test the individual intermediate results. Those values which do not satisfy the criterion are rejected, and a new mean

flow is computed.

To date, a prototype system, developed under contract to Columbia

Gas System Service Corporation, has been tested over a period of several months on an in-service gas line of 2 ft diameter. The measurement site included a large orifice metering station, thereby permitting

direct comparison of the outputs of the two measuring systems over

periods as short as 30 seconds and as long as several days. The standard deviation among the individual ultrasonic flow measurements, made

at 1/25 second intervals, was used to evaluate the very short term

fluctuation in the diameter averaged flow, while the 30 second and longer

intervals were compared with the orifice measurements averaged over

the corresponding time periods,

2. The Required Measurement

The geometry of the measurement is sketched in Fig, 1, which

shows an ultrasonic propagation path at an angle of 45° to the pipe axis,

and intersecting the axis. Ultrasonic energy is consecutively transmitted upstream and then downstream. The flow velocity is determined by

means of accurate measurement of the upstream and downstream transit times, as well as the time difference between upstream and downstream transits.
CD

CO<

C/3

LLI O

o

_l

LL

UO

_1

LU>

296
The basic equation which is solved by the flowmetering system is

given below.

Q= (v)(PCF)(^ ^) (F^^)' (1)

s f

In the above equation, Q is the area averaged gas flow rate in units

of standard cubic feet per second and v = diameter averaged flow as measured by the ultrasonic flowmeter. PCF represents the flow profile correction factor, and takes into account the fact that the area averaged

flow is not generally the same as the diameter averaged flow, P^, T£,

Pg and Tg are the flowing and "standard" pressure and temperature,

respectively. The quantity Fpy is the supercompre ssibility of the gas

and arises from its non-ideal properties insofar as the gas law is concerned. This quantity depends upon Pf, Tf, and the mole fractions of

the various constituents of the gas mixture.

3. Theory of Operation

Electronics

:

The basic mode of operation of the flowmetering system is such

that most of the desirable features of narrow band coherent cw operation are obtained, while many of the shortcomings of such operation are

avoided. As mentioned earlier, the system measures flow by means of

measuring the absolute phases of and phase differences between various

frequency components of a composite waveform which is consecutively

transmitted upstream and downstream along a 45° path which intersects

the pipe axis. In considering the operation of the system, it is convenient to represent the transmitted signal as a cw wave of frequency

00 , modulated by a rectangular wave whose repetition frequency is to^.

Since the modulation waveform can be represented as a Fo\jrier series,

we can express the transmitted waveform as follows:

CO

E^ ~ sin t E^[A^ Sinn t -Sr cos ncoj^ t]. (2)

Since the modulation waveform has zero amplitude during the "off"

period and finite amplitude during the "on" period, its mean value is

non-zero and therefore the^Bp Fourier coefficient is non-zero. Therefore, rewriting (1), we have

E^ ~ sinoOot|B^ + ^?^[^n ^innco^ t + B^ cos nco^ t]j> (3)

From the above equation, it can be seen that contained within the transmitted waveform is a spectral component of frequency co^. Side band

components at f requencie s given by (co^ + nco^) also occur. The received

297
waveform is passed through a bandpass filter having a sufficiently high

Q so that these sideband frequency components are rejected, and only

the component is passed. Thus, a replica of the original (unmodulated) cw wave is retrieved. Of course, the phase shift due to the transit through the pipe will be present in the retrieved ooq component of the

signal. A switching arrangement is provided, which passes the "downstream" waveforms to one coq filter, and the "upstream" waveforms to

a second filter. If the phases of these waves are

01 = ^o^

02 = ""oh

CO D

o 1 sinQ c-v COS0

CO D

o 1 sind C+VCOS0 I-

(4)

where tj^ and t2 are the upstream and downstream transit times respectively, D = pipe diameter, and v = average flow velocity over the ultrasonic path, and c = sound speed in the gas.

The phase difference is

w D . o V cost/

c -V cos U

It can be seen from the above equation that, although the phase difference - is proportional to the flow velocity, it is also proportional to the reciprocal of the sound speed. This dependence is eliminated

by another narrow baJid technique which is described below.

The modulation waveform (the Fourier series of equation (2)) has a

large Fourier component (com) at its fundamental frequency. This waveform is obtained by means of an amplitude detector, and is passed

through a bandpass filter of sufficiently high Q to reject all of the harmonics ncoj-Q. Thus, in a manner analogous to the above co^ case, a cw

wave of frequency co^^ emerges from the bandpass filter. In the flowmetering system, the detected modulation waveforms of the received

downstream and upstream transmissions are (by means of a sequential

switching system) passed to two co^^ bandpass filters. The phases of

these two waves are, respectively,

CO D m 1 •A = CO t

1 ^1 sinQ C-VCOS0 ,

CO D ^

/ m 1 0 = cj t = 2 m 2 sine C+VCOS0

298
We next combine equations (5) and (6) to obtain

which is independent of sound speed.

The phases 0^ and 02 are measured by means of separately comparing the filtered oo^ signals with an internally generated reference wave

at frequency a)m» whereas the (0^-02) phase difference is determined by

direct comparison of the filtered upstream and downstream oJq signals.

Thus, three independent measurements are made. Each of these measurements is conducted on a cycle-by-cycle basis on the continuous

waves emerging from the four oo^ and oa^^ filter s.

From the above, it is clear that all measurements leading to the

computation of flowing gas velocity are made under substantially coherent, narrow band conditions. This type of operation is most advantageous in the case of ultrasonic measurements in a gas because the

accompanying improvement in signal-to-noise ratio goes far to offset

the decrease in received power due to the large impedance mismatch at

the gas/ transducer interfaces.

It turns out that the optimum (oo^) frequency for the phase difference

(01 - 02) i^sa-surement is about 5 kHz for a 2 foot diameter pipeline, and

a maximum natural gas flow rate of ~30 ft/ sec. Physical and practical

constraints limit the transducer active diameter to less than 1.5".

Since the sound speed in the gas is on the order of 1300 ft/ sec(~1.6 x

10 in/ sec), the wavelength X of a 5 kHz ultrasonic signal is 3.1 in.

Since the (disc) radiator is only I / 2X to 1/ 3X in dimension, it would

exhibit a very wide radiation pattern. This excessive beam width would

result in a disastrously low power level at the receiver. This problem

was solved by means of utilizing a carrier wave (100 kHz) which has the

desired beam width of about 10° for a ~1" diameter transducer. The

"information" waveform (i..e. the modulated 5 kHz wave) amplitude

modulates the 100 kHz carrier. The received composite waveform is

prefiltered through a 100 kHz filter (see Fig. 2) which is just wide

enough to pass the +5 kHz modulation components of the information

waveform. Following demodulation of the prefiltered 100 kHz signal,

those frequency components of importance are retrieved. In this way,

the ultrasonic energy is efficiently transmitted through the gas and the

pertinent information is recovered at the receiver.

The implementation of the approach described above is shown in

block diagram form in Fig. 2. A 10 MHz crystal controlled clock is the

source from which all waveforms, switch timing, and reference signals

299
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are derived. The (solid state) switching is arranged so that, when

transducer Tj^ is transmitting (coupled to the waveform generator),

transducer is receiving (coupled to the 100 kHz bandpass filter, and

vice versa. After detection of the 100 kHz carrier, the information

waveform, with T2 receiving, is coupled (via a solid state switch) to the

lower set of 5 kHz and 25 Hz bandpass filters. When Tj^ is receiving,

the information waveform is coupled to the upper set of 5 kHz and 25 Hz

filters. As discussed earlier, these filters are sufficiently narrow that

only the pure frequencies oa^ and '^j^ sinusoidal waveforms are passed.

The digital phase ineasurements are accomplished by means of

passing the above sine waves to zero-cross detectors. In the case of

the0-j^ ~ ^2 ^^a-surement, the 5 kHz phase difference is digitally measured by means of counting the number of 5 MHz pulses which fit in the

time slot equal to the overlap time of the two 5 kHz square waves. In

the case of 0-]^ and these phases are individually measured by counting the number of 5 MHz pulses which fit in the time slot equal to the

overlap between each ooj^ square wave and a 25 Hz reference square

wave. The outputs of the phase measuring system are therefore three

counter outputs. Let AT, T-^, and T2 be the number of counts associated with the A0, ^i, and ^2 nieasurements, respectively. The counter

outputs, scaled down by 2x, are related to the phases as fdllows:

AT = 2.5 X 10 6 /A0

6 (

^1

\ CO

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T^ = 2.5 X 10

where the factor of (co^/ ooj-^) has been included in the first of the above

equations to take into account the fact that the AT count occurs more

frequently than the Tj^, T2 counts by this ratio.

Combining equations (8) and (7), we obtain the equation for diameter

averaged flow velocity:

The AT, Tp and T^ counts are transferred to the computer at a rate of

25 times per second, and separate "v" computations are made at this

rate. After a predetermined statistically large (e.g. 100) number of

individual flow (v) computations are made, the statistical mean and the

standard deviation among the v's are computed. Chauvenet' s criterion"^'^ gives an optimum deviation from the mean, beyond which a data

301
point should be considered to be invalid. This criterion depends upon

the number of data points as well as the standard deviation. For 100

points, it turns out that (assiaming a normal distribution) all points deviating from the mean by the statistical limit of error (3ct) should be

disregarded. This criterion is applied to the one hundred accumulated

v's, and a new mean value, v, is computed.

After this computation, the sequencing of the switching (Fig. 2) is

automatically modified so that the inputs to the 5 kHz filters are reversed. This "configuration change" has the effect of providing a measurement of |02 -

</>li > rather than - ^zl* ^ phase shifts of each

of the 5 kHz filters is identical, then of course the two phase measurements will be the same. But, if a difference in phase shift exists between the two filter circuits (including associated zero-cross detectors),

an error in velocity measurement will occur. The following operation

= 1/2{|</)^ - 0^1^ +|(/)^ - (^^l^l (10)

removes the effect of differences in phase shift between these two filters,

as long as the (assumed small) phase disparity, )6</)2 - is less than

the magnitudes {(j)^ - 02 I a I 01 ~ ^zIb* product Tj^T2 is unaltered by the configuration change, cind the overall cycle time, including

all computations, is slightly less than 30 seconds. Therefore, the

simple averaging of the measured velocities suffices to null any filter

drift. Thus, if "A" and "B" represent the two configurations, the computation of the averaged velocity,

v= l/2(v^ +v^), (11)

provides the desired result, v^ and Vg are each independently the

result of the statistical analysis already discussed.

Microcomputer:

The computing and data handling functions of the flowmetering system are accomplished with an 8 bit microcomputer which utilizes the

well known 8080 CPU chip, along with 7000 words of ROM and 8000

words of RAM.

The software utilizes a priority interrupt routine with data inputs,

teletype l/O and power failure all being interrupt driven. The supercompressibility calculation is the normal background program, being

replaced by the data processing routines when a complete set of input

data has been accepted. A teletype readout routine is provided, and

serves the dual function of providing information useful for troubleshooting, as well as detailed short term data analysis. Any of several

302
output formats is selected by typing a single key. For example, continuously updated arrays of all AT, Tj^, and data and intermediate velocity computations for up to 100 sets in "A" and "B" configurations may be

printed. Three LED readouts are provided, each of which can present

final and intermediate computations of any one of 38 (thumbwheel) selectable) parameters.

4. Prototype System Fabrication

Figure 3 is a photograph showing the complete electronics package.

The portion of the system schematically described in Figure 2 is contained in the four circuit boards in the upper left of the cabinet. All

other boards constitute the microcomputer. The power supply is housed

at the bottom of the cabinet.

The front panel layout is shown in Figure 4, A resettable electromechanical counter can be seen at the upper left. Its function is to integrate the flow velocity, and its unit of output is millions of standard cubic feet. Three LED displays are provided. The thumbwheel switches

to the right of these allow any one of 38 different quantities to be displayed, including error codes in case of equipment malfunction or maladjustment of some thumbwheel inputs. The center of the panel contains

thumbwheel switches which input the mole percentages of the various

gas constituents, while the lower left column of switches input site parameters such as pipe diameter, base temperature and pressure, and gas

specific gravity (in case its constituents are not known). The right

column of thumwheel switches inputs the various statistical quantities to

be utilized in the automatic data analysis.

Note that the front panel is provided with a lockable door having a

transparent window. This provides an element of safety from tinkering

by unauthorized personnel.

The electronics package is designed to be housed up to 250 feet

from the pipeline.

5. Supercompressibility

Since the important unit of measurement of natural gas flow is

standard cubic feet, and since the gas under measurement is at elevated

pressure, it is necessary to compute the supercompressibility of the gas

in order to provide instrumental readouts in units of standard cubic feet.

The computation utilizes the base equation adopted by the American Gas

As sociation. ^

The computer inputs needed to perform the computation are flowing

gas pressure and flowing gas temperature. These are both measured at

303
FIGURE 4 ULTRASONIC GAS FLOWMETER FRONT PANEL.

304
the pipeline just downstream, of the ultrasonic transducer locations and

appear as DC analog inputs to the microcomputer. Also used in the calculation are the mole percentages of the following gas constituents:

^"4- C^H^. C3H3, C^H^^, CjH^^, H^, O^, N^, He, and H^S.

6. Energy Computation

Since the system depends upon a priori knowledge of the proportions

of the above gas constituents, it is a relatively simple task to combine

this information with the known energy content (BTU/Std cu. ft. ) of each

of the constituents, and to thereby provide outputs of energy flow rate

and integrated energy flow down the pipeline. If and vvhen reliable online gas constituent measurement means become available, the present

system could be easily adapted to accept such inputs and thus provide

measured energy flow and its time integral.

7. Profile Correction

The correction of the measured diameter averaged flow for velocity

profile variations with Reynolds number has not constituted a main objective of the work to date. Instead, the primary objective thus far has

been to first provide a flowmetering system having the required stability

and reproducibility. The field tests to be described were conducted at

fairly constant flowing gas pressure and temperature, the flow rate

varied over less than one order of magnitude, and its composition was

constajit. Therefore, a total variation in Reynolds number of less than

one order of magnitude vvas encountered. Since this represents a very

modest perturbation in flow profile at a nominal Reynolds number of 10^,

a relatively simple correction was utilized. This assumes that the flow

can be expressed as

v(r) = v(o)

n

rR

(12)

and is the well known Power Law approximation for flow profile. In the

above equation, n is determined by the Reynolds number, and varies

by about 10% over the measured flow range. Although the merits and

deficiencies of the Power Law assumption have been discussed at length,

we believe that its use is appropriate under the stated flowing gas conditions.

A calibration program, planned for the near future, will involve

comparative measurements between the present system and an orifice

station over the entire range of about three orders of magnitude in

Reynolds number. Since Reynolds number is directly computed by the

flowmeter, it is believed that a comprehensive calibration effort will

305
provide the profile correction data required for operation under essentially straight pipeline conditions. Effects which raust be considered are

pipe vvall roughness as well as possible perturbations due to the small

discontinuities at the pipe wall caused by the ultrasonic transducers.

Should the results of the above raeasurements demonstrate the need

in certain applications of ultrasonic flow measurement over more than

one path, this can be mechanically and electronically implemented within the framework of the present design.

8. Insertion Mechanism

The ultrasonic transducer insertion mechanism used in this program is shown in Fig. 5. It consists mainly of stainless steel or nickel

plated standard hardware and fittings, and an adapter to the ball valve.

It accommodates the required seal-off s and explosion- proof equipment,

yet allows for probe rotation and translation adjustments which are useful in the set-up procedure to maximize signals.

9. Ultrasonic Transducer

Requirements include: center frequency, 100 kHz; 3 dB bandwidth,

+5°; beam orientation, 45° to axis of insertion mechanism or pipe axis;

holder O.D. less than '^50 mm; center frequency and bandwidth to be retained from -20 to +150°F and 30 to 1500 psig; means to be included for

damping of vibrations potentially induced by shed vortices.

A variety of transducer shapes was fabricated and tested, such as

truncated cones, chamfered discs and elliptical discs, in attempts to

control and broaden bandwidth by geometrical, non-attenuating means.

Since these geometrical approaches did not provide an adequate solution

in the available time, a more conventional combination of a radial-mode

resonator epoxied to a circuit board disc was used. This combination

was silicone-grease-coupled to an epoxy-impregnated graphite impedance-matching disc.

The reason for selecting the radial mode follows. Low-frequency

piezoelectric transducers, if operated in the thickness mode, are too

large for the present application. However, theory and experience with

acoustic emission transducers in the 100 kHz range have shown that

cross coupling between radial and axial modes in piezoelectric materials

suchas PZT enables one to make thick disc transducers in approximately

the same diameter as common contact transducers in the megahertz

range. The thick disc resonant frequency equation derived by Lucey''' is

f = Np/D [l - l/3(Np/N33)(]VD)^ (12)

306
where L = length, D = diameter, N„ = planar frequency constant and

N33 = axial frequency constant. This equation is valid for 0<L/D<0.7.

For a typical PZT element, if D = 19 mm and L = 7.6 mm, equation (12)

predicts f = 100 kHz.

10. Field Test Setup

The field tests were conducted at the Alexandria, KY orifice metering station operated by Columbia Gas Transmission Corporation. The

facility is a custody transfer station for the Cincinnati, Ohio area and

contains ten individual tubes. A photograph of a portion of the facility

is shown in Fig. 6.

The ultrasonic flowmeter was housed in a small trailer on the site.

The location of the ultrasonic, pressure, and temperature transducers

was about 400 ft upstream of the orifice meters. All pertinent pressure

and temperature data from the orifice station, as well as profile corrected flow data from the ultrasonic system, are fed to a data logger.

This device reads all inputs at approximately 30 second intervals and

stores the (digital) records on magnetic tape. In this manner, about

three days accumulation of data can be collected before a new tape reel

is inserted. The taped data were processed and analyzed with an IBM

370 computer by Columbia Gas System Service Corp. personnel at

Columbus, Ohio.

Coaxial cables were used to couple the ultrasonic transducers to

the electronics. These were approximately 200 ft in length and introduced an overall attenuation of about 10 dB.

11. Experimental Results

The ultrasonic flowmetering system, as delivered to the test site,

was uncalibrated. The purpose of the tests at Alexandria was to determine the short term and long term tracking between the ultrasonic meter

and the orifice meter. However, since the outputs of both meters were

simultaneously recorded on magnetic tape and computer processed, it

was a relatively simple matter to use a small portion of the comparative

data for the purpose of calibration of the ultrasonic meter, and to treat

the remainder as experimental data. This was done for two consecutive

periods. Each period was approximately 65 hours in duration and included more than 8000 individual measurements of flow velocity.

The ultrasonic system calibration was done under the assumption

that the meter has linear response, but that there may be a zero offset

and a slope error. These potential errors are taken into account by

means of experimentally determining the values of and in the

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309
following equation

V'p=Vp(l+K^)+K2. (13)

where V and V' are the uncorrected and corrected flow velocity readings, respectively. The values of Ki and K2 for the two above mentioned

^65 hour periods are given below:

First Period

= 0. 1603 € = -0.16%

= 0. 1343 ft/sec N = 8100

Second Period

= 0. 1792 e = 0.061%

= 0. 0059 ft/sec N = 8178

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orifice data and the corrected ultrasonic data over the interval of N data

points. Thus,

N N

- ZVp

' - 30 "

where Vq = flow velocity measured by the orifice meter.

The factor of 100/ 30 is introduced because the full scale output is

30 ft/ sec and € is expressed as percent of full scale.

It should be noted that £ represents the agreement between the two

integrated flowmeter readings over each 65 hour period, following onsite calibrations. We consider the first value 6 - -0.16% to be reasonable, considering the long integration period. However, the second

value e = 0.061% is probably not representative of the instrumental stability, reproducibility, or linearity. Rather, this small a variation must

be considered the result of a lucky choice of calibration point sets.

To get an estimate of the shorter term statistical deviations between the tv\/o flowmeters, the two above data sets were each divided

into sets of 1000, 100, 10, and single points, and the standard deviation of the average of each of these point sets was computed. The standard deviations were relative to the orifice station. The results are

310
plotted in Fig. 7. The data for the Second Period are more representative of the instrumental stability, since we have already shown that e for that period is vanishingly small. Therefore, the standard deviations

for the second period do not include any significant calibration error.

(The (T's of the First Period include a calibration error which results in

an offset of 0.16%, and the a's would therefore be expected to be somewhat higher than those of the Second Period.)

A direct comparison between (5 minute averaged) computed flow

velocity readouts of the ultrasonic and orifice meters is shown in Fig. 8.

This particular segment was chosen because of the abrupt 2:1 change in

flow velocity. Also plotted in the figure are the differences (% of full

scale) between the two meter readouts.* The abruptness of the above

change in flow is further exemplified in Fig. 9, which is a direct comparison between the 30 second averaged flow velocity readouts of the two

meters. Note that the statistical fluctuations in the data are somewhat

greater than those of Fig. 8. This is because the data of Fig. 8 have

been averaged over 5 minutes, while the data of Fig. 9 are taken at 30

second intervals. We are virtually certain that the observed short term

(30 second) fluctuations are the result of the very short term (^40 millisecond) fluctuations due to turbulence. The short term fluctuations of

the electronics system, as measured in the laboratory, was of the order

of 0.02% of full scale. Although such a discontinuous change in flow

velocity is rare in occurrence, these data segments were chosen for

presentation for the purpose of demonstrating the tracking between the

two flowmeters.

Finally, we present the (100 point averaged) data representing the

two entire ~65 hoar periods. These are shown in Figs. 10 and 11. The

large (~4:1) excursions in flow velocity are due to the periodic daily

energy demand variations of the city of Cincinnati.

12. Discussion

It is important to stress the fact that the calibration of the ultrasonic flowmeter was done at the test site and utilized data from the orifice meter. Therefore, the agreement between the two measurements

cannot be interpreted in terms of absolute accuracy. It should also be

pointed out that the time duration between "calibration" and "test" data

was in no case more than about 65 hours.

-l^Note that + 1% of full scale corresponds to +0. 3 ft/ sec in flow

velocity.

**It should be noted that, although nearly perfect tracking is evident in

the data shown, two other similar abrupt flow changes (<30 seconds)

did not demonstrate such good tracking.

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COMPARISON OF ULTRASONIC AND

ORIFICE METER VELOCITIES

TEST PERIOD NO. 1

100 POINT AVERAGED DATA (50 min)

J I I I I I I I

8 16 24 32 40 48 56 64

APPROXIMATE TIME (HOURS)

FIGURE 10

315
O MEASURED VELOCITY (ORIFICE)

8 16 24 32 40 48 56 64

APPROXIMATE TIME (HOURS)

FIGURE 11

COMPARISON OF ULTRASONIC AND ORIFICE METER VELOCITIES

TEST PERIOD NO. 2

100 POINT AVERAGED DATA (50 min)

316
The conclusions vv'hich one can draw from the above results are the

following

:

1. With on-site calibration, the ultrasonic gas flowmetering system

has demonstrated a potential overall accuracy of about +0.4% of

full scale (standard deviation) over an integration period of one

hour, and of about +0.25% of full scale over an integration period of between 8 hours and 65 hours.

2. Long term stability over periods of weeks and months has not

yet been demonstrated. A calibration program planned for the

near future will provide numerical results in this regard.

3. The tracking agreement between the ultrasonic and orifice flowmeters was demonstrated to be very good over integration periods of 30 seconds and five minutes. It appears as though local

turbulence was the limiting factor in this agreement, and gave

rise to relative measurement fluctuations of the order of

^20.6% of full scale (+0.18 ft/ sec) for a 30 second integration

period, and about 0.5% (_+0.15 ft/ sec) for a 5 minute integration

period. Since these (assumed random) local fluctuations appear

at both meter locations, we could attribute the effects due to

local turbulence of about +0.13 ft/ sec and +0.10 ft/ sec for the

two time periods, respectively.

4. Velocity dependent changes in flow profile were compensated

adequately by assuming the well known Power Law flow profile,

and making the appropriate Reynolds number dependent correction in the single path integrated flow velocity measurement.

The necessity and/ or desirability of using multiple cords will

be another subject of study in the calibration work.

5. Modifications to the present ultrasonic flowmeter are underway

to improve its temperature stability, as well as the phase measurement accuracy. A second system is presently being fabricated and will include these improvements. An improved ultrasonic transducer is also under development and is expected to

result in a low pressure limit of less than 50 psig.

13. Acknowledgments

A large portion of this work was supported by Coliimbia Gas System

Service Corporation. In addition, significant technical contributions were

made to the program by Mr. James Hager,* Mr. William Munk, and Mr.

Richard Colburn, all of Columbia Gas System Service Corporation. The

contributions of Mr. Ken Fowler and Mr. Dana Patch in the transducer

development are also acknowledged.

Mr. Hager is presently with McFadden Sales, Inc. , Columbus, Ohio,

317
14. References

1. E. Moffatt and K. Fetterhoff, "Large Volume Flow Measurement bySonic Techniques", Proc. A. I. P. Symposium on Flo\v, Pittsburgh,

PA, May, 1974.

2. F. B. Cook and E. M. Moffatt, "Flow Meter for High Pressure Gas

Using Sonic Pulses", Proc. I. S.A. Conference, New York, NY,

October 1974.

3. N. E. Pedersen, L. C. Lynnworth, J. E. Bradshaw, "Improved

Ultrasonic Fuel Mass Flowmeter for Army Aircraft Engine Diagnostics", USAAMRDL TR-75-8, June 1975.

4. Y. Beers, "Theory of Errors", Chap. 7, Addison-Wesley Publishing Company, Reading, MA, 1953.

5. W . Price, "Nuclear Radiation Detection", Chap. 3, McGraw-Hill

Book Company, New York, NY, 1958.

6. A.G.A. Gas Measurement Committee Report No. 3, "Orifice Metering of Natural Gas", 1955, 1956.

7. G. K. Lucey, Jr. , Resonant and Antiresonant Frequencies of Thick

Discs and Thick Rods, J. Acoust. Soc. Amer, •43(6), 13Z4-1328

(May 1968).

318

I
National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

A NEW NON-INTRUSIVE FLOWMETER

R. S. Flemons

Canadian General Electric Co. Ltd.

Peterborough, Ontario

Canada K9J 7B5

A new type of flowmeter is described which can be clamped

to the outside of existing pipes and can measure liquid

velocity without calibration. Two diametral ultrasonic beams

are used to interact with the liquid and to detect passage of

inherent tracers such as turbulent eddies. Cross-correlation

analysis is used to determine the transit time of the eddy

pattern and thus the liquid velocity. The operating

principles of the signal processing electronics and the

digital cross-correlator are outlined. Comparisons with

accurate gravimetric flow measurements and field applications

are briefly described.

Instruments to measure fluid flow have a long history of development

and application. In spite of this they possess limitations which can

seriously limit their suitability for use in many situations. This

situation was encountered in 1971 with conventional instruments installed

on a Heavy Water separation plant, where the process fluid is an extremely

corrosive and toxic solution of Hydrogen Sulphide. The high cost of

shutting down the plant to check and replace the flowmeters forced an

assessment of alternative measuring systems. All systems known to us at

that time required penetration of the piping, which introduced potential

leak sites and exposed the instruments to deterioration from the

corrosive fluid. In addition, routine inspection and calibration of the

primary elements entailed shutting down the plant to gain access to the

exposed parts. This resulted in a serious loss of production and an

increased possibility of leakage of the toxic fluid.

The work of Dr. M.S. Beck and his colleagues at the University of

Bradford, England [1] suggested that using a combination of ultrasonics

to interact with the liquid through the pipe walls and a cross-correlation

analyser to determine the liquid transit time, a new type of flowmeter

might be developed with none of these limitations. The development of

a practical instrument of this type was carried out by Canadian General

Electric with technical support from Dr. Beck and later with financial

assistance from Atomic Energy of Canada Limited.

319
Features of the Ultrasonic Cross-Correlation Flowmeter

The transducers* of the ultrasonic cross-correlation flowmeter

(USCCFM) may be clamped to the outside of existing pipes ranging from

4 to 30 inches CD. Pipe wall thickness and material impose no

problems because the ultrasonic energy propagates across a pipe

diameter and is not refracted at the pipe wall. Since one set of

transducers may be used on all sizes of pipe, the instrument is

particularly attractive for use on large pipes where the installed

cost of conventional instruments is often very high. The range of flows

which maybe measured is indicated in the following table.

Pipe Size 4" 8" 16" 30"

Flow Range

(USGPM)

25 - 500 200 - 4,000 800 - 13,000 8,000 - 80,000

With some minor modification these ranges can be somewhat extended.

Installation is rapid and Inexpensive. Pipe preparation is

confined to removing loose paint or scale from the outer surface. Since

the pipe wall is not penetrated there is no leakage hazard and the proces

fluid cannot corrode or erode the sensors. Calibration is never required

since the instrument simply measures fluid transit time over a known

distance. For the same reason the instrument accuracy is unaffected by

changes in fluid properties such as temperature, density and viscosity.

Fluid conditions such as turbulence and traces of gas or particulate

matter, which often cause errors in other instruments, form the signals

which are utilized by the USCCFM. Long straight pipes upstream are not

required, in fact the preferred location is within 5 pipe diameters

of an elbow.

Relationship to the Ultrasonic Travel Time Difference Meter

At the time this development started (1971) we were not aware that

a form of the ultrasonic travel time difference meter [2] had been

developed in which the transducers are placed on the outside of a pipe

wall [3]. Since this meter may appear to be similar to the USCCFM, it

is important to contrast the principles of operation. The travel time

difference meters send ultrasonic energy through the flowing fluid in

two directions. The effective propagation velocity of energy travelling

in the direction of fluid flow is the sum of the sonic velocity and the

fluid velocity, whereas for energy travelling against the fluid flow

the effective propagation velocity is the sonic velocity minus the fluid

velocity. By comparing the ultrasonic transit time in the two directions

the average fluid velocity may be determined. Since the sonic velocity

is far greater than normal flow velocities the fractional changes are

small and sophisticated electronic circuits are required to achieve

precise fluid velocity measurements.

* In this paper, "transducer" means simply an element which transforms

electrical to acoustic energy, or vice versa. This transducer is not

a calibrated component which affects the accuracy of the instrument.

320
The form of the travel time difference meter in which the transducers

are mounted on the outside of the pipe has another feature which

distinguishes it from the USCCFM. As is apparent from the operating

principle outlined above, it is ultrasonic energy propagating in the

direction of the pipe axis which is affected by the fluid velocity. To

provide a component of energy flow in this direction, the ultrasonic

beams are sent diagonally across the pipe. This causes the beams to be

refracted at the interfaces between the transducers and the pipe wall and

between the pipe wall and the liquid. The angle of refraction is a

function of the liquid properties and of the pipe wall material and

thickness. For this reason the transducer system has to be carefully

designed to match pipe and liquid parameters.

Interaction of the Ultrasonic Beams with the Flowing Liquid

The operating principle of the USCCFM is illustrated in Figure 1.

Two continuous-wave ultrasonic beams are directed diametrally across the

pipe at locations spaced a known distance apart, typically between i and

2 pipe diameters. As in the case of the travel time difference meter the

sonic propagation is affected by velocity although in this case it is the

transverse velocity components of turbulent eddies that are of interest.

The eddy represented in Figure 1 will first accelerate and then decelerate

the upstream beam. It will be swept downstream by the fluid flow and will

then produce a similar effect on the second beam. In each beam the eddy

pattern interacts with the ultrasonic wave causing its phase to be

disturbed or "modulated".

Arrangement of Ultrasonic Transducers

The ultrasonic energy is generated by a simple thickness-mode

piezo-crystal which is supported in a cylindrical mount and pressed

against a silicone rubber coupling block. The coupling block is in turn

pressed against the pipe wall. A drop of heavy silicone oil is used on

both surfaces of the block to exclude air from the interface. Identical

crystals and mounts are used to receive the ultrasonic energy on the

opposite side of the pipe.

Figure 2 shows the mounting arrangement. The cylindrical mounts

containing the crystals are supported normal to the pipe surface and

confined by the white nylon caps shown surrounding the coaxial connectors.

Rotating the caps advances the cylindrical mounts and presses the 'coupling

block against the pipe surface. The only critical dimension is the

spacing in the direction of the pipe axis between the upstream and

downstream pairs of transducers. This is established by four spacers

which link the pairs of mounts. Longer spacers are used when the mounts

are used on larger pipes.

321
Electronic Circuit Elements

Figure 3 shows a block diagram of the signal circuits. The upstream

and downstream channels are identical up to the point where their signals

are fed to the correlator. Each has a low power sinusoidal oscillator

operating between 0.7 and 1.5 MHz at 1 watt level. Coaxial cables,

which can be at least 100 feet long, couple the oscillator to the

transmitting transducer. Similar cables couple the receiving

transducers to a broad-band R.F. amplifier which presents amplitudelimited square waves to a synchronous demodulator. The received R.F.

signals are normally about 0.1 volt, although the amplifier will provide

full limiting for signals of 1 mV.

The phase demodulator consists of two balanced mixers which are

synchronized by direct and quadrature signals from the associated

transmitting oscillator. Four output signals are generated which are

direct and inverted sine and cosine functions of the instantaneous

phase difference between the transmitted and received signals. A

selector circuit chooses the signal having a suitable polarity of

change-of-voltage with respect to the change-of-phase-delay . Voltages

so selected are fed via a high pass filter to the correlation analyser

from each of the upstream and downstream channels. The selector

circuit overcomes the ambiguity caused by the large and somewhat

unstable phase delay (typically 50,000 degrees) which exists in each

ultrasonic transmission path. Note that a complete set of electronics

from oscillator to selector is provided for each channel. To avoid

interference the two oscillators operate on slightly different

frequencies

.

The signals fed to the correlator from the demodulators are low

frequency random voltages which are replicas of the phase modulation

imposed on the ultrasonic beams by the passage of turbulence patterns

through them. There is a significant probability that a pattern which

intercepts the upstream beam will also intercept the downstream beam.

Therefore the output signals from both demodulators will contain

similar voltage variations but the voltage pattern from the downstream

channel will tend to lag in time by an amount corresponding to the

transit time of the turbulence patterns between the beams. It has

been found that this closely corresponds to the liquid transit time.

Although turbulence patterns have been found to be the source

of most of the phase modulation, similar modulation is produced by

minute variation of temperature and density which also travel at the

liquid velocity.

The Basic Cross-Correlation Analyser

In the initial development stages of the USCCFM, the demodulated

signals were fed to a Hewlett Packard 3721A Correlation Analyser [4]

which displayed the signal delay time as a peak position on an

oscilloscope. Although this instrument was invaluable for circuit

322
development, its size, cost and the lack of an electrical output

representing peak position made it impractical as a component of an

industrial flowmeter.

It has proved possible to build a correlator on four 4" x 6"

circuit cards which provides a continuous readout of flow velocity

and other features suitable for use in a flowmeter. The operating

principle is indicated in Figure 4. The demodulated analogue signals

from the upstream and downstream channels are applied to zero-crossing

detectors which generate binary pulse trains, indicating the

instantaneous polarities of the signals. The serial pulse train

derived from the upstream signal is applied to a serial-in parallel-out

shift register where it is shifted under control of a sampling clock.

At each output a serial pulse train appears which duplicates the input

pulse train delayed in time by a known amount. Each of these outputs

is applied to a separate binary multiplier where it is compared with

the non-delayed pulses derived from the downstream signal.

If both inputs to a multiplier are truly random, its output will

be in the 0 state as often as in the 1 state. If both inputs are

always coincident, the multiplier output will always remain in the

1 state. For lesser degrees of coincidence, the 1 state will

predominate to a degree which indicates the degree of coincidence

(or correlation) between the multiplier inputs. Each multiplier feeds

a separate multi-bit binary counter which fills to capacity

(i.e. overflows) at a rate dependent upon the proportion of I's at its

input. Since all counters are reset to zero whenever one overflows,

the ordinal position of the counter which overflows indicates the

channel where the greatest correlation exists. The upstream signal

applied to this channel has been delayed by a time which is determined

by the number of shift register stages multiplied by the time delay

per stage, i.e. the product of the ordinal position of the channel,

multiplied by the period of the sampling clock. Since this delay

produced the maximum correlation with the downstream signal, it

corresponds to the travel time of the fluid between the ultrasonic

beams

.

This basic principle has been used in a number of cross- correlation

analysers [4,5]. To enable such an analyser to be constructed

economically, a LSI chip was developed at the University of Edinburgh

which contains a 12 stage shift register with corresponding multipliers,

counters and auxiliary circuits which determine the overflow position.

The correlator which forms part of the USCCFM uses 5 such chips, which

largely accounts for its compact form.

Features of the Correlator used in the USCCFM

The conventional cross-correlation analyser described above is not

very suitable for use in a flowmeter. One problem is due to the

discrete steps of delay time represented by the successive output of

the shift register. Particularly for short delay times, adjacent steps

323
can differ by a large fractional amount. Another undesirable feature

is that the output is in terms of delay time which is the reciprocal

of the desired output, flow velocity. To overcome these problems,

the frequency of the sampling clock has been made variable and is

servo-controlled to hold the average overflow position constant, in

a chosen channel. If the overflow actually occurs in a lower numbered

channel (indicating a time delay which is too short) the clock

frequency is increased so that each stage of the shift register

represents a shorter delay and more stages are required to match the

transit delay. If the overflow occurs in a higher numbered channel the

opposite control action occurs and the clock frequency is lowered until

the overflow occurs in the chosen channel. If the flow velocity

changes, the instrument adjusts the clock frequency to follow the

changing transit delay. The control loop is an analogue circuit with

integral feedback, providing a continuously variable clock frequency

with infinite resolution and avoiding the discrete steps of the

overflow channels. Since the clock frequency is inversely proportional

to the delay in the shift register, the readout can be easily made

directly proportional to flow velocity. The time delays provided by

the overflowing counters and by the integral servo feedback provide

smoothing of the statistical nature of the delay measurement.

Of course many auxiliary circuits are necessary in such an analyser

which will not be explained in detail. One such circuit gives the

correlator the ability to recognize and lock-on to correlatable signals.

Another circuit determines the degree of correlation between the inputs

(or the cross-covariance) and indicates to the operator the quality of

the signals. Three types of output are provided; a pulse train

related to the sampling clock frequency which is directly proportional

to the flow velocity, a digital readout of the sampling clock frequency

measured against a crystal controlled standard and a scaleable, isolated

analogue current output suitable for interconnection with conventional

plant instrumentation.

Sources of Measurement Error

An important feature of the USCCFM is that it measures flow velocity

in terms of two fundamental quantities, T, the time required for the

fluid to travel d, the distance between the two beams. The velocity

V = d/T. Thus the instrument does not depend upon or require

calibration. If volumetric flow Q is required, the pipe area A must be

determined and Q = Mass flow, of course, entails multiplying by the

fluid density.

The beam separation d is set by the mechanical construction which

introduces a negligible uncertainty, and by the acoustic symmetry of the

transducers which is estimated to be +2 mm. At the minimum value of d

(0.1 m) used on small pipes this could introduce a +2% error. It becomes

proportionally less on larger pipes.

324
The pipe area A may be calculated as

ir/A { (circumference/ir) - 2 x wall thickness}^

(If the pipe is out-of-round , the circumference measurement effectively

averages the O.D.)« Permissible thickness variations from handbook

values for standard pipes can cause a 3% error in the calculated area.

It is preferable to measure the wall with an ultrasonic thickness gauge

which will probably reduce the error in area to 1% or less.

The error in determining the transit delay T has several components.

The time delay in the shift register is obtained by measuring the

frequency of the sampling clock which can easily be done with negligible

error. The factors which relate this time delay to T are more difficult

to assess. It can be shown mathematically, and we have demonstrated

in practice that, on the average, the time delay which produces the

maximum number of coincidences in the multipliers corresponds to the

time delay inherent in the signals from the two receiving transducers,

providing that the bandwidth and phase response of the two electronic

channels are identical for the correlatable components of the signals.

The use of a simple polarity-correlator does not affect this

relationship (5,6).

Because of statistical scatter due to both the axial velocity of

the turbulence patterns and presence of uncorrelated noise in the signals,

there is a variance in the measured flow rate. This can be reduced by

increasing the averaging time at the expense of response time. The

averaging time is adjustable and an acceptably steady output can usually

be obtained with an averaging time in the order of 10 seconds.

The other factor which affects the relationship between the measured

time delay and the average liquid velocity is the flow profile. The

meter measures the average velocity across a diameter of the pipe

whereas the volumetric flow is proportional to the average velocity

integrated over the pipe area. The error due to this cause has been

calculated by Birger [7] for the ultrasonic travel time difference meter

in fully developed turbulent flow. His analysis also applies to the

USCCFM. It is presented in Figure 5 as a flow coefficient k, which is

a weak function of Reynold's number. True flow is k times the indicated

flow.

In practice, the USCCFM has been found to operate better in a more

turbulent flow than the "fully developed" condition. A suitable location

is about 5 diameters below an elbow. This additional turbulence flattens

the velocity profile somewhat and the effective value of k differs

slightly from Birger 's calculations. The effective value as determined

by our laboratory calculations is indicated by a broken line in Figure 5.

Since the flow near an elbow has a somewhat uncertain velocity profile

a further error source is introduced probably amounting to +2%.

325
The error sources are independent so it is reasonable to combine

them as a root-sum-square. Taking as an example an 8 inch pipe with

measured wall thickness

Maximum error in d +1%

Maximum error in A +1%

Maximum error in k +27o

1 Net maximum error (2^ + 1 + 1)^ = 2.4%

No allowance has been made for statistical scatter since it is assumed

the averaging time is long enough to make this negligible.

Laboratory Tests of System Performance

To investigate the instrument performance a series of careful

comparisons has been made between flow as measured by prototype USCCFMs and as

determined by precise mass/time measurements. The first such tests were

conducted at the hydraulic laboratories of the National Research Council in

Ottawa, Canada, before the correlation analyser section of the instrument was

designed. The transit times were measured with a Hewlett Packard 3721A

Correlator. These tests demonstrated the ability of the ultrasonic, signal

electronics and demodulation systems to produce correlatable data without

introducing erroneous time delays. A scatter in the value of k of up to

+.04 from Birger's value was observed in tests covering a wide range of flow

conditions. This included the human error in reading the delay time from

the oscilloscopic display, which is not a problem with the present instrument.

A more recent series of tests at the Chalk River Nuclear Laboratories

of AECL, using the final form of the instrument with its own correlation

analyser and digital readout of flow velocity, has shown that accuracies

of +.02 (2%) of the measured value (at 1 a limits) can be maintained over

a 10 to 1 range of flow without adjustment. A full report of these tests

is in preparation [8] . This performance is particularly encouraging when

it is realized that there are no calibrated transducers or other sources

of progressive deterioration in the system. The linear response and the

fact that accuracies do not have to be expressed in terms of the full scale

value are also advantages.

Field Experience

The instrument has been used on numerous occasions at Heavy Water

production plants at Port Hawkesbury, Nova Scotia and Tiverton, Ontario.

Since these plants are built out-of-doors to avoid possible concentration

of toxic Hydrogen Sulphide in confined areas, the conditions of application

are often very difficult, entailing rain, snow and difficulties of access.

It has been demonstrated that the meter can be installed in a new location

and measure flow within an hour or two even in adverse conditions. In these

locations, comparisons were made with plant instrumentation. In all cases

the measurements agreed within the likely uncertainties of the existing

instruments

.

326
On another occasion the instrument was used to investigate the

partitioning of flow in a part of the moderator system during commissioning

of the Bruce Nuclear Generating Station. The only accessible measuring

locations were short pipes between valves and other sources of flow

disturbance. Nevertheless, useful measurements were obtained which enabled

modification of the hydraulic design.

Summary

The USCCFM utilizes an old principle of flow measurement, that of

timing the passage of a tracer a known distance along a flow path, but it

does this in a new way. The tracer used is the inherent turbulence

pattern present in fast flowing liquid. The tracer is monitored from the

outside of existing piping without penetration. A compact correlation

analyser is used to continuously determine the transit time of the tracer

pattern and provide a real-time readout of flow in velocity, volumetric

or mass units. Because of the use of the tracer principle, calibration

is not required and a small correction for velocity profile provides

sustained accuracies of +2% of the reading over a 10 to 1 range of flow

in a given location. The existing hardware can be applied to pipes ranging

from 4 to 30 inches, which accommodate a flow range from 25 to 80,000 USGPM.

Acknowledgements

The author wishes to acknowledge the contributions to this

development made by Dr. M.S. Beck and Dr. C.N. Wormald of the University

of Bradford, Dr. K.J. Serdula of AECL Chalk River, Ontario, the staffs of

the Port Hawkesbury and Bruce Heavy Water Plants and of his colleague

Mr. D.L. Freeman.

327
References

Beck, M.S., Calvert, G., Hobson, J.H., Lee, K.T., Mendies, P.J.,

"Total Volume and Component Flow Measurement in Industrial Slurries

and Suspensions using Correlation Techniques", U<ia^a/i2Jn<Lnt avid ContAoZ,

Vol. 4, No. 8, pp T133 - T138, August 1971.

McShane, J.L., "Ultrasonic Flowmeter Basics", lYl^t/iumtvvtoutiOYl TucknotOQLj

,

pp 44 - 48, July 1971.

Torrance, J.W., "Confusion and Delusion in Sonic Flow Metering", Paper

presented at 28th Annual Instrumentation-Automation Conference and

Exhibit, I.S.A., Houston, Texas, October 15 - 18, 1973.

Anderson, G.C., and Perry, M.A., "A Calibrated Real-Time Correlator/

Averager /Probability Analyser", HwiloXt-VoickoJid JouAnaZ, Vol. 21,

No. 3, pp 9 - 15, November 1969.

Jordan, J.R., "A Correlation Function and Time Delay Measuring System

Designed for Large Scale Circuit Integration", Paper presented at

International Symposium on Measurement and Process Identification

by Correlation and Spectral Techniques, Institute of Measurement and

Control, University of Bradford, England, 2-5 January, 1973.

Boonstoppel, F., Veltman, B., and Vergouwen, F., "The Measurement of

Flow by Cross-Correlation Techniques", Paper presented at lEE Conference

on Industrial Measurement Techniques for On-Line Computers, June 1968.

Birger, G.L., "Certain Problems in Calibrating Ultrasonic Flowmeters",

1 zmoAAX^' naya T2.(ikyilka, No. 10, pp 53 - 55, October 1962.

Bazerghi, H., and Serdula, K.J., "Evaluation of the Performance of the

CGE Ultrasonic Cross-Correlation Flovraieter", Paper to be presented at

Canadian Nuclear Association Conference, Montreal, Canada, June 1977.

328
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333

National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS

,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

RECONSTRUCTING THREE-DIMENSIONAL FLUID VELOCITY VECTOR

AND TEMPERATURE FIELDS FROM ACOUSTIC TRANSMISSION MEASUREMENTS

S, A. Johnson, J, F, Greenleaf, M. Tanaka and G, Flandro*

Biodynamics Research Unit

Department of Physiology and Biophysics

Mayo Foundation

Rochester, Minnesota 55901

A theory with supporting experimental

evidence is presented for reconstructing the

three-dimensional fluid velocity vector

field and temperature field in a moving medium from a set of measurements of the acoustic

propagation time between a multiplicity of

transmitter and receiver locations on a

stationary boundary surface. The inversion

of the integrals relating the acoustic propagation path to the propagation time measurements is affected by linearization and discrete approximation of the integrals and

application of an algebraic reconstruction

technique (ART) . The inversion of these

integrals provides reconstructions of both

acoustic refractive index and vector fluid

velocity. Temperature reconstructions are

obtained from the temperature dependence

of acoustic refractive index. The problem

of the presence of certain invisible fluid

functions is treated. Since this technique

does not require the presence of scattering

centers or the optical transparency of the

medium, it may be applied in many cases

(i.e., turbid, opaque, or chemically pure

media) where Doppler or optical (e.g., laser

holography) methods fail,

1. Introduction

Methods for the measurement and description of the flow

of fluids (both liquids and gases) may be divided into two

main classes: invasive techniques and non-invasive techniques.

Invasive methods make use of devices such as probes, the introduction of markers, the measurement of pressure changes across

restrictions, etc. The non-invasive methods make use of the

external measurement of some flow-dependent property such as

* ^fechanical Engineering Department, University of Utah, Salt Lake City, Utah.

335
those relating to changes in optical properties, acoustic

properties, electro-magnetic properties, etc. At the present

time, several optical methods exist for gaining information

about fluid flows. Laser Doppler, Schlieren optics,

interferometric , holographic, etc., methods are known to the

art. These methods require that the fluid be nearly transparent to the incident light. In addition to near perfect

fluid transparency, Doppler methods require the presence of

scattering centers or particles. Natural or artificial dust

particles or fluctuations in density can serve this purpose.

Low scattering center concentration provides longer ranges

but sensitivity suffers. Higher scattering center concentratiQn provides higher sensitivity but limits the range (depth)

of measurement. The use of higher transmit power levels

increases range and provides a stronger scattered signal but

power is usually restricted by practical upper limits.

Schlieren, interferometric , and holographic methods work well

in general only for gas flows where density variations may be

larger than for liquids.

Acoustic methods have also been proposed and applied

which make use of either the Doppler scattered sound or transmitted sound along a single bean. The use of pulsed Doppler

methods allows measurement of the component of fluid velocity

(not the true fluid velocity) along the acoustic beam. The

same trade off between sensitivity, which governs laser

Doppler measurements, also exist for acoustic doppler

techniques.

The average fluid velocity component along a transmitted

beam can be determined from the measured time of propagation

along the beam. Flow meters designed around this principle

are well known.

All of these methods suffer from several common weaknesses: first, only the component of flow parallel to the

beam is determined; second, only that region of the flow

traversed by the beam is sampled. Thus, none of the previously

mentioned methods in their simplest form measure true threedimensional flow.

This paper will demonstrate that three-dimensional fluid

flow can be determined by a new transmission method which overcomes the range and sensitivity problems of doppler methods

and which overcomes the averaging problem (determining only

the average parallel component) of the single beam transmission

delay method.

It has been suggested that fluid flow within a measurement region may be determined by transmitting and receiving

acoustic energy through the measurement region along a plurality

or rays such that each volume element is traversed by a set

of rays having components in each direction for which flow

336
components are to be reconstructed [1] . The propagation time

of the acoustic energy along the plurality of rays constitutes

the only measurements required by the method [1]

.

Each ray propagation time measurement is an integral of

a function of acoustic speed and fluid velocity along the

ray and the set of such measurements constitute a simultaneous

set of integral equations which may be inverted (solved) to

obtain the unknown fluid velocity vector field [1] . The equation which relates the propagation time along each ray from a

source a to a receiver b is given by (see Figure 1 and 2)

^ab "-^a "^^/l |T^C(?)+V(r) | | (1)

where x is the unit tangent vector along the acoustic ray as

seen frSm the moving mediumt, C(r)_^i§ the local acoustic speed

as seen in the moving media, and V(r) is the fluid velocity

vector as measured in the laboratory.

In general the actual ray path taken by acoustic energy

from the source a to the receiver b is not known a priori even

though the time of propagation may be measured quite accurately.

It will be shown that the actual paths may be found either by

appropriate simplifying assumptions or from proper mathematical consideration of the complete set of propagation times

between many sources and receivers on a boundary surface

surrounding (but not necessarily enclosing) the flow. This

paper will proceed with the simplest case of fluid velocities

much less (10% or less) than the speed of sound and then generalize to higher fluid velocities.

When the velocity of the fluid is everywhere much less

than the speed of sound, the denominator in Equation 1 can be

approximated by the expression (C + x 'V) where it is assumed

that the ray tangent vector x as seerJ from a coordinate system

embedded in the fluid is almost identical to the ray tangent

vector X as se^n in the laboratory coordinate system. In most

fluids, when |v| <<C, the variations in C(r) are also correspondingly small. With these assumption. Equation 1 can be

written as:

^ab " ^a^l/C) (1-?^V/C)ds (2)

The assumption that x is nearly equal to x allows the further

assumption that the acoustic rays are nearly straight lines.

Thus, the ray paths may be found in terms of the knov/n source

point a and receiver point b. The use of straight line rays

Figures in brackets indicate the literature references at

the end of this paper.

337
provide a further benefit, namely, a simple method for separating the dependence of the time t^j^ on both C(r) and V(r) , This separation is obtained by forming the linear combinations

(t , + t, ) and (t^j^ - t, ^) where now t, ^ refers to the propagation tlmfe between b ana a when b serves as the source and a

as the receiver,

*ab + ^ba

= 2/^(l/C(?))ds (3)

tab - ^ba = -2/aC?-V/c2)ds (4)

Thus, even if C(r) is a function of position r it may be

found by the inversion of the multiple set of equations (3)

corresponding to a well chosen set of ray paths. This solu2 tion may be used to define C m Equation 4 so that V may be

found from the same or a similar set of ray measurements used

to solve Equation 3.

It is important to choose a set of ray paths which provide

a unique solution to Equations 3 and 4, One such good arrangement of transducers which provide paths with linear independent

integrals is shown in Figure 1, Flow is assumed in a cylinder

formed by translating a closed plane curve, nowhere convex inward, in a straight line (square and circular pipes are members

of this set) as shown in Figure 1. The source and receiver

transducers are placed on the intersection of this cylinder and

two parallel planes. These planes are shown arranged such that

the axis of the cylinder is normal to both plane a and plane b.

Ray paths are used from any transducer a. in plane a to any

other transducer aj in plane a or to any transducer bj in plane

b. As shown in Figure 1, both intraplane rays and interplane

rays are utilized in this arrangement.

Geometric Arrangement of Transducers

Ring of Transducers

Wall of Cylinder

Ring of Transducers

I 02"- °i

\ \ \

Flow

338
Figure 1 Illustrates a geometry in which three-dimensional

fluid flow can be reconstructed with a few simplifying assumptions. Two of m.any planes a, b, c, .containing transducers a^^,

a2f...a2, b^,b2*...b2/ etc. are shown,

-> -> The mathematical form of t^V can be derived by reference

to Figure 2 which shows a coordinate system in the fluid flow

field with ^he z axis in the direction of the axis of cylindrical

symmetry. t is the uni;^ tangent vector to a acoustic ray at an

arbitrary point, i.e., t = dr/ds, while i, j", R are unit vectors

along the x, y, z axes which are fixed in direction. Let

r = F + z_^k, then = dS/ds and then and dR are the projection^ of T and ds respectively onto the xy plane. The magnitude

of dr is ds, the magnitude of dR is dR. Here a,3,Y the

direction angles and (R ,0) are the polar coordinates of ^ .

Coordinate System in Flow Field

(C-Axis in Direction of Cylindrical Channel)

X

Figure 2 Coordinate system in flow field with_^z-ax^s in direction of cylindrical channel axis (C-axis) , t = dr/ds is unit

tangent vector to acoustic ray at an arbitrary point P, but AAA

i, j, k unit vectors along a,y,z axes are fixed in^direction. Note

|dr| - ds and the projection gf ds is dR, i.e., |dR| = dR. Note

also the fluid flow vector V(r) at point P.

With this notation, an expression for V'T may be written

in terms of comgenents V , V , V of V and the cylindrical a

coordinates of t. Noted^ th^t ^dR =^(§in Y)ds where t = Rq + k

cos Y. With this substitution for t-V Equation 4 becom.es

(note, in this paper the symbol = will indicate a definition)

339
I(a.b.)=t , -t, o'li' a.b. b.a.

1 J j 1

b 2 -2/ (1/C ) [(V cose+V sine)sinY+V cosylds (5) a X y z

which may be written

b.

^2 = [Io^^i^i^'"^l^^i^i^^ " -2cosY/^^ (1/C'')V dR (6)

where

b.

p

I,lij (a.b.) = -2/ ^ Cl/C ) (V cose+V sine)dR (7) a. x y

The advantage of writing Equations 6 and 7 in place of

Equation 5 is made more clear by noting that from Equation

7 is independent of y , Thus, V and V may be found by taking

a sufficiently large set of ray^ in ^either plane a or plane b

(not from plane a to plant b) and solving the corresponding

set of Equations 7 for V and V . In other words, I (a. a.) =

^ X y '013

I, (a. a.) when y = Tr/2 and where a. is the projection of b. onto

J 3 D the a-plane.

The function V may be found without solving Equation 7

for V and V if the value of can be found for each a. and

X y 2 1

b • . If a. and b. are in different planes and if a. and a. are

their projections along the cylindrical axis onto one plane, then

I.j^(a^bj) is identical in value to I^(a^aj). With I2 determined

for each ray in a sufficiently large set of rays from I and I^,

it is then possible for Equation 6 to be inverted. Thus,

Equation 6 may be replaced by the following practical equation,

relating V to actual time measurements,

z

lo = [ (t K

- t, ) - Ct - t ) ] =

2 a.b. b.a. a. a. a. a. ID Di ID Di

a . J

-2COSY/ ^ (V /C )dR (8)

a . z 1

Thus, V is found from data consisting of measurements

between transducer pairs both in the same plane and between

distinct planes. The integral in Equation 7 or Equation 8

may be written in terms of an approximate discrete sum by

subdividing planes a or b into finite elements or pixels.

Then dR corresponds to the length of the ray in each pixel.

Using this notation. Equation 7 and 8 become

I^(a,i,j)g = -2Z(1/C^) [V^(k)cos 6^ + V (k)

k

^s^^sk

340

(9)
l2Ca,b,i,j)g = -2cos I Cl/C^) (k) L^j^ (10)

where L , is the length of ray s in pixel k. Method.s for

solving Equations 9 and 10 are well-known [2]

.

The case of nonconstant speed of sound C is also described

by Equations 3,6, and 7 or their extensions, such as Equation

9. Thus, both V and C can be reconstructed even in those cases

where C is not a constant. The case where flow changes rapidly

with coordinate z can be treated by the use of more than two

measuring planes, i.e., m.ore planes than a and b, and by

spacing those planes closer together.

The ART method is usually applied to reconstruct scalar

quantities from their projections. In the operation of this

technique, the reconstruction of ^^ctor quantities is required.

This is accomplished by writing (t«V) as (V cos a + V cos 6 +

cos Y) . At each point along the ray cos^a, cos 3, ^nd cos Y

are known. The quantities V^, V^, and are each sought

in the pixels (picture elements) in a square array with N

pixels on a side. One way to accomplish this task is to modify

the ART algorithm to reconstruct a scalar U in a rectangular

2

array of 3N pixels where the value of U is V in the first

square sub array of N by N, where U is V in the middle square

sub array of N by N, and where U is V ^in the last square sub

array of N by N. In the case of cylindrical si^nmetry such as

used in Equation 9 and 10, only two such square sub arrays are

used to solve Equation 9 and only one such square sub array

is required to solve Equation in.

2. Fast Flows

The problem of reconstructing fluid flows where the velocity

of the flow is not significantly small compared to the speed of

sound requires a more accurate treatment of the ray paths than

the straight line approximations of Equation 9 and 10 [1]

.

Thus, the ray paths may bend significantly due to both the

transformation from moving to laboratory coordinate systems

and from acoustic speed variations due to density variations

(the latter being more common in gases than liquids) . If a

complete set of acoustic ray paths could be determined, than

Equation 1 could be inverted notwithstanding its nonlinear

nature. Techniques are known for finding the ray paths when

Vlr) and C(r) are known. One such technique is called ray

tracing and has been previously described [1,3] . Thus, it

seems that either the ray paths or V and C can be found independently even in fast flows.

While man]^ methods could be suggested for finding both the

ray paths and V and C, for purposes of illustration only,

an iterative technique will be described. First, linear ray

341
paths are assumed on a sufficiently coarse g^rid (i.e., large

pixels) , Second, the speed C and velocity V are reconstructed.

Third, more accurate ray paths are found by ray tracing. It

may be necessary to smooth the reconstructions obtained from

the previous step before ray tracing. Fourth, Equation 1

is solved again for new values of C and V using the ray path

from step three. Steps three and four may be repeated many

times as the grid size is made smaller.

The case for supersonic flow requires special treatment

because there may be transducer pairs for which it is impossible to find linking rays (sound cannot travel upstream in

supersonic flows) , Although it seems probable that a suitable set of invertable equations may be obtained in many

cases of this type when proper consideration is given to

transducer placement, a proof of this conjecture is not presently known to the authors.

3, Invisible Fluid Flow

Under some circumstances. Equations 3 and 4 cannot be

inverted. A simple example is the case of radial flow in a

plane from a point source in that plane. If this flow has

angular symmetry, than t^j^ - t^^ = 0, Any such flow in a

plane which contains t (i,e,, t has no components perpendicular

to ^his plane) is invisible. However, the flow may be detected

if T has components perpendicular to the plane. See Figure 3

for the geometry of this example.

Figure 3 Invisible radial flow. See text for explanation.

From Figure 3, it is seen that a decrease in speed on the

left of symmetry axis L-j^ L2 is equal to the increase in speed

on the right of axis L^^ It should be noted that radial

flows are not invisible to the second order terms in the expan2 sion of Equation 2 since (t'V/C) does not change sign upon

342
reflection about although the contribution from the

second order term is usually negligible for Iv|< C/5,

Under some circumstances the integral in Equation 8 may

not vanish, as it does in the case of angularly symmetric

radial flow, yet the system of Equations 7 and 8 are still

not invertable because each integral in a set corresponding to

different rays is not linearly independent, A simple example

of this situation is the case of two-dimensional flow in a

channel with rays taken between points on opposing walls of

the channel. The geometry of this case is given in Figure 4

below,

ULTRASONIC QUANTITATION OF

AVERAGE TWO-DIMENSIONAL FLOW IN A CYLINDER

Linear T ransducer Array

bi b2 bfi^i b„

a, 02 On

Linear T ransducer Array

Figure 4 Two-dimensional flow in a channel cannot be reconstructed by linear approximation with the geometry shown. However, the average flow is determined by forward and backward

measurements along each ray. See text for explanation.

In Figure 4 transducers a^ a^ are located on the lower

wall and transducers b^ b^ are located on the upper wall.

Let a (a., b.) = a. . be the angle which the ray between a. and I'j i,] ^ ^ 1

bj makes with the x-axis. The x-axis is parallel to the parallel

channel walls. Let the shortest distance between the channel

walls be L, The integrals (Equation 4) can be evaluated by

changing the parameter of integra^ion_^by the formula ds = (sin a. . ) - dy and noting that V • x =^ V cos a. .. Here we

assume e 0 for steady state flow.

Thus Equation 4 becomes

b

.

u - t, , = -2 cot a. . /

^ ^x dy (11) a.b. b.a. x,3 a.

343
2 If the fluid is incompressible then C is constant. The

continuity equation and the assumption V = 0 implies that

8 V =0. Thus the integral in Equatio^i 11 is a constant

independent of the path a.b. and all such equations are linearly 13 2 dependent. Note t , -t, a.=-2<V >X ^/C where ^ a.b. b.i X a.b.

< V > is the average flow and X , is the x-axis distance

X ^ a.b.

1 3

between a . and b .

.

1 : The problem of possible invisible fluid flows may be

minimized by two approaches: first, the second order terms

in the expansion of Equation (1) or (2) may be used, second,

three-dimensional (3-D) rather than two-dimensional (2-D)

geometries may be used. This latter approach could make use

of the notion that many invisible flows in 2-D are detectable

in 3-D. An additional help may be found in requiring the 3-D

reconstruction to be consistent with fluid dynamic equations

such as the continuity and momentums equations.

4, Reconstruction of Temperature Fields

The theory developed in the preceding sections may be

applied under certain conditions to the determination of the

three-dimensional distribution of temperature. If, for

example, it is known that the material to be probed by the

measuring acoustic fields is homogeneous then the reconstruction of the velocity field may be related to the temperature

of the material by a simple function which maps velocity of

sound to temperature, A similar method has been proposed by [4]

Sweeney using optical rather than acoustic properties of matter.

If the substance is pure degassed water, then the velocity

of sound in the neighborhood of 19°C is given by the formula

c[m/sec] = 1461 [m/sec] + 3,44 (T-ig^C) +

,0185 (T-19°C) ^ + , . , (12)

where T is measured in °C, Thus, the inverse function giving

T as a function of c is the required mapping function, A . difference of 20°C for water produces about the same percent

change in velocity of sound as the difference between striated

muscle and water. Reconstruction accuracy may be increased if

the difference in temperature is required. Thus, circulation

and metabolic heat generation may possibly be reconstructed,

A map of mixing of two identical fluids of different temperatures may be also be reconstructed. Thus, a reconstruction

of c can be transformed to a reconstruction of T. The reconstruction of c is obtained by solving the system of Equations

^' 344
5, Use of a Priori Information

The fluid dynamic equations may be written in finite

difference form to provide addition equations of constraint

to the previous equations. This makes possible more accurate

reconstructions or reconstructions from fewer projections,

^or example, the equation of continuity (here d = density,

V = velocity)

V • (d^) +

^ = 0 (13)

produces the following set of finite difference equations in

two dimensions

d.^T . . V .

, T . -d. . ^ V . .+

2.,3+l,t y, 1,3+1 i,D,t y,i,3

where here Ax = Ay = 1 by a choice of coordinates. Note k = (i,

when comparing V , V , and d in the two notations (e.g., V , = V . ., etc.) ^ ^ ^'^

In a similar manner the momentum and energy equations

could be written in finite difference form and used with equation. The use of Laplaces ' equation as a priori information

in reconstruction has been reported by Radulovic [5]

,

6. Computer Simulation Studies

Digital computer simulation studies were conducted in

FORTRAN language on a CDC 3500 computer to test the previously

described theory. Computer simulation permits the testing a

a wide range of flow configurations and acoustic path geometries. In our case, both two-dimensional parallel ray projection and fan beam projections were simulated. Such projection data were made either by integrating Equation 2 ^y_^analytic

or numerical means for a specific velocity function (V(r), The

set of acoustic propagation times "Ct^j-j} thus obtained was then

\^sed as measurement data to reconstruct the velocity function

V (r) . The simularity of V (r) and V(r) is a measure of the

accuracy of the reconstruction method. A modified algebraic

reconstruction technique (ART) with an underrelaxation parameter

of 0.75 was used [6],

The angular velocity of several vortex models with angular

symmetry were reconstructed from simulated parallel projection

data. The velocity of a natural fluid vortex measured in a

plane perpendicular to the vortex axis may be modeled by

^P^CS) = ^

Cl-e-«^''^^e (15)

345
where 0 is a unit vector defined to be perpendicular to ^ and

A

to k. This function behaves like wR for very small R and like

2 wA /R for very large R. The maximum value occurs near R = A,

In order to simplify the simulation and reconstruction

algorithms it is desirable to study a function which vanishes

outside a finite radius B and yet corresponds closely to the

behavior of Vp^^ (^) defined in Equation 15. The function

(S) defined by Equation 16 was chosen to meet these requirements

(w - ^^|)Re, R < A

2 2 ^ ^P2^^) = \ - ^) e, A < R < B (16)

0 , B < R

The three-dimensional velocity V($) for a point in a

plane is given b^ V(R) = Vp (R) + V^(j^)k. In these simulations,

the function V (R) was defined to have a parabolic shape

representing a downward flow given by

-K[(R/B)^-1] , R < B

V„(R) = I (17)

z R > B

The constants of geometry and flow were given values of

A = 4,24 cm, B = 9.55 cm, w = 50 radians/sec, K = 91 cm./sec

aiid simulated projection data were computed. A graph of

|Vp2(R) 1 vs. R along a diameter and the corresponding generated

fan beam projection data (t_^j^ - tj^^) are shown in Figure 5.

Also shown in the same figure is the function V (R) which is

_y R

a reconstruction of |Vp2(R) I from the set projection data

t^j^ - tj^Q. The geometry of the flow and transducer locations

(the geometry corresponds to that given in Figure 1) is also

described in Figure 5.

346
GEOMETRY FOR FLUID FLOW MEASUREMENT

OE = E = 15cm

3 cm

Figure 5 Experimental geometry for vortex reconstruction.

Line I: shows the source and detector geometry transducers

at a and b on circle of radium OE, Square C-j^C2C2C^ is area

reconstructed. Radius OB is 0.9 of OD. OA is 0.4 of OD. For

simulation, transducers on radius Oa = Ob. For experimental

data collection source at a and receiver at b and receiver

moves on arc of circle b MF, with a at center. Line II:

shows the simulated time difference (t , - t, ) ordinate cal- ab ba

ibration shown for values of A,B,w,k given in this figure.

The abscissa R is the perpendicular distance from 0 to ray

ab. Line III: shows assumed planar fluid speed function

(solid line) and reconstructed values sampled along line ODE

(solid dots) . Line IV: shows assumed perpendicular (V^) fluid

component (solid line) and reconstructed values sampled along

line ODE (solid dots)

.

347
Gray scale pictures of the three components V , V , V

of the reconstructed function V (^) and of the planar liriagnitude

2 2 1/2 2

(V^ + ) ' are shown in Figure 6. The picture of (V^ +

2 1/2 V ) shows the circular nature of the concentric flow lines.

^In these studies, 153 views each with 153 rays were used.

RECONSTRUCTION OF SIMULATED FLOW

Figure 6 Reconstruction of vector components of simulated

fluid vortex. Top left shows x component V . Top right

shows y component V . Bottom right shows z component V . In these fluid velocity component reconstructions black is

negative, gray is zero and white is positive in values. Bottom

2 2 1/2 left is the magnitude of the xy planar component (V^ + V ) Note circular equal speed contours. Images are 64 ^ pixels per side.

7, Experimental Flow Studies

The success of the computer simulation studies prompted

an experimental verification of these results in the laboratory. Accordingly, an experiment was conducted with a flow

and transducer geometry which matched the simulation studies

as closely as possible. Although a circular transducer array

like that shown in Figure 1 was not available, a fan bean

348
geometry corresponding to the fan F^aF2MF, shown in Figure 5

was possible by using an ultrasound scanner designed for

breast cancer detection studies [7] . The source moved on a

circle of radius 21 cm while the detector moved around the

source on a circular arc of radius 2 6 cm on the opposite side

of the center. All source intervals are equal.

The difference (t

ments of t

- t.

"ab

ab ^ba^ digitized, separate measure

and t^^ provided by a time of flight detector.

The values of (tab - tj^^) were determined for 60 views with

150 rays per view and processing by the same algorithm used

in the simulation studies. All ray intervals are equal.

The raw data for all rays in all views is shown in Figure

7. Also shown are the gray scale pictures for the components

and V and the magnitude (speed) (V ^ + V ^)'^^^. No measure

ments of-^the component V were made. ^ ^

RECONSTRUCTION OF EXPERIMENTAL FLOW

Top View

Side View

(Isometric

)

Figure 7 Experimental data and reconstruction of vector

components of fluid vortex. Top left shows experimental

data and is an image of the difference between the time of

flight with flow and without flow (fast arrival = white, no

change = gray, show = black) vs. scan position (left to right)

vs. angle of view (top to bottom). Top right shows reconstructed planar fluid speed (V^^ + V^)-""/^, black is zero,

white is positive. Bottom left shows'^ x component of velocity

V^. Bottom, right shows y component of velocity B . In V

V^, black is negative, gray is zero, and white is ^ositive^

349
Reconstructed flow is maximum (73 cm/sec) at a radius of

0,62 cm. Reconstructions are 64 pixels per side. Geometry

of vortex, scanning plane, and vortex generator are shown in

top and side views in right margin.

8. Experimental Temperature Tomography

The theory presented allows the reconstruction of temperature fields in moving or static fluids. This process might be

termed temperature tomography or thermotomography . A reconstruction of a temperature static fluid was made before attempting

the reconstruction of temperature in a flow field. A static

temperature field was produced by circulating water from a constant temperature bath through three long sausage shaped

balloons. The three balloons were arranged to form an equalateral triangle with sides of about three inches in a second

constant temperature bath containing the ultrasound scanning

transducers (5 MH , wide band) , A fourth balloon with no

water circulation, filled with dilute sodium chloride located

near the center of the triangle provided a constant nontemperature dependent refractive index area. The horizontal scanning

plane passed through the center of the four balloons mounted

at equal heights with their axis parallel and vertical. Sixty

projection views of about 150 samples each were collected for

each configuration of a preset constant temperature difference between the scanning bath and the balloons. The bath

temperature was kept at 22 °C for each configuration. The salt

solution filled balloon provided a large refractive index constant temperature object to test the hypothesis that a reconstruction of only temperature changes in a nonhomogeneous

object is possible. Figure 8 shows the result of the reconstruction of refractive index for four different balloon temperatures. The high contrast central salt solution filled

balloon dominates the four temperature variable balloons. In

the temperature difference tomograms in Figure 9, the evidence

of the presence of the salt solution filled balloon is nearly

removed by subtracting a reconstruction of all four balloons

at 22*'C from each reconstruction of different configuration.

This single, initial experiment indicates this technique holds

promise for reconstructing temperature differences in inhomogeneous materials.

350
Figure 8 Initial temperature reconstruction experiments with

water and saline filled balloons. Three water filled balloons

and one saline balloon at 22**C are shown in top left. Water

balloons at 23°C in top right. Water balloons at 21,5''C in

bottom left and at 20. 5 "C at bottom right. Images were made

with a convolution reconstruction algorithm from 60 views. The

scan time was five minutes. in all images the saline filled

balloon is in the center.

351
Figure 9 Reconstruction of temperature differences. This

figure uses the data of Figure 8 to subtract out constant

temperature background from the changing temperature objects.

In each case, the 22 constant temperature reconstruction

shown in top left of Figure 8 is subtracted from:

top left - top left of Figure 8 (not complete cancellation)

top right - top right of Figure 8

bottom left - bottom left of Figure 8

bottom right - bottom right of Figure 8

Note the almost complete suppression of the saline balloon

compared to Figure 8; thus, only temperature differences are

reconstructed. (The particular subtraction routine used here

actually gives the absolute value of the differences.)

352
i 9, Experimental Reconstruction of Mixing and Fluctuation

The previously described theory provides a means for

' obtaining a tomographic image of the mixing of two fluids or

the fluctuation in fluid velocity of one fluid. If the data

collection period is shorter than fluctuation time of regions

larger than or equal to the spatial resolution of the instrumentation, then a "snap shot" or time resolved fluctuation

reconstruction is possible. Repetition of this process as

short time intervals would result in a time and spatial reconstruction of flow and fluctuation in flow. Such high speed

instrumentation would require parallel data collection techniques.

Statistical characterization of mixing and fluctuation of

flows with stationary statistical properties is possible with

nonsimultaneous sampling techniques. The mixing of a

tracer element in a flow has been characterized by som.e writers

[8] by a stream segregation function I defined by dividing

the variance of the percent concentration of a tracer at a

point in the flow by the square of the mean of the percent

concentration at that same point. The parameter I thus defined is a unit-less measure of mixing or fluctuation.

The reconstruction of the statistical properties in a

plane must be obtained from the statistical properties in the

projections. It can be shown that the mean value at a point

in a reconstruction is the reconstruction of the mean values

of the projection data (this follows because reconstruction and

the mean are linear operators) . It can also be shown under

certain conditions that in a series of experiments the variance

in refractive index at each point in a cross-section in the

fluid can be reconstructed by applying the reconstruction

algorithm to the variance of the projection data (this follows

if the fluctuations from pixel to pixel along a ray are independent) ,

353
An experiment to collect data for such a statistical

analysis of mixing and fluctuation at each point in a reconstructed tomographic plane has been performed. A spinning

vortex plume of warm (24**C) water was directed downward into

a tank of cold (15°C) water. Fluctuations are enhanced by

the angular velocity and the more bouyant nature of the warm

plume. Data for the reconstruction of refractive index crosssections at the same spatial position was collected.

The reconstruction of the mean of these cross-sections

of refractive index (i.e. temperature) is shown in Figure 10.

The reconstruction of the variance of this set shown in

Figure 11.

Figure 10 The top image shows a two-dimensional reconstruction

of the mean temperature (the acoustic refractive index is temperature dependent) in a spinning warm vortex of water in a tank

of cold water from eleven experiments. The image on the bottom is the data used to make the above reconstruction following

the same format used to describe the data displayed in Figure 7.

354
Figure 11 The top image shows a two-dimensional reconstruction

of the variance in refractive index (temperature dependent)

from eleven experiments. The image on the bottom is the profile data used to make the reconstruction at the top. The profile data shown represents the variance of eleven profiles,

each corresponding to one of eleven experiments. See Figure 7

for data format.

355
10. Potential Applications

One application of this method is the design and construction of improved flow meters for liquids and gases. The

ability to reconstruct the local speed of sound allows inference of local density and temperature changes. The combination

of velocity and density reconstructions permits the reconstruction of local and net mass flow rates. The inclusion of percent mass per volume of trace materials or pollutants permits

the calculation of total mass flow of such materials in pipes,

flues, or smoke stacks. Also, the total energy flow may be

calculated in regions around wind power extraction machines

such as windmills, wind turbines, or vortex towers or vortex

power devices. In some circumstances, the temperature of a

gas flow may be reconstructed. That this is possible may be

seen by observing that the speed of sound in a monoatomic gas

is given by

C = ((Cp/C^)kT)^/^ (15)

where C is the speed of sound C and C are the heat capacities

at constant pressure and constant volurlne respectively, k is a

constant proportional to the molecular weight of the gas, and

T is the temperature of the gas in degrees Kelvin. Thus, a

reconstruction of C when squared and rescaled provides a reconstruction of the temperature T (or the molecular weight, if

T is known)

,

If the gas has no sources or sinks, then the continuity

equation

V- (pV) = 0 (16)

where p is density and V is_^velocity, may be integrated through

out the region where C and V were previously reconstructed.

This integration will require some assumptions on the value on

some suitable boundary. The integral of Equation 16 is a reconstruction of the density function p. The pressure distribution may be obtained by using the gas law, P = p RT/M (here,

M is the gas molecular weight, R is the gas constant, P is the

pressure, p = density, T = absolute temperature) and the reconstructions of p and T. Thus, a reconstruction of pressure i

possible. The technique of measuring temperature might be

extended to mixtures of gases using average molecular* weights

.

As has been seen, the data collected by the apparatus may

be processed by algorithms of various levels of sophistication

(e.g., straight lines or curved line reconstructions). One

further level of improvement would involve the use of certain

fluid dynamic equations such as the continuity equation, the

momentum equation, and/or the energy equation (or their equivalents) as constraints in the solution of Equation 1. This

may lead to greater accuracy in reconstruction or allow reconstruction with less data [1]

.

356
One further potential application is the reconstruction

of the statistical moments of flow at each voxel (cubic volume

element) in space.

Biomedical applications such as measuring flows of blood,

respiratory gases, and other fluids both in and out of the

body are possible when suitable measuring boundary conditions

exist. For example, fluid carrying prosthetic devices could

be tested by this noninvasive technique. However, where a suitable measurement collecting boundary does not exist (e,g,, flows

in the heart can only be viewed from a narrow view angle due

to shielding of the lungs) , Doppler methods will remain unchallenged.

11, Image Display Methods

The capability to reconstruct the value of a scalar or

vector physical parameter in two or three dimensions calls for

the use of compatible display methods. Gray scale pictures are

useful for 2-D scalar display. The display or vectors sampled

4- X X \ V

Figure 12 Gray scale (magnitude) and vector array (direction)

computer generated image of a reconstructed vortex in a water

tank. The array of vectors simulates a square array of yarn

tufts, each tuft attached at one end to a stationary button.

Thus, the tufts are inclined downstream in the clock-wise flow

shown above.

357
on a 2-D plane may use a separate gray scale images for each

component. This techniques was used in Figure 6. Another

possible display method could color code the vector field with

a color here governed by direction and with brightness or saturation governed by magnitude. Another method, which works

well for fluid flow, is to display the stream lines or to display a 2-D array of arrows aligned with the stream lines, A

display of the reconstruction of a fluid vortex using a field

of arrows superposed upon a gray scale image of flow magnitude

is shown in Figure 12, The display of functions in 3-D may be

done by use of set of parallel cross-sections or by the use of

special three-dimensional display hardware.

SUMMARY

A mathematical theory and computational algorithms are

presented for an acoustic method for reconstructing fluid

velocity vector fields. Computational FORTRAN programs were

written and tested with simulated and real data for the case

where the fluid speed is much less than the speed of sound.

These tests indicate the method has the capacity for reconstructing three-dimensional fluid flows and net flows. It

seems likely the method can be extended to reconstruct mass

flow fields (variable density) and temperature fields. The

method is capable of reconstructing flows with or without the

presence of scattering centers [9]

,

ACKNOWLEDGEMENTS

The authors wish to thank Miss Pat Snider and associates

for the secretarial and graphic assistance in preparing the

manuscript. The photographic help of Mr, Leo Johnson is

appreciated. The use of Mr, Willis VanNorman's gray scale

and color computer graphic programs is appreciated. The

support from Dr, Earl H, Wood, Chairman, Biophysical Sciences

Unit, Mayo Clinic, and Dr, Erik Ritman, also at Mayo Clinic,

is appreciated. Discussion with David T, Pratt, Ph,D, of the

University of Utah, M, E, Department, on fluid mixing is

appreciated. This research was supported by Grants HL-00170,

HL-00060, RR-00007, HL-04664, NOl-HT-4 -2 904 from the National

Institutes of Health, United States Public Health Service;

NOl-CB-64041 from the National Cancer Institute.

358
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F. A. , and Sjostrand, J. : Reconstruction of three-dimensional velocity fields and other parameters by acoustic

ray tracing. 1975 Ultrasonics Symposium Proceedings,

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[2] Johnson, S. A., Greenleaf, J. F., Chu, A., Sjostrand, J.,

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Reconstruction from projections: Theory and practice in

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Papers, August 4-7, 1975, Stanford, California, pp TUB 2-1 - TUB 2-4.

[3] White, R, W. : Acoustic ray tracing in moving inhomogeneous

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.

[4] Sweeney, D. W. , Attwood, D. T., and Coleman, L. W.

:

Interferometric reconstruction of electron-number densities

in laser-induced plasmas, Proc. Image Processing for 2-D

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[5] Radulovic, P. T. : Use of a priori information in

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fields. Proc. International Symposium on 3-D Reconstruction, Brookhaven, New York, July (1974)

.

[6] Herman, G. T. , Lakshminarayanan, A. V., and Rowland, S. W.

:

The reconstruction of objects from shadowgraphs with high

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[8] Pratt, D. T.: Mixing and chemical reaction in continuous

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[9] Johnson, S. A.: Patent pending.

359

National Bureau of Standards Special Publication 484, Proceedings of the Symposium on Flow in Open Channels and Closed Consults held at NBS

,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

VALIDATION OF USE OF DYE-DILUTION METHOD FOR FLOW

MEASUREMENT IN LARGE OPEN AND CLOSED CHANNEL FLOWS

W. H. Morgan

Peerless Pump

An Indian Head Company

Montebello, CA 90640

D. Kempf

CEL Laboratories

FMC Corporation

Santa Clara, CA

R. E. Phillips

Turner Designs

Mountain View, CA

The use of water soluable tracers for determining flow rates of rivers

and streams has been practiced for several decades, both in the United States

and abroad. During the last 15 years the economical availability of water

soluable dyes, coupled with the considerable technical improvement in fluorometry equipment, has made feasible the use of dye-dilution methods for in situ

calibration of flumes and other flow measurement devices.

In July 1976 a validation of the precision of flow measurement by dyedilution techniques was undertaken at the St. Anthony Falls Hydraulic Laboratory,

University of Minnesota, by comparing volumetrically measured flows over the

range of 50,000 to 80,000 GPM. Subsequently, the method was applied in a field

acceptance test of a large variable speed vertical turbine pump handling

effluent from a wastewater treatment plant.

In both the validation and acceptance tests a dye of the Rhodamine family

was continuously injected at a low precisely controlled rate into the flow

stream. Several precise dilutions of the injected dye concentrate were prepared

so as to bracket the anticipated 'dilution expected of the measured flow. A

fluorometer was used to detect levels of dye concentration in the fully mixed

flow at downstream sampling point as well as that of the prepared volumetric

diluents

.

The most serious of the measuring problems encountered, at the field test

site, was the variation in turbidity which was overcome by a change in the type

of Rhodamine dye used and through carefully controlled volumetric dilution of

samples

.

This paper describes the apparatus and procedures used during the

validation test as well as those of similar activities pursued during the

conduct of verification tests of hydraulic performance for the two installed

variable speed pumps. Results of the validation tests are demonstrated to

be within 3% of volumetrically determined flows and the field application

of the dye-dilution method for determining hydraulic performance are shown

to be within 1.5% of the predicted head-capacity pump curve.

361
1. Introduction

In late 1972 a Mid-West headquartered pump company was awarded

a contract to furnish four large single stage mixed flow vertical

turbine pumps for installation in a new addition to aii existing wastewater treatment plant serving metropolitan Louisville, Kentucky. These

pumps handle secondary treatment effluent. The design point of each is

101 MGD (70,000 GPM) at 93 ft. TDH when operating at 600 rpm. Two

pumps are each driven by vertically mounted direct connected 2000 HP

synchronous motors. The remaining two pumps are each driven through a

right angle gear and horizontal Eddy-current coupling connected to a horizontal 2000 HP synchronous motor. Each of the above furnished two

types of mounted units are illustrated in Figures 1 and 2. Though the

pump manufacturer has facilities to test full size and complete pumping

equipment, an agreement was reached with the Consulting Engineers to

provide hydraulic performance data for the prototype units based upon

the results of model testing.

A brief description of certain aspects of pump location within

the Station's pump house will provide background to better appreciate

subsequent discussion. A combination of one constant and one variable

speed unit is installed in one of two interconnected sump bays. Each

unit is served by a 54" diameter steel discharge line in which neither

flow measuring devices nor pressure regulating mechanism is installed.

Beginning at the pump house the inverts of discharge piping are buried

some eight feet below grade. The total length of discharge piping for

each variable speed unit is approximately 665 feet (Bl) and 717 feet

(B2) as measured between pump house and outfall located in the levee

adjacent to the Ohio River. At the top of the levee each 54" line

contains a siphon breaker made from 20" diameter pipe and extending

approximately 10 feet above grade. The elevation difference between

centerlines of the 54" diameter pipe at pump discharge and at levee is

59.95 feet. After primary treatment the effluent passes through chlorine contact basins and then through a Parshall Flume located just

ahead of the sump bays. Though provisions were included for chlorination treatment of the effluent, this process has not as yet been activated. Station layout, Parshall Flume, and discharge line profile are

illustrated in Figures 3-5.

During September 1975 Station personnel conducted quasi hydraulic

performance acceptance tests on each of the installed pumps. These

tests were run to verify performance at design point conditions and to

establish as well the maximum capacity obtainable under field head

conditions without cavitation. The Parshall Flume, with ball- float

integrating flow totalizer, was used to measure the effluent in- flow

to individual pump bays. Because the Station at that time was not in

full operation, only limited quantities of effluent were available

which precluded sustained high volume flow rates. This condition

caused significant variations of water level both in the flume and

sump bays during testing periods. Results of these tests indicated

that the pumps were delivering considerably less flow than the

specified capacity. Subsequent analysis of forwarded test data

362
strongly suggested numerous areas of uncertainty in the flow data.

In large measure these discrepancies centered around the effect on

capacity measurement from undesirable entrance conditions to the

flume, inadequate flow and head measuring equipment, and suspect

accounting of flow- volume diverted into latent cisterns. Even

though the foregoing test data was obscure in certain vital areas,

it was left to the pump manufacturer to demonstrate that the installed pumping equipment did, in fact, meet the specified hydraulic

conditions.

Throughout a period extending into early 1976 an extensive search

was carried on to determine and evaluate the suitability of commercially available flow measuring devices other than the already existing

flume. Initially, search was restricted further to flow devices sanctioned by the Standards of the Hydraulic Institute. In addition to

instrument accuracy, the cost of the total package required to perform

verification testing was of paramount importance to the pump manufacturer.

A variety of flow metering equipment was investigated. These included velocity sampling devices such as the pitot tube, insert turbine

meter and insert vortex shedding meter. Single and multi-plane response

ultrasonic meters, magnetic and magnetic resonance meters were considered. Devices such as the foregoing were either too costly or were

dependent upon assuming a profile for fully developed flow. In this

latter connection, recent investigations had demonstrated that even for

straight pipe of adequate upstream length random and significant variations from fully developed profiles occurred under steady flow conditions (1). From among differential producing meters the venturi type,

flow tube, and orifice plate (end-cap or otherwise) were entertained.

Here again initial cost and/or expense of calibration, excavation or

installation tended also to rule out these devices as suitable substitutes for the flow measuring flume.

During the foregoing quest, a number of references were encountered

citing the use of tracer dyes for flow measurement of streams, rivers,

and estuaries (2, 3, 4); calibration of ultrasonic and large venturi

meters (5, 6); establishing of flow patterns in forebays of large generating plants as well as the determination of hydraulic performance (7, 8).

It became rapidly apparent that only one group of tracers, fluorescent

dyes, deserved serious consideration. Such characteristics as low dye

cost, being environmentally acceptable, and with portable equipment

easier to make quantitative measurements of flow rate in any shape and

size of conduit. This method was suited ideally for our purpose with

one exception,^ i.e., the method was not sanctioned by the Hydraulic

Institute.

After a number of discussions between representatives having job

site vested interests, agreement was reached for the use of the dyedilution technique at the job site for measuring pump flow. Acceptance

of this proposed method was qualified, however, by the proviso that a

measuring accuracy to within 3% of true flow be demonstrated through

validation testing.

Figures in parentheses indicate the literature references at the end

of this paper. o^^o
To meet the requirements for a validation test proved to be a

formidable challenge. Foremost of these was in assessing the degree

of accuracy obtainable for controllable flows bracketing the 70,000

GPM design point of the pump. A further prerequisite was for a large

reservoir capacity of unrecirculated water that would provide an

approximate constant head during capacity draw off. An additional

desirable condition would be to have a water composition that contained a minimum amount of suspended solids. Obviously, for reasons

of economy, the location should be within the continental United States.

Fortunately, these requirements were realized at the University of

Minnesota's St. Anthony Falls Hydraulic Laboratory under the direction

of Professor John F. Ripken. Figures 6 and 7 show certain of the

facilities of this Laboratory used for validation testing.

This paper deals not only with the activities and results connected with the validation test, but also includes the procedures used

and results from the use of dye-dilution techniques during subsequent

hydraulic performance verification tests of installed equipment.

2. Validation Tests

A special physical phenomenon called fluorescence characterizes

certain tracers and it is this distinctive property upon which the dyedilution technique depends. A fluorescent substance is one which absorb

light of one wavelength (color) and emits light of a different wavelengt

a. Dye : From a number of fluorescent tracers available, two from

the organic Rhodamine family were chosen as being suitable for

use at either test location. They are Rhodamine B and/or WT.

At the validation test site only Rhodamine B was used. Each

of these dyes has a detection limit of 10 ppt or less.

Rhodamine B, in powder or liquid form, is more available commercially than the liquid Rhodamine WT. In liquid form

Rhodamine B comes in an acetic acid solution of 407o (by wt . ) , while Rhodamine WT is supplied in a lOTL aqueous solution.

These tracers are highly persistent and great care must be

exercized during their handling and in the preparation of

dilutions. It is essential that once a flask or pipette is

used for a particular dilution that those pieces of glassware

be thereafter dedicated for that same dilution. Both of the

Rhodamines absorb readily to some plastic containers. Therefore it is recommended that flasks, pipettes, and sampling

bottles be of borosilicate glass (Pyrex) , polyethylene, or

polypropylene materials (9)

.

b . Equipment

1) Fluorometer: The fluorometer measures the strength of

the fluorescence in a diluent. A schematic diagram of

the Turner Designs Model 10 Series portable fluorometer

used for both the validation and verification tests is

shown on the following page:

364
OPTICAL SYSTEM

With this instrument set in automatic mode, should a reading fall below 20% of full scale, a more sensitive

range is automatically selected. Adjustment for background fluorescence is made only on one range and, if

during operation other ranges are selected, the proper

correction will be made automatically. Zero light also

gives zero reading. Its sensitivity permits detectability to less than 10 parts/trillion and provides

response linearity to within +17o. The instrument is

portable and can be operated on conventional AC or

powered by battery. Because the instrument is waterproofed, it can be used in a high moisture- laden environment without affecting its operation or accuracy of

response. The instrument contains telemetry outputs for

driving recorder equipment.

2) Injection Pump: The constant-rate injection pump contains a 1/4" diameter polished and ground piston operating within an accurately honed cylinder block. The

pump is driven by a 1/8 HP synchronous motor. This type

driver assures that the established flow of injected dye

will be maintained at a constant rate of delivery over

extended periods of continuous operation. The desired

flow rate is obtained by micrometer adjustment of stroke.

During validation tests, the injection rate was set at

78.7 X 10"^ cfs. The injection pump recirculated dye

stock for a minimum of 1/2 hour to insure that pump parts

reach a stable operating temperature. After warm-up the

rate of injection flow is determined by the time required

to fill a two- liter volumetric flask.

3) Digital Voltmeter: A digital voltmeter, Model 5900 with

BCD output is plugged into telemetry outputs of the fluorometer. This displayed visual response to relative change

in fluorescence of the diluent.

365
4) Digital Printer: A Model 5505A printer was used to

make a permanent record of the change in millivolts - a function of a change in fluorescence.

5) Assorted Glassware:

a) Volumetric Flasks - Class A; six two- liter and two

one- liter

.

b) Volumetric Pipette - Class A; three each of 5, 10,

20, 50, and 100 millileters.

c) Sample Bottles - Polyethylene Plastic with screwon Polypropylene caps.

6) Mixing Vessel for "Stock"; 30 gallon plastic container.

Definition of terms used in connection with the dye-dilution

technique described in this paper.

1) STOCK A concentration of dye contained in the mixing

vessel that is not only injected into the flow path,

but is used also to make the STANDARDS dilutions.

2) BASE SOLUTIONS Intermediate dilutions of the STOCK

from which STANDARDS dilutions are prepared.

3) STANDARD A "standard" is a known concentration or

dilution of the injected dye. The dilution is made

using precisely measured volumes of the flow media and

dye being injected. Dilutions, known here as alpha,

beta, and gamma standards are of such a concentration

as to bracket the anticipated concentration level of the

SAMPLE.

4) BLANK The amount of fluorescent materials occurring

naturally in the system expressed as a concentration of

the fluorescent material being measured. The fluctuation

of blank during a measurement period is the important

factor and not the absolute magnitude.

5) SAMPLE An amount drawn from the flowing media, downstream from the point of dye- inj ection , that contains a

concentration of dispersed dye.

6) DILUTION RATIO This is the entire basis of the technique.

The ratio is formed by the relative concentrations of

injected dye and that of the SAMPLE collected.

7) CUVETTE A glass sample holder of specific dimensions

inserted into the fluorometer for subsequent analysis of

the fluorescence level of contained media. The cuvette

was used to hold quantities of BLANK, STANDARD, and

SAMPLE from each test run.

366
PREPARATION OF DILUTIONS : One of the most critical preparatory

phases in the use of the dye-dilution method is the preparation of the

STANDARDS. It is essential that all flasks and pipettes be thoroughly

rinsed with undiluted media and that each flask or pipette once used

for the preparation of a specific dilution be dedicated thereafter to

the particular dilution for which the glass hardware was first used.

This precaution cannot be overly emphasized since Rhodamine dye is

extremely persistent.

1. In succession and proper sequence, approximately fill each

2- liter flask to graduated line with undiluted flow media

obtained from the Station at which SAMPLES will be drawn.

2. Dilute the STOCK concentration by preparing at least two BASE

dilutions. Using dedicated flasks and pipettes draw a 20 ml

volume of the STOCK and slowly drip into 2- liter flask of first

BASE. Using a dedicated pipette top off to graduated line with

undiluted flow media. The second BASE is prepared similarly

using again a dedicated 20 ml pipette and 2- liter flask.

However, the second BASE is prepared by withdrawing 20 ml of

the resultant first BASE dilution.

3. Again in a similar manner the three STANDARDS, alpha {Cf ),

beta (^jS ), and gamma ("V ) are prepared, except that specific

volumes used in their preparation are each drawn from the

second BASE.

4. A diagram for preparing these separate dilutions is shown in

Fig. 8.

5. It will be noted that each of the STANDARDS bear to each other

a concentration ratio of 1, 2, and 4.

DETERMINATION OF DISCHARGE FLOW - CONSTANT RATE INJECTION : When

using the dye-dilution method it is not necessary to know the exact

concentration of either the dye injected or that of the sample collected downstream. It is necessary only to know the dilution factor.

In both validation and verification tests the constant-rate injection

method was used. The flow rate to be determined (Q) is related to the

dilution factor and the rate of dye injection as follows:

Q = q ^-^^ where (1)

Q = Flow rate in test channel (or discharge flow)

q = Flow rate of the constant-rate injection pump

C = Concentration of the dye being injected

c = Concentration level of dye dispersed throughout

the flow media and collected as SAMPLE downstream

The point at which samples are drawn must be located sufficiently

downstream of the point of dye injection to assure complete mixing

across the plane of flow.

367
To account for background fluorescence a BLANK of undiluted media

is drawn at the sampling point before and after each test run. A

numerical value of the BLANK'S fluorescence is determined by the

fluorometer. In our case, this value is transmitted through the instrument's telemetry output terminals and converted into signal levels to

drive the printer. The numerical value of the BLANK (B) is subtracted

from each term in the DILUTION RATIO before determining discharge flow

rate. To account for the fluorescence of the BLANK the foregoing

relationship is modified to read as follows:

Should a temperature difference exist between quantities in the

dilution rates, each should be temperature compensated before the

determination of discharge flow. A temperature correction curve is

shown below (10):

TEMPERATURE DIFFERENCE (BASE TEMPERATURE MINUS SAMPLE TEMPERATURE), IN °F

368
TESTS AND RESULTS : A sketch of the test setup is shown in Fig. 9.

Two sampling Stations were used in the validation test. Station No. 1

was located approximately 190 feet downstream from the point of constantrate injection. A second Station was subsequently set up out on the

walkway between the two volumetric tanks. An approximate flow rate was

established by adjusting the variable orifice plate at upper river pool.

The final flow rate was determined by flow measurements in the Hydraulic

Laboratory's two calibrated volumetric basins. A continuous and constant

rate of flow was assured by virtue of the large upper pool reservoir

being a flow branch of the Mississippi River. By means of a unique system

of diverter valves located upstream and just preceding the volumetric basins

the established flow rate in the test channel can be switched rapidly (in

less than 2 seconds) from one volumetric basin to another and without flow

interruption.

After flow is established in the test channel, BLANKS are drawn before

the dye is injected. After dye injection a period of 10 minutes was

allowed for complete dispersion of dye in the flow stream and before the

start of filling volumetric basins as well as simultaneous taking of

SAMPLES

.

The first few days of testing were unproductive. The dye technique

was consistently yielding calculated flows approximately 8% lower than

the volumetric calibration. A common technique was being used in which

the sample water from the flume was continuously pumped through a flow

cell in the fluorometer. The fluorometer was calibrated by pouring

standard dilutions of the stock solution dye through the flow cell. The

water for the standards was collected from the river upper pool just prior

to entry into the flume. The intention was to provide an automatic correction for any masking effect of suspended solids by preparing the standards

in the same water whose flow rate was being measured. It was some time

before it was recognized that the water at the sampling point was grossly

different from the entry water. The extreme turbulence in the flume introduced massive amounts of air bubbles, some of which were microscopic, and

rose very slowly. These air bubbles scattered the exciting light in the

fluorometer and this scattered light was causing the instrument to effectively add a constant 0.08 ppb of dye to the measurement. Had the dye

concentration been 50 ppb, instead of 1.0 ppb, the error introduced would

have been only 0.167o. Lack of dye prevented increasing the concentration,

so the effect was largely overcome by collecting dilution water at the

sample point, and measuring some thirty discrete samples. Standards and

samples were allowed to stand for as closely as possible the same length

of time in the hope that the scattering effect would be the same. At this

point, time and available storage vials did not permit letting the samples

stand long enough to permit the bubbles to completely dissipate. Had it

been possible to use higher dye concentration, or permit overnight standing,

it is believed correlation with the volumetric calibration would have been

even better than reported. From a practical standpoint accuracies of

better than 1.% are difficult to obtain with this method in other than

laboratory conditions.

By adopting the above control techniques the resultant determination of discharge flow by the dye-dilution method correlated well with

that determined by volumetric basin measurements. The accompanying Percent Deviation Curve, Fig. 10 displays the results of test runs 9 through

15, shown in Table 1.

369
TABLE 1. DATA AND RESULTS FROM SIX TEST RUNS

DATE 7/30 7/31 7/31 7/31 7/31 7/31

TEST

RUN

9 10 11 12 13 14

INJECTION

FLOW

cfs X IQ-^

78.7 78.8 78.8 78.8 78.8 78.8

BLANK 7.5 15.0 17.0 125.5 106.33 125.0

lOROMETER

TUT

-

mv

STANDARDS

Alpha 2653 2679 8624.16 *•>'-

Beta 1123 1314 1348.25 4332.92 4530 4370

1-1 W o

Gamma * 665 683 2197.5

SAMPLES 2433.4 3869 1945.17 3678.84 4162.2 4477.35

DISCHARGE

FLOW; DYEDILUTION, cfs72.377 53.11 198.81 186.61 171.89 153.71

DISCHARGE

FLOW; VOLUMETRIC BASINS, cfs

74.031 54.463 106.012 198.352 175.096 155.793

PERCENT

ERROR

-2.23 -2.48 +2.64 -5.92 -1.83 -1.34

"These STANDARDS compromised

^'"'-These STANDARDS not read due to curtailed test schedule

The scatter pattern of results, with the exception of run #12,

fell within a tolerance zone of +3%. The regression line for all

points, including test run #12, lies almost wholly within the -3%

error zone which, together with its correlation coefficient of +.998,

supports the contention that, under prevailing conditions at the

validation test site, the dye-dilution method for determining discharge flow will produce results at least within the objective of a

+3% error.

A sample calculation for determining the discharge flow by the

dilution method follows.

370
SAMPLE CALCULATION

From Run #10

Injection Flow Rate (q)

:

1. Time to fill 2- liter flask = 14.95 minutes

2. q = 2.000 1 X 0.2642 gal/1

14.95

q = 78.8 X 10"^ cfs

= 35.34 X

10"-^ GPM

Flow Rate in Channel by Dye-Dilution Q;

1. Q = q

C - B

c - B

where C = B Standard; C = 1314 mv at ratio 2 x

B = Blank B = 15 mv

c = Sample

2. Q = (78.8 X 10"^)

c = 3869 mv

1314 - 15

t

l314 - 15~

| 3869 - isj

= 78.8 2

[

"1299"

]

3854J

(2 X 10+^)

Q = 53.11 cfs

371
CONCLUSIONS : The following observations pertain to the procedures

used and results obtained during validation testing:

1. At the outset, turbidity of the Mississippi River was believed

to be the prime factor affecting correlation between volumetrically determined discharge flows and those obtained by dyedilution. Due to the change in preparatory procedures prior

to fluorometric analysis, the subsequent correlation between

each method of flow determination was greatly improved. Therefore, it was felt that turbidity had negligible effect on the

correlation results.

2. Highly turbulent flow created by the waterfall effect at the

bottom of the dye- injection shaft created microscopic bubbles

that caused a positive measurement error. This effect was the

most significant factor in not being able to correlate better

between flow rates obtained by the two measuring methods. The

detrimental effect on the data from this condition was largely

overcome by procedures adopted during the last day's activities.

3. Taking undyed water for the preparation of STANDARDS, as

well as taking of BLANKS and SAMPLES, all from the same test

Station, improved data correlation.

4. Placing STANDARDS, BLANKS, and SAMPLES in a constant temperature

bath for at least 30 minutes greatly contributed toward improvement in the results. This time period also allowed a share of

the entrained bubbles to rise from solution and partially overcame their undesirable effect on data.

5. The overall conclusions, based on data and activities associated

with validation testing, confirmed the contention that discharge

flows could be determined within the permissible error of +37o

provided that extreme care was used in the preparation and

handling of dilutions; adopting proper procedures for collecting,

storing, and handling of the various dilutions prior to fluoro--

metric analysis; having on hand an adequate amount of dye concentrate to, if necessary, adjust downstream concentrations for

extended periods of time to levels in excess of 50 ppb ; and the

availability at a test site of the services of a chemist or

chemical engineer.

3. Verification Tests

Job- site verification tests began the week of September 26. However, on the evening of September 27, the day prior to the start of the

testing, the area was drenched by torrential rainfall which did not

abate till mid-day October 2. Postponement to a more favorable period

was ruled out due to customer scheduling and prior commitments of

personnel involved in the tests. Climatic conditions created a number

of untoward problems previously unencountered . The greatest of these

was the direct passage through the treatment plant of raw sewage and

372
heavy silt loads from overflowing catch basins. As a result, some

procedures established during the Minnesota validation test were

modified, of necessity, to procedures that require more time-consuming

dye-dilution techniques before obtaining final fluorescence values of

specific dilutions.

The Pump Station contained four pumping units that were designated

by the Station as Bl and B2 for variable- speed pumping units. Fig. 2

and by B3 and B4 for the constant- speed units. Fig. 1. As mentioned

previously in the Introduction, there were no provisions for controlling

pump discharge pressure so that only by speed regulation could flow

rates and discharge pressure be varied. Therefore, field performance

tests were performed only on the variable speed units. Capacity was

determine'd by the dye-dilution method while discharge head was observed

by pressure indication on a calibrated pressure gage whose sensing line

was connected to a pressure manifold tapped into the pump column.

a. Dye ; Both Rhodamine B and Rhodamine WT tracer dyes were used.

After the first test, only Rhodamine WT was used due to its

lower absorption characteristic.

b. Equipment :

1) The same Fluorometer , Injection Pump , Digital Voltmeter , and Digital Printer were used during job- site tests as

were used at the validation test site. Volumetric flask

glassware, of similar make and size, and STOCK mixing

vessel were duplicated.

2) Submersible Pump: 3 gpm capacity.

3) Calibrated pressure gage: Range 0-200 feet H2O; graduated in 2/10 foot increments capable of interpolation to

1/10 foot; 8: dial.

4) Digital Counter: Model 521A; interfaced with an electromagnetic tack- generator pick-up having 60 tooth gear head.

5) Cuvettes: #9820 rimless Pyrex culture tubes, 13 mm x

100 mm.

PREPARATION OF DILUTIONS : Precautions similar to those practiced

during validation were applied in the preparation of dilutions used in

the verification test. Concentrations of these dilutions were prepared

by procedures described earlier.

1. STOCK

a) For run #0 only a STOCK solution was prepared by diluting

40 percent active Rhodamine B to approximately 1.6 percent.

b) For runs #1, 2, 3 and 4 a STOCK solution was prepared by

diluting 20 percent active Rhodamine WT to approximately

4.0 percent.

373
2. STANDARDS

a) The Alpha, Beta, and Gamma STANDARDS were prepared from

the second BASE by diluting 30 ml, 20 ml, and 10 ml

respectively to 2000 ml.

b) For all tests the relative concentrations of Alpha, Beta

and Gamma were in the order of 3, 2, and 1 respectively.

All dilutions were prepared using undyed effluent drawn from

the downstream sampling test station.

DETERMINATION OF DISCHARGE FLOW : Constant-rate injection was

used. Therefore, as described previously, the DILUTION RATIO, BLANK,

and dye- injection rate were related in the same manner to obtain discharge flow.

TEST AND TEST PROCEDURE : Fig. 11 and Fig. 5 illustrate the test

setup and test station locations.

The siphon breaker, in the discharge line of test pump, was

unbolted at grade flange and pivoted out of the way. The sump pump

fastened to its 3- inch d iameter support pipe was inserted through the

exposed opening down to approximately the center line depth of the

54- inch discharge pipe. The support was then anchored to siphon breake

flange. Through use of the sump pump a continuous stream of the discharging effluent was brought to grade. One pint polyethylene bottles

were rinsed several times with the effluent, then filled. Thus, from

the sampling station, undyed effluent was drawn for use not only in

preparing the STANDARDS but also in detecting background fluorescence

of the flow stream (BLANKS). All of the SAMPLES of the effluent containing injected dye were collected in similar manner and from the same

sampling station. Preceding each test run, and at the sampling station

a number of BLANKS were drawn as well as a quantity of undyed effluent

for the making of STANDARDS.

For test runs #0 and #1 stop gates in the Station's flow- through

channel were regulated so that a portion of effluent passed through

the flume and into the test pumps' sump bay. A stabilized flow rate

was obtained by a combination of stop gate regulation and pump speed.

Sump levels were read on a tide board that was installed by transit

in each sump. Stabilization was achieved when the sump level remained

essentially constant. The 10 foot mark on the tide board was used as

the reference datum and corresponded to Pump Station's elevation 400.0.

When flow into pump sump had stabilized and with pump speed

steady dye was injected through the delivery tube and into pump

suction. After a minimum of 5-minutes had elapsed, to allow for pipe

surface coating and diffusion of dye in the flow path, a number of

SAMPLES were drawn from the downstream sampling point. Thirty SAMPLES

were obtained for each test run except #4 in which only nine SAMPLES

were collected.

374
For each test run, and covering the same time period during which

SAMPLES were being obtained, pressure gage levels, pump speed and sump

elevation were being monitored and recorded. Table 2 includes a

sample of such data.

After SAMPLES were collected, an interval lasting approximately

20 minutes, dye- injection was stopped. It is to be noted that the

injection pump is not stopped. The dye is diverted to the mixing

vessel and then recirculated. A ten minute period was considered

sufficient to allow the system to be purged of residual dye after

which final BLANKS and undyed effluent for the making of new STANDARDS

were drawn. All SAMPLES and BLANKS were transported to the Treatment

Plant's chemical laboratory and there placed in a temperature stabilizing bath. STANDARDS were placed also in the same bath.

On October 3 the Wastewater Treatment Plant's capacity could keep

up with incoming flow without having to "pass- through" untreated effluent.

Thus, for test runs #2, #3, and #4 secondary treatment effluent entered

sump bays

.

In order to complete the tests as scheduled, which had been complicated and delayed by weather conditions, BLANKS were not drawn nor

STANDARDS prepared between runs #2 and #3 and between runs #3 and #4.

LABORATORY PROCEDURES : The effect of the high and variable

turbidity on the accuracy of discharge flow was directly dependent

upon the amount of dye that might be absorbed by substances in the effluent.

Filtration and absorption recovery tests were made of the Rhodamine B

dye from the effluent. Dilutions were prepared with a 5 ml solution of

Rhodamine B dye using distilled water. By comparing fluorometric readings

of these filtered and unfiltered dilutions it was noted that Rhodamine B

dye was partially lost by sorption on the filter medium. Further recovery studies on equal dilutions in the effluent and in distilled water

indicated an apparent loss of 27% in the effluent. An automatic correction factor applied to the data could take this into account were it

not for the distinct possibility that a variable solids load existed.

One can appreciate the error introduced if, for example, a 107o change in

solids (undiluted) would yield a 2.77o error. Filtration of the test

dilutions was unacceptable since the resulting fluorometer readings

would be lower, indicating higher than actual flow rates.

An additional recovery study was made consisting of further specific

dilutions to test specimens of dye-effluent solutions. Ten milliliters

of the specimens were diluted to 100 milliliters with distilled water.

The result of this study showed that the effect from variable turbidity

on the determined flow rate would introduce an error less the 0.3%. It

was also proved by this study that the tracer was masked by the solids

and not absorbed on the solids. Therefore, for each test run all BLANKS,

STANDARDS, and SAMPLES were diluted tenfold with distilled water before

analysis.

A portion of each BLANK, STANDARD, and SAMPLE was poured into

individual cuvettes that had been previously washed. These cuvettes

375
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were then placed in a room temperature water bath whose level was at

approximately 1/4 inch above the fluid level in the cuvette. After

a 15 minute period to allow the fluid to temperature stabilize, each

cuvette was placed in the warmed-up fluorometer and analyzed. For

each cuvette sample 10 readings were recorded at approximately 0.8

second intervals initiated after a 10 second stabilizing period. An

average of the printout readings of SAMPLES and the before and after

BLANKS and STANDARDS were used in the dye- dilution relation as described previously.

RESULTS : This section is divided into two parts. First, the

determination of discharge flow rate (capacity - gpm) and second,

the determination of the pump's total field head (TDH - feet) as

measured across the plane of pump discharge.

1. Capacity : Hand calculations done in the field were refined

later by statistical methods. Steps in the statistical

analysis included the following operations:

a) A mean and standard deviation of ten readings for each

SAMPLE, BLANK, and STANDARD were calculated.

b) A calibration curve was generated for each set of SAMPLES

by plotting the calibration standard means as a function

of relative dye concentration. A least squares linear

regression analysis was performed. Utilizing the calibration curve, each SAMPLE mean was converted to a relative

dye concentration, and a flow value was calculated.

c) A statistical analysis was performed of the flow values

obtained for each test run. This includes calculations

of mean flow values, standard deviations and 95 percent

confidence limits. See Appendix for a sample calculation.

d) The capacity determined by the foregoing analyses is

tabulated below:

Mean Flow + 95%

Run

No. Date

Pump

No.

Confidence Limit

(GPM)

0 9-30 Bl 51.2 (+3) +1.4 (+3)

1 10-1 ^1 54.2 (+3) +0.3 (+3)

2 10-3 B2 63.0 (+3) +0.5 (+3)

3 10-3 B2 59.8 (+3) +0.5 (+3)

4 10-3 B2 67.1 (+3) +1.6 (+3)

2. TDH(Field ) : In the test set-up drawing, Fig. 11, a distance

is given from the floor to centerline of gage for unit Bl and

B2. Converted to Station elevations these are 419.52 and 420.79

377
respectively. The average of water surface elevations for a

given test run was subtracted from the respective gage elevation

to obtain the vertical distance Z in the equation:

H gage + v2/2 + Z = Pressure at manifold location, feet (3)

The value for V^/2g is calculated using an inside diameter of

54 inches for both column and discharge piping. The value

used for acceleration of gravity, g, is 32.17 ft./sec.^.

To obtain total field discharge head, at outlet face of discharge elbow, the head loss for the column (above pressure

manifold) and that of the discharge elbow are each subtracted

from the results obtained in (3) above, i.e.

TDH (Field) = 11 gage + 1 1 + Z - (Hcol + ^Disch elbow), (4)

feet ^

Table 3 gives the result of combining these head values

for each pump flow rate as was determined by the dyedilution method.

The values shown in Table 4 result from applying corrections

to Table I values for RPM, GPM, and TDH so as to obtain the

pump's head-capacity characteristic at the constant design

speed of 600 rpm. The corrections applied to the data are

based on the use of certain of the Affinity Laws which

relationships are used throughout the pump industry and

which are also in accordance with the Standards of the

Hydraulic Institute (11)

.

Figure 12 provides a plot of pump field performance

based on the separate results of the above two steps.

Because of an electrical power failure to the Bl variable speed

unit (requiring more time to repair than was available within

the time allocated for testing) only runs #0 and #1 were made

with this unit. The remaining test runs #2, #3, #4 and #5 were

made using the B2 variable speed unit.

During the dye-dilution tests. Station personnel took comparative flow readings using the Parshall Flume's instrumentation.

Based on the results from this parallel activity Station operators determined that their integrating flow totalizer equipment

was out-of-calibration. As a consequence their previously

obtained flow measurements were erroneous, producing flow

readings that were low by as much as 15 percent. It was determined also that this error was not constant throughout the flow

range

.

The capacity shown for test run #5 was obtained after the dyedilution tests were completed. The purpose of this test was

to determine the maximum flow rate available to the Station

under one pump operation should such a contingency arise.
TABLE ,3

TOTAL DYNAMIC HEAD-FIELD

TEST

RUN

NO. RPM GPM

GAGE

PRESS.

FEET

tOF GAGE

TO WATER

SURFACE

"Z"

FEET

VELOCITY

HEAD

v2/2g

FEET

ELBOW

PLUS

COLUMN

LOSSES

FEET

TDH

FIELD

FEET MGD

n SI 900 4S R 99 S 0 7QS 0 1 9 w o . o 73 7

L M-O . i. 99 S w . oyo 0W , J JSOw ftQ 9 78/ O . 0W

J DVJO S Q 800 M-D • J Z90W 9 . Z 0 A9S 9 8A 1

2 517 63,000 46.3 20.8 1.213 0.474 .67.8 90.7

4 532 67,100 46.6 20.3 1.376 0.538 67.8 96.6

5 602 84,000 46.2 21.8 2.156 0.840 69.3 120.9

"To Water Surface: Elevation of Gage^Mlnus Elevation of Water Surface in

"^Calculated : Based on Procedures Contained in References (12) and (13)

Wet Well

TABLE 4

TABLE 3 DATA CONVERTED TO 600 RPM

TEST

RUN

NO. RPM GPM

TDH

FIELD

FEET RPM GPM

TDH

FIELD

FEET MGD

0 492 51,200 68.8 600 62,439 102.3 89.9

1 495 54,200 69.2 600 65,697 101.7 94.6

3 506 59,800 67.2 600 70,909 94.5 102.1

2 517 63,000 67.8 600 73,114 91.3 105.3

4 532 67,100 67.8 600 75,677 86.2 109.0

5 602 84,000 69.3 600 83,720 68.8 120.5

379
To perform this test both of the chlorine contact tanks were

filled to almost overflowing as well as each of the interconnected sump bays. The pump speed was adjusted to approximately design speed and the decrease in surface level of the

channels recorded. This test was based on volumetric changes

in the artificially created reservoir. These latter readings,

plus the Parshall Flume's readout values were used to approximate the capacity. The previously determined error in the

Parshall Flume's instrumentation was taken into account in

arriving at this approximate value for capacity.

CONCLUSIONS : The following observations pertain to the procedures

used and results obtained during pump verification testing.

1. Extreme turbidity of the effluent and its variable solids

content required the use of dye-dilution techniques different

from those practiced at the validation test. Dye recovery

studies demonstrated that Rhodamine B was absorbed on the glass

filter medium during attempts at filtration. It was demonstrated

that by using distilled water to make tenfold dilutions of the

various samples that 99.57o of full recovery was assured.

2. It is advisable to use high concentrations of injected dye,

consistent with EPA regulations, so as to ensure that the dye

concentration of the SAMPLE is sufficiently high to nulify background fluorescence and the effects from minor turbidity on

fluorometric analysis. The change to Rhodamine WT, obtained

locally, permitted a 2-1/2-fold increase in the concentration of

the injected dye over that used with Rhodamine B, which had been

brought to the job site in limited quantity.

3. Before fluorometric analysis, place all STANDARDS, BLANKS, and

SAMPLES in a room temperature water bath for as long a temperature stabilizing period, as is consistent with the test

schedule, but for not less than 1/2 hour.

4. It is recommended that a chemist be retained as consultant

during the first three or four times the dye-dilution method

is applied for the determination of flow rate.

5. The dye-dilution technique based on the procedure and results

outlined in this paper is an economical method for field use

where metering and valving are not included in a buried or

otherwise inaccessible discharge line. By using this technique,

the field performance of the tested units met specified hydraulic

design conditions and was within 1-1/2% of predicted H-Q curve.

ACKNOWLEDGMENT is given to Messrs. G. W. Allen, S. G. Sharp, and Ms. L. A.

Enriquez, R&D, PG&E, Ca. ; and also to Messrs. G. W. Crabtree, and F. Martin

(illustrations). Peerless Pump, for their expertise and valuable assistance

during the course of one or both of the tests.

380
4 . REFERENCES

(1) Alden Research Laboratories, Worcester Polytechnic Institute,

Evaluation of the Westinghouse Leading Edge Flow Measurement

System, Report ARL M71-83.

(2) Pritchard, D. W. , and Carpenter, J. H. , Measurements of

Turbulent Diffusion in Estuarine and Inshore Waters, in International Assoc. Sci. Hydrology Bulletin 20, 37-50 (1960).

(3) Nickerson, D. H. , Glouster-Forced Circulation of Babson

Reservoir, in Sanitalk, Quarterly Review, Mass., Division of

Sanitary Engineering, 1-10 (1961).

(4) Kilpatrick, A. M. , Flow Calibration by Dye-Dilution Measurement, in Civil Engineering, 74-76 (1968).

(5) BATTELLE, Pacific Northwest Laboratories, Performance Evaluation

of 96 Inch, 900 cfs Venturi Flowmeter, Report (1973)

(6) Allen, G. W. and Sharp, S. G., Moss Landing Dye-Dilution Pump

Test, Report (1976).

(7) Hutton, S. P. and Murdock, Comparative Flow-Measurement Tests

at Finlarig Power Station, 395-404 (1962).

(8) U. S. Dept. of Interior, Bureau of Reclamation, Pump Acceptance

Test - Dos Amigos Pumping Plant, Report HM-17 (1973).

(9) Turner Designs Brochures; Fluorometric Facts-Flow Measurements;

and Flow Measurements in Sanitary Sewers by Dye-Dilution.

(10) U. S. Dept. of Interior, Techniques of Water- Resources Investigations of the United States Geological Survey, Fluorometric

Procedures for Dye Tracing, Book 3, Chapter 12A, 28 (1968).

(11) Hydraulic Institute Standards, 13th Edition, Test Code, 62 (1975).

(12) Hydraulic Institute Standards, Pipe Friction Manual, 24 (1961).

(13) Miller, D. S., Internal Flow - A Guide to Losses in Pipe and Duct

Systems, Part I, Design Data, 19 (1971).

381
APPENDIX

SAMPLE CALCULATION, RUN 3b

1. CAPACITY : SAMPLE #22

a. Results of ten telemetry readings, (mv) from fluorometer analysis

3644, 3739, 3599, 3554, 3585

3592, 3395, 3569, 3519, 3505

b. Mean Response (R) of above data = 3579 mv

c. Standard Deviation (S) =44.9

d. Linear Regression Equation - based on Run 3b calibration curve

R = 143.1413C + 158.99 Intercept = 158.99

Slope = 143.1413

Where C = relative dye concentration, ppb, of the SAMPLE

to STOCK concentration

c = (R - 158.99) = (3579 - 158.99) = 23.89 x 10^^

143.1413 143.1413

f. Ci = Concentration of injected Rhodamine WT (0.04)

g. q = Mean flow rate of injection pump (based on time to fill a

2- liter volumetric flask)

q = 2.000 L X 0.2642 gal/L = 35.65 x lO'^ gpm

14.82 min

h. Discharge Capacity:

Q = (0.04) (35.65 X 10-3) ^ ^ 23.89 x 10-9

Note: If the mathematical mean of all 29 SAMPLES was used, the tabulated

discharge capacity of 59,800 GPM would be obtained.

2. TOTAL DYNAMIC HEAD-FIELD :

a. By proper substitution of applicable values from Run 3, Table 3,

into the head equation (4) , the total field head at outlet face

of discharge elbow for 506 rpn pump speed will be obtained.

b. The right hand portion of Table 4 gives the CAPACITY and

TOTAL DYNAMIC HEAD-FIELD for the pump's design speed of 600 RPM.

These values are obtained by using the following Affinity Law relations

«TM2 = GTM1 mf andH2 = Hi

^

Where the subscript 1 relates to the capacity and head obtained at

test speed while subscript 2 relates to the change in those parameters when the test speed is changed to design speed.

382
2000 HP -600BPW -4 160V0LTS VERTICAL SOLID SHAFT

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FIG. I

383
FIG. 2

384
385
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15-10 TYP

FINAL AFFLUENT

PUMPINS STATION

PLAN VIEW OF FLUME

PARTIAL VIEW OF CHLORINE CONTACT TANK

CROSS-SECTIONAL VIEW OF FLUME

^PUMP ROOM FLOOR

EL. 4 15JO

REF , 34.75

-7-0 —

SUMP FLOOR '-eO - SLUICE GATE

EL 36075 INV. EL 362.00 TYR

FIG. 4

386
387

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392
FIG. 1

1

393

National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersb urg, MD, February 23-25, 1977. (Issued October 1977)

DILUTION METHOD OF DISCHARGE MEASUREMENT IN PIPES

E. R. Holley

Department of Civil Engineering

University of Illinois at Urb ana-Champaign

Urbana, Illinois 61801

The dilution method, or the method of mixtures, is a tracer

technique for discharge measurement in steady flow. The method

is based on the mass balance of a tracer injected at a steady

rate into the pipe flow and depends on measuring the concentration at one or m.ore points downstream of the injector in order to

evaluate the average concentration of the tracer. The accuracy

of the method depends on the tracer, on the injection and

measurement equipment, and on the locations of the sampling ports

since these locations determine the accuracy with which the

concentration measurements represent the actual average

concentration.

Minimizing the errors associated with the locations of the

sampling ports depends on being able to predict the rates at

which mixing takes place for the applicable injection and pipe

flow conditions. The mixing can be viewed as being either

initial mixing associated with the injection or ambient mixing

associated with the pipe flow. Both analytical work and

experimental work on ambient mixing are presented with attention

being focused on injection and sampling at the pipe wall. If

the available mixing distance is restricted, multiple-point

injection or multiple-point sampling can be used. However,

analytical work and the limited available data indicate that the

inherent asymmetry of the flow may have a strong influence on

the rates of mixing for multiple-point systems.

Relatively little definitive information is available on

initial mixing. Analytical and experimental work on one type of

jet injection is reviewed.

Equipment errors can be analyzed if a sufficient number of

experiments are done in advance of the discharge measurement.

Based on data from the literature, an error analysis is

presented for a situation where a radioactive tracer was used.

The total error in the calculated discharge was determined to be

approximately 0.5% at the 95% confidence level.

Some possible areas for additional research are presented.

1. Introduction

1.1. Basic Concepts

The dilution method, which is also called the method of mixtures, is a tracer technique for discharge measurement. The method is based on the

395
mass balance of a tracer which Is injected into the flow. For a steady

mass rate of injection and for steady flow, the mass balance of a conservative tracer gives

c.q + c^Q cudA = c^CQ+q) (1)

where c, is the injection concentration of the tracer, q is the volumetric

injection rate, c^ is the background concentration, Q is the pipe discharge,

c and u are the concentration and velocity distributions at a downstream

cross section, A is the area of the pipe, and c is the average concentration. From eq. (1)

,

Q

c.-c

c^-

(2)

By knowing the injection conditions (c. and q) and by measuring the background concentration and the concentration downstream of the injector in

order to determine c^, the pipe discharge Q can be calculated using eq. (2)

If c. >> c. >> (2) can be simplified to

Q = ~ q

c^

(3)

which gives the pipe discharge in terms of the dilution of the injected

tracer after thorough mixing with the flow,

defined as

As implied by eq. (1), c is

1_ cudA (4)

with Q = Q + q. If Q » q, then

concentration may also be defined as

Q in eq. (4). An area-average

cdA (5)

1.2. Modus Operandi

There are many possibilities for types and locations of injection and

subsequent measurement of concentration in the pipe, and these in turn

affect the accuracy with which c can be approximated at a specified

396
distance downstream of the injector. The simplest injection is normally a

single injection at the pipe wall. If enough flow length is allowed or is

available for the tracer to become thoroughly mixed with the flow, any

single measured value of c will be representative of c_^ and the accuracy

will depend primarily on the random errors.

In situations where the flow length available for mixing is restricted

so that uniformity of the concentration distribution can not be obtained by

mixing associated with the pipe flow, there are at least two possible

alternatives for evaluating c . One alternative is to accelerate the

mixing. This may be done by injecting the tracer at more than one point at

a particular cross section and/or by using high velocity injection. In

some cases, mixing vanes or other types of mixing devices downstream of the

injection location might also be practical. Essentially any desired degree

of mixing can be obtained with properly designed mixing devices, but

devices which produce a high degree of mixing may also cause an unacceptably

large head loss.

A second alternative is to use multiple-point sampling. In many cases,

this alternative is less expensive than multiple injection or mixing

devices. The ASME Research Committee on Fluid Meters [l] recommended that

thorough mixing should take place between the injection and sampling

locations. This recommendation apparently assumes that only one sampling

port will be used and therefore may be overly restrictive if multiple-point

sampling is used. However, even if sampling points are distributed

throughout the pipe cross section, multiple-point sampling may not be

desirable at very short flow distances where the concentration distribution

is highly non-uniform within a cross section because multiple-point

sampling leads to an evaluation of c^, which may be a poor approximation to

c for highly non-uniform distributions. This distinction is normally

important only near the injection location. See the end of section 2.1.2.

In this paper, attention will be directed primarily toward circular

pipes where both injection and sampling are done at the pipe wall and where

no mixing devices are used. This type of application is often the most

convenient because it requires no access to the interior of the pipe and

sometimes it can be implemented without interrupting flow in the pipe.

1.3. Some Advantages and Disadvantages

Some of the potential advantages of the dilution method are that (a)

it can be used to obtain either intermittent or continuous measurement of Q,

(b) it can^be applied to a range of discharges differing by a factor of as much as 10 using essentially the same equipment [2] , (c) in most cases it

causes negligible head loss, (d) in many cases it does not require

interruption of flow in the pipe, (e) it can be used for either liquids or

gases, (f) it is independent of the size or degree of uniformity of the

pipe, and (g) it does not require full pipe flow. Some of the potential

Figures in brackets indicate the literature references at the end of this

paper.

397
disadvantages are that (a) it requires the addition of a foreign substance

(the tracer) into the flow, (b) the tracer mast be conservative or must

have a well-defined or measurable behavior in the flowing fluid, (c) a significant length of pipe may be required between the injection and

measurement stations, (d) the method has only been investigated for steady

flows, and (e) it would appear to have restricted applicability in

entrance sections (e.g. penstocks) which are too short for complete

development of the turbulent boundary layer. The dilution method apparently could be applied to some types of unsteady flows; however, this

possibility has apparently not been investigated in detail and will not be

considered further in this paper.

1.4. General Accuracy Considerations

The accuracy of the dilution method is affected by several factors.

For example, it is essential that the sampling downstream from the injector

location provide an accurate representation of c . The selection of the

appropriate sampling location(s) depends on being able to analyze and

predict the mixing of the injected tracer. These factors are considered in

section 2. In addition, the accuracy of the method depends on the random

errors associated with the injection equipment, the measurement

instruments and the procedures which are used. These factors are considered in an example in section 3.

2. Analysis of Mixing

Errors can be introduced into the determination of Q if the sampling

does not provide for an accurate evaluation of c . The accuracy with which

c can be obtained with a particular set of sampling points will depend on

the concentration distribution at the sampling cross section. If no mixing

devices are used, the concentration distribution or the degree of mixing at

any cross section can be viewed as being a result of initial mixing

associated with the injection system and ambient mixing associated with the

flow in the pipe.

Some analytical solutions for concentration distributions will be

presented to gain insight into the ambient mixing problem, then some data

related to ambient mixing will be presented. The degree of initial mixing

which can be obtained for various types of injectors is not well defined.

Some analytical and niimerical work and some data which are available for

one particular type of injection will be presented after the discussion of

ambient mixing.

2.1. Ambient Mixing

2.1.1. Analytical Representation

An analytical representation of ambient mixing in an idealized steady,

uniform, turbulent pipe flow can be obtained by assuming that the velocity,

radial diffusivity, and circumferential diffusivity are each constant and

equal to their cross sectional averages which are respectively u, e , and

398

^
Bg. These assumptions are supported by the work of Evans [3] who investigated the effects of variable diffusion coefficients on concentration

distributions, by the agreement of the calculated results with numerical

simulations using variable u, e , and e^ [A], and by the agreement between

the analytical results and the Sata, as discussed in section 2.1.2. The

assumption of a constant velocity implies that c^ = c . Thus, the subscript

will be dropped and c will be used in this section. In obtaining the

analytical solution, let r = radial coordinate, R = pipe radius, D = pipe

diameter, x = longitudinal coordinate, 0 = circumferential coordinate,

c = concentration, f = Darcy-Weisbach friction factor, Sc = turbulent

Schmidt number and ~ = cross sectional average. Define t^e dimensionless

variables

n

p

X

c

= e ,/e

r/R

e x/uR^ 0.0480 7l (x/D)/Sc^ [5]

(6)

r

c/

c

The concentration distribution for a conservative tracer released at a

constant rate from a point source located at X = 0, p = p' , and 6=6'

can then be written [5] for p' 4- 0 and X > 0 as

CO J (A p')j (X p)

C = l+ y W cos n(e-e') y expC-A^ X) ^

^ n nm n=0 m=l 2 „ (1 - \ )J (A ) z V nm

nm

(7)

in which W = 1 for n = 0, W =2 for n > 0, v = n/iT, A = the mth

positive root of J' (A) = 0, and J is a Bessel function of the first kind

of order v. For an undisturbed source at the pipe centerline (p' = 0),

the analogous solution is

0=1+1 expC-A^ X)

° (8)

m=l j2(A ) o om

For multiple sources, the principal of superposition can be used to obtain

the concentration distribution. If there are k sources and the j-th

source is located at p! ,6'. and has a strength or mass injection rate of

M, , then the nomalizei concentration distribution is

399
c =

k

1 M.C.

k

2 = 1'

(9)

with C. being given by eq. (7) for each p'. ,9' or eq. (8) if p ' = 0.

Similarly, if as a result of injection ani iilitial mixing, a Concentration

distribution given by

C = C (p' ,9')

o o (10)

exists at X = X , then the mass flux through each incremental area

(p'dp'd9*) at X can be considered to be a source with a mass injection

rate of u C (p ' , 9

' )p ' dp ' d9 ' . Again, the principal of superposition can be

applied and the concentration resulting from all of the incremental sources

or from eq. (10) can be obtained by integrating over the cross section at

X^ to give

C =

rl-n fl _u C (p',9') C(X-X ,p,e,p' ,9')p'dpM9'

o o

o

27T (11)

u C (p' ,9')p'dpM9'

o

where C(X-X ,p,9,p',e') is the concentration distribution given in eq. (7)

or eq. (8) if p' = 0. In both eq. (9) and eq. (11), the denominator is

included to normalize the calculated concentration so that the average

value will be unity.

For the assumed flow conditions, these analytical expressions can be

used to obtain the concentration distributions, the coefficient of variation (C ) of the concentration distribution, and the maximum variation of

the concentration within a cross section. In order to perform these

calculations, it is necessary to specify the value of n j and in order to

relate the calculations to actual distance in a pipe, it is necessary to

specify the value of Sc . Both n and Sc are defined in eq. (6). Holley

and Ger [5] have shown lhat a good representation of data from various

investigators is obtained for r\ = 1.8 and Sc = 1.0, so these values will

be used in the following computations. All of the results will be

presented in terms of dimensionless variables and parameters. For

reference, X = 1.0 corresponds to x/D = 170 for f = 0.015 and as shown in

eq. (6), x/D for a given X varies inversely with the square root of f. It

was also shown [5] that the value assumed for Sc^ affects the value

calculated for r) when analyzing data on mixing in pipes. Thus, one of

these values can not be changed independent of the other. For example, the

value Sc can not be changed in the definition of X without also changing

the value of n. In all of the calculations, it is assumed as mentioned

previously that both injection and sampling are at the pipe wall.

400
The coefficient of variation is frequently used as a statistical

measure of the degree of mixing and is defined by

C = V

("2 IT

1

o

ni/2

(C-l) pdpde (12)

with C = c/c. Calculated values of C are shown in figure 1 for three injection conditions, namely a single wall source, two sources of equal

strengths spaced 180 apart at X = 0, and three sources of equal strengths

spaced 120 apart. By definition, a mathematical source has no initial

mixing associated with it. Corresponding curves for N, > 4 (where N, is

the number of equally spaced sources at X = 0) give curves of C vs. X

which are essentially the same as for N = 3. Figure 1 shows a large

decrease in the mixing distance required to obtain a given value of C

when the two sources are used instead of just one. However, mixing

distances as short as those indicated in figure 1 for > 2 can not be

obtained in the laboratory without extreme experimental care and certainly

would never be obtained under normal operating conditions. The reason is

that the steeper slope of the C - X curves for N_j_ > 2 is a characteristic

of the analytical evaluation of C for mathematical symmetry with respect

to the centerline of the pipe. Symmetry in this case means symmetry of

both the flow and the concentration distributions as represented by

sjnnmetrical (or unifo^) distributions of velocity and diffusion

coefficients and by ^ M. cos(e-0') identically zero for all 9, where k is

i=l the number of sources as in eq. (9). For k > 2, this expression is

identically zero for any number of equally spaced sources of equal

strengths. This type of mathematical symmetry will never actually exist in

either the velocity distribution or the injection of the tracer into the

pipe. The flatter slope for N_j_ = 1 is a characteristic of the asymptotic

slope for increasing X for all asymmetrical conditions [5].

A qualitative indication of the effect of asymmetry on the rates of

mixing can be obtained from the analytical solution. The nature of the

solution is such that it would be extremely difficult to obtain an

analytical solution analogous to eq. (7) for a variable velocity and

diffusivity. However, using eq. (9), it is rather straightforward to introduce asymmetry into the mixing by varying the strengths of the sources.

Figure 2 shows the results of this type of computation for two sources

where the strengths of the sources differed from the average strength by

+P, with P varying from 0% to 20% as indicated on the figure. The figure

also shows the curve for N = 1 for comparison. It should be emphasized

that variation of source strengths is being used to indicate the general

effect of as3nMnetry which would be present for any of a number of reasons

and not because it is felt that variations of 20% in source strengths are

likely to actually occur. Figure 2 shows that the C - X curves break

away from a common curve at values of X which depend on the amount of

asymmetry (i.e. on P) and that each curve approaches or attains the

asymptotic slope for asymmetrical conditions. It will be seen in figure 5

(section 2.1.2) that this is the type of behavior found in measured

concentration distributions.

401
If the behavior shown in figure 2 does indeed represent the effects of

asymmetry of any type, then this behavior would have definite implications

both on analyzing laboratory data to study mixing rates and on trying to

predict mixing distances for discharge measurements with multiple-point

injection. If the asymmetry of the flow and mixing (rather than the mean

flow characteristics or the injection condition) has a dominant influence

on the C - X behavior, it will be difficult to find consistency in data

collected by different investigators and it will be equally difficult to

predict mixing distances with any degree of assurance.

These comments above refer only to multiple-point injections. As

mentioned again in section 4, research is needed to define the effects of

multiple-point injections and multiple-point sampling on the accuracy of

the dilution method. For a single injector, the inherent asymmetry of the

injection is apparently more dominant than any flow asjrmmetry, at least for

laboratory flows. This matter is considered further in the discussion of

the experimental data in section 2.1.2 (figures 5 and 6).

While C is indicative of the degree of mixing or the uniformity of

the concentration distribution at a cross section, it does not give a

direct indication of the maximum possible error using a specific set of

concentration measurements to approximate c. This maximum error can be

evaluated for the analytical solutions. It will be assumed that samples

are taken from S ports spaced equally around the pipe wall and that Cg is

the average of the S measurements of c. Then, by letting the sampling

ports be in the "worst" position relative to the concentration distribution

so that the difference between c^ and c is a maximum, it is possible to

evaluate AC where

(13)

max

AC is a direct indication of the maximum possible error in calculating c ana Q from concentration measurements obtained at the pipe wall.

Figure 3 shows values calculated for AC. and AC^/C for N, =1 and 2,

1 1 V +

including the effects of asymmetry. The general appearance of the AC^

curves is similar to the corresponding C curves. The characteristic

slopes on figure 2 are the same as those on figure 1. However, the values

of AC^ are larger than the corresponding C values, with the ratio of AC^^

to C being shown at the top of figure 3. ^For X > 0.13, AC^/C varies

between 1.76 and 2.90 depending on the injection conditions an3 on X.

For large X, AC^/C for all asymmetrical conditions approaches 1.76. As

the figure shows, if enough pipe length is available to provide a mixing

distance of X = 1.25 (x/D = 220 for f = 0.015), then any concentration

measured at the pipe wall will be within a maximum of 0.5% of c (except for

random measurement errors) . It is only for situations where the available

mixing distance is limited that multiple-point sampling or multiple-point

injection needs to be considered. As shown in figure 3, the mixing

distance required to obtain a given value of AC can be reduced significantly by using two sources , but the amount of reduction again depends on the

asymmetry of the mixing.
Since the principle of superposition applies to the calculated concentration distributions, it can be shown that AC for = 2 is the same as

AC^ for = 1. Thus, the curves in figure 3 for AC^ for two asjanmetric

sources can also be viewed as indicating the behavior to be expected when

two samples are taken from an asymmetrical flow with one source. This

fact will be useful in the discussion of some of the data in section 2.1.2

and particularly in conjunction with figure 6.

2.1.2. Analysis of Data

Data from three sources as presented in Table 1 were analyzed for

comparison with the material presented in section 2.1.1.

Table 1. Sources of Data on Ambient Mixing

Source

Pipe

Diam- Reynolds

eter Number

inches Range

Pipe

Material Fluid Tracer

Sampling

Points per

Cross Section

Clayton,

Ball, and

S packman

[6]

4 5x10^-

5x10

Smooth Water Radioactive 10

Ger and

Holley

[4,7]

7.7x10 - Galvanized Water Dilute NaCl

1x10 Iron

37

Filmer and

Yevdjevich

[8]

36 5.8x10^-

4.5x10

Steel Water Rhodamin WT 28

Clayton et^al [6] performed experiments at four Reynolds numbers (Re).

For Re = 5 x 10 , 1 x 10 , and 7.7 x 10 , they used four injection conditions, namely a single wall injector (N_^ = 1) , a single center injector,

four wall injectors (N = 4), and four injectors at p' = 0.63. For

Re = 5 X 10 , they used a single edge injector and a single center injector.

A radioactive tracer was used. Concentration measurements were made at

several cross sections on 3 radii spaced 90 apart. The sampling ports

were at p = 0, 0.31, 0.63, and 0.98, giving a total of 102samples. The

radioactivity measurements were analyzed by calculating x (relative to

well-mixed conditions) for each group of ten results from a given cross

section. The x values were apparently obtained by giving the same weight

to each of the 10 samples, while eq. (12) for calculating C would imply

the use of weighting factors proportional to the pipe subarea represented

by each sampling point. Nevertheless, it was assumed that this areaweighting was negligible, as is usually the case except very near the in^

jection location. Then C could be calculate^ directly from the given x values just by comparing definitions of x and C . The individual

concentration measurements were not reported, so it was not possible to

obtain AC for this data.

403
Ggr and Holley [4,7] reported two experiments at Re = 7.7 x 10 and

1 X 10 with one wall source (as well as data on jet injections to be considered in section 2.2). A NaCl tracer was used with concentration

measurements being made at 37 points within each measurement cross section.

The 37 points were distributed to sample the center of 37 approximately

equal subareas of the pipe. values were reported.

Filmer and Yevdjevich [8] performed five experiments with an injection

tube on the pipe centerline. Although these experiments do not conform to

the specified condition of having wall injection, they do provide data

which can be used to gain insight into the ambient mixing process since by

definition ambient mixing is associated with the flow in the pipe and not

with the injection conditions. The Reynolds numbers for the experiments

were 5.8 x 10 , 25 x 10 , 32 x 10 , 39 x 10 , and 45 x 10 , Even though

the injector was on the pipe centerline, the mixing was asymmetrical

primarily because part of the tracer is normally drawn into the wake of the

arm supporting an injection tube. A fluorescent tracer was used. Concentration measurements were made on 4 radii spaced 90 apart. One set of

measurements was made using 12 radii spaced 30 apart, but the authors

concluded that the 4 radii provided sufficient accuracy. There were 7

sampling tubes placed on a rotatable radial arm so that each sampling tube

was at the center of equal subareas of the pipe cross section. Thus, C

in eq. (12) could be calculated directly as the coefficient of variation of

the concentration measurements. These measurements were tabulated by

Filmer and Yevdjevich [8] and provide what is apparently one of the largest

sets of available data for a single injection condition.

The values of C calculated from measured concentration distributions

are influenced both Ey the degree of non-uniformity of the concentration

distributions and by the accuracy with which the individual values of c are

measured. For the smaller values of X, the dominant influence is the concentration distribution within the pipe and the empirical values of C and

AC should be expected to agree closely with the analytical work presented

in section 2.1.1. However, for the larger values of X where the

concentration is relatively uniform within each cross section, empirical C

values do not approach zero; rather they approach an approximately constant

value which is indicative of the random errors in the individual concentration measurements. Similarly, for the smaller distances, the values of AC

for multiple-point sampling are indicative of averaging of unequal

concentrations from non-uniform distributions within the pipe. Therefore

AC^ should be significantly less than AC. . However, as X increases and

AC^ decreases, a point is reached where the magnitude of AC^ relative to

AC^ is indicative simply of sampling statistics. The "critxcal" value of

X at which the values of C and AC cease to represent the mixing and begin

to represent random errors depends on the injection conditions, the number

of samples, and the random errors in the measurement of c. (See figures 4,

5, and 6, which are discussed below.)

Figure 4 shows empirical values of C for N = 1. The solid points

are from the four applicable experiments of Clayton et al [6] and the open

points are from the two experiments of Ger and Holley [4,7]. There is good

agreement between experiments of the different investigators and between

the experiments and the analytical curve for = 1 which is also shown on

figure 4. The results for Clayton et al also indicate the approximately

404
constant value of C for large X. For the radioactive counting used byClayton et al, they ^ndicated a 90% confidence Interval for random errors

in their values of x • The corresponding 90% confidence interval for

random errors in C is 1.7 x 10 < C < 4.6 x 10 . In figure A (and in

figure 5 to be considered later) , the values of G for large X do fall within

this confidence interval. The dashed lines which are shown in figures 4

and 5 to separate the regions of the influence of mixing_in the pipe from

the regions of random errors correspond to C = 4.6 x 10

V

Clayton et al [6] also reported three experiments with four equally

spaced wall injectors (N_^ = 4). The C values for these experiments are

shown in figure 5 which also includes some calculated curves analogous to

those shown in figure 2 for N, =2. The curve for N, = 1 is also shown for

comparison. The values of P shown in figure 5 indicate the range of

injector strengths relative to the mean. Around the pipe circumference

the injector strengths were equal to the mean value plus P, plus P/2,

minus P/2, and minus P. There is general agreement between the calculated

curves and the data, especially if it is assumed that the degree of

asymmetry in the flow may have not been the same for all of the experiments.

It must be emphasized that the comparison shown in figure 5 is not intended

to imply that the strength of the injectors actually varied by 10% or 20%.

Rather, the major part of the asymmetry undoubtedly was inherent in the

flow, but as mentioned earlier it is easier to obtain asymmetry of mixing

in the analytical solution by varying the source strengths than by varying

the flow characteristics. In figure 5, as in figure 4, the C values for

large X fall within the previously given confidence interval ¥or random

errors.

Figure 6 shows several types of information from Filmer and

Yevdjevich's [8] experiments with a (disturbed) centerline injector.

Specifically, the figure shows C^ values and AC^, AC^ , and AC^. The AC

values are based on the four concentration measurements obtained nearest

the wall (p = 0^96) . AC^ is the one of the four values with the greatest

deviation from c. AC^ is based on the "worse" one of the two averages

of the diametrically opposed samples, and AC^ is the average of the four

values. First, consider the values of C , as indicated by the open

symbols. There is good agreement between the five sets of data as long as the C values are large enough not to be significantly influenced by

random errors. (The dividing line of C = 0.035 between the effects of

mixing and random errors was obtained by a visual injection of the data in

figure 6.) The line drawn through the data points for C is the visual

best-fit line having a slope corresponding to the asyinptotic slope for

asymmetric conditions shown in the previous figures.

The solid points above the C line are for AC^. The line drawn

through these points is based on Ihe line drawn for C and corresponds to

the previously cited value of AC-./C = 1.8. Having drawn these two lines,

J it was possible to see that the data were the same as if a point source

injection had been used at a virtual origin of X = - 0.08. Assuming this

condition, it was possible to use figure 3 and the equivalence between AC^

I for N_j_ = 2 and AC^ for = 1 to draw the curve for AC2 on figure 6. The

value of P = 5% was chosen as the one providing the best envelop curve for

the AC^ values in the mixing region on the figure. Considering that only

405
four values were available at each cross section for evaluating AC and AC

the data show excellent agreement with the characteristics obtained from ^ the analytical solution.

The values of AC^ at a given X are naturally smaller than the

corresponding values of AC , but a mixing distance based on AC. would not

be very much shorter than that based on AC^. Thus, the installation of

more than two ports for sampling might not be justified, but this statement

is based on very limited data and needs more study, as mentioned in section

4. Certainly, both the_data and the analytical work indicate that the

accuracy of evaluating c based on the mixing characteristics does not continue to increase as the niimber of sampling ports increase. Thus, the use

of 10 sampling ports at the wall does not give any better average of the

mixing characteristics than say 3 wall ports. Of course, multiple samples

ftom the 3 ports would reduce the random errors.

The data of Filmer and Yevdjevich [8] were also used to compare values

of c^ and c by assuming a power law velocity distribution and using eqs.

(4) and (5). For X > 0.3, it was found that there was less than 0.1%

difference in the two values of c.

2.2. Initial Mixing

For any number of injection ports and any number of sampling ports,

the mixing distance required to obtain a specified C or AC value can be

reduced compared to those considered above by using initial mixing. If

attention continues to be restricted to injection at the pipe wall with no

mixing devices mounted inside the pipe, then initial mixing can still be

achieved by having jet injection instead of the source-type injection

considered throughout section 2.1.

With jet injection, there are a number of parameters which can be

varied to control the amount of initial mixing. Among the significant

parameters are the number of jets, the angle of injection relative to the

flow in the pipe, and the initial jet diameter and velocity relative to

the pipe diameter and flow velocity in the pipe. There are some

indications [4,7,9] that the diameters and velocities can all be combined

and replaced by the ratio (M) of the initial momentum in the jet to the

momentum in the pipe flow.

There are many publications on topics such as jets in cross flows and

other similar topics which have some relation to jet injections into pipes.

However, there are relative few publications that directly address the

topic of the mixing of a tracer with a jet injection at the pipe wall. Ger

and Holley [4,7] performed both analytical and experimental studies of a

single jet injected perpendicular to the pipe wall. The analytical work

led to a numerical model to simulate both the initial jet mixing and the

subsequent ambient mixing. It was found that there was an optimum

momentum ratio (M) which produced the shortest mixing distance. The

optimum corresponds to the penetration of the bent-over jet to approximately

the pipe centerline, as shown in figure 7. For smaller M, the jet does

not reach the centerline and a larger mixing distance results. For | larger M, there is an overpenetration with an associated increase in the

406
mixing distance. The experiments indicated that the optimum value of M is

approximately 0.016. The two experiments conducted with this value of M

and the corresponding results of the numerical model are shown in figure 8.

(After completion of the previous publications [4,7], a slight error was

found in the value of n which was used in the numerical model. As a result,

the curve shown in figure 8 is slightly different from the similar curves

shown in the previous publications.) Figure 8 shows excellent agreement

between the calculations and the data and also shows a significant reduction

in the mixing distance compared to that for a single source as shown in

figure 1. However, if mixing distances on the order of that shown in

figure 8 are needed, in some cases it might be more practical to use

multiple-point, low-velocity or source injection (figure 2) or multiplepoint sampling (figure 6). If even shorter mixing distances are required,

it still remains to be determined how much initial mixing can be obtained

by using multiple jets. This topic is currently being investigated by the

author.

3. Random Errors

In the previous section, it was shown that some general conclusions

could be obtained about the magnitude of possible errors resulting from

the selection of a particular set of sampling points to approximate c^.

After a high degree of mixing has been obtained in the pipe so that no

errors are expected because of inappropriate sampling locations, random

errors will still be present. It is usually not possible to draw general

conclusions about the magnitude of random errors since these errors depend

on the particular tracer, procedures, equipment and instrumentation being

used. Therefore, in this section, an example is presented to demonstrate

the evaluation of random errors when using a particular tracer, namely a

radioactive tracer, and to show that the errors can be kept within

generally acceptable limits if sufficient care is used. The example is

based on measurements reported by Clayton and Evans [2] , who also presented an error analysis and discussed a number of other aspects of radioactive tracer techniques. The present error analysis has the same general

purpose as that of Clayton and Evans and therefore has some similarities to

their analysis. However, most of the presentation and discussion given

here is different from that of Clayton and Evans since a different approach

is used.

3.1. Experimental Procedure

Clayton and Evans [2] used a^radioactive tracer (sodium. 24) to measure

a discharge of approximately 11 m /s (388 cfs) of water. A unidirectional

piston pump which was driven by a synchronous motor was used to inject the

tracer into a secondary injection circuit which withdrew water from the

pipe and then flowed back into the pipe through a single port. The

volumetric flow rate (q) from the Injection pump was determined from

previous calibrations. Downstream of the distance required for mixing,

nine samples from the pipe (after a steady state concentration distribution

developed) were used for evaluation of c . Also, five diluted samples of

the injection solution were used to determine c. . All of the instruments

were calibrated before and after (and sometimes during) the testing to

407
eliminate systematic errors. The background concentrations were determined.

The densities of the injection solution of the water in the pipe and of the

samples were also measured.

3.2. Error Analysis

The purpose of this error analysis is to evaluate the confidence

interval for the calculated mean discharge Q. In doing this type of

analysis for an expression such as eq. (2) , normally some assumptions must

be made. The assumptions which are made in this analysis are different

from the assumptions made or implied by Clayton and Evans [2] in their error

analysis; this difference is the reason that this analysis is different from

theirs. In their analysis, Clayton and Evans apparently used several

approximations and methods that had been presented in earlier reports [10,

ll] which they cited, but the author did not have access to those reports

while preparing this paper.

The present error analysis generally follows the method presented by

Ang and Tang [12] and is based on standard techniques for the analysis of

propagation of errors when the mean value (indicated by an overbar) of x

is a function of the mean values of other variables such that

X = (fi (u, v,w, . . . ) (14)

Assuming that the errors in u,v,w,... are small relative to the mean values

and are independent of each other, then

_ - 2 - 9x ^ - 9x

^ var(x) = var(u)(—) + var(v) (—) + var(w) (-rrr) + ... (15) dU dV dW

where var( ) is the variance of the variable indicated. The square root

of the variance is the standard deviation which will be indicated as s( ).

If the variable is a mean value (indicated by an overbar) then s( ) is the

standard deviation of the mean or the standard error. If the function (j) in eq. (14) involves only products and quotients (no sums, exponentials,

etc.), then dividing eq. (15) by the square of eq. (14) gives —var(x) var(u) , var(v) , var(w) , :2~ = ~T~"^~^2~'^~T~"^ ••• ^^^^ X u V w

The standard error for any of the variables (for example, u) is

s(u) = ^(17)

where s(u) is the estimated or known standard deviation of the sample and

n is the sample size. If the standard deviations are known or can be

evaluated for samples of u,v,w,..., then the standard error for x can be

calculated from eq. (15).

408
The analysis of the propagation of errors will be applied to eq. (2)

for Q written in the fom

_ c,

Q = q~ (18)

d

where it has been assumed that << c, and d is defined as

Q 1

d = c_ - c (19)

Q o

The overbar on c previously indicated an integration over the pipe cross

section as shown^in eq. (4). However, throughout section 3, the overbar on

c indicates the mean value of the measured values used to evaluate the

average concentration in the pipe. From eqs . (14) and (18), the standard

error in Q can be expressed as

var(Q) var(q) var(c.) var(d)

-rr-

- -TT- +

-rr- + (20)

Q q d

Each of the three terms on the right-hand side of eq. (20) will be_evaluated and the results combined to determine the standard error in Q.

Throughout the evaluation of the errors, the standard errors will be divided

by the mean values and presented as percentages of the mean values.

One of the major assumptions in the analysis will come in the specification of a confidence interval for Q from the standard error in Q. If the

distribution of Q were known, then the relation between s(Q) and a confidence interval would be known or could be calculated. Knowledge of the

distributions of all of the variables included explicitly or implicitly on

the right-hand side of eq. (18) does not give a direct indication of the

distribution of Q, although it would allow the determination of the

distribution of Q (for example, by a Monte Carlo analysis). In the present

error analysis, it will simply be assumed that Q is normally distributed.

Then the confidence interval for the calculated value of Q is As(Q) where

X for the 95% (two-sided) confidence interval is 1.96 and for the

99% confidence interval is 2.58.

3.2.1. Standard Error in q

The pump was calibrated to determine q^^ at a line frequency (in the

United Kingdom) of 50 Hz. In order to correct for variations in line

frequency (f) , the actual discharge was given by

^ = 50 ^50 ^21)

Application of eq. (16) gives

409
var(q) var(f) var(q )

-ZT- ' + -TI^ (22)

3 From the calibration of the pump, it was known that = 2.0996 cm /s and

s(q^Q)/q^Q = 0.05%. It was determined that f = 49.96 Hz during the discharge measurement, and the electric utility company quoted s(f)/f

= 0.04%. Substituting these values into eq. (22) and taking the square

root gives

•4^ = 0.06% (23)

q

3.2.2. Standard Error in d

Each of the concentrations in d was determined from counting the

radioactive emissions from collected samples over a given time interval.

In order to compensate for the radioactive decay of the samples being used

to determine c , an adjustment factor was applied to determine the equivalent count rate at a base time. The adjustment factor involves integration

of the exponential decay term over the counting period. However, in order

to provide some simplification in the presentation of the error analysis, it

will be assumed following Clayton and Evans that just for the error analysis

the adjustment factor can be represented by a single exponential decay term

based on the mean time of counting the samples. This assumption seems

justified since the error in the adjustment factor does not make a large

contribution to the total error.

If K is the equivalent count rate at the base time, and k is the

measured count rate, then

k = K exp(-a = ) (24)

h

where a = 0.693, t is mean time measured from the base time and Tj is the

half-life. The concentration is

- - ^1 - t ^1

c = K 3^ = k exp(a | ) J=- (25)

P2 \ '2

where is the density of the water under the conditions (temperature and

pressure) for which the concentration is desired (e.g. in the pipe) and

is the density of the water during counting. Equation (25) applies to

both c and c^ (as well as to c^ to be considered later, when p^ must be

the density in the injection pump) . It will be assumed that the relative

standard errors were the same in all density measurements. The use of

eq. (19) for the definition of d, eq. (15) for the standard error and eq.

(25) for both c^ and c with subscripts on K and k corresponding to the

subscripts on c gives

410
var(d) var(k^)

+ a

var(Ti ) var(p) ^+ 2 -2

P

(26)

o var(K ) o_

o

+ 2

var(p)

P

-2 - - after dividing by d_ and cancelling the p^/p2 terms since the densities

multiplying K and K are equal in this case. In eq. (26), the first set

of bracketed terms comes from c^. The second set of terms comes

from c . It has been assumed that there is no error in the measurement of

o time and that the background measurement is not subject to a decay

adjustment.

The following data were given by Clayton and Evans

:

= 80488 counts (average of nine 4-min unadjusted counts)

= 20150 counts/min (adjusted for decay)

= 25.8 min.

K = 396 counts/min (average of two 10-min counts)

0.1%

)

T, = 15.0 hr = 900 min

o

s(p)

s(V-1 = 0 . 1 hr = 6 min

p^ = 1.002 g/cm (for both c^ and c^)

p„ = 0.9982 g/cm^ (for both c^ and c ) 2 Q o Based on the Poisson distribution for counting radioactive emissions, the

dimensionless standard deviation of the count rate for determining c^ is

fi!^, 100% = 0.35%

Eq. (17) for n = 9 gives the standard error as

(27)

s(k ) _ ^ = 0.12% (28)

411
In conjunction with the sampling to obtain c , one background count

was obtained immediately before taking the samples and another one about

k\ hours later. The actual counting of the samples to determine k took

about one hour. The two background count rates differed by about j% over

the 4^ hour period. In the calculation of Q, K was assumed to be constant.

In conjunction with taking water from the pipe ?or diluting the samples of

the injected solution (see section 3.2.3), five 16.67-minute background

counts were taken over a period of 11 hours. For these samples, s(K ) /K

= 2.49%. An F test showed at a confidence level greater than 99% that ° this value of s(K ) was not due just to counting statistics and therefore

that the background concentration was varying. The use of s(K ) /K

= 2.49% and eq. (17) for n = 5 gives ° °

s(K ) = 1.11% (29)

Because of the 11-hour time interval and the tendency for background concentrations to vary slowly in many cases, the value in eq. (29) do doubt

over-estimates the error in during the actual one-hour sampling period.

Substitution of the appropriate values into eq. (26) and taking the

square root gives

= 0.19% (30)

d

Even with the apparent over-estimation of the error in_the background

count, the error associated with the determination of c_ accounts for

of the value given in eq. (30)

.

3.2.3. Standard Error in c.1

The injection concentration was determined from counting five dilutions of the injection solution. The dilution was done to reduce the

concentration so that the counting could be done on the same equipment used

to count the samples from the pipe. If D is the dilution ratio, then

c. = K. tr^ = K D =^ (31)

where K. is the count rate of the undiluted injection solution, K^^ is the

count rate of the diluted solution and and are as defined for eq.

(25). It was not practical to dilute all of thg samples by thg same ratio.

The five dilution ratios varied from 2.676 x 10 to 4.265 x 10 . The

dilutions were counted for 16.67 min with the adjusted count rates varying

from 29392.3 to 46847.5 counts /min. The products of and D for the five

samples were approximately equal (as they should be) wxth the mean value of

being

K. = 1.25206 X lO"*""*" counts/min (32)

1

412
The standard deviation of the 5 values of K^D or K. was

D 1

s(K.)

1

K. = 0.13% (33)

From eq. (17) with n = 5,

s(K.)

K

= 0.06% (34)

From eqs. (16) and (31)

var(c^)

c.X

var(K^)

1

+ 2

var(p)

=2

P

(35)

As previously, i^ is assumed that both p values (p^ = 1.016 g/cm and

P2 = 0.9982 g/cm ) were subject to the same relative error as given in

section 3.2.2. Substitution of the appropriate values in eq. (35) and

taking the square root yields

s(c.)

= 0.15% (36)

3.2.4. Confidence Interval

From eq. ^18) and the experimental values given in the above sections,

the discharge Q was calculated to be 11.02 m /s (389.2 cfs). Eqs. (20),

(23) , (30) , and (36) give the standard error in Q as

^= 0.25%

Q

(37)

Defining y as the true mean discharge and utilizing the previously stated

assumption that Q is normally distributed, it can be estimated from eq.

(37) and the previously cited values of A that

- 1 ^ 0.49% at 95% confidence level (38)

^-1 ^ 0.64% at 99% confidence level (39)

413
It should be noted that the magnitude of error represented in eqs. (38)

and (39) normally can not be obtained by casual application of the experimental procedures and techniques presented by Clayton and Evans. Nevertheless, they indicated that this accuracy can be obtained routinely through

well planned and carefully executed measurements with radioactive tracers.

Any errors associated with possible unsteady flow in the secondary

injection circuit (section 3.1) were not included in the error analysis

since the effects of this unsteadiness on the concentrations within the pipe

will normally be damped out in the pipe "if the transit time of the tracer

in the secondary circuit is short compared with the transit time" in the

main pipe [2]. In addition, if any of the errors in q or c. produce an

unsteadiness in the mass injection rate, these may be damped out during

flow in the pipe. Thomann [13,14] has presented an analytical method

based on the Fourier frequency response function to allow a quantitative

evaluation of the damping of temporal variations in rates of injection into

fluid flows. Although his analysis was presented in terms of open channel

flows, it is equally applicable to pipe flow with the proper selection of

parameters

.

4. Possible Research Areas

The potential and accuracy of the dilution method for discharge

measurement can be increased by additional research in several areas. A

few such areas for possible further work are indicated by the following

comments

:

1) While the available data for steady uniform flow provide some indication of the advantages of multiple-point sampling, the results are sparce

and research is needed in this area since multiple-point sampling can

greatly reduce the flow distance required to achieve a given accuracy and

thereby increase the potential applicability of the method.

2) Data are available on mixing in uniform flows (with single-point injections) , but study is needed on the effects of non-uniformities (elbows,

contractions, valves, swirl in the flow, etc.) and on the effects of the

associated flow asymmetry on mixing. This is especially needed for

multiple-point injection and sampling systems.

3) There are some unpublished results indicating that pumps may provide

essentially complete mixing even for tracers injected at the suction port.

However, there needs to be a definitive investigation of the mixing

efficiency of various types and sizes of pumps.

4) It is apparent that high velocity injection even at the pipe wall can

increase the initial mixing and thereby reduce the required mixing distance

as compared to a low velocity or source-type injection. Research is needed

on various types of high velocity injection in order to evaluate the

amount of initial mixing which can be obtained and to define the orientation and other characteristics of the injection which will provide the

highest degree of mixing under a given set of conditions. The benefits of

injection systems with high-initial mixing need to be compared with the

benefits of multiple-point sampling. This comparison needs to consider the

414
power requirements of the injection system and the possibility of the injection system causing a head loss in the pipe.

5) Both analytical and experimental work is needed to evaluate the

potential applicability of the dilution method to unsteady, full-pipe flows.

6) The dilution method is applicable to determining the discharge for steady,

partially-full pipe flows in sewers [15]. The characteristics of various types

of tracers when used in both storm and sanitary sewers needs to be established.

Information similar to that for full pipe flow is needed on rates of mixing,

effects of multiple-point injection and sampling, and means of accelerating

mixing when the sewer length between manholes is not sufficient to obtain the

desired degree of mixing. Applicability of the dilution method to unsteady,

partially-full pipe flow as exists in sewers needs to be investigated.

7) For other situations besides sewer flow, the stability, reliability,

accuracy and potential application of various tracers need to be established.

8) Similarly, the random errors for various types of equipment and instruments need to be evaluated.

9) The available information. Including information on cost and power

requirements for injection systems needs to be made available in readily

usable form.

10) Standards for use of the dilution method need to be developed based on

recent and on-going research results in order to recognize and utilize the

potential of the method.

5. Conclusions

Errors in the use of the dilution method to measure discharge in a

pipe may come from the use of inappropriate sampling points to determine

the average concentration or from random errors in the equipment, instruments, and procedures being used.

It is customary to represent the completeness of mixing by using the

coefficient of variation (C ) of the concentration distribution or by

using some other closely reYated parameter. However, C does not give a

direct indication of the accuracy with which a given set of measured

concentrations will represent the average concentration. AC was introduced

for this purpose, where AC was defined as the maximum difference between

the average of a set of measured values and the true average concentration

(c), with the difference being normalized with respect to c. For a single

injection port at the pipe wall and a single measurement port at the pipe

wall, it is possible to predict the mixing distance which must be allowed

in order to obtain a specified value of AC. For example, for distances

greater than about 220 pipe diameters, any measured concentration is within

a maximum of 0.5% of the average concentration (except for random measurement errors). When the available mixing distance is restricted, the use of

either multiple-point injection or multiple-point sampling can reduce the

415
required distance by as much as 70%, but the amount of reduction apparently

depends on the amount of asymmetry in the flow and is therefore difficult

to predict accurately.

The use of a jet injection perpendicular to the pipe wall can provide

initial mixing and significantly reduce the required mixing distance.

However, the amount of reduction depends on the ratio of the initial

momentum of the injected jet to the momentum of the pipe flow. For the

optimum momentum ratio of about 0.016, the mixing distance with a jet injection can be reduced by about 35% compared to a single wall source.

Study is needed to define the initial mixing that can be obtained with

other types of high velocity injections.

Based on data from the literature, it was shown that at the 95% confidence level a discharge calculated using the dilution method with a radioactive

tracer was within 0.5% of the actual discharge when the concentration measurements were made after a sufficient mixing distance.

Several areas for possible additional research were presented.

6. References

[l] ASME Research Committee on Fluid Meters (H.S. Bean, ed.). Fluid

Meters - Their Theory and Application^ ASME, 6th ed. (1971).

[2] Clayton, C. G. and Evans, G. V., The cons tant-rate-lnjectlon and

velocity methods of flow measurement for testing hydraulic machines.

Report AERE-R-5872, Wantage Research Laboratory, Berkshire, U.K.,

50 p. (1968).

[3] Evans, G. V., A study of diffusion in turbulent pipe flow, J. Basis

Engrg. ASME^ 89D . 624-632 (1967).

[4] Ger, A. M. and Holley, E. R. , Turbulent jets in crossing pipe flow,

Hydr. Engrg. Ser. 30, Univ. of 111., 197 p. (1974).

[5] Holley, E. R. and Ger, A. M. , Circumferential mass dlffuslvity in

pipes, J. Hydr. Div. ASCE^ in preparation.

[6] Clayton, C. G. , Ball, A. M. , and Spackman, R. , Dispersion and mixing

during turbulent flow of water in a circular pipe. Report

AERE-R-5569, Wantage Research Laboratory, Berkshire, U.K., 31 p.

(1968)

.

[7] Ger, A. M. and Holley, E. R. , Comparison of Single-Point Injections

in Pipe Flow, J. Hydr. Div. Am. Soo. Civil Engr. 102 , 731 (1976).

[8] Filmer, R. W. and Yevdjevich, V. M. , The use of tracers in making

accurate discharge measurements in pipelines. Report CER66RWF-VMY 38,

Colorado State Univ., Fort Collins, 90 p. (1966). See also:

Experimental results of dye diffusion in large pipelines, Pros.

Twelfth CongressJ Intern. Assn. Hydr. Res., Fort Collins, _4, 115

(1967).

416
Beltaos, S. and Rajaratnam, N. , Circular turbulent jet in an opposing

infinite stream, Dept. of Civ. Engrg. , Univ. of Alberta, Canada, 18

p. (1973).

Clayton, C. G. , Spackman, R. , and Ball, A. M. , The accuracy and

precision of flow measurements by radioactive isotopes, Symp. on

Radioaot-tve Tracers in Industry and Geophysics 3 IAEA, Prague (1967) .

Smith, D. B. , Weam, P. L., and Parsons, T. V., Accuracy of OpenChannel Flow Measurement using Radioactive Tracers, Report AERE-R-5676,

Wantage Research Laboratory, Berkshire, U.K. (1968).

Ang, A. H-S. and Tang, W. H. , Probability Concepts in Engineering

Planning and Design^ (John Wiley and Sons, Inc., New York, 1975).

Thomann, R. V. , Systems Analysis and Water Quality Management,

(Environmental Research and Applications, Inc., New York, 1972).

Thomann, R. V., Effect of longitudinal dispersion on dynamic water

quality response of streams and rivers, Water Resources Research ^ 9_,

355 (1973).

Shelley, P. E. and Kirkpatrick, G. A., Sewer flow measurement - A

state-of-the-art assessment, Report EPA-600/2-75-027 , EPA Muncipal

Environmental Research Laboratory, Cincinnati, Ohio (1975).

417
418
419
LTV

420
ZERO

(wall

source)

small optimum

(underpenetration)

LARGE

(overpenetration)

Figure 7. - Penetration of Jet Perpendicular to Pipe Wall

Figure 8, - Coefficient of Variation for Optimum Jet Injection

Perpendicular to Pipe Wall

421

National Bureau of Standards Special Publication 484, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

APPLICATION OF A FLUORESCENCE TECHNIQUE TO DYE-CONCENTRATION

MEASUREMENTS IN A TURBULENT JET

Hsien-Ta Liu, Jung-Tai Lin, Donald P. Delisi, and Frank A. Robben

Flow Research Company

A Division of Flow Industries, Inc.

Kent, Washington 98031

A laser dye-fluorescence technique for measuring the concentration of Rhodamine-6G dye in a turbulent water jet is

presented. The measurement device consists of a 1-watt argon

laser, which was used to excite the fluorescent dye, and an

optics assembly} which was used to detect the radiated energy

from the dye.

The dye-fluorescence technique is a remote-sensing method,

which does not interfere with the flow field. Thus, a unique

feature of this technique is that it provides the ability to

conduct simultaneous measurements at many positions, which are

as close as desired, parallel to the jet axis. Measurements

show that the detecting device has an extremely high frequency

response and that the sensing volume can be as small as 0.1 mm . This technique can measure a wide range of concentrations with

ratios larger than lO'*. In the concentration range from 10 ^

to 10 ^ g/ii, the calibration of the photomultiplier tube output

to the dye concentration is linear.

Measurements of the mean and fluctuating dye concentrations

in a round, turbulent water jet are reported. The Reynolds

number of the jet was approximately 1600. The jet was tripped

inside the nozzle to ensure a turbulent jet. Results show that

the mean and rms concentrations on the jet axis decrease with the

distance from the nozzle as x ^

. The profile of the axial

mean concentration agrees very well with the profiles other investigators have measured in an air jet traced by oil smoke. The

axial rms concentratiqn, however, tends to have a relatively

high value in the case reported here. Spectra of the concentration fluctuations are also discussed.

Simultaneous measurements of mean and fluctuating concentrations parallel to the jet axis were conducted to demonstrate the

unique features of this technique. The results for tests made

with two sensors focused at the same point show that the two

sensors indeed measured the same signal. Both on- and off-axis

measurements were conducted at x/D = 30 , with three sensors

separated by 0.3 and 0.7 cm. Cross-correlations and the convective velocity estimated from the cross-correlation are

presented.

Key Words: Auto-correlation; convective velocity; crosscorrelation; exit concentration; exit velocity; flow visualization; fluorescence; fluorescent dye; laser; laser dye-fluorescence

technique; mean concentration; mean velocity; measuring volume;

remote sensing; root-mean-square concentration; simultaneous

measurements; Schmidt number; spectra; water jet.

423
1. Introduction

A laser dye-fluorescence (LDF) technique for measuring concentration

fluctuations of dye in liquids has been developed [l].* The measurement

device uses a 1-watt argon laser as the light source to excite fluorescent dye, which is the tracer. An optical assembly detects the radiated

energy or fluorescence from the dye. The LDF technique is remote-sensing

and, therefore, does not interfere with the flow field. The optical

detector has an extremely high frequency response, which is limited only_

by the electronic circuitry. In the concentration range from 10 ^ to 10

g/£, the calibration curve of the detector output to the dye concentration

is linear [ij . In our investigation, we applied the LDF technique to

measure mean and turbulent dye concentrations in a turbulent water jet.

Previous investigators have used various techniques to measure concentrations of scalar quantities. For example, Corrsin and Uberoi [2 J used

thin-wire probes to measure the temperature in a heated air jet. Kiser [s]

used conductivity cells to measure the salinity concentration in a water

jet. Rosensweig, et al. [a] was the first to use a light-scatter technique

to measure the material transport of oil smoke in a turbulent air jet.

The light-scatter technique was further developed by Becker, et al. [s]

,

Liu and Karaki [6], Yang and Meroney [?] , and Gad-el-Hak, et al. [s] for

measuring concentrations in various turbulent flows.

Using the principle of light absorption, Lee and Brodkey [9] developed

a light probe to study turbulent mixing in a pipe flow, where water was

the working fluid. Nye and Brodkey [lOj refined the light probe to improve

its spatial resolution and frequency response.

In this paper we first describe the experimental setup and the LDF

technique in section 2. In section 3 we present the results of our experiments and compare them with results obtained with other techniques. Finally,

concluding remarks are given in section 4.

2. Experimental Setup

Figure 1 is a schematic of the experimental setup. The system consists of a water jet with a diluted solution of a fluorescent dye as the

effluent, an argon- ion laser beam, and a photomultiplier detector assembly.

The photomultiplier detector, which is focused onto a small volume of the

laser beam, measures the dye fluorescence excited by the laser. Since

the intensity of the fluorescence is proportional to the dye concentration,

we can measure the dye concentration accordingly. Figures la and lb show

the orientation of the laser beam for measurements taken perpendicular

and parallel to the jet axis.

2.1 The Water Jet

For the experiments, a submerged water jet was injected through a

0.32-cm-I.D. stainless-steel tube into a water tank. The tank was 7.5 m

long, 20 cm wide, and 80 cm deep and had glass side walls to permit flow

visualization. The effluent of the water jet was a dilute solution of

Rhodamine-6G dye, manufactured by Kodak, and its concentration was 30

*Figures in brackets indicate the literature references at the end of this

paper.

424
llg/i. The exit velocity of the effluent was 51 cm/s, and the corresponding

Reynolds number was 1620. To ensure a turbulent jet, we tripped the water

jet with a small nozzle, 0.8 mm in diameter, installed about 2 cm upstream

of the jet exit. The effluent was stored in a chamber in which the pressure

was regulated. The flow rate was regulated to within 2-3% with a needle

valve and was measured with a f lowmeter-valve assembly (Gilmont Model F7360)

.

For simultaneous measurements perpendicular to the jet axis, the jet

was oriented downward (fig. la). To reduce reentrainment into the jet,

we added Nacl solution to the effluen_t to increase Its specific gravity to

1.001. The resulting Froude number (I/Itx Ap/p D) was 17,000, where

g is the acceleration of gravity, Ap is the density difference between

the effluent and the ambient fluid, p is the density of the ambient

fluid, and D is the diameter of the jet. The buoyancy of the effluent

was negligible compared to the inertia of the jet at such a high Froude

number

.

For simultaneous measurements parallel to the jet axis, the water jet

was oriented horizontally (fig. lb). In this case, the effluent was

neutrally buoyant, and we closely monitored the ambient dye concentration.

The water in the tank was drained and refilled after every few measurements.

2.2 The Laser

The laser used in the experiment was a 1-watt argon-ion laser manufactured by Lexel (Model 85) . The spectrum of the laser light was composed of two major lines at wavelengths 514.5 and 488 nm, and several

other minor lines.

In the experiment, the laser was operated in a single-line mode (488

nm at 500 mW) with the use of a prism wavelength selector. The laser beam

was projected horizontally through the glass side walls and onto a mirror

inside the tank. The laser beam and the water jet were aligned by adjusting

the angle between the incident and reflected beams on the mirror. An

achromatic lens (approximately 0.6-m focal length) was used to reduce the

diameter of the laser beam at the measuring points to about 1 mm.

2.3 The Photomultiplier Detector

The photomultiplier detector assembly consists of an achromatic

lens (48-mm effective aperture and 193-mm focal length), a filter, a pinhole, and an RCA 931A photomultiplier. For the experiments the lens was

mounted in one end of an aluminum tube. The pinhole, which had a diameter

of 0.6 mm, was mounted in the other end, and it was imaged on the laser

beam. Therefore, the measuring volume was formed by a cylinder with a

diameter of 1 ram, which crossed the 1-mm-diameter laser beam at a right

angle. The geometry gave an effective measuring volume of about 1 mm .

2.4 Optical Characteristics of the System

Upon excitement by the laser light, the molecules of the Rhodamine6G dye absorb photons of radiation (peaked at 530 nm) , and, subsequently,

some of the energy absorbed is reradiated (peaked at 560 nm) . Figure 2a

gives the absorption and stimulated emission characteristics of Rhodamine6G, as published by Kodak. A Corning glass filter, stock no. CS 3-67,

is installed in the detector assembly to block the laser light scattered

425
by particulates or air bubbles in the water jet. The filter transmits

very little light with wavelengths shorter than 550 nm, and the RCA 931A

photomultiplier , which has an S4 response, determines the high end cutoff

(approximately 600 nm) . Figure 2b shows the window that results from

this system. We also measured the attenuation of the laser along the path

of the Rhodamine-6G solution. For a concentration of 30 yg/£, the attenuation was as low as 0.1%/cm.

2.5 Calibration

For calibration, we prepared dye solutions of known concentrations

in 15-mJl pyrex beakers. The beakers were positioned, one at a time, at

the measuring point in the tank, and the fluorescence intensity was measured

for various dye concentrations. The output from the photomultiplier was

first amplified and fed into a Nova 800 minicomputer (see section 2.6).

Figure 3 shows a typical calibration curve, which is linear for dye concentrations, between 10 ^ and 10 ^ gm/£.

2.6 Data Acquisition and Analysis System

The outputs of the photomultiplier detectors were recorded with a

Nova 800 minicomputer (Data General) . This system can handle multiple

channels (up to 32 channels) for direct on-line data acquisition and processing. For the experiment each channel was equipped with a signal conditioner, which has the functions of D.C. offset, differential amplification, and low-pass filtering.

In our experiment, we chose a cutoff frequency of 400 Hz. The filtered

signals were digitized with an Analogic A/D converter at a sampling rate of

800 Hz for each channel. The signal was first recorded on an lOMEC magnetic

disk and then stored on digital tapes for future processing. A set of

computer programs is available for calculating statistics, such as mean

and rms values, auto- and cross-correlations, skewness and flatness factors,

and auto- and cross-spectra.

3. Experimental Results

In this section, we present the results of flow visualizations and

concentration measurements made in a turbulent, round water jet and compare our results with those obtained by other investigators with other

techniques

.

3.1 Flow Visualization

The experimental setup described in section 2 was slightly modified

to permit flow visualization of the turbulent water jet. With a glass rod

2 mm in diameter placed in frort of the laser beam, the beam spreads into

a sheet of light. We used the sheet of light to observe the interior

structure of the jet. Figure 4 shows two cross-sectional views of the jet,

one along and the other perpendicular to the jet axis. These two-dimensional

cuts reveal the turbulent structure of the jet, such as the eddy sizes and

the entrainment through the jet boundary. These results also provide useful information for designing quantitative measurements.

426
3.2 Mean and Root-Mean-Square Concentrations

Figures 5 and 6 are, respectively, the profiles of mean and root-meansquare (rms) concentrations measured at four stations, x/D = 25, 50, 100,

and 160 downstream of the jet exit. The measurements were made with three

phot omul tiplier detectors, separated by 0.3 and 0.7 cm. At each station,

we made six to ten sets of measurements, with a total of about 20 or 30

data points. Figures 5 and 6 show that the maximum mean concentration

occurs on the jet axis, while the maximum rms concentration occurs off the

axis. These findings are consistent with those of similar measurements [2,ll]

Figure 7 shows the decay of the mean concentration along the jet axis,

along with two other sets of data measured in an air jet with heat and oil

smoke as the tracers [2,ll]. All three sets of data indicated an x decay

law. There was excellent agreement between our data and those of Becker, et

al. [llj, and the decay of the mean concentration along the jet axis can be

approximated by

C _ = 6.2 (^) ^ for 25 < ^ < 160 . (1)

C

o Because of the relatively high sensitivity of the LDF technique, we were

able to measure dye concentrations at a distance about four times greater

than that reported by Becker, et al. [ll] . The advantage becomes apparent

when we examine the self-preserving state of the jet later in this section.

In view of the excellent agreement of the results, we compare the

half-radius of the same two sets of data in figure 8. Although both sets

of data show that the half-radius increases as x , the virtual origins

were different. The negative virtual origins of our data may be attributed

to the tripping nozzle, which is installed 2 cm upstream of the jet exit.

Taking this into account, we can approximate our data by

^ = 0.1 (| - 10) for 25 < I

< 100 . (2)

The decay of the rms concentration along the jet axis of the same three

sets of data are shown in figure 9. Our data can be approximated by

(3)

No agreement was observed between any two sets of the data. Notice that

the tracers used in these experiments were altogether different. Corrsin

and Uberoi [2] used heat that had a Prandtl number of 0.74 as a tracer.

Becker, et al. [ll] used oil smoke as the tracer. ^The oil smoke had an

"equivalent" Schmidt number estimated to be 6 x 10 [a]. Here, we used

Rhodamine-6G dye, which has a Schmidt number of the order of 10^ as the

tracer. The effect of the Schmidt number may be one of the major reasons

accounting for the differences among the three sets of data. More

research is required, however, to resolve these differences.

To examine whether the jet has reached the^ self-preserving state, we

plotted, in figure 10, the ratio of Jet to C as a function of x/D

427
The data of Becker, et al. [ll] and that of Corrsin and Uberoi [2] are

also shown in that figure. When the self-preserving state is reached, the

value of \/c2/ C^^j^ should reach an equi^librium value. The data of Becker,

et al. [llj shows an equilibrium \/T^r^ax about 0.22, and the data

of Corrsin and Uberoi [2] shows a value of about 0.18. Our data show

that the self-preserving st ate i^s reached at x/D ^ 60 and that the

equivalent value is about vC^/C^^^^ = 0.44 . Again, we speculate that

the effect of the Schmidt number could have a significant effect on the

self-preserving state of the concentration field in a turbulent jet. More

research is required to clarify this point.

3.3 Simultaneous Measurements Along the Jet Axis

Because the LDF technique is remote-sensing and does not interfere with

the flow field, we can conduct simultaneous measurements at many positions

along the jet axis. These measurements cannot be performed with a nonremote-sensing technique because the presence of the upstream sensor would

interfere with the measurements of the downstream sensor. To demonstrate

this capability, we conducted two series of experiments. In the first series,

we carefully aligned two photomultipliers so they were focused, as close as

possible, at the same measuring volume. Measurements were made at x/D = 20,

50, and 100 , along the centerline of the jet. In the second series, we

measured the centerline concentrations simultaneously with three photomultipliers at x/D = 30 and at 50 . The measuring volumes, parallel to

the jet axis, were separated by 0.3 and 0.7 cm.

In figure 11a, we show tha auto- and cross-correlations of the concentration fluctuations measured with the two photomultipliers focused at the

same measuring volume (x/D = 50). As shown, they are prac ticaj. lv identical

to each other. The coefficient of cross-correlation \[c^c^/'J^'4^^ is as high

as 0.998. In addition, examination of the individual spectra shows that

there is no noticeable difference. Up to 100 Hz, the data measured by the

two photomultipliers show high coherence and little phase shift (figure lib)

.

Beyond 100 Hz, the coherence begins to decrease, and the phase shift increases

probably because of the imperfect matching of the measuring volumes.

The cross-correlations,

T

R^2(^) = ^

1

Cj^(t)c2(t - T)dt , (4)

with three photomultipliers focused at measuring volumes separated along

the jet axis by 0.3 and 0.7 cm, are shown in figure 12. Here the upstream

measuring volume was at x/D = 30 . In figure 12, we see that, as the

separation between the measuring volumes increases, the maximum R-]^2('^)

decreases, and the time at which it occurs increases. The envelope of the

cross-correlations defines the space-time correlation, which is represented

by the heavy line in figure 12. Furthermore, from the spatial separation

Ax between the sampling volumes and from the corresponding time separation

At between the times at which the maximum occurs, we can estimate

the convective velocity U(^ = Ax/Ax . This convective velocity should be

very close to the mean velocity of the jet, except in the region near the

edge of the jet. In figure 13 we plot the convective velocity, normalized

by the exit velocity, as a function of x/D . Also plotted are the results

measured in an air jet [2] and in a water jet [l2]. It should be noted

428
that the convective velocity is evaluated indirectly from the crosscorrelations of the concentration fluctuations, and, yet, there is

excellent agreement among the three sets of data.

3.4 Spectra

Spectra of concentration fluctuations were calculated by transforming

the auto-correlations with an FFT algorithm flSl . There were a total of

2 points, with a maximum sampling rate of At = 0.0013 second. The

statistical error was about 30 percent.

According to Batchelor [l4], the spectrum of concentration fluctuations

in the viscous-convective subrange follows

E^(k) = A(|)^ ^ exp(-A ^) (5)

c

for a scalar quantity of large Schmidt number, where k = (g/Vk ) and

e is the dissipation rate of the turbulent energy, V is the kinematic

viscosity, < is the diffusivity, and is the dissipation rate of

the concentration fluctuations. According to Batchelor [14], the constant

A has a value of 2, but experimental data show that the value of A

varies from 0.72 to 13.2 [is] . Since our measurement was made for a dye

solution with a large Schmidt number (-10^) , we decided to test the theory

Batchelor expressed in eq. (5).

In figure 14, we plot the spectra measured at x/D = 20, 50, and 100

The frequency is given in Hz,_and it can be related to the wave number by

Taylor's hypothesis k = 2'n"f/U . For the station at x/D = 50 , we show

two spectra calculated from two ensembles to indicate the statistical error.

In this figure, we see that, in the low-frequency range, all three spectra

contain a f ^ region over about one decade of frequency. This range tends

to increase with the distance. In figure 15 we rep lot the spectra presented

in figure 14 as log fS(f) versus f . For the f

^ range, the data

should present a constant of fS(f) . Also, if equation (5) is valid, the

plot should show a linear region. In figure 15, we see that linear regions

occur for all three spectra. For each spectrum, we indicate a cutoff

frequency f , which is estimated by f = k U/27T , where k is the cutoff

wave number corresponding to the dimension of the measuring volmne (~ 1 mm)

and has a value of about 6.28 mm ^ and the convective velocity U is

estimated from the results shown in figure 13. The spectral measurement

beyond this cutoff frequency will therefore be invalid. These results

support the theory of Batchelor expressed in eq. (5) although we cannot

estimate the value of the constant A for lack of information concerning

the measurement of the dissipation rate e

4. Concluding Remarks

In this paper, we have successfully demonstrated the usefulness of the

LDF technique for measuring dye concentrations in a turbulent water jet.

The present system, however, is by no means optimized; there is still

considerable room for improvement. For example, by suitably collimating

the laser beam and using a smaller diameter pinhole, the measuring volume

could be reduced to as small as 0.1 mm with some sacrifice of the intensity

of the fluorescence signals. The reduction of the fluorescence intensity

429
could be compensated by optimizing the optical characteristics of the system.

Perhaps one optimization would be the use of the 514.5-nm line Instead of

the 488-nm line of the laser, if we could find a filter that would efficiently

block the laser light from the fluorescent light.

By comparing our results with those obtained from heat and oil-smoke

transport [2,ll], we see various degrees of agreement. The centerline mean

concentration, which varies as x

'• , agrees excellently with that of the

oil-smoke transport [ll] and has a higher value than that of the heat

transport [2]. The centerline root-mean-square concentration, which also

varies as x , is, however, consistently higher than those of the other

two [2,11]. The differences could likely indicate the significance of the

Schmidt-number effect.

We also show that, with the LDF technique, we can make simultaneous

measurements along the jet axis or at the same point. This result demonstrates the system's remote-sensing capability. Finally, our spectral

measurements validate Batchelor's theory on the spectrum of concentration

fluctuations for a scalar quantity of large Schmidt number.

5. References

[1] Robben, F.A., Lin, J. T., Delisi, D. P., and Liu, H. T., Feasibility

study of a dye-fluorescence technique for measuring concentrations in

fluids. Flow Research Note No. 91, Flow Research Company, Kent,

Washington (1976)

.

[2] Corrsin, S. and Uberoi, M. S., Further experiments in the flow and

heat transfer in a heated air jet, Natl. Advisory Comm. Aeronaut.

Tech. Notes No. 998 (1950).

[3] Kiser, K. M. , Material and momentum transport in axisymmetric turbulent jet of water, AI.Ch.E. Journal 9, 386 (1963).

[4] Rosensweig, R. E., Hottel, H. C, and Williams, G. C, Smokescattered light measurements of turbulent concentration fluctuations,

Ch. Eng. Sci. 15, 111 (1961).

[5] Becker, H. A., Hottle, H. C, and Williams, G. C, On the lightscatter technique for the study of turbulence and mixing, J. Fluid

Mech. 30, part 2, 259 (1967).

[6] Liu, H. T. and Karaki, S., An optical system for measurement of mean

and fluctuating concentrations in a turbulent air stream, J. of Physics

E: Scientific Instruments 5, 1165 (1972).

[7] Yang, B. T. and Meroney, R. N. , A portable laser light-scattering

probe for turbulent diffusion studies. Rev. Sci. Instrum. 45 , no. 2,

210 (1974).

[8] Gad-el-Hak, M. , Morton, J. B., and Humphris, R. R. , An investigation

of turbulent diffusion. Tech. Rept. No. ESS-3837-102-75, Univ. of

Virginia (1975).

[9] Lee, J. and Brodkey, R. S., Light probe for the measurements of

turbulent concentration fluctuations, Rev. Sci. Instrum. 34 , 1086

(1963).

430
Nye, J. 0. and Brodkey, R. S., The scalar spectrum in the viscousconvective subrange, J. Fluid Mech. 29 , 151 (1967).

Becker, H. A., Hottel, H. C, and Williams, G. C, The nozzle-fluid

concentrations field of the round turbulent free jet, J. Fluid Mech.

30, 285 (1967).

Albertson, M. L. , Dai, Y. B., Jensen, R. A., and Hunter Rouse, M.

,

Diffusion of submerged jets. Trans. ASCE 115 , 639 (1950).

Bendat, J. S. and Piersol, A. G. , Random Data (John Wiley & Sons, Inc.

New York, 1971).

Batchelor, G. K. , Small-scale variations of convected quantities like

temperature in turbulent fluid, J. Fluid Mech. 5, 113 (1959).

Hinze, J. 0., Turbulence - An Introduction to Its Mechanism and Theory

(second edition, McGraw-Hill Book Co., Inc., New York, 1975).

431
E

(13 (A

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700 650 600 550 500 ^

450

16 18 20

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24

(b) Effective Window for the Receiving Optics

Fig. 2 Optical Characteristics of the Laser Dye-Fluorescence System

433
25

Voltage (volts)

Fig. 3 A Calibration of LDF System

434
(a) Cross-Sectional View Along the Jet Axis

(b) Cross-Sectional View Perpendicular to the Jet Axis

4 Flow Visualization of the Turbulent Water Jet With a Sheet

of Laser Light

435
436
437

439

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441
Fig. 11 Measurements of Concentration Fluctuations With Two

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442
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445
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446
National Bureau of Standards Special Publication A84, Proceedings of the

Symposium on Flow in Open Channels and Closed Conduits held at NBS,

Gaithersburg, MD, February 23-25, 1977. (Issued October 1977)

THE APPLICATION OF MONTE CARLO AND BAYESIAN PROBABILITY

TECHNIQUES TO FLOW PREDICTION AND DETERMINATION

Dr. Frank J. Berte'

Reactor Thermal Design and Analysis

COMBUSTION ENGINEERING POWER SYSTEMS, INC.

Windsor, Connecticut 06095

The prediction of flow rate through a flow network, which

is in the design stage, requires a detailed analysis of

component resistance and pump performance uncertainties.

In order to evaluate the predicted best estimate flow rate

and its associated probability distribution, a combination

of the Monte Carlo and Stratified Sampling techniques can be

utilized. The use of these techniques permits the required

results to be obtained even when non-gaussian probability

distributions are associated with network flow resistances

and/or pump performance parameters. The calculations necessary

to produce the best estimate predicted flow rate and its

associated probability distribution, are performed using

a computer code, which embodies the Monte Carlo and Stratified

Sampling techniques. After the flow network has been constructed,

and flow measurements have been made, the problem becomes one

of determining the best estimate true flow rate and its

associated probability distribution. Ordinarily if a number

of different techniques were available for measuring the

flow in the network, the one with the smallest uncertainty

would be utilized to determine the true flow rate. An

alternate technique exists whereby the predicted flow as

well as the flows measured by the available techniques are

combined to produce one best estimate flow rate and its

associated probability distribution. The statistical technique which can be used to accomplish this combination is

Bayes' rule. This combination of predicted and measured

flow rates and their associated uncertainties is performed

by a computer code which embodies Bayes' Rule as well as

the Monte Carlo and Stratified Sampling techniques.

Key Words: Flow prediction, flow determination, Monte

Carlo; Stratified Sampling; Bayes' Rule; statistics

447
1 . Summary

The evaluation of both flow prediction and flow determination uncertainties

require the generation and manipulation of probability distributions. The

generation of these probability distributions has been accomplished by

using a combination of Monte Carlo' and Stratified Sampling techniques [2][3] ^

.

A detailed discussion of the methodology employed to generate these probability

distributions is presented in section 2. The use of the probability distribution generation methodology in the calculation of predicted flow rate and its

associated probability distribution is presented in section 3. This section

also contains an example of the application of the discussed analysis

techniques to a sample problem. In section 4, the analysis technique

for the combination of flow prediction and measurement values according

to Bayes ' Rule is discussed, again a sample problem is presented.

2. Probability Distribution

Generation

Before probability distribution generation is discussed per se, a brief

discussion of how these distributions are used is in order. Let us assume

that we are dealing with a problem in flow prediction. The flow prediction

code input is made up of various parameters related to the flow network

characteristics. Each of these parameters is assumed to have a known

probability distribution associated with it. In order to utilize the

Monte Carlo technique, a set of data for each of these parameters must

be generated, such that the set conforms to the input specification of

its probability distribution. Once the sets have been generated, (we

will discuss this generation in detail shortly) one value for each parameter

is chosen at random, for the respective data set for that parameter, and put

into the functional relationship equations relating these input parameters

to the predicted flow rate. This process is repeated "N" times, where "N"

is the number of data points in each data set, and "N" values of the resulting

predicted flow rate are generated. From this set of predicted flow rates

the probability histogram for the predicted flow rate is generated.

It is obvious that the larger the number "N" the more computer time is

used by the code performing the calculation. The objective of the probability distribution generation scheme presented here is to minimize "N",

while still achieving the most accurate and precise results for the predicted

flow rate probability distribution.

The Monte Carlo-Stratified Sampling technique employed in both the flow

prediction and determination methodology requires the division of a probability distribution into a finite number of intervals. If one

wishes to generate a probability distribution for a variable X, that

is, if one wants a set of X-j values having a specified probability

distribution, the following steps would be required. It is assumed

^Figures in brackets refer to the numbered references at the end of

this paper.

448
that the values of X^- are bounded by a lower limit "a", and an upper

limit "b". First, the range from "a" to "b" is divided into a finite

number of intervals of equal width.* The probability that a value of

X falls within each interval is then used to find the number of X

values required in the interval, given a fixed total number of X values

required in the complete set of X values. The interval probabilities

can be supplied from analytical algorithms for standard probability

distributions, such as the gaussian, or they can be supplied specifically,

on an interval by interval basis, in the code input. Non-symmetrical or

discontinuous probability distributions can thus be readily accommodated.

However, most problems encountered have only required a gaussian or a uniform probability distribution for code input variables.

For gaussian probability distributions the code utilizes the following

analytical expression for cumulative probability as a function of the

normal deviate [5].

P(x) =1-1/2 (l+d^x+d2X^+d3X^+d^x^+d5X^+dgX^)"^^+e(x) (1)

where e(x) is the error associated with the result, and is

bounded by |e(x)| £ 1.5 x 10"^

0.04986 73470

0.02114 10061

0.00327 76263

0.00003 80036

0.00004 88906

0.00000 53830

is the normal deviate, or

X ={deviation/standard deviation of the gaussian distribution}

Once the number of data points in each interval has been determined for the

variable of interest, a pseudo random number generation scheme [1] incorporated

into the computer software is utilized to generate the required number of data

points in each interval. This process is repeated until all intervals are

filled. The data set for the variable X now has the properties of the input

probability distribution specified. Only one more step remains, that is to

ramdomize the order in which the data points appear in line matrix; Data

*Note, in the generation of data sets having gaussian distributions, which

theoretically should span the region from minus to plus infinity, it has

been found, for our purposes, that a range from minus to plus five (5)

standard deviations about the mean value was satisfactory.

449
Point 1, Data Point 2..,. Data Point "N". This is necessary because the

code generates the data for each interval starting at the lowest and

working to the highest numbered one. Therefore, although the data is

random within each interval, on an interval to interval basis it is not.

After the randomization of the whole data set has been completed, one

has a set of randomly ordered X^. data with the statistical properties

specified in the code input.

The scheme for generating a set of data with the required statistical

properties has been presented. The question now becomes, how do you

decide on the optimum number of intervals and data points to generate

a set of data whose statistical properties are in good correspondence

with the specified input properties, and yet assure the minimum use

of computer time to accomplish the task? One needs an optimization

procedure for determining the number of intervals, and number of data

points to be used. This procedure has been devised for gaussian distributions and is based on the optimization parameter given in equation (2).

The object then becomes to achieve an optimization parameter with a value

close to unity, while still minimizing the number of data points and

hence computer running time used.

A number of test cases were run to determine the number of intervals and

the number of data points required to achieve an optimization parameter

approximately equal to unity. Two sets of test cases were run, first

the number of intervals was held constant and the number of data points

was varied, and second, the number of data points was held constant and

the number of intervals was varied. For each of these cases the optimization parameter was plotted vs. the varied parameter. These two plots

are shown in Figure 1 and 2, respectively. Figure 3 illustrates the combination of the results from Figures 1 and 2, which is an optimization

parameter surface. A number of deductions can be drawn from these figures,

they are as follows.

Optimization

Parameter

for Gaussians

standard deviations

of code generated data set

specified input

standard deviation

(2)

450
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452
453
From Figure 1

1. As the number of data points increases the optimization parameter

approaches unity as expected.

2. The curve is expected to lie beneath the optimization parameter equals

unity line, since one would need an infinite number of points and intervals

to get the gaussian tails completely filled.

3. The optimization parameter is sufficiently close to unity when the

number of data points used is greater than about 2000. The variation

observed beyond this number of data points is believed due to slight

random fluctuations in the data distributions within each interval of

the generated probability distribution.

4. The optimum number of data points for 100 intervals is therefore about

2000, this gives an optimization parameter close to unity (error in

generated standard deviation is less than one percent) with the minimum

required computer running time. Note that the optimum number of data

points per interval is approximately 20.

From Figure 2

1. The optimum number of intervals is approximately 40, for 800 data

points utilized.

2. As the number of intervals gets too small relative to the optimum 40,

the gaussian distribution changes, until a uniform distribution is achieved,

when the lower limit of two intervals is reached.

3. As the number of intervals increases beyond the optimum 40 value, the

generated data set standard deviation falls off until at a large enough

number of intervals one expects a zero value for it. This is due to the

fact that the probability that a data point falls in any interval must

decrease as the number of intervals increases, assuming that the total

number of data points is fixed.

4. The optimum ratio of the number of data points to number of intervals

again appears to be about 20.

For uniform distributions it is obvious that the optimum number of intervals

must equal the number of data points required. It is usually sufficient

though, to make the number of data points divisible by the number of intervals used, an integral number of times.

If other probability distributions are required, which are not gaussian

or uniform, the same procedure can be utilized, except that higher moments

may have to be considered in devising the optimization parameter.

454
Another method of generating a set of data with a given probability

distribution, involves the use of the Monte Carlo technique in conjunction

with the cumulative distribution function. This method, which does not

rely on Stratified Sampling is deficient in two respects, relative to

the combination of Monte Carlo and Stratified Sampling techniques.

First, when one generates a probability distribution, one has no assurance

that the tails, say for a gaussian distribution, will be adequately represented by the data. Second, for the same number of points one can expect

a wider variation in statistical properties of the resultant data set.

These deductions were verified by a brief study whose results are presented

in Table I. In order to generate a set of data having a normal distribution

using the cumulative distribution function technique, the following calculation is required. First, a random number is generated between 0 and 1.

This value, considered to represent a probability, is then utilized to

determine the corresponding value of the normal deviate. This process

is carried out "N" times in the computer code by using a rational approximation [5] given in equation (3).

2

C +C-, t+Cpt

xp = t - {-^ ^

^} + e(p) (3)

l+d.t+dof^+d^t^

where t = -/ f^ ^

p is the input cumulative probability, this value must be

between 0 and 0.5 in magnitude.

xp is the normal deviate for a gaussian distribution with

a standard deviation of unity

Cq= 2.515517

c^= 0.802853

C2= 0.010328

d^= 1.432788

d2= 0.189269

d^= 0.001308

e(p)= is the error in the approximation |e(p) | <4.5xl0~^

455
TABL E I

Comparison of the Standard Deviation of a Gaussian Distribution as Generated by the Combined Monte Carlo-Stratified Sampling (MCSS) Techniques

and by the Monte Carlo Technique Using the Cumulative Distribution

Function (MCCDF)

Note

Seven runs were made using each technique, for MCSS runs, 26 intervals

and 500 data points were used; for the MCCDF runs, 500 data points were

used. In both cases the specified input was a gaussian distribution with

a standard deviation of unity.

Mean Standard Deviation Standard Deviation of

Technique of the Seven Runs the Seven Standard

Deviations

MCSS 0.987981 0.003955

MCCDF 0.955134 0.027560

456
3. Calculation of the Predicted Flow

Rate and Its Associated Probability

Dis tri bution

Prediction of flow rate, in a flow network with parallel loops,

requires the solution of simultaneous equations. It is the author's

opinion that it is difficult, if not impossible, to analytically determine predicted flow uncertainties, under these conditions, without

resorting to some type of Monte Carlo or finite difference approach.

The calculation of the predicted flow rate and its associated uncertainty

distribution for a flow network in the design stage, is usually based

on the following information.

1. A complete description of the flow network geometry,

2. The uncertainty distribution and best estimate predicted values

for each network component flow resistance,

3. The best estimate values, and the uncertainty distributions, for the

coefficients of each pump head vs. flow characteristic equation,

4. The network flow equations relating pump head to flow head

losses.

The calculation sequence is as follows, first the data sets for each

flow resistance and pump characteristic equation coefficient are generated

according to the input probability specifications. Second, the first random

value of each resistance and pump characteristic equation coefficient is

inserted into the network equations. These equations are then solved for

one possible value of the predicted flow rate. This step is repeated until

all values, in each data set, have been utilized once. The final step is the

printout of the probability histogram for the predicted flow rate and the

relevant associated statistics. The nature of the Monte Carlo-Stratified

Sampling technique is such that functional relationship between the input

uncertainties and the predicted flow uncertainty can be accounted for

accurately.

An illustrative example of the application of the above procedure is

presented below.

Flow Prediction Uncertainty Analysis; Illustrative Example

Objective : Find the best estimate predicted flow rate 'and

its associated probability distribution, in the

flow network.

Given : 1. Flow network schematic diagram, see Figure (4).

457
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458
2. Flow network simultaneous equations

A2Q2^+B2Q2+C2

A3Q32+B3Q3+C3

Rl[Q^+Q2+Q3+Q4]^ +

Rl[Ql+Q2+Q3+Q4]^ +

Rl[Ql+Q2+Q3+Q4]^ +

Rl[Q^+Q2+Q3+Q4]^ +

R

R

R

R

2

2

3

3

[Ql+Q2]^

+

[Q3+Q4]^ +

[Q3+Q4]^ +

[R4+R8] [Qi]^

[R5+Rg] [Q2]^

[r^+r^q] CQ33'

(4)

where Ai,Bi,C.j are the coefficients of the number one pump head vs flow

characteristic equation.

A2,B2»C2J etc are the same coefficients for the number two pump, etc.

Rl,R2--R-]-| are the network component flow resistances.

Q-j is the flow through number one pump, Q2 through number two pump, etc.

3. The best estimate and standard deviation values for the coefficients

of each pump head vs flow characteristic equation, see Table 2 below.

459
TABLE 2

I

Pump Characteristic Equation Coefficient Uncertainties

Pump

Number Coeffi cient Mean Value Standard Deviation

-1.700 1.213 xio'^

1 5.977 1.360

C^ 106.8 3.621

-1

A2 -1.448 0.8572 xlO

B2 3.280 0.9995

C^ 113.7 2.788

A^ -1.398 1.625 xlO"^

B3 3.355 1.885

C3 112.3 5.200

A^ -1.672 0.8707 xlO'^

4 B^ 6.028 0.9918

C4 106.4 2.690

Note (1) all uncertainty distributions are assumed to be gaussians

(2) the dimensional units of "A-j" are [M/(M^/sec)^]

the dimensional units of "B-" are [M/(M"^/sec) ]

the dimensional units of "C^-" are [M]

(3) See Figure (4) for locations of each pump in the flow network.

(4) All gaussian distributions were generated using 2000 data points

and 100 intervals.

460
4. The network loop component flow resistances best estimate and

standard deviation values, see Table 3 below.

TABLE 3

Component

Number Mean Value • Standard Deviation

1 3.381x10"^ 1.690x10"^

2 3.047x10"^ 1.524x10'^

3 3.046x10"^ 1.523x10'^

4 1.015x10"^ 5.075x10"^

5 1.015x10"^ 5.076x10"^

6 1.015x10"^ 5.075x10"^

7 1.015x10"^ 5.076x10"^

8 2.292x10"^ 1.146x10"^

9 2.292x10'^ 1.146x10"^

10 2.292x10"^ 1.146x10"^

11 2.292x10"^ 1.146x10"^

Note: (1) the dimensional units of each "R." are sec) ]

(2) see Figure (4) for locations of each network component flow

resi stance

(3) All resistance probability distributions were assumed to

be gaussian and each was generated using 2000 points and 100

intervals

.

461
Code Results

1. Best estimate predicted flow rate is 24.6627 (M /sec) or 120.4 percent

of the reference flow rate.*

2. Standard deviation of the predicted flow rate probability distribution

is 0.5469 (M^/sec), or 2.67 percent of the reference flow rate.

The probability histogram for the predicted flow rate is shown in Figure

5, also shown in this figure is an ideal gaussian having the same standard

deviation and mean value as the code result.

The correspondence of these two curves indicates that the predicted flow

probability distribution is indeed a gaussian. In order to qualify the

correspondence level of the code generated histogram and an ideal

gaussian histogram, a chi-squared statistic was used, which compared the

actual to the ideal data point population in each interval. This statistic

has the property that when the correspondence is ideal, its value is zero.

As the level of confidence decreases the value of this statistic increases.

This single indici of the degree of correspondence has been found useful.

Equation (5) defines this statistical index mathematically [4].

^2 _

" [Pop, (actual) - Pop, (ideal )]^

^

{

~

} (5)

Pop. (ideal)

where, n is the number of intervals into which the predicted probability

distribution is divided

n-1 is the degrees of freedom of this chi-squared distribution

Pop. (actual) is the number of data points which fall within

the ith interval

.

Pop. (ideal) is the ideal number of data points expected in the

ith interval, if the distribution is a gaussian.

There are two requirements for using this statistic correctly, first

none of the Pop^- (ideal) must be less than 1, and not more than 20

percent of these values should be less than 5.

The code results for the predicted flow rate probability distribution

are shown in Figure 5, note that 100 intervals were used for the printout of the result distribution. In order to apply the chi-square test

of this resultant the 100 intervals were reduced to 20, this was done

in order to fulfill the first requirement for the use of this test.

*For simplicity, flow rates will be expressed as a precentage of the reference

flow rate of 20.4839 (M^/sec), in the remainder of this paper.

462
TABLE 4

Actual and Ideal Interval Populations for

the Predicted Flow Probability Distribution

Interval Actual Population Ideal Gaussian Population

1 2 1 2 2 5

3 7 12

4 24 26

5 42 50

6 100 86

7 120 135

8 158 188

9 211 235

10 289 262

11 256 262

12 255 235

13 195 188

14 137 135

15 94 86

16 52 50

17 28 26

18 13 12

19 9 5

20 3 1

The confidence limits for the chi -square can be found from the following

approximation, for a large number of degrees of freedom [5].

= V [ 1 - I- + Z„, /|^] 2 (6)

where a, is the probability level associated with the confidence limit

calculated.

V is the degrees of freedom of the chi-square statistic, which

is the number of intervals minus 1.

Z is the normal deviate corresponding to the "a" probability level.

2

For a = .99, and v = 19, is 36.21, from equation (6). From equation (5),

the predicted flow rate probability distribution chi-square value is 30.66.

This value falls below the confidence limit, and the hypothesis, that the

predicted flow probability distribution is a gaussian, is confirmed.

463
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464
4. Determination of the True Flow

Rate and its Associated Probability

Distribution

When flow measurements are made on the actual flow network, these

measurements can be combined with the predicted flow using Bayes' Rule.

The resulting true flow rate probability distribution thus reflects all

available flow data. The following information is required to apply the

analysis technique.

1. The predicted flow rate uncertainty distribution and best estimate value

2. "J" independent flow measurements", uncertainty distributions and

best estimate values.

The calculation of the true flow rate uncertainty distribution involves

the following sequence of events. First, the data sets for the predicted

and measured flow rates, according to the input probability specifications,

are generated. Second for each flow rate interval, Bayes' rule as shown in

equation (7), is applied [2].

P(M,&M.&...M IT.).P(T.)

P(T.|M^8,M2& ..Mj) =

J

' ^ ^

(7)

where ' {PCM^SM^S. . .M^ | T. ) . P(T.)}

P(T.) is the predicted probability that T-j

is the true flow rate,

i is the flow interval indici,

I is the total number of flow intervals used,

P(T. |M,8iM2&. . .M ) is the probability that T-j is the true flow

rate given that the independent measurements

M,&Mp&...M have been made.

P(Mi &M2^' • M j

) 1s the probability of measuring the M-1&M2

^ &...M flow rates given that the true

flow "^is T^.

.

and P(M^&M2&...Mj|T. ) = '^^ P(Mj|T.)

j=l

J is the number of independent flow measurements

made.

The true flow rate probability histogram is found by plotting P(T^. IM1&M2. • .Mj)

»

for each possible true flow rate T-. Further development of this

method is planned to associate a confidence level with the true flow rate

probability distribution calculated.

*Independent flow measurement refers to the method, for example, heat balance,

and venturi Ap flow measurements, are independent methods.

465
An illustrative example of the above procedure for evaluation of the

true flow best estimate value, and associated probability distribution,

is presented below.

True Flow Probability Distribution Determination; Illustrative Example

Objective : Find the best estimate true flow rate and

its associated probability distribution given

the predicted and measured flow rate information

requi red.

Given:

TABLE 5

Flow Prediction and Measurement

Probability Distributions

Source of Flow

Rate Data

Predi cted

Measured, Method 1 Measured, Method 2

Best Estimate Value

120.4%

123.1

125.3

Standard Deviations*

1.81

3.75

*A11 probability distributions were assumed to be gaussian in form, 2000

data points and 200 intervals between 100 and 140 percent design flow were

used.

Code Results :

1. Best Estimate True Flow Rate is 122.6 percent of the reference flow rate.

2. Standard Deviation of True Flow Rate Probability Distribution is

1.389 percent of the reference flow rate.

3. The true flow rate probability distribution is a perfect gaussian

as can be seen from the coincidence of the actual and ideal

gaussian distributions presented in Figure 6.

466
467
One interesting result of the application of Bayes' Rule to continuous

probability distributions, is that if all the distributions involved are

gaussian in form, the result can also be found without recourse to a computer code by the application of the method called minimization of

the chi-squared. In this analysis the flow prediction is treated as

another independent measurement of the flow rate. The derivation of the

standard deviation and the mean value of the true flow rate probability

distribution by this method are presented below, and the code and

analytical results are compared in Table 6.

If one is given a number of independent variables which are measures of

the same parameter and each of the variables has a gaussian probability

distribution associated with it, the most likely value of this parameter

can be found by the following technique. First a chi-squared variable [2]

can be generated using equation (8)

where, n is the no. of independent measures of the parameter in question

is the mean value of the itji independent measurement

a. is the standard deviation of the ith^ independent measurement

about its mean value.

J is the unknown value of the measured parameter which will

minimize the magnitude of the chi-square statistic.

One can find the value of X which minimizes the v^alue of the chi-square

statistic by differentiating "X^ with respect to X and setting the resulting

expression equal to zero, as in equation (9).

2 2 2 2 ^ = 0 = -2 (i

) (X - X) - 2 (I-

) (X. - X) - 2 (i-) (X3 - X) (9)

or, X = X, )^ + X. (i- )^ + X3 (i- f

The standard deviation of X can be found by utilizing expressions (10) and

(11).

468
Thus, a(XX )

) = /" 1

1 2 1 2 -,2

) + (f-) + (7- )

(12)

X analytical value

using equation (10)

,

and data from

Table 5.

112.56 percent of volumetric

reference flow

a(X) analytical value

using equation (12),

and data from

Table 5.

1.391 percent of volumetric

reference flow

TABLE 6

Comparison of Bayes' Rule Result and

Minimization of the Chi -Square Statistic

Analytical Technique

Best Estimate

True Flow Rate

Bayes' Rule

122.67

Chi -Square Minimization

122.56

True Flow Rate

Standard Deviation 1 .389 1 .391

469
REFERENCES

[1 ] Knuth, D.E., The Art of Computer Programming Vol. 2 , Semi -Numerical.

Algorithms (Addi son-Wesley 1969).

[2] Meyer, S.L., Data Analysis for Scientists and Engineers , (John Wiley

and Sons, New York, 1975).

[3] Clark, Jr., M. and Handson, K.F., Numerical Methods of Reactor Analysis

(Academic Press, New York, 1964).

[4] Dixon, W. J., and Massey, Jr., F.J., Introduction to Statistical

Analysis, (McGraw Hill Inc., 1969).

[5] Abramonitz, M. , and Stegun, I. A., Ed. Handbook of Mathematical Functions , (National Bureau of Standards Applied Mathematic Series 55, Ninth

Printing, November 1970).

470
AUTHOR INDEX

Abernathy, F. H. 957

Allion, H. 33

Annett, John 145

Appell, Gerald 109

Bajura, R. A. 523

Baker, Milton 803

Ball, John M. 847

Baumoel, J. 267

Berte' , Frank J. 447

Bradshaw, J. E. 293

Brennan, J. A. 881

Brower, W. B. Jr. 719

Bruner, Ronald F. 277

Clark, M. E. 479

Cullen, J. T. 549

Dang, Clement 187

Davis, L. M. 55

Davis, R. W. 219, 491

DeCarlo, J. P. 549

Delisi, Donald P. 423, 783

Dijstelbergen, H. H. 935

Durgin, William W. 471

Dussauge, J. P. 649

Edgerly, D. E. 921

Fabris, Gracio 659

Flandro, G. 335

Flemons, R. S. 319

Gad-el-Hak, Mohamed 571

Gaviglio, J. 649

Gold, D. S. 621

Grant, D. M. 91

Greenleaf, J. F. 335

Guthrie, D. L. 173

Halmi, Dezsoe 61

Haney, Robert A. 755

Hayward, A. T. J. 1

Holley, E. R. 395

Irwin, L. K. 895

Jennings, Roger 821

Johnson, S. A. 335

Kempf, D. 361

Kirchhoff, Robert H. 783

Koenig, H. L. 145

Kondic, N. N. 765

Kulin, G. 123

Kullman, C. G. 25

LaNasa, Paul J. 871

Lin, Jung-Tai 423

Liu, Hsien-Ta 423

Lowell, Francis C. Jr. 243

Lynnworth, L. C. 293

Mann, D. B. 881

Martig, Kenneth W. Jr. 83

Mattingly, G. E. 33, 491

McQuivey, S. 109

Meagher, Thomas F. V. 687

Mero, Thomas 109

Miller, R. W. 549

Moore, E. 33

Morel, P. R. 293

Morgan, W. H. 361

Neale, Lawrence C. 471

O'Brien, Joseph F. 145

Peckham, Vernon D. 687

Pedersen, N. E. 293

Pellegrin, M. T. 523

Phillips, R. E. 361

Pierce, F. J. 621

Pontius, P. 33

Replogle, John A. 201

Robben, Frank A. 423

Robertson, J. M. 479

Rudland, R. S. 705

Ruff, James F. 187

470-A
Sachs, Donald C. 687

Saxton, Keith 187

Scottron, V. E. 737

Servoz, Alan 719

Seward, Wallace N. 929

Singley, G. Wayne 145

Skrldulis , James 145

Spencer, E. A. 947

Suhoke, Robert B. 597

Tabler, J. H. 25

Tanaka, M. 335

Vlncenty, C. 173

Washington, D. R. 173

West, James A. 945
N«S-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

I^UiM.K ATION OK RKPORTNO.

SP 484 -- Volume 1

2. Gov't Accession

No.

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4. TITLE AND SUBTITLE

Flow Measurement in Open Channels and Closed Conduits

Volume 1

5. Publication Date October 1977

6. Performing Organization Code

7. AUTHOR(S) 8. Performing Organ. Report No. Lafayette K. Irwin , Editor

9. PERFORMING ORGANIZATION NAME AND ADDRESS

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Library of Congress Catalog Card Number: 77-14243

16. ABSTRACT (A 200-word or less [actual summary of most si^ilicant information. If document includes a significant

bibliography or literature survey, mention it here.)

The wide range and complexity of problems and potential solutions that must be considered for useful flow measurements are emphasized by the papers contained in these

proceedings. Fifty-three presentations cover: characteristics of new and improved

instruments; applications of traditional and new measuring devices in field environments; procedures for identifying and analyzing errors or uncertainties in data under

specific conditions; uses of physical and numerical models; politico-economic changes

that affect international standards for flow measurement; and philosophical bases for

making measurements. The fluids of most interest are water and waste water, petroleum

and related refined products, air, natural gas and stack gas.

Experimental and analytical investigations on instrument performance and interpretation of results include innovative applications of traditional and new flow measurement

techniques to fluid flows in open channels and closed conduits. The traditional devices

or techniques include weirs, flumes, current meters, orifice plates, turbines, hot-wires,

pitot-static tubes, velocity traverses, dye-dilution, and others. More recent instrumentation developments and procedures such as laser doppler anemometry, acoustic and

thermal imaging, acoustic pulse velocity and doppler anemometry, numerical modeling,

vortex shedding and digital computation are covered for particular measurement purpose.?.

The most significant trend reflected in these presentations is the general awareness

that uncertainties in measured quantities at the lowest point in the measurement chain,

i.e., in the field or plant, are more important than accuracy statements derived from

controlled laboratory studies. Other trends in evidence are the rising importance of

turbine meters for use as transfer standards and in-line measurements of liquids and

gases in filled pipes and the increasing number of applications for acoustics and laser

technology for flow measurements in both open channels and closed conduits.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Acoustic flow meters; closed conduit flows; current meters; dye-dilution methods; errors

in flow measurement; flow measurement; fluid flow modeling; fluid velocity; flumes; gas

flov7 standard; hot-wire anemometry; international flow standards; laser anemometry; open

channel flows; nrifir.p. meters; pi rnt-sfarir mptprs; turbine meters; weirs

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