^T ° F *cu v NBS TECHNICAL NOTE 831

1974

153

H

Introduction to Liquid Flow Metering and

Calibration of Liquid Flowmeters

Lief O. Olsen

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

*< ">%

\

\

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued June 1974
Library of Congress Catalog Canl Number: 74-600090

National Bureau of Standards Technical Note 831

Nat. Bur. Stand. (U.S.), Tech. Note 831, 60 pages (June 1974) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1974

For sale hy the Superintendent of Documents, U.S. Government Printing Office, Washington, IXC. 20402 (Order hy SI) Catalog No. C13.46:831). Price 95 cents.
CONTENTS

Page

1

.

INTRODUCTION 1

2 . UNITS AND SYMBOLS i

3

.

LIQUIDS 4 3.1. Liquids Defined 4 3.2. Density of Liquids 5 3.2.1. Hydrometers 5 3.2.2. Test Procedure with Hydrometers 5 3.2.3. Application of ASTM-IP Petroleum Measurement Tables 6 3.2.4. Density of Water 9 3.2.5. Compressibility of Liquids 9 3.3. Viscosity of Liquids 10

3.3.1. Dimension and Units of Viscosity 11 3.3.2. Effect of Pressure on Viscosity 12

3.3.3. Effect of Temperature on Viscosity 13

3.4. Vapor Pressure of Liquids 14

4

.

FLOW IN CLOSED CONDUITS 16

4.1. Conservation of Mass 16

4.2. Energy of Liquids in Motion 17 4.3. Differential Head Meters 18

5

.

MEASUREMENTS REQUIRED 21 5.1. Temperature 21 5.1.1. Illustrative Examples 21

5.1.2. Temperature Measurements „ 21

5.2. Pressure Measurements 22

5.2.1. Manometers 23

5.2.2. Corrections and Use of Conversion Factors for Manometers 26

5.2.3. Illustrative Examples 27 5.2.4. Mechanical Pressure Gages 27 5.3. Volume Measurements 28

5.4. Time Measurements 28

5.5. Mass Measurements 28

6. METHODS OF CALIBRATION 29

6.1. Introduction 29

6.2. Gravimetric Calibration Systems 29

6.2.1. Static Weighing Calibrators 29

6.2.2. Dynamic Weighing Calibrators 30

6.3. Volumetric Calibration Systems 33

6.4. Meter Installation 36

6.5. Illustrative Examples 38

7

.

TYPES OF FLOWMETERS 40

7.1. Introduction 40

7.2. Differential Pressure Type Meters 40

7.2.1. Orifice Meters 41

7.2.2. Flow Nozzles 42

7.2.3. Venturi Flowmeters 43

7.2.4. Laminar Meters 45

7.3. Variable Area Meters 46

7.4. Turbine Flowmeters 47

7.5. Positive Displacement Meters 50

iii
Page

7.6. Magnetic Flowmeters 51

7.7. Magnetic Resonance Flowmeters 52

7.8. Ultrasonic Flowmeters , 53

8 . REFERENCES 55

ILLUSTRATIONS

1. Temperature - apparent density relation of a typical hydrocarbon 8

2

.

Laminar flow between parallel plates in relative motion 11

3. Effect of temperature on the viscosity of a typical hydrocarbon liquid 15

4

.

Flow in a closed conduit 16

5

.

Flow through several meters in series 17

6. Coefficients of discharge vs Reynolds number for typical orifice and

nozzle meters 20

7

.

Manometers 24

8. Recommended connections between meters and pressure sensing elements 25

9

.

Static weighing calibrator 30

10. Dynamic weighing calibrator 31

11. Dynamic weighing calibrator 33

12

.

Standpipe calibrator 34

13. Ballistic calibrator 35

14

.

Typical flow straightener 37 15. Thin plate orifice meter 42

16

.

ASME long radius flow nozzles 43

17

.

Classical (Herschel) Venturi 44

18

.

Laminar flowmeters 45

19

.

Variable area flowmeter 46

20. Turbine flowmeter characteristics 49

21. Positive displacement flowmeter 50

22. Electromagnetic flowmeter 52

23. Nuclear magnetic resonance flowmeter 53

24

.

Ultrasonic flowmeter 54

TABLES

1

.

Comparison of Force-Mass-Length-Time Systems 2 2. List of Symbols 3 3. Density of Water 9

4

.

Isothermal Compressibility of Liquids 10

5. Dynamic Viscosities of Liquids at 20°C 12

6. Effect of Temperature on the Dynamic and Kinematic Viscosities of Water 13

7

.

ASTM Viscosity - Temperature Charts 14

8. Densities of Water, Mercury and Air 26

9

.

Recommended Minimum Lengths of Straight Pipe 37

10. Maximum Thickness for Thin Plate Orifices 41

iv
Introduction to Liquid Flow Metering and Calibration of Liquid Flowmeters

Lief 0. Olsen

These notes are intended to serve as an instruction manual for technicians and engineers engaged in metering liquids and calibrating liquid flowmeters. It is a condensed

review of the properties of liquids and the mathematical relations required in this work.

References to more complete sources of properties of liquids, theoretical relations and instructions for metering liquids are included. Separate chapters discuss liquids and their

properties as they affect flow, the theory of incompressible flow of liquids and the mea- surements required in the metering of liquids. One chapter describes several different apparatus and their use in the calibration of liquid flowmeters. The last chapter contains

brief descriptions of the many types of flowmeters such as differential pressure, positive

displacement, electromagnetic and ultrasonic. It also includes a discussion of the physical principles involved in their design and use.

Key words: Calibration; flow measurement manual; liquid flow; liquid flowmeters; metering.

1. Introduction

These notes have been written primarily to serve as an instruction manual for military

personnel engaged -in the metering of liquids and calibration of liquid flowmeters. The intent has been to include a condensed review of the fundamental quantities and relations en- countered in this work. References to more complete sources of properties of liquids, theoretical relations and instructions for metering and calibrating have been included.

Separate chapters discuss liquids and their properties as they affect flow, the theory

of incompressible liquid flow and the measurements required in the metering of liquids.

Other chapters are concerned with the methods of calibration of liquid meters and brief descriptions of the various types of meters.

Throughout this paper the system of units adopted by ASME has been used. This system

uses pound for both the unit of force and unit of mass, which is at variance both with

international practice and with the International System of Units (SI) which have been adopted officially by the National Bureau of Standards. However, the ASME system is followed

because this is the system used by almost all involved in flow metering in the U.S.

2. Units and Symbols

Throughout the world, in both science and industry, two fundamental systems of dimensions are in general use. They are the "gravitational" system based on force, length and

time and the "absolute" system in which mass, length and time are the primary units. In any coherent system of dimensions only three primary dimensions may be used and all other

dimensions are defined in terms of these three. Both of these systems are in use in the

United States. In the engineering profession customary or "English" units are commonly

used. In most scientific work throughout the world the absolute system is generally used;

originally in the C.G.S. system the mass length and time units were the gram, centimeter

and second, respectively. The International System of Units (SI) was defined and given official status by the General Conference on Weights and Measures in 1956. It is an absolute

system based on the kilogram, metre and second. Confusion resulting from this multiplicity

of systems and units is not surprising.

At its annual meeting in 1963, the ASME Research Committee on Fluid Meters voted to use "absolute" dimensions in the equations and physical data presented in the Sixth Edition

of its report [1]**. In that issue the pound is used to name both the unit of mass and

This work was sponsored by the Department of Defense, Calibrating Coordination Group

under Project 73-85 of DoD/CCG, FY 73 Flow Project, National Bureau of Standards Project 2130467.

Figures in brackets indicate literature references at the end of this paper.
unit of force. To prevent confusion the abbreviations for "pound-mass" and "pound-force"

are lbm and lbf. In the case of pressure abbreviations psi, psia, and psig "f" is omitted because it is obvious that pressure is force per unit area and "p" in these abbreviations is pounds force.

The major reason for using an absolute system based on mass rather than force in the

gravitational system is the constancy of mass. The mass of a substance or body does not

change with its location; it is the same on the surface of the earth or in space. Force,

as represented by weight, is a measure of the gravitational pull of the earth upon the body;

consequently the force may vary as the local value of the acceleration due to gravity, g Q.

The system and units adopted by the ASME will be used exclusively in this work. This

is done so as not to add to the current state of confusion. Table 1, a comparison of some

of the systems of dimensions, is included only for information and in the hope that it and

the following discussion will be of help in the understanding and use of the various systems.

The reader is cautioned to confine his dimensions and units to those prepared by the ASME.

Table 1. Comparison of Force-Mass-Length-Time Systems

Absolute Gravitational Absolute International

Item customary

M,L,t

customary

F,L,t

F,M,L,t System of Units [13]

(SI) M,L,t

Force, F poundal,pdl pound force,

lbf

pound force,

lbf

newt on, N

Mass, M

Length, L

pound mass,

lbm

foot, ft

slug

foot, ft

pound mass,

lbm

foot, ft

kilogram,

kg

metre, m

Time , t Velocity L/t

Acceleration L/t2

Pressure F/L2

second, s ft/s

ft/s2

pdl/ft2

second, s ft/s

ft/s2

lbf/ft2

second , s ft/s

ft/s2

lbf/ft2

second, s m/s

m/s2

N/m2

The relation between mass and force is derived from Newton's second law that force is

proportional to mass times acceleration,

F <x Ma = ML/t 2 (1)

The force F due to the gravitational attraction between the earth and a mass, M, is numerically

F = M gr,/gr (2)

where g^ is the local acceleration due to gravity and gc is a gravitational conversion

factor, the numerical value of which depends upon the units of F, M, and gj, as follows:

gc = 1, for F in poundals, M in pounds mass and g£ in ft/s2,

gc = 1, for F in pounds force, M in slugs, and g£ in ft/s2,

gc = 32.174, for F in pounds force, M in pounds mass, and g£ in ft/s2

gc = 1, for F in newtons, M in kilograms and gj, in m/s2.

Thus, in the absolute customary or "English" system based on the pound mass, the poundal is defined as the force required to accelerate a mass of one pound at a rate of one ft/s 2 and

F = Ma (poundals) (3)

In the absolute F, M, L, t system advocated by the ASME and used in the Sixth Edition

of Fluid Meters [1] and herein, the unit of mass is the standard pound mass (lbm), and the

pound force (lbf) is defined as the force required to accelerate a mass of one pound at g ,
the standard acceleration of gravity of 32.174 ft/s 2 . Stated another way, the lbf is the

force required to support a standard lbm in vacuo against standard gravity. Thus in pounds

force,

Ma/gf Ma/32.174

From which it is apparent that

1 lbf = 32.174 poundals

(4)

(5)

Although gc can be considered to be dimensionless, with only its numerical value a function of the units used, it is sometimes convenient to use the dimensions derived from

equation (4) so that

g. = 32.174 lbm.ft/lbf-s 2 (6)

This selected unit of force lbf will be used only as a measure of pressure or weight, never as a measure of the quantity of matter (mass). The use of the proportionality factor

gc will be required in computing weight of 1 lbm in the earth's gravitational field as

Weight = lbf = lbm g /gc (7)

where g. is the local acceleration of gravity.

"Weighing" is the process of determining either the mass or weight of a body depending on the apparatus employed. Weighing on a lever balance is a comparison between an unknown

mass and selected standards of mass. Effects of variations in the acceleration of gravity

are equal on both known and unknown mass; hence mass, not weight, of the unknown is determined directly. Spring balances and load cells will also indicate directly in mass if they

have been calibrated with known masses at their place of installation. If calibrated elsewhere a correction for local gravity must be applied. Pressures as indicated by columns

of manometric liquids will also require correction for local gravity. This will be shown

in a later section.

A complete list of the various parameters used in this work is presented in Table 2. The letter symbols, definition, dimensions, customary units, (SI) units and conversion factors are included. Every effort will be made to confine all calculations, examples, and

discussion to the customary units as listed in Table 2.

Symbol Definition Dimensions Customary Multiply To get

Units by SI Units

A Area L

2 ft 2 .092903 m2

c Velocity of sound

in fluid

L/t ft/s 0.3048 m/s

C Coefficient of

discharge

None

D Diameter L ft 0.3048 m

f Frequency fl hertz

gc Proportionality

constant

None 32.174

H Local acceleration

of gravity

L/t2 ft/s 2 0.3048 m/s2

9.80665 m/s2 So International standard L/t^ 32.174 ft/s2 acceleration of gravity

h Height of liquid

column

L ft 0.3048 m

Ah Differential height

of liquid column

(h rh2 )

L ft 0.3048 m
Symbol Definition

K Meter factor

a Characteristic length

L Length

m Mass rate of flow

M Mass

P Gage pressure

Ap Differential

pressure (p^-P2)

P Absolute pressure

ql Volume rate

of flow

R Reynolds number

s Time

T 2 Temperature

V Velocity w Specific weight

B Diameter ratio

e Angular displacement of

u3 Viscosity

v3 Kinematic viscosity

P Density

T Shearing stress

Dimensions Customary Multiply To get

Units by SI Units

(pulses per gallon)

L ft 0.3048 m

L ft 0.3048 m

M/t lbm/s 0.45359 kg/s

M lbra 0.45359 k8 , M/Lt 2 lbf/ft 2 47.880 N/m2 M/Lt 2 lbf/ft 2 47.880 N/m2

M/Lt 2 lbf/ft 2 47.880 N/m2 L3/t ft3/s .0283169 m3/s

None

t s - s - °F or °R see 5.1. °C or K

L/t ft/s

lbf/ft3

0.3048 m/s

D2 /D! (throat /upstream)

weigh b<jam, radians

M/Lt lbm/s- ft 14.8816 g/cm'S

L2/t ft 2 /s 9.2903 x 102 cm2 /s

M/L3 lbm/ft 3 16.0185 kg/m3

M/Lt2 lbf/ft 2 47.880 N/m2

Note:

1. q is also used as ft3/min and gallons/minute. To obtain U.S. gallons per minute

multiply ft3/s by 448.8311.

2. Conversion of temperature units is covered in another section.

3. Since viscosity and kinematic viscosity in the reference literature are usually

presented in poise and stokes, where poise is grams per centimeter second and

stoke has the units of cm2/s, it is more convenient to convert to the customary

units as follows

:

p, in lbm/s -ft = 0.067197 x (number of poises), and

v in ft 2 /s = 1076.39 x (number of stokes).

Liquids

3.1. Liquids Defined

Since this manual is concerned with the metering of liquids and the ca of liquid flowmeters, a short discussion is in order about what is a liquid

perties of liquids which affect the flow. For most engineering purposes a defined as a substance in such a state that its particles have sufficient c main in one mass but are free to move about within the mass by the applicat

force. This fluid property causes the liquid to conform to the shape of it

When at rest the mass of the liquid is confined by the bottom and sides of

presents a level upper surface except for the effects of surface tension at

libration

, and the proliquid can be

ohesion to reion of slightest

s container,

the vessel and

the boundaries.

An ideal or perfect liquid is one which for purposes of developing theory is

assumed to be both nonviscous and incompressible. Such liquids do not exist and theory

based upon such assumptions is subject in its application to corrections for the effects

of physical properties that have been neglected. This definition of liquids is seen to in- clude a wide variety of substances of diverse properties. It will include light and heavy

substances such as water and mercury, and it will also include materials of very different

viscosities such as gasoline and heavy lubricating oils.

It must be obvious that density and viscosity of liquids have a profound effect

upon the metering of liquids through conventional meters such as nozzles, orifices,
variable area meters, turbine-type meters and others. These effects will be discussed

later in Chapter 4. Definitions and measurements of density and viscosity will be treated

in this chapter.

3.2. Density of Liquids

The densities of flowing liquids are frequently required to convert between mass and

volumetric units of flow and flowrate. They are also required in the manometric measurement of liquid pressures.

The density of liquids is influenced only to a relatively small extent by variations

in temperature and pressure; however, for accurate determinations of flowrate these small

effects must be considered. The accepted definition of density is the mass of a substance

contained in a unit volume. In customary or "English" units this will be pounds mass per

cubic foot (lbm.ft~3) an(j ±n si units kilograms per cubic metre (kg.m~3).

Specific gravity is an abstract number expressing the ratio of the density of the test liquid to that of some other standard liquid at a stated temperature. Water is usually

used as the standard liquid. The temperature of both the test and standard liquids should

be indicated.

Density and specific gravity, although fundamentally different, can be determined in

the same way. A great number of methods of determination are indicated by Hidnert and

Peffer [2]. Of the eleven methods described, use of the hydrometer is the most commonly

used in flow laboratories and will be discussed herein.

3.2.1. Hydrometers

The accuracy of hydrometer observations depends upon the cleanliness of the instruments and of the liquids in which the observations are made. Extra care should be taken

to keep the liquid surface free of contamination. In order that the readings shall be uniform and reproducible the surface of the hydrometers, and especially of the stem, must be

clean, so that the liquid will rise uniformly and merge into an imperceptible film on the

stem.

In order that a hydrometer may indicate correctly the specific gravity of a specified

liquid, it is essential that the liquid be uniform throughout both in temperature and com- position. It is also necessary that the hydrometer be at the same temperature as the liquid. For most accurate results, the observations should be made at the temperature specified on the hydrometer. If the observation is made at some temperature other than that for

which the hydrometer is designed, the reading will be in error. The magnitude of the error will depend upon the thermal expansion of the hydrometer and of the liquid used.

If the latter properties are known, tables of corrections for temperature may be prepared for use with hydrometers at various temperatures. Such a set of tables [3] has been

prepared for liquid petroleum by the American Society for Testing Materials.

3.2.2. Test Procedure with. Hydrometers

Pour into a glass cylinder enough of the test liquid to float the hydrometer.

Pour the liquid slowly along the side of the cylinder to prevent the formation

of air bubbles.

Stir the liquid to assure that both liquid and hydrometer are at a uniform temperature and near that of the room. Stir carefully to avoid the formation of air

bubbles.

Immerse the hydrometer slightly below the point where it floats naturally, then

allow it to float freely.

Observe a point slightly below the plane of the liquid surface, then raise the

line of vision until this surface seen as an ellipse becomes a straight line.

The point where this fine line cuts the hydrometer scale is the reading of the

instrument. Observe the temperature indicated by a thermometer immersed in the

liquid.

5
e. Repeat c and d, average the readings and apply the scale correction to the average

reading.

3.2.3. Application of ASTM-IP Petroleum Measurement Tables

1. It must be emphasized these tables are applicable only for liquid petroleum products.

2. Tables involving the reduction of a specific gravity reading to a standard temperature assume the reading has been made with a glass hydrometer. The hydrometer

can be either a 60/60°F or an API type. Corrections for the thermal expansion of

the glass as well as for the liquid hydrocarbons have been incorporated in the

tables. When instruments other than glass hydrometers are used for the measurement of specific gravity, additional corrections will be required in reducing

to the standard temperature. The magnitude of the correction will depend upon the

temperature at which the observation is made and the expansion characteristics of

the particular instrument involved.

3. In the United States, all commercial weights are required by law to be weights in

air. The values listed in these tables are for weighings made in air and are in

pound mass (lbm) units rather than pound force (lbf ); however, as discussed herein, pounds per gallon (lb/ gal) will refer to pounds mass per gallon as weighed in

air (apparent density). Density as weighed in vacuo will be lbm/ gal. The density

of standard air for purposes of conversion from weights in vacuo has been taken as 0.001217 g'cm-3 at 60°F, 760 mm Hg pressure and 50 percent relative humidity.

Weighings are assumed to be made with brass weights having a density of 8.40 g«cm~3

at 32°F. When density in vacuo is required, a correction for air buoyancy must be

applied.

4. The density of air at standard conditions is, for practical purposes, 0.00915

lbm/gal. Its exact value can be determined, if required, by procedures outlined

in paragraphs 3059 and 3060 of API Standard 1101. On a percentage basis, the cor- rection for this effect depends on the density of the liquid. For liquids having

specific gravities in the range 0.7 to 0.8, the correction is about 0.14 percent.

5. The density of water in vacuo at 60°F is 0.99904 g/ml. The mass of a gallon of

water at 60°F in vacuo is computed as 0.99904 x 3785.3/453.59 = 8.3372 lbm in vacuo. The assumed apparent density of a gallon of water at 60°F in air of standard conditions against brass weights is 8.3282 lb/gal.

6. Since gravimetric calibrators weigh in air, for conversion of the data from gravimetric to volumetric units it is convenient to use lb/gal in air as obtained from

the tables. No correction for air buoyancy need be included in following this procedure.

7. As an illustration of one application of the ASTM-IP Tables assume the following

example. A volumetric flowmeter indicating in gallons is calibrated on a gravimetric calibrator using MIL-F-7024A, Type II at test temperatures in the range 80

to 85°F. It is desired to find the apparent density in lb/gal in air of the test

liquid throughout this temperature range.

The specific gravity of a sample of the liquid is measured with a 60/60°F glass

hydrometer. A previous calibration of the hydrometer shows this particular instrument has a scale error of -0.0008.

The observed reading is 0.7654 at 74°F

Applying the scale error correction -0.0008

The corrected observed specific gravity reading is 0.7646 at 74°F

Table 23, page 239 [3] gives a corresponding 60/60°F specific gravity of 0.7705.

Table 26, page 494 [3] gives an apparent density of 6.4146 lb/gal in air at 60°F

for this liquid. Table 24, page 426 [3] gives the expansion factors for this

60/60°F specific gravity hydrocarbon between the temperatures of 60°F and the

calibrating temperatures:
Calibrating Expansion Apparent*

Temp. °F Factor Density

lb/gal in air

74 .9922 6.3646

80 .9888 6.3427

81 .9883 6.3395

82 .9877 6.3357

83 .9872 6.3325

84 .9866 6.3286

85 .9860 6.3248

* The values of lb/gal in air at the listed temperatures were obtained by multiplying expansion factors times 6.4146, lb/gal in air at 60°F.

8. The above method is convenient in that apparent densities of the liquid hydrocarbons can be determined accurately at any temperature from a single specific

gravity observation made at any known temperature. The temperature of the observation must be known to within 0.5°F.

9. The above method is inconvenient in that it requires the use of many tables and

a calculating machine. When considerable calibration work is confined to one liquid, such as MIL-F-7024A, Type II, Tables 24 and 26 [3] can be given in chart

form. An example, lb/gal in air versus 60/60°F specific gravity for temperatures in the range 60 to 100°F is shown in Figure 1.

This chart requires that the 60/60°F specific gravity be known through the use of

Table 23 and the observed specific gravity reading at an observed temperature.

It will be found that the chart gives the same values of lb/gal in air vs. temperature as in the previous example for a liquid hydrocarbon having 60/60°F

specific gravity of 0.7705.

10. Another method can be used to compute apparent density provided the specific

gravity observation with the 60/60 °F hydrometer is made at the same temperature

at which apparent density is desired. Two factors must be considered in using

this method.

First, there is the thermal expansion of the glass in the hydrometer. A correction C must be made to the observed specific gravity, SG , to obtain the true

specific gravity SGt at temperature t; C = 0.000025 5/9 SGQ (60-t) is the cor- rection for the expansion or contraction of the hydrometer from the reference

temperature of 60 °F. For liquid hydrocarbons having a specific gravity range

0.7 to 0.9 the approximate value C = 0.0001 per 10°F is usually adequate. Note

that C has a negative value for temperatures above 60°F.

The second factor is that an additional correction for the buoyancy force of the

air is necessary when apparent density is desired, because true specific gravity

is the ratio of the density of the liquid at the observed temperature in vacuo divided by the density of water at 60°F in vacuo.

Consider the previous example of an observed specific gravity of 0.7646 at 74°F,

corrections for the hydrometer scale error having been previously applied.

SGQ

= 0.7646 at 74°F

C = -.00014

SGt = 0.76446 at 74°F

Density in vacuo = 0.76446 x 8.3372 = 6.37345 lbm/gal at 74°F

Correction for air buoyancy (par. 4)= -.00915

Apparent density = 6.3643 lb/gal at 74°F
6.40

Pi

l-H <

M

B

I

piw

p-l

o

6.38

6.36

6.34

6.32

6.30

6.28

.765 .770

60/60°F Specific Gravity P& '' 'V

Fig. 1 Temperature - apparent density relation of a typical hydrocarbon.

775
This is in good agreement with the value obtained previously using the tables.

Using the above procedure, specific gravity observations can be taken over the

temperature range of interest with any particular liquid. The apparent density

can be computed for each observation. A plot of apparent density versus tem- perature will tnen give apparent density ac any desired temperature within the

range investigated.

3.2.4. Density of Water

The density of pure water is usually determined from the measurement of temperature.

Table 3 presents both mass in air and mass in vacuo for one gallon of pure water at a pres- sure of one atmosphere. The weights in air are for dry air at the same temperature as the

water up to 40°C and at a barometric pressure of 760 mm Hg and against brass weights having a density of 8.4 g-cm~3. Above a water temperature of 40°C the temperature of the air

is assumed to be 20°C, i.e., the water is allowed to cool to 20°C before the weighings are

made. The volumetric computations are based on a water density of 1 kg per liter at 4°C; one liter = 1 dm3 = 10~3 m3 . The density in grams per milliliter (g/ml) is included in

the table.

Table 3. Density of Water

Temp Temp Density Density Density Apparent density

°C °F g/ml lbm/ft 3 lbm/gal lb/gal

32 0.99987 62.418 8.3441 8.3346

2 35.6 .99997 62.424 8.3450 8.3355

4 39.2 1.00000 62.426 8.3452 8.3359

6 42.8 .99997 62.424 8.3450 8.3357

8 46.4 .99988 62.419 8.3442 8.3350

10 50 .99973 62.409 8.3430 8.3338

12 53.6 .99953 62.397 8.3413 8.3322

14 57.2 .99927 62.380 8.3391 8.3301

16 60.8 .99897 62.362 8.3366 8.3277

18 64.4 .99863 62.340 8.3337 8.3248

20 68 .99823 62.316 8.3305 8.3216

22 71.6 .99780 62.289 8.3269 8.3181

24 75.2 .99733 62.259 8.3229 8.3142

26 78.8 .99681 62.227 8.3187 8.3100

28 82.4 .99626 62.193 8.3141 8.3054

30 86 .99568 62.157 8.3092 8.3006

32 89.6 .99506 62.118 8.3040 8.2955

34 93.2 .99440 62.076 8.2985 h.2900

36 96.8 .99372 62.034 8.2927 • 2842

40 104 .99225 61.942 8.2805 8.2723

50 122 .98807 61.681 8.2457 8.2369

60 140 .98323 61.379 8.2054 8.1966

70 158 .97779 61.040 8.1600 8.1514

80 176 .97182 60.667 8.1101 8.1015

90 194 .96534 60.262 8.0560 8.0474

100 212 .95838 59.828 7.9979 7.9894

3.2.5. Compressibility of Liquids

For most metering of liquids the effect of pressure on the density can be safely

ignored. However, for best accuracy at high pressures this effect should be considered

and if necessary accounted for. A pressure of 1 psi (lbf/in 2 ) results in the com- pression of water, one part in 300,000; thus at a pressure of 1000 psi the water is com- pressed approximately 1 part in 300 or about 0.3 percent. The compressibility of typical

hydrocarbons is approximately three times that of water; certainly the effect of pressure

on density should be included at these higher pressures.
If p]_ and P2 are the densities of liquids at pressures of p^ and P2 in atmospheres,

respectively, at any temperature the ratio of densities is related to the coefficient of

compressibility, b, by the equation

P1/P2 = l-b(p2 - pi) (8)

Table 4 is a tabulation of b for various liquids at a temperature of 20°C.

Table 4. Isothermal Compressibility of Liquids

Liquid b, atm- l

Acetone

Benzene

Carbon disulphide

Carbon tetrachloride

Ethyl alcohol

Glycol

n-Heptane

n-Hexane

n-Octane

Water

Illustrative example:

Find the density of water at a pressure of 500 atmospheres and a temperature of 20°C.

From Table 1 the density of pure water at 20°C and one atmosphere pressure is 8.3305 lbm

per gallon in vacuo. The coefficient of compressibility b for water is 46 x 10-

6, thus

128 x 10-6

95 IT 93 II 105 II 111 II 38 II 144 II 163 II 121 II 46 II

8.3305

= 1 - 46 x 10-6 (500 - 1)

P2

P

2 = 8.5262 lbm/ gallon

3.3. Viscosity of Liquids

Liquids at rest may be regarded as having perfect fluidity; however, in a viscous

slow-moving liquid internal shearing resistance arising from viscosity must be considered

and allowance made for its effects. In such so-called laminar flow, viscous forces usually predominate. In a laminar flowmeter, for example, streamline flow occurs in the metering element; the pressure drop for a given volume rate of flow is proportional to viscosity. Thus in these applications viscosity should be determined with the best possible

accuracy.

In turbulent flow, inertia forces resulting from liquid density will predominate, and

viscous forces have only a rather small secondary effect. Orifice meters and nozzles are examples of instruments in which turbulent flow usually occurs, and values of viscosity to an accuracy of a few percent are usually adequate under these conditions.

Some flowmeters may have either laminar or turbulent flow at the metering element

depending upon the size of the meter, the rate of flow, and the viscosity of the fluid.

In this case, laminar flow is approached as meter size and rate of flow decrease or as viscosity of the fluid increases. Small variable-area meters are an example, especially

when they are used to meter fluids of high viscosities.

Dynamic viscosity u is a measure of the resistance which a fluid offers to change of

shape, i.e., to shearing. The nature of this quantity u. sometimes called the coefficient

of viscosity, can be indicated clearly by considering the laminar motion of a viscous

fluid bounded by two flat parallel plates, one of which is stationary and the other moving

parallel to its surface with a constant velocity V as shown in Figure 2.

10
Moving Plate

Stationary Plate

Fig. 2 Laminar flow between parallel plates in relative motion.

With no slip at the boundaries, the fluid layer at the lower plate will have zero velocity

and the layer at the upper one a velocity equal to V. The particles in the uppermost layer

are carried along with the moving plate and this layer in turn imparts a motion to the one immediately below it. At any point at a distance y above the lower plate the velocity is

u Vy/b where b is the distance between plates. The rate of shear is 3u/9y = V/b and in

this case is constant throughout the fluid. The shearing stress at any point is

T = ^ = vV/h (9)

The force necessary to move the plate is simply the value of t multiplied by the area of

the plate.

This arrangement of two parallel plates serves as the basis for the definition of the

coefficient of viscosity when equation (9) is solved for p

u = t b/V

A definition for the more general case is

T U ~ 3u/3y

(10)

(ID

In some problems it is convenient to introduce a modified form of the coefficient of

viscosity, to be obtained by dividing u by the density of the fluid, p. This quantity denoted by v has the value

v = w/p (12)

and is commonly known as the kinematic coefficient of viscosity.

3.3.1. Dimension and Units of Viscosity

The coefficient of viscosity Eq. 10 and the coefficient of kinematic viscosity Eq. 12 are not dimensionless quantities, and their values will depend on the units employed for

their measurement. Introducing the fundamental dimensions of mass, length and time in

Eq. 10

U = T b „. Force Length ML < 1 V

" Area ' Velocity t? * L2

hi

L M_Lt

(13)

11
thus u has the dimension M/Lt . In the customary or "English" system p has no assigned name but is usually expressed in pounds mass per foot second (lbm/ffs). In the International system of Units (SI), the unit is the poise, which is 10"1 kg/m-s or one gram

per centimeter second (g/cm*s). The conversion between these units of coefficient of

viscosity is

1 lbm/ft-s = 14.8816 g/cm-s or poise

Similarly, introducing the fundamental dimensions into Eq. 12

, M . M L

2 V = VI /pM = / —T = Lt ' L-* T (14)

Thus in the "English" system of units the kinematic viscosity has the dimensions of ft 2 per second (ft 2 /s). The unit in SI is the stoke (St) [13] which is equal to 10~^m2 /s or cm2 /s. The conversion between these units is

1 ft 2 /s = 929.03 cm2 /s or stokes

For more manageable numbers the coefficients of dynamic and kinematic viscosities

are tabulated in centipoise and centistoke where

centipoise = poise x 10 2 and

centistoke = stoke x 10 2

The coefficients of dynamic viscosity in centipoise for a few common liquids at 20°C are tabulated in Table 5.

Table 5. Dynamic Viscosities of Liquids at 20°C

Liquid Viscosity, Centipoise

Acetone

Benzene

Carbon tetrachloride

Ethyl alcohol

Ethylene glycol

Glycerin

Heptane

Hexane

n-Octane

Turpentine

Water

Gasoline

Kerosene

Light lubricating oil

Medium lubricating oil

Heavy lubricating oil

3.3.2. Effect of Pressure on Viscosity

The effect of changes of pressure on the viscosity of liquids is practically negligible under ordinary conditions. Some liquids such as ether and benzene show a slight in- crease while water exhibits a decrease with increasing pressure. However, the change can- not be ignored when high pressure is used or when viscosity-sensitive meters are used.

Results of viscosity tests by Oilwell Research, Inc. on MIL-F-7024A, Type II naphtha and

MIL-H-5606A hydraulic oil show the following changes in viscosity with pressure:

12

0.33

.65

.97

1.20

19.9

490.

.41 .33

.54 1.49

1.00

.3 - .6 2.0

2.5 - 150

150 350

350 2000
For MIL-F-7024A Type II

y =v * (1 + °- n

f§00> (15)

where v a is the kinematic viscosity at atmospheric pressure and v is the viscosity at the

elevated pressure in psig.

For MIL-H-5606

v - va(l + °- 13 m> (16)

3.3.3. Effects of Temperature on Viscosity

The viscosity of liquids decreases with increasing temperatures. The effect of temperature on the viscosity of water is shown in Table 6.

Table 6. Effect of Temperature on the Dynamic and Kinematic Viscosities

of Water

Temperature

°C DynamicViscosity

Centipoise

Density

g/cm3

Kinematic

Viscosity

Centistokes

5 10

15

20

2530

35

40

45

5055

60

65

707580

85

90

95

100

1.787

1.519

1.307

1.139

1.002

0.8904

0.7975

0.7194

0.6529

0.5960

0.5468

0.5040

0.4665

0.4335

0.4042

0.3781

0.3547

0.3337

0.3147

0.2975

0.2818

0.9998

1.0000

0.9997

0.9991

0.9982

0.9970

0.9956

0.9940

0.9922

0.9902

0.9880

0.9857

0.9832

0.9806

0.9778

0.9748

0.9718

0.9686

0.9653

0.9619

0.9583

1.787

1.519

1.307

1.140

1.004

0.8931

0.8031

0.7237

0.6580

0.6019

0.5534

0.5113

0.4745

0.4421

0.4134

0.3879

0.3650

0.3445

0.3260

0.3093

0.2941

Unfortunately there is no such simple table for the hydrocarbon liquids used in military calibrating laboratories. These are not pure compounds but consist of various blends

of liquid hydrocarbons. As a consequence, viscosity measurements of these liquids are

required. For best accuracy the viscosity should be determined at the temperature of the

flowing liquid. Where a lesser accuracy is permitted, such as its value for Reynolds

number determinations, the temperature viscosity relationship can be shown on ASTM Viscosity-Temperature Charts [4]. These charts are a convenient means for the determination of

viscosity of petroleum oils at any temperature, within a limited range, provided viscosities at two temperatures are known. The charts are so constructed that for any given

petroleum oil the viscosity-temperature points lie on a straight line. There are six such

I charts whose viscosity and temperature ranges are shown in Table 7.

13
Table 7. ASTM Viscosity-Temperature Charts

Chart Viscosity Range Temperature Range

A 33 - 1x108 SAYBOLT -30 to +450°F

UNIVERSAL SECONDS

B 33 - 1x105 " "

" -30 to +350°F

C 2 - 2xl07 Centistokes -30 to +450°F

D 0.40 - 100 " -30 to +450°F

E 2.0 - 2x10? " -100 to +450°F

F 0.40 - 2xl0 7 " -100 to +700°F

A portion of Chart D is shown in Figure 3 which also includes an example of its use.

Illustrative example:

Find the kinematic viscosity at 75°F of a petroleum oil which has viscosities of 70

and 1.95 centistokes at -30 and 100°F, respectively. Plot the two known viscosity temperature points on the chart as shown. Draw a sharply defined straight line through them.

The desired viscosity at 75°F at the intersection of the straight line with the 75°F

line is shown to be 2.75 centistokes.

If viscosities at two temperatures are not known they should be determined in accord- ance with the Method of Test for Saybolt Viscosity (ASTM Designation: D 88) or the Method

of Test for Viscosity of Transparent and Opaque Liquids (Kinematic and Dynamic Viscosities)

(ASTM Designation: D 445).

Conversion of centistokes to Saybolt seconds may be obtained from the Method for Conversion of Kinematic Viscosity to Saybolt Universal Viscosity or to Saybolt Fural Viscosity (ASTM Designation: D 2161).

More accurate measurements of kinematic viscosity are made with capillary-tube viscometers such as the straight tube (glue-type), Ostwald, Cannon-Fenske , Sil, Ubbelohde and

Cannon master (this is not intended to be a complete listing of types). Directions for

the use of these viscometers are available in [5] and [6].

3.4. Vapor Pressure of Liquids

Vapor pressure is defined as that pressure existing above a liquid in equilibrium with

its own vapor. Vapor pressure is a function of the liquid and of its temperature. For instance the vapor pressure of water at 20° and 100°C is 0.339 and 14.696 lb/in2

, respectively. The maximum Reid vapor pressure of JP-4 Jet Fuel is specified to be 7 lb/in 2 at 100°F

and that of MIL-F-7024A, Type I is 2 lb/in2 . The vapor pressure of MIL-F-7024A, Type II, a less volatile liquid, is considerably lower.

A minimum operating back-pressure for any flowmeter installation should be maintained

to preclude a change in the meter calibration factor due to two-phase (liquid and air or vapor) conditions. This pressure is a function of the vapor pressure of the liquid and

the presence of dissolved gases, and can be experimentally determined under actual test conditions if necessary. For turbine flowmeters it has been defined as the line pressure

at which the calibration factor (pulses per gallon) at 125 percent of the nominal maximum

flowrate increases 1/2 percent over the corresponding calibration factor obtained at the same flowrate but with a pressure 10 lb/in2 higher. The operating back-pressure is mea- sured four pipe-diameters downstream of the turbine flowmeter.

Where these tests are not made, the back-pressure should be set to a pressure not less

than the sum of the vapor pressure plus three times the measured pressure drop across the

flowmeter. For example, a turbine flowmeter having a pressure drop of 7 lb/in2 at the

maximum rated flow of JP-4 jet fuel should have a back-pressure of not less than

14
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TEMPERATURE, °F

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Fig. 3 Effect of temperature on the viscosity of a typical hydrocarbon liquid.

15
7 + (3 x 7) = 28 lb/in2 psia

Other flowmeters such as orifice, nozzle and Venturi meters are also subject to error due to cavitation. The pressure at the throat or downstream pressure tap should always be

in excess of the vapor pressure of the liquid. Although it is conservative, whenever possible the minimum pressure recommended for turbine flowmeters should be used for these

other types of flowmeters.

4. Flow in Closed Conduits

4.1. Conservation of Mass

Consider a steady state of flow of liquid through a conduit such as that shown in

Figure 4, which might be a Venturi meter.

Fig. 4 Flow in a closed conduit.

If there is no flow through the walls either into or out of the conduit and further if

there is no liquid accumulated or destroyed within the conduit, then the rate of mass flow

passing any cross section of the tube must be the same. Referring to Fig. 4

m = pAV = PiAjVi = P2A2V2 = P3A3V3 (17)

having the dimensions M/t such as lbm/s, where p represents the density of the liquid in

lbm/ft3, A the area of the cross section in ft2 and V the average velocity of the liquid in ft/s at that section. The subscripts 1, 2 and 3 refer to the particular cross sections of

the tube. Equation 17 represents the general statement of the conservation of mass or con- dition of continuity which must be satisfied in most problems in liquid flow. In flows in

which the liquid can be considered incompressible (i.e., the density p is a constant) the

equation above may be written

q t = A]Vi = A2V2 = A3V3 (18)

having the dimension L^/t such as ft^/s.

Calibrations of fluid meters usually are based on either Equation 17 or 18. Gravimetric (mass vs time) calibrations utilize Eq. 17 while volumetric (volume vs time) calibrations may employ Eq. 18; however, for most accurate results especially where there may

be temperature differences between stations Eq . 17 should be used.

Extending this reasoning to a calibration problem, consider a steady mass rate of

flow through a system consisting of several flowmeters in series with a weigh-time calibrator shown in Fig. 5.

16
Valve

Orifice

T

Nozzle

"TTTurbine

r

Weigh Tank

Diverter

Fig. 5 Flow through several meters in series.

If there is no leakage in the entire system and a steady state of flow is established past

section 1 the conservation-of-mass equation states

m^ = m2 = m3 = M/t (19)

where mj_, m£ and 1113 are the mass rates of flow through the three sections and meters and

M is the mass collected in the weigh tank during collection time t. Thus either the orifice meter, nozzle or turbine meter or all of them can be calibrated with respect to the

weigh-time calibrator. Another method of calibration is to use one of the three meters

as a transfer standard and calibrate the other two with respect to it.

4.2. Energy of Liquids in Motion

Liquids either at rest or in motion must conform like all other matter to the law of

conservation of energy. In the case of liquids in motion three forms of energy are of

greatest importance. They are kinetic, pressure and potential energies. Thermal or internal energy in general is of no great consequence and in the case of liquid flows in which

there is no addition or extraction of heat it can be safely ignored. Energy in all forms

is defined as the ability to do work. Each of the three forms kinetic, pressure and

potential described herein has that ability and the same fundamental dimensions.

Kinetic energy is that energy which a mass possesses by virtue of its velocity and is

given by the relation, for the absolute F, M, L, t units,

MV2 Total kinetic energy = 2gc For the case of liquids flowing through a conduit and assuming a uniform velocity across

a section and applying the same dimensional units

V

2 Kinetic energy per unit mass = =

which has the dimensions of ft«lbf/lbm.

The pressure maintained in a confined liquid also represents the ability to do work.

If a vertical tube is connected to the conduit, the liquid will rise to a height h equal to p/w. In other words, the liquid flowing within the conduit could be lifted to height h by

virtue of its pressure p. In the dimensions and conventions applied herein

17
Pressure energy = p/p ft-lbf/lbm

The potential energy is that energy a mass possesses because of its elevation h

above some arbitrary datum and will have the same dimensions as kinetic and pressure

energies

Potential energy = hw/p = h gjj/gc fflbf/lbm

Notice that all three forms of energy have the dimensions of fflbf/lbm which is proportional to the height of a column of liquid. To determine the total energy of the liquid simply multiply by the pounds mass involved.

The general energy equation is simply an accounting of the energy passing stations

1 and 2 shown in Fig. 4. In the case of liquid flow through a meter or meters in series

where there is neither thermal nor mechanical energy addition or subtraction between station 1 and 2 the energies at the two stations must be equal.

2gc £1

V 9 2

n 2 , 22 + £ + ^1 g £ /gc = 2g7 +

P^ + h 2 *l'*c (20)

Equation 20 was first proposed in 1738 by Daniel Bernoulli and is now known as Bernoulli's theorem.

4.3. Differential Head Meters

Since the liquid is virtually incompressible and the temperature does not change,

P]_ = P2 and from equation 18

or Al vl = A2 V2

Vl = % v2 (21)

If we assume the axis of the meter run to be horizontal, the elevations of section 1 and

2 are equal and the potential energy terms can be deleted. Substitute Eq. 21 in Eq. 20

to get

V2

2 = 2 g c p l"p 2 A„

(22)

Let the diameter ratio

then

and V2

2 = 2gc

^' 6

B* - <gi

Pl-p?

, p \ (23)

The factor P1-P2 is equal to the difference between the static pressures and stations 1 and

2; if these pressures are measured with columns of the flowing liquid then

21ZE-2- = (h

x - h 2 ) g£/g c

This difference in the height of the two columns is generally referred to as the "head"

of the meter and is thus responsible for the term "differential head meter." The term

1/(1-6*) is called the velocity of approach factor. The theoretical velocity at section

2 from Eq. 23 is

18
v2 =

2 s c

( pi-p?)

p

1/2

1 1-B 4

1/2

(24)

and the theoretical volumetric rate of flow from Equations 18 and 24 is

2gr (p i-p ?,) P

1/2

q t = A2V2 = A2

and the theoretical mass rate of flow from Eqs. 17 and 24

,1/2 f 1

1

. 1-8 4

i t = pA 2 V2 = A2 [2gc p(p!-p2)] 1-6 4

1/2

1/2

(25)

(26)

The actual flowrate, either mass or volumetric, is rarely if ever equal to the theoretical

as predicted by the equations above. To obtain the actual flowrate it is necessary to introduce a factor called the "Coefficient of Discharge" defined by the equation

C = actual rate of flow

theoretical rate of flow

The actual volumetric and mass rates of flow through Venturi tubes, nozzles and orifices

are then

q = CA2

m = CA2gr (Pl~P?)

P

1/2

1 1-B 4

1/2

and

2gcP(Pl_ P2)

1/2

ll-B'

1/2

(27)

(28)

Equations 27 and 28 can be used to solve for the rate of flow through orifices,

nozzles and Venturi meters provided the discharge coefficient is known. Unfortunately

this coefficient C is not a constant for a given meter, but is found to vary with density,

viscosity and velocity of the flowing liquid. However, it can be shown that C is a function of Reynolds number R, which is a dimensionless ratio of the aforementioned parameters

plus a characteristic length £, usually a diameter, either D^ or D2

R =

p V£

= viN (29)

Of course the dimensions of all parameters must be consistent so that the numerical value

of R is independent of the system of units used. For instance

p - lbm/ft^ or gram/cnH

V - ft/s or cm/s

£ - ft or cm

Vi - lbm/ffs or gram/cm*

s

Thus, with either system of units R is the same and is truly dimensionless.

Typical curves showing the variation of C with R for orifice and nozzle meters are

shown in Figure 6, and numerical values of C vs R for a wide variety of orifices and nozzles

are presented in [1]. Because of the functional relationship between C and R it is evident

that the calibration of such a meter requires measurements over the entire range of

Reynolds numbers. In a weigh-time calibration, C can be determined from Eq 28 and R from Eq. 29 rewritten as

19
COEFFICIENT OF DISCHARGE

o o

00o o

o

o

o

en

©en

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R= £V2Dz = ^E_ (30)

U TIUD2

It is evident, for a given meter, R can be increased through an increase in the mass rate

of flow and by a reduction in the viscosity of the flowing liquid.

5. Measurements Required

5.1. Temperature

Measurement of the temperature of flowing liquids is necessary for the accurate

determinations of their densities and viscosities. For a typical hydrocarbon such as MIL-F-7024A Type II, an error of 1°F in the measurement of its temperature will result

in an error of about 0.06 percent in density and 0.8 percent in viscosity.

The units of temperature commonly used are degrees Fahrenheit (°F) , degrees Celsius

(°C) and the corresponding absolute temperatures degrees Rankine (°R) and Kelvin (K) . In liquid flow problems, absolute temperatures are seldom required, however, they are included

for information and completeness. In engineering work in liquid flow problems, the Fahrenheit temperature scale is the one most frequently used. However, a working knowledge of

converting from one scale to the other is required because properties of the various liquids found in the literature are frequently referred to the Celsius scale of temperature.

The relative sizes of the degree Fahrenheit or Celsius is a result of the defined fixed points on the temperature scales. The temperature of equilibrium between ice and airsaturated water is (32°F) (0°C) and the temperature of equilibrium between liquid water and

its vapor (steam point) is (212°F) (100°C) . Both fixed points are at a pressure of one standard atmosphere (14.6959 lbf/in2 ), (101325 N/m2)

.

Thus

Temperature °F = 9/5 Temperature °C + 32 (31)

Temperature °C = 5/9 (Temperature °F - 32) (32)

Temperature °R = Temperature °F + 459.67 (33)

Temperature K = Temperature °C + 273.15 (34)

5.1.1. Illustrative examples:

1) A thermometer reads 72.5°F. What is the absolute temperature in °R?

T = °F + 459.67 = 72.5 + 459.67 = 532.17 °R.

2) A thermometer reads -72.5°F. What is the absolute temperature in °R?

T = °F + 459.67 = -72.5 + 459.67 = 387.17 °R.

3) A thermometer reads 35°C. What is the temperature in °F and absolute temperature

in °R?

T = 9/5 (35) + 32 = 95 °F. T = 95 + 459.67 = 554.67 °R.

5.1.2. Temperature Measurements

Usually the temperature of liquids is measured with immersion sensors such as thermometers, thermocouples and thermistors. In general, the accurate measurement of temperature

of liquids with calibrated sensors of these types is not difficult providing a few precautions are taken. Any of these sensors, when immersed in a fluid, attain a state of

thermal equilibrium with the fluid but do not necessarily arrive at the same temperature

as the fluid. At equilibrium the transfer of thermal energy between the sensor and

surrounding fluid by convection is equal to the transfer of thermal energy between the

21
sensor stem and its surroundings by conduction and radiation. The difference between the

temperatures of the sensor and liquid is reduced by increased convective transfer between

sensor and liquid and reduced thermal transfer through the sensor stem. Both of these are attained through intimate thermal contact between sensor and liquid and through increased

depth of immersion of the sensor. In meters equipped with thermometer wells the thermometer should not be placed in a dry well. To improve thermal contact the well should be

filled with liquid.

When measuring the temperature of liquid with a thermometer as when hydrometer observations are being made, throughly stir the liquid to eliminate thermal gradients and wait

several minutes before making observations to ensure thermal equilibrium between liquid,

hydrometer and thermometer.

5.2. Pressure Measurements

The unit of pressure to be used in the calculations of this work is pounds force per

square foot (lbf /f

t

2 ) . Pressure gages are usually calibrated in lbf/in2

; however, lbf/ft 2 is determined by multiplying lbf /in2 by 144 in 2 /ft 2 . It will be necessary to designate

whether the value of pressure is

a) Pressure difference between two points in a flow system, (Ap) in psid (pounds per

square inch differential);

b) A pressure relative to the existing atmospheric pressure, using negative values

to indicate pressures below atmosphere, (p) in psig (pounds per square inch gage);

c) Absolute pressure, (P) in psia (pounds per square inch absolute ) where

P = p + barometric pressure.

In the direct measurement of pressure difference Ap, it is necessary that both sides

of the pressure sensing element be connected to pressure taps in the flow system. The

difference in pressures at these two taps will be derived from the indication of the sensor. Instruments used for the measurement of pressure difference include manometers,

differential pressure gages of the corrugated diaphragm capsule or Bourdon tube design,

and differential pressure transducers of the pre-stressed diaphragm type.

Instruments used for the sensing of pressure p relative to atmospheric pressure must

have one side of the pressure sensing element vented to atmosphere. Instruments used for

this purpose include manometers, Bourdon tube gages, and aneroid capsule-type gages.

For the direct measurement of absolute pressure P, it is necessary that one side of

the pressure sensing element be exposed to zero absolute pressure. Barometers are one form of such instruments. However, the absolute pressure of a point in the flow system

usually is not measured directly. Rather, gage pressure and barometric pressure are measured separately and the absolute pressure is the sum of gage and barometric pressures.

In nearly all work in metering liquids, and in calibrating liquid flowmeters the

measurement of Ap usually will be the only pressure measurement of interest. Gage and

absolute pressures will be needed only when these pressures are sufficiently great to alter the properties of the flowing liquid, or when two-phase flow is possible at low

pressure.

It was shown in Fig. 4 and Equations 27 and 28 in Chapter 4 that both the volumetric

and mass rates of flow through a calibrated meter (orifice, nozzle and venturi types) can be determined from the measurements of pj and p£ or Ap. Conversely, in the calibration of

such meters either the mass or volumetric rate of flow and Ap must be measured in order to determine C, the coefficient of discharge.

Gage pressures p± and P2 can be measured separately and the difference obtained

through subtraction; however, generally it is more accurate and convenient to directly measure the pressure difference with a differential gage such as a manometer.

22
5.2.1. Manometers

Manometers contain a liquid such as water or mercury in which the difference in liquid

levels increases with increased pressure difference. The positions of the liquid levels

are indicated by a scale, calibrated in units of length, placed beside the manometer column(s). Thus, this device senses pressure directly, but its indication, the difference

in liquid levels, is only proportional to pressure. The manometer like the pressure gage

will measure pressure difference, gage pressure, or absolute pressure depending upon its

design.

If the pressure gages shown in Fig. 4 are replaced with vertical tubes, the liquid

will rise in the tubes to heights h^ = pi/w and h2 = P2/w shown in Fig. 7a where w is the

specific weight of the liquid in lbf/ft-*, p is the pressure in lbf/ft* and h is the column height in feet, thus

Ap = wAh

and if the density of the liquid is known, as is usually the case,

Ap = Ahpg £ /gc This is a very important relation; unfortunately, it also is one that is frequently

neglected. To reiterate, conversion of a measured column of liquid to pressure or differential pressure requires both the density of the liquid and the local acceleration of

gravity.

Obviously at high pressures these columns will be too long to measure conveniently.

Two forms of manometers that circumvent this difficulty are shown in Figures 7b and 7c.

Fig. 7b shows a U-tube as used in practice as a differential gage with heavy liquid such

as mercury in the bend of the U-tube, as compared to the light liquid flowing through the

Venturi meter. Fig. 7c shows an inverted U-tube used as a differential gage, with a light

liquid (non-miscible) or a gas (air) in the bend of the U and the heavier liquid flowing

through the Venturi meter. It can be shown that in figure 7b

Ap = Ahm (pm

- Pl ) g £ /gc (35)

where Ahm is the differential height of the manometric liquid, pm and p]_ are the densities

of the manometric and flowing liquid respectively, and in figure 7c

Ap = Ahm ( P]_ - p a ) g £ /gc (36)

where p ais the density of the lighter liquid or gas above the manometric liquid which is

the same as that flowing through the meter. In this case with gas in the upper portion of

the inverted U-tube, the absolute pressure and temperature of the gas will be required

to determine its density.

Errors will be introduced in the pressure measurements if any air or vapor is trapped

in any part of the pressure connecting lines between the pressure holes in the meter and

the manometer or differential pressure gage. These lines including manometer or pressure

gage must be completely filled with the flowing liquid. In the case of meters installed

in a horizontal line the pressure holes or taps should be in the horizontal plane of the

center line of the pipe, or any other convenient position in the lower half of the pipe.

Recommended arrangements for piping between the meters and pressure sensing elements

are shown in Fig. 8. The piping should include air or vapor vents, a by-pass or equalizer

valve and the piping should be sloped as shown to facilitate purging the lines of air or vapor and to minimize the possibility of trapping same.

23
X

Vents

Ah.m

u

Fig. 7 Manometers.

24
wH$W«

O^

wH983

Of

25
5.2.2. Corrections and Use of Conversion Factors for Manometers

In converting an indicated level difference or height Ahm

of a manometer column to pressure in lbf/ft , it is necessary to consider the following:

a) The units of length in which the height Ah is expressed (inches, centimeters,

etc.).

b) The manometer liquid (water, mercury, hydrocarbon, etc.) and the flowing liquid

(water, hydrocarbon).

c) The density of the manometer liquid (s) and the density of the gas above the

liquid as in Fig. 7c.

d) The local acceleration of gravity g£-

The pressure differential Ap in lbf/ft can be determined from Eq. 35 as follows:

When the scale of the manometer is calibrated in units of length other than feet, i is necessary to convert to the equivalent of length in feet with appropriate conversion

factors. When the scale is graduated in centimeters divide the observed Ah by 30.48

to obtain Ahm in feet; when graduated in inches divide the observed Ah by 12.

Densities required in Eq. 35 of three common manometric fluids-water, mercury and

air are shown in Table 8. The density of hydrocarbons can be determined as discussed

in Chapter 3.

Table 8. Densities of Water, Mercury and Air

Temperature Water Mercury Air at 14.696 psia

°F lbm/ft3 lbm/ft 3 lbm/ft3

32 62.418 848.719. .08073

34 62.422 848.547 .08040

36 62.425 848.375 .08008

38 62.426 848.203 .07976

40 62.426 848.031 .07944

42 62.425 847.860 .07912

44 62.423 847.690 .07880

46 62.420 847.520 .07849

48 62.415 847.349 .07818

50 62.410 847.178 .07788

52 62.403 847.008 .07757

54 62.395 846.837 .07727

56 62.387 846.666 .07697

58 62.377 846.494 .07667

60 62.367 846.324 .07638

62 62.355 846.154 .07609

64 62.343 845.984 .07579

66 62.330 845.814 .07551

68 62.316 845.644 .07522

70 62.302 845.473 .07494

72 62.286 845.303 .07465

74 62.270 845.133 .07437

76 62.253 844.964 .07410

78 62.235 844.794 .07382

80 62.217 844.623 .07355

82 62.198 844.452 .07328

84 62.178 844.282 .07301

86 62.157 844.112 .07274

88 62.136 843.942 .07247

90 62.114 843.772 .07221

92 62.091 843.603 .07195

26
Temperature Water Mercury Air at 14.696 psia

°F lbm/ft3 lbm/ft3 lbm/ft3

94 62.068 843.434 .07169

96 62.044 843.265 .07143

98 62.020 843.095 .07117

100 61.995 842.926 .07092

5.2.3. Illustrative examples:

1. Consider first a manometer of the type shown in Fig. 7b in which the manometric

liquid is mercury and the lighter liquid flowing through the meter and filling

the manometer above the mercury is water. Let the temperature of the manometer

liquids be 72°F, the differential mercury column 52.8 cm of mercury and the local

acceleration of gravity 32.120 ft/s 2 . Ap in lbf/ft 2 is determined by solving

Eq. 35 with the densities obtained from Table 8 as follows

Ap = ~~ (845.303-62.286) ff^ff = 1354.13 lbf/ft 2

2. A water manometer of the type shown in Fig. 7c is used to measure the differential

pressure between the two stations in a water meter. Temperature of the water in

the manometer is 72°F. The differential height is 36.30 inches of water, the

pressure of the compressed air above the water columns is 151.690 psig and gn is

again 32,120 ft/s 2 . Find Ap in lbf/ft 2 .

Everything is known for the solution of Eq. 36 except the density of the air (p a ) above the water columns. It can be determined from the density at 14.696 psia

as given in Table 8 and the principle that the density of ideal gases is directly

proportional to the absolute pressure.

p Thus, p a = p s -a

r s where Pa

is the absolute pressure above the water columns and p s is the density

of air as listed in Table 8 at the absolute pressure Ps of 14.696 lb/in2

;

n-,,, c 151.690 + 14.696 . ..... ,, .. o p a

= .07465 x . 7 . = 0.84518 lbm/ftJ

and Ap = ^~^- (62.286 - .845) ff^ff = 185.55 lbf/ft 2

5.2.4. Mechanical Pressure Gages

When a mechanical pressure gage of the Bourdon tube or corrugated capsule type is used

to measure either pressure or pressure differential in a liquid flow system, no correction

is necessary for the local acceleration of gravity provided the gage was calibrated in

force units. If the gage is calibrated with a dead-weight gage at a location having other

than a standard acceleration of gravity (32.174 ft/s 2 ), allowance must be made for the

local value of the weights of the calibration masses.

In the measurement of pressure at a pressure tap with a mechanical pressure gage, the

indicated pressure must be increased or decreased by a pressure equivalent to the hydrostatic head of the vertical distance of the gage above or below the pressure tap. This

correction is

P = hp g£/gc

However, this correction is valid only if the pressure tubing between the tap and gage

is filled completely; if not, the proper correction cannot be determined.

27
In the measurement of differential pressure with a mechanical gage no such correction is to be applied. The connecting tubes should be installed as in Fig. 8 and must

be completely filled with liquid.

5.3. Volume Measurements

Obviously an initial determination of the volume of volumetric calibrators or provers

is necessary. Subsequent measurements of the volume are required in the event of damage

or suspected damage to the prover. Finally, good calibration practice requires routine

repeat measurements of this volume at fixed time intervals. Two general methods for the

calibration of volumetric provers are used. These are: 1. calibration by means of test

measures, and 2. calibration by means of master meters. The first is the preferred method

in which the volume of the liquid is withdrawn from the prover into certified test measures or the prover is filled with liquid from the test measures. The procedures for call

brating volumetric provers is treated in detail in [7].

Test measures used for calibrating volumetric provers are ordinarily of 1, 5, 10,

50 and 100 gallon capacity. These measures can be calibrated either "to contain" or "to deliver." Measures certified "to contain" will contain the volume specified, regardless

of the type of liquid used. Measures certified "to deliver" will deliver accurately only

when water is used and must be wetted and drained before each use. The drainage time for measures 10-gal and smaller is 10 seconds from the time flow ceases and dripping begins,

and 30 seconds for measures exceeding 10 gallons.

The volume of the standpipe calibrator is determined from the volume of the test measures required to fill the volume between the two level sensors. The volume of the ballistic calibrator is measured by the volume of liquid discharged into test measures by the

passage of the piston or spheroid between the two position sensors.

In these calibrations it is necessary to accurately determine the volume of a partially filled test measure. This may be done with glass graduates or gravimetrically.

Temperatures of the liquid in the test measures and in the prover must be measured

to enable volume corrections due to changes in density of the liquid.

Detailed instructions for these volume determinations are beyond the scope of this

manual. When these measurements are required the instructions in [7] should be followed.

5.4. Time Measurements

Electronic-counter timers are preferred for measuring the time interval of either

gravimetric or volumetric collection of the liquid. Many makes are available which operate on the principle of counting the pulses from a quartz-crystal oscillator. They usually indicate in units of 0.001 second and have provision for electronic actuation. They

are considered to be preferable to synchronous clocks and stop watches actuated mechanically. Standards of time and frequency are broadcast by the National Bureau of Standards

over station WWV near Fort Collins, Colorado. These signals, accurate to a few parts in

10-^-3, are convenient references for the calibration of timers.

5.5. Mass Measurements

Gravimetric-type calibrators use a number of different types of scales. For the

smaller capacities equal-arm balances and straight lever systems are used. Because of

their inherent stabilities torsion-tape and flexure-plate balances are also used. In general, free-swinging lever scales of conventional design are preferred; however, quickweighing and automatic indication devices are sometimes employed. Before using the scale

for calibration purposes a test extending over the load range at which it will be used

should be conducted. For large weighing scales this calibration should be done with test

weights that meet class F tolerances and not by the comparison of one scale with another.

28
Test weights may be secured from state or city Weights and Measures Departments. For

the smaller weighing scales and balances and for substitution weighing procedures, the more accurate laboratory weights such as class P or Q are preferred.

Before applying the test weights make sure the scale is free from binding of any

of its knife edges and calibrator connections. Following the calibration make a sensitivity measurement at full load of the scale. Any reduction in sensitivity in future

measurements is probably caused by binding or damaged knife edges.

Weighings are usually made in air; consequently, a correction for air buoyancy

must bo made to determine the mass of the material being weighed. In the case of water

such corrections have been made and the apparent density of water weighed in air is available in tables such as Table 3. When the specific gravity of a hydrocarbon is known the

apparent density of the liquid can be determined from Table 26 [2] or as described in 3.2.3.

6. Methods of Calibration

6.1. Introduction

Liquid flow measurements having the best accuracy are usually made in apparatus that

collects and measures the total flow delivered during a finite interval of time. The

quantity of the collected liquid is measured either gravimetrically or volumetrically and

the liquid density is usually required in either case. In the gravimetric system, measurements of the collected mass and time of collection yields mass rate of flow and a density

determination permits conversion to a volume rate of flow. In a volumetric apparatus,

measurement of collected volume and time of collection yields the volume rate of flow

directly. In this case a measurement of the density of the flowing liquid is required

to convert to a mass rate of flow.

Uncertainties in the measurements of mass, volume, time and density can be estimated;

however, the uncertainty in the flowrate is usually greater than the combined uncertainties

in the parameters listed above. One source of uncertainty common to all methods of metering liquids is introduced by the inability to set flowrate absolutely steady and to meter

without error. As some meters have measurement precision equal to or better than the calibration precision, it becomes impossible to separate the imprecision in the meter indication from that of the calibration flow standard. Another source of imprecision in the

indicated flowrate of the meter is the influence of the character of the upstream flow.

Variations in Reynolds number, velocity profile and degree of swirl are usually responsible

for decreased precision in the measurement of flowrate.

6.2. Gravimetric Calibration Systems

In a gravimetric calibrating apparatus, calibration of a liquid flowmeter involves

the determination of the time interval required for the measured mass of liquid to pass

through the meter, at a constant indicated rate, under the specified conditions of calibration. Since extreme accuracies are possible in the measurement of mass and time, the

limiting factors in the procedure are the technique, sensitivity of the meter, constancy

of the indicated flowrate, method of collecting the liquid and the method of timer actuation.

There are two distinct types of gravimetric calibrators, these are "static" weighing

calibrators and "dynamic" weighing calibrators.

6.2.1. Static Weighing Calibrators

Gravimetric calibrations of liquid flowmeters of the best accuracies are made in

apparatus in which the collected liquid is weighed under "static" conditions of no flow

into the collection tank. Such an apparatus is shown schematically in Fig. 9.

29
VENT

CONTROL VALVE

0-— PRESSURE GAUGE

-THERMOMETER

-METER

SUPPLY

RETURN TO STORAGE

Fig. 9 Static weighing calibrator.

The primary function of the flow supply and control system is to deliver filtered

liquid of known physical properties at a constant controlled and measured temperature to the meter under test. It should also serve to eliminate surges and pulsations and to provide a sensitive method of control over the entire range of flowrates. This system must

also prevent any gas or vapor from entering the meter. Special care should be taken to prevent the formation of vapor during the throttling action of the flow control valve,

and whenever possible this valve should be located downstream from the meter. The piping

between the meter and nozzle of the diverter should be sized as small as possible to minimize the possibility of storage of gas and/or vapor during the collection period of a flowrate measurement. During this period the flowrate discharging from the nozzle must be

equal to that through the meter.

The diverter directs the flow either to storage or to the weigh tank without disturbing the flow through the meter. The diverter should actuate the timer in both directions

when the splitter bisects the nozzle. It should pass through the liquid stream as rapidly

as possible (30 milliseconds or less) to minimize possible diverter error. This can be

accomplished by rapid diverter travel through a thin liquid sheet formed by the nozzle

slot. Generally, this liquid sheet has a length 25 to 50 times its thickness.

Calibration of a meter at a selected rate of flow is performed by a tare weight

measurement with the diverter in the "return to storage" position. The diverter is then

actuated to direct the liquid into the weigh tank and to simultaneously start the timer.

When an appropriate quantity of liquid is collected, the diverter is returned to its

original position automatically stopping the timer. The gross weight is then recorded

along with the indicated time interval.

6.2.2. Dynamic Weighing Calibrators

The static weighing method requiring tare and gross weight measurements is time con- suming and thus not well-suited for applications where convenience and speed of operation

is important. For these applications "dynamic" weighing is frequently utilized. In this

30
method the time interval required to collect a preselected weight of liquid is measured;

the weighing is performed while the liquid is entering the tank. Experience has demonstrated that accuracies as good as in the "static" calibrators can be achieved; however,

additional dynamic factors must be considered.

A typical dynamic weighing calibrator is shown schematically in Fig. 10.

- TIMER SWITCH

LIQUID IN RATE = m

RETURN TO STORAGE

Fig. 10 Dynamic weighing calibrator.

In this apparatus, the weight of the liquid in the tank increases until it overcomes the

resistance of the counterpoise weights on the end of the weigh beam which then rises and

actuates the timer. At this time an additional counterpoise weight is added to the pan

thereby depressing the weigh beam. When it rises again the timer is stopped.

During the weigh cycle, three dynamic phenomena take place and their effects must

be considered. One, there is a change in the impact force of the falling liquid between

the initial and final weigh points as a result of the reduced distance between the point

of discharge and the liquid level in the tank. Two, there is a collection of a greater

mass of liquid than predicted by mt . The additional amount is equal to the mass con- tained in the falling column between the liquid levels in the tank at the initial and

final weigh points. Three, the change in inertia of the scale and weigh tank with the

resultant change in the time required to accelerate the weigh beam past the timer trip

point. It can be shown that the decrease in impact force is equal and opposite to the

additional weight collected from the falling column. Thus, even though these effects can

be appreciable they completely cancel each other.

The effect of change of inertia between the initial and final weigh points can be- come appreciable when the mass of collected liquid is a significant part of the total mass of the weigh system and/or a large travel of the weigh beam to the point of timer

31
actuation is required. The error caused by this effect is shown in the difference At

between the time t2 required for the deflection of the weigh beam at the final mass M2

and the time t^ for the tare mass Mj_. This error has been shown to be approximately

,

Tfl 1/3 M

1/3 M

1/3

*« - «2 - *! - C^> <"? -?

3

) (37)

where L is the length of the weigh beam as shown in Fig. 10, 6 is the angular displacement of the weigh beam in radians from the bottom rest position to the timer trip position, and m is the mass rate of flow discharging into the weigh tank. In terms of percentage error

... 1/3 1/3

(At) 1Q0 . 10Q ( 6Li

} 1/3 m

2/3 ( M

? gl ) (38)

fc C g£

where t c is the time for the collection of the mass (M2 - M^) . It is obvious from the

relation above that for a given scale the percentage error in flowrate determinations

is a function of beam deflection 9, mass rate of flow m and gross and tare masses M2 and

M^ of the liquid, tank and other masses supported by the scale. In an existing dynamic

weighing apparatus, in which the maximum possible flowrate is required, the only practical means to reduce dynamic error is through the reduction of which should be made as small as practicable. Shafer and Ruegg [8] have shown good agreement between the inertia error determined from Eq. 38 and the error as determined by comparing the dynamic calibrator with a static calibrator. Such an evaluation should be made when possible; cer- tainly some attempt at evaluation is necessary whenever Eq. 38 predicts a large error.

The quantity (M2 - Mi) for both "static" and "dynamic" weighings is an apparent mass,

i.e., it is the mass determination uncorrected for the buoyant force of the air displaced

by the collected liquid. To determine the mass rate of flow this correction along with a possible correction for error in the time of collection can be made as follows:

VM

i

t c ±At

i-P a /P r 1-Pa/P

(39)

where p, p a and p p are the densities of the liquid, of air and of the weights. The quantity At is the error in the measured time of collection. It may be ± At in the case of

static weighing and must be determined by experiment. In dynamic weighing it is -At as shown in Eq. 38. The volume rate of flow can be determined from

q = m/p (40)

Where the apparent density of the liquid is known as in Table 3 for water and Table 26 [3]

for hydrocarbons, the volume rate of flow can be determined directly from

q = (M2 -Mi)/p(tc ±At) (41)

Another dynamic calibrator, shown in Fig. 11, collects the liquid in a tube of known

cross-sectional area and uses the pressure existing on the bottom as a measure of the mass

of liquid collected. The vertical rise in liquid between the initial or tare and the final

weigh positions is usually in the range of 2 to 20 feet. Thus it is necessary to measure

this pressure difference to an accuracy of better than 0.1 percent. The apparatus illustrated uses a mercury manometer as the pressure measuring device; electrodes spaced to correspond to different collected weights provide the impulse for the actuation of the

timer.

32
STANDPIPE

TIMER

ACTUATORS

FLOW

FROM METER

DRAIN VALVE

LEAK DETECTOR

_3

RETURN TO STORAGE

Fig. 11 Dynamic weighing calibrator.

For accurate calibrations using this type of apparatus several factors must be

considered:

1. The thermal expansion of the standpipe cross-sectional area.

2. The density of the manometer liquid.

3. Complete drainage of the cylinder walls before starting the weigh time

determination; particularly important with high-viscosity liquids.

4. Air pressure within the cylinder or manometer column must not change during

the timing interval.

5. Air or vapor in the connecting line between cylinder and manometer must be

eliminated

.

6. Dynamic considerations require that the pressure measuring device has com- pleted its acceleration and attained a constant velocity prior to starting

the timing interval.

6.3. Volumetric Calibration Systems

There are a great variety of volumetric calibration systems; however, all of them

employ either the standpipe or ballistic methods. Although the many calibrators utilizing the standpipe method vary in details, the essential features of all of them are included in the schematic diagram of Fig. 12.

These systems consist essentially of a supply tank, flowmeter test section, flow

control system, calibrated receiver (standpipe) and instrumentation. Some are completely self-contained and include liquid storage, pumps, filters and heat exchangers for

temperature control. The standpipe is equipped with liquid level sensors at levels

bracketing accurately known volumes.

33
STANDPIPE

LIQUID LEVEL

SENSORS

CONTROL VALVE

PRESSURE GAUGE

DUMP VALVE

LEAK DETECTOR

SUPPLY TANK

FLOW

STRAIGHTENER

PUMP

Fig. 12 Standpipe calibrator.

34
Calibrations with this system are made in the following manner. With the dump valve

open and the standpipe drained, the pump is started and a flow through the meter is adjusted with the flow control valve. After a condition of equilibrium in flow and temperature has been established the dump valve is closed. The liquid rises in the standpipe,

starts a timer at the first liquid level sensor and stops the timer at the upper sensor. During this timing interval the output of the flowmeter under calibration is observed and

recorded. In the case of turbine-type flowmeters the liquid level sensors also gate the

counter which is used to count the pulses during the calibration interval. Other required

measurements are the temperature of the liquid both in the standpipe and flowing through

the meter, a density determination of the liquid and a viscosity measurement.

The "ballistic" or "piston" calibrator is the other common form of volumetric calibrator. One variation of this form is shown in Fig. 13.

FLOW REVERSING VALVES

PUMP

Fig. 13 Ballistic calibrator.

It consists of a liquid reservoir, pump, flowmeter test section, control valves and a calibrated prover section. The prover is either a straight tube or a U-tube of essen- tially constant inside diameter equipped with a piston or ball. Pistons are generally

used in provers wi"h straight tubes while liquid filled elastomer spheroids are used in

the U-tubes. In either case there must be no leakage past the piston or ball. Detectors

are located near the ends of the prover tube to detect the passage of the piston. The

volume of the tube between the two detectors is accurately calibrated, usually with calibrated containers. The passage of the piston or ball across the detectors gates electronic timers and/or counters.

35
Calibrations of meters in this system are performed in the following manner. The

entire system must be filled with liquid; all air and vapor must be eliminated from prov-

-_ ind all piping and valves. With the valves to the prover closed, the desired flow is

established through the meter to the reservoir through the bypass valve A; at this time

valves B and C are closed. After an equilibrium condition is attained and with valves D and E open with F and G closed, A is closed while simultaneously opening B and C. Thus,

the flow is directed into the prover driving the ball, in the indicated direction, which

actuates the timer or counter as it passes the two location detectors. A complete proving requires an additional travel in the opposite direction by reversing valves D, E, F and G.

This system is particularly suitable for calibration of meters with high vapor pressure liquids because it is a closed constant volume system. It also lends itself well to the calibration of turbine type flowmeters which because of their high speed of response

can tolerate a somewhat nonuniform flow during the calibration interval. For meters having a lower response rate (variable area and head types) constant flow and meter output

must be reached before triggering the first detector.

These calibrators must be monitored frequently to ensure against leakage past the

piston or ball during the proving interval. Any such leakage will result in more flow

through the prover than indicated. This monitoring can be done by recalibrating the

prover volume. A more convenient method consists of frequent provings with a reference

turbine meter.

In general the measurements required during calibrations include temperature, density

and viscosity of the liquid flowing through the meter. The temperature of the prover is

required to determine its volume.

In the case of calibrations of turbine flowmeters the total count of pulses between

the two detectors is required. For volume rate of flow, density measurements will not

be required unless there is a difference between the densities of the liquid in the flowmeter and prover.

A variation of the ballistic calibrator is a type in which the piston is driven by

some means other than the flowing liquid. Some are driven with precision screw threads

and a variable speed motor while others are driven by compressed gas. In either case the

piston travel is tracked with some form of shaft encoder which generates a train of electrical pulses which can be counted on electronic counters. Thus, each pulse is equivalent

to the passage of a discrete volume of liquid. Apparatus of this type is well-suited for

the calibration of turbine-type flowmeters. Calibrations can be made quickly and conveniently with small volumes of liquid. Where calibrations are required with several liquids

of diverse properties this is a decided advantage.

Calibrations of turbine meters with these calibrators can be performed in several ways.

Generally, the counter recording the turbine flowmeter pulses is gated by two sensors actua- ted by the passage of the piston. The spacing of the sensors brackets an accurately known

volume. Thus meter factor K is

v - Total pulses ,~. Q ^ Volume, gallons

One manufacturer uses the period (the time between two successive pulses) or multiple

periods of the flowmeter to gate the pulse counter receiving the pulse train generated by

the piston. Since each pulse represents a discrete volume of liquid, the total count will

be proportional to the volume per pulse of the meter or the reciprocal of K.

6.4. Meter Installation

Variations in the piping layout of flowmeter installations can have a large effect

on the accuracy of flow measurement. The rate of flow derived from the outputs of orifice,

36
nozzle, Venturi and turbine flowmeters may be In error If the piping arrangements tend to produce distorted flow conditions. Helical swirls, vortices and skewed velocity distributions in the flow entering flowmeters have been found to degrade the accuracy of flow measurements. Upstream elbows, tees, valves, projecting gaskets, misalignment and projecting

burrs on pressure taps can cause appreciable errors.

For best results the liquid should approach the flowmeter with a fully developed ve- locity profile (usually turbulent) and free of swirls or vortices. This condition can be

achieved through the use of adequate lengths of straight pipe both upstream and downstream

from the flowmeter. Eight diagrams depicting different piping installations are shown in

[1] with the recommended straight lengths of pipe in terms of number of diameters, L/D.

For most installations 20 diameters upstream and 5 diameters downstream are adequate to hold errors due to piping configurations to less than ± 0.5 percent. The upstream length

can be reduced when straightening vanes are used. In this case 10 diameters between the

straightener and meter is adequate. A typical straightener is shown in Fig. 14. A length

of two diameters is required upstream of the straightener which normally has a length of

2D; thus, the overall length of the straightener section is 14D.

2D

14D

•L 10D

Fig. 14 Typical flow straightener.

Additional lengths of straight pipe either with or without straightening vanes should

be provided whenever possible. The control valve should always be located downstream from

the flowmeter. If an upstream shut-off valve is required it should be a full ported gate

or ball valve so as not to disturb the flow when full open.

When a calibration is required for a flowmeter installed with straight lengths less

than these minimums it should be calibrated with its own or duplicated piping including

upstream and downstream elbows, tees, valves, etc.

Appendix C of [9] also states that 20 diameters of meter-bore piping upstream and

5 diameters downstream of the meter provides effective straightening in many installations. Shown also are five meter installations with the recommended lengths of straight

pipe between the meter and upstream disturbance. The recommendations are summarized in

Table 9.

Table 9. Recommended Minimum Lengths of Straight Pipe

Upstream disturbance

Concentric reducer

Sweeping elbow

Two sweeping elbows in the same plane

Two sweeping elbows in planes at right angles

Valve partially open

Pipe length L/D

15

20

25

40

50

This table clearly shows the reason to avoid upstream valves and compound elbows.

32
6.5- Illustrative examples:

A. Calibration of a turbine-type flowmeter with a gravimetric dynamic calibrator.

Consider a turbine flowmeter Model ANC8-4 installed on a dynamic gravimetric calibrator. The meter is installed horizontally with the meter pickoff coil pointing upward. The meter assembly consists of an upstream 5-inch length of tubing.

The pickoff coil was connected to a frequency meter and a totalizing counter

gated by the dynamic weigh beam. The following measurements were made:

1. An observed specific gravity of a sample of the liquid at 70°F was 0.76505.

2. Temperature of the flowing liquid was 80°F.

3. Mass of liquid collected was 25 lbm.

4. Frequency of meter was 1400 hertz.

5. Total count of pulses during collection of 25 lbm was 113233. Find "K factor"

(pulses per gallon)

.

a. From the observed specific gravity at 70°F the Specific Gravity 60/60°F

is found to be 0.7693 from Table 23 page 239 [3]. Since the calibrator

weighs in air, the weight in air of a hydrocarbon having a Specific

Gravity 60/60°F of 0.7693 is found to be 6.4046 lb/gal from Table 26

page 494 [3]. To find the weight in air at 80°F multiply 6.4046 by the

volume reduction factor of 0.9888 for 80°F found in Table 24 page 426 [3].

6.4046 x .9888 = 6.3329 lb/gal

b. Thus the number of gallons collected is

25/6.3329 = 3.9476 gallons

c. And

113233 pulses ,.,„, .. , ,

,

K = , -.-,, v

-, = 28684 pulses/gallon

3.9476 gallons

At the National Bureau of Standards the usual calibration of turbine flowmeters consists of experiments at five frequencies. The reported values

are the arithmetic means of ten separate observations at each of these

frequencies. They are taken in groups of five successive runs on each

of two different days. Also a statement of accuracy is usually given,

such as

:

Reported Calibration Values

Pulses/Second Pulses/Gallon

120 28801

350 28791

700 28703

950 28716

1400 28684

These reported values have an estimated overall uncertainty of ± 0.13

percent based on a standard error of 0.01 percent and an allowance of

± 0.1 percent for possible systematic error.

B. Calibration of two turbine flowmeters in series with a dynamic gravimetric calibrator.

Two turbine flowmeters can be calibrated simultaneously by counting the total

pulses generated by each during discharge of a weighed quantity of liquid con- verted to gallons.

38
Consider two turbine flowmeters Model ANC 8-4, Serial numbers 1A and IB. As received for calibration the assembly consisted of a 5-inch length of tubing containing straightening vanes, meter 1A, two 5- inch lengths of tubing with

the second containing straightening vanes, meter IB, and a 5-inch length of exit

tubing. The assembly, mounted horizontally with the meter pickoff coils pointing upward, was connected to the calibrating bench with an additional 5-inch

length of straight tubing downstream from straightening vanes. The fluid was

the same as that used in Example A. The following measurements were made:

1. Temperature of the liquid flowing through the meters was 80°F.

2. Mass of liquid collected was 8 Ibm3. Frequencies: Meter 1A 720 hertz, Meter IB 700 hertz.

4. Total count: Meter 1A 37239, Meter IB 36258.

The weight per gallon in air is the same as in Example A because the same liquid is used and metered at the same temperature; thus the liquid weighs

6.3329 lbs per gallon in air at 80°F. Since 8 lbm was collected the volume

collected was 8/6.3329 = 1.2632 gallons.

Thus

for Meter 1A K = 37239/1.2632 = 29480

for Meter IB K = 36258/1.2632 = 28703

Again the results for the arithmetic mean of ten separate observations are listed below:

Meter 1A Meter IB

Pulses/Second Pulses/Gallon Pulses/Second Pulses/Gallon

120

360

720

960

1440

29539

29477

29480

29467

29490

120

350

700

950

1400

28801

28791

28703

28716

28684

Once these flowmeters are calibrated either of the meters can be used as a transfer standard to calibrate other meters having approximately the same flow range.

Before such a meter is used as a transfer standard the locations of the two flowmeters should be interchanged to determine the effect, if any, of the upstream

flow pattern on its K factor. If the change due to reversal is less than 0.1

percent either of the meters can be used as a transfer standard. If the change

is greater, additional straight tubing and/or straightening vanes should be inserted ahead of the upstream meter and between the two meters.

Calibration of a turbine flowmeter with a second calibrated turbine flowmeter.

Assume the downstream meter IB in Example B is the calibrated meter and has a K factor of 28791 pulses/gallon at a frequency of 350 hertz. The pickoff coils

are connected to frequency meters and preset counters. The liquid hydrocarbon

is the same as in previous examples. The flow through the meters was adjusted

so that the frequency of meter IB was 350 hertz. The following measurements

were made:

Meter No. 1A Meter No. IB

Frequency, hertz

Total pulses, 100 seconds

K factor

360

35877

350

35042

28791

39
Find the K factor for meter 1A

K Total pulses 1A

K1A K1B Total pUi S es IB

Kia = 28791 x Hill = 29477 pulses/gallon

Since the calibration is in terms of volume and since the same liquid at the same temperature is flowing through both meters a determination of the density

of the liquid is not required. However, if there is an appreciable change in

temperature of the liquid between the two meters allowance for the change in

density will be required.

In the last Example C let the temperature of the liquid flowing through Meter

1A be 80°F and 85°F through Meter IB. Let the frequencies and total count

for the two meters be the same as in Example C. We know the total count for

100 seconds and the K factor of Meter IB from which we can find the volume

flowing through IB in 100 seconds as follows:

35042 total count , „,-,, ,

,

^ -, ni ;t—r, = 1.2171 gallons

28791 pulses/gallon

Since the liquid flowing through 1A is cooler and more dense the volume flowing

through it in 100 seconds is found by use of Table 24 page 426 [3] where the

factor for reducing volume is 0.9888 and 0.9860 at 80 and 85°F, respectively, for

a liquid hydrocarbon having a specific gravity 60/60°F of 0.7693.

Thus the volume flowing through 1A is

1.2171 x tHjUf " 1-2137 gallons

0.9888

and

K

35877 total Count onc^n i ; n

1A = 01 __ r^ = 29560 pulses/gallon ia 1.2137 gallons

It should be noted that the tabulated frequencies are nominal and should not

be used in the calculations.

7 . Types of Flowmeters

7.1. Introduction

Practical commercial liquid flowmeters making use of a wide variety of physical

principles are available. Differential pressure types such as orifice, nozzle, Venturi

and laminar meters depend on the conversion of potential energy to kinetic energy of

the liquid. Positive displacement meters such as quantity and turbine-type flowmeters

are driven by the liquid. In the electromagnetic flowmeter the basic principle is

similar to that of an electric generator. The ultrasonic flowmeter depends upon the

speed of sound in the liquid. Thermal meters make use of change of enthalpy in the

flowing liquid.

7.2. Differential Pressure Type Meters

There are perhaps more meters of the differential pressure type in present use than

all other liquid flowmeters combined. The theory and equations common to orifice, nozzle

and Venturi meters were discussed in Chapter 4.

40
3/32 + 1/32

5/32 + 1/32

1/4 + 1/16

3/8 + 1/8

7.2.1. Orifice Meters

Of the differential pressure type flowmeters, the orifice is the most widely used.

The principle advantages of the orifice flowmeter are low cost, easy installation and re- placement, and fairly well established coefficients of discharge.

On the debit side, the orifice has a high nonrecoverable head loss. Many different

configurations of orifices such as the quadrant edge, double bevel, conical inlet, eccentric and segmental described in [ l] are used; however, this discussion will be confined to thin plate concentric orifice flowmeters.

The orifice plate is usually made of Monel, stainless steel or other nonrusting steel

alloy. The thickness of the plate for low temperature liquid flows should not exceed those

shown in Table 10.

Table 10. Maximum Thickness for Thin Plate Orifices

Pipe Size, Inches Orifice Thickness, Inches

Up to 3

4 to 6

7 to 8 10 and over

The orifice hole should be concentric with the pipe. The width of the cylindrical edge

of the orifice hole should be between 0.01 and 0.02 times the diameter of the pipe. If

the thickness of the plate exceeds this limit the orifice outlet edge should be beveled

at an angle of 30 to 45 degrees from the pipe axis. The inlet edge of the orifice

should be square and sharp, free from burrs or rounding. The orifice diameter should be

measured in four meridian planes at 45 degrees. The diameter ratio should not be less

than 0.25 or more than 0.70.

A typical installation of a thin plate orifice with pressure connections is shown in

Fig. 15. Pressure taps are located and defined as follows:

1. Flange Taps. The centers of the pressure holes are one inch upstream and one

inch downstream from the upstream (inlet) and downstream (outlet) faces of the

orifice plate.

2. Taps at D and D/2. The center of the upstream pressure tap is located one pipe

diameter upstream of the inlet face of the orifice plate. The center of the

downstream tap is one-half pipe diameter downstream from the inlet face of the

orifice plate.

3. Vena Contracta Taps. The center of the upstream pressure tap is usually located

one pipe diameter upstream from the upstream face of the orifice plate. The

downstream tap is located at the place of minimum pressure. This location is

shown in Figure 1-5-5 of [1].

4. Corner Taps. The pressure holes are located in the corners formed by the pipe

walls and orifice plate. Corner taps are generally used for the smaller meter

tubes where either the flange tap or D/2 tap would place the downstream tap in

a varying pressure recovery region.

Flow through thin plate orifice flowmeters can be calculated with Equations 27 and 28 and the coefficient of discharge. It is always preferable to determine

this coefficient through calibration when the orifice is installed in the pipe

section in which it is to be used. When this is not possible the coefficients

can be obtained from Tables II-III-2, II-III-3 and II-III-4 reference [1] for

flange taps, taps at D and D/2 and vena contracta taps, respectively.

41
D and D/2 TAPS

\ss"ssssss<.ss\s ^AJ3

Fig. 15 Thin plate orifice meter.

7.2.2. Flow Nozzles

Flow nozzles are more costly than orifice flowmeters but cost less than equivalent

Venturi meters. For the same flow they have about the same head loss as orifice meters.

The principle advantage of the nozzle over the orifice is its higher coefficient of discharge and one that is less subject to change due to erosion. The coefficients of discharge of orifice flowmeters have been found to increase appreciably due to rounding of

the upstream edge.

Flow nozzles are made of aluminum, brasses, bronzes and stainless steel. All of

these are satisfactory for liquids free of corroding or abrasive substances. Stainless

steel should be used for high temperature, corrosive and abrasive environments.

Flow nozzles having a wide variety of configurations have been manufactured, calibrated and are in use. This discussion will be confined to the "long radius" or elliptical inlet nozzle as recommended by the ASME, not because it is inherently a better nozzle

but because more reliable information on discharge coefficient versus Reynolds number is

available for this configuration.

The curvature of the inlet to the nozzle throat is the quadrant of an ellipse.

The proportions of the ellipse with respect to pipe and throat diameters are shown in

Fig. 16. The low S nozzle is recommended for diameter ratios d/D below 0.50 and the high

8 nozzle for 8 between 0.45 and 0.80. There is a small range of 8 between 0.45 and 0.50

where either can be used. All important dimensions for both low and high 8 nozzles are

shown in Fig. 16.

These flow nozzles are used in continuous pipe lines, at the end of a pipe line discharging to the atmosphere, or at the outlet of a plenum chamber. The inlet pressure tap

is located one pipe diameter upstream from the inlet plane of the nozzle when the nozzle

is installed in a continuous pipe line or at the open end of a pipe section. When the

nozzle is installed at the exit of a plenum chamber, the inlet pressure is the pressure in

the plenum chamber.

42
ZZZZZZL

D Di

1 _l

yszzz

r i - r Lt -*

\\\\\\\V^

High 3 Nozzle

r } = 1/2D

r2 = 1/2 (D-d) L

6 - 0.45

y = 0.6D or = 1/3D

Low 6 Nozzle

ri - d

= 0.5

< 5/8d = r 2 = 2/3d

0.6d =Lt = 3/4d

Fig. 16 ASME long-radius flow nozzles.

With high 6 nozzles the outlet pressure connection is located D/2 downstream from

the entrance plane of the nozzle. For a low 3 nozzle the pressure tap (either pipe wall

or throat) is located 1.5d downstream from the entrance plane of the nozzle. When a nozzle is installed at the outlet end of a pipe section or the outlet of a plenum chamber,

the outlet pressure is measured with a barometer located near the nozzle outlet. For liquid flow this measurement is usually not required. The differential pressure drop across

the nozzle is then the upstream pressure measured against atmospheric pressure.

Flow through these nozzles can be calculated from Equations 27 and 28 and the co- efficients of discharge listed in Table II-III-5 reference [1]. As in the case of orifice

meters it is preferable whenever possible to determine the nozzle coefficient through calibration of the nozzle when installed in the pipe section in which it is to be used.

7.2.3. Venturi Flowmeters

The Venturi tube flowmeter is the most costly and the largest of the differential

pressure meters. Its principle advantage is its low overall pressure loss. The efficient

conversion of velocity head to pressure head in the conical expansion section is responsible for its low pressure loss. The classical (Herschel) Venturi tube flowmeter is shown

in Fig. 17. A number of modifications of the Venturi are available. Some involve a com- bination of a nozzle inlet with the conical expansion (the Nozzle-Venturi) . Others employ

modifications in or near the throat section to obtain a high differential pressure with as low an overall pressure loss as possible. All of these modified Venturis require calibration as the relation between coefficient of discharge and Reynolds number is not well

known.

The classical Venturi shown in Fig. 17 consists of a cylindrical inlet, a convergent

conical section, a cylindrical throat and a divergent conical discharge section. The

dimensional proportions as recommended in [1] are tabulated in Fig. 17.

These tubes are usually made of cast iron or cast steel in the smaller sizes. In the very large sizes some are fabricated of welded sheet steel. The throat section may

43
m

rH

T3 UV

m CM eg 8

*? iH o • II V

CM o

+1 +1 r-~ TJ o Q T3 m

iH

rH m m +1 00

r». CM II v

CO \D o II V

• • CO rH CM

r-l CO Pi CM a

II II II V II II V —

i

CM T3 ^H

pei PO m 8 r-*

CM

CO

-

II V

vO /^ &

V O

o nv II V T3 *T3 iH e CO + <r

~

rt rH

*d- u

NO -

• o

Q o u ^*

v—' 4-1 o M3 o

V •* m

**"N» CM CM «

•H Q «**^ rH

hJ

+1 Q MV • o

M II V KD O CM CO V ^, N ^^, v£> II V O Q

II V

TJ

T3 =

«o IV »v

<r HA

A

CMCO

•H •*^ 4J ^^ J N Q rJ >% m

s

4Jc

111 >

4)O

COu

PC

00

44
be lined with bronze or other corrosion-resistant material. The inlet and throat sections

contain piezometer rings or annular chambers for measuring static pressures at these

points.

As with orifice and nozzle flowmeters the mass or volume rate of flow for the classical Venturi flowmeter can be calculated from Equations 28 or 27. The coefficients of discharge can be obtained from page 64 reference [1] for rough cast, machined and rough welded sheet steel convergent sections with tolerances varying from 0.7 to 1.5 percent. For more accurate flow determinations a calibration of the Venturi meter should be performed.

7.2.4. Laminar Meters

Another form of head meter is one which has a linear relationship between the pressure drop and the flowrate from zero up to some maximum rate, depending on the geometry

of the meter and the viscosity of the liquid flowing through it. In its most common form

the laminar or linear meter consists of a capillary tube or a bundle of capillary passages.

It is not necessary for these passages to be circular or even uniform in cross-section.

Both the single tube and bundle form of meters are shown in Fig. 18.

The linear relationship between pressure drop and rate of flow requires that the

Reynolds number be below about 2300, the value at which laminar flow begins to become

turbulent. In the case of laminar meters the Reynolds number is difficult to determine

because the characteristic length is usually not known. It may be measureable in the case of a single tube of circular cross-section but not for a bundle of tubes including

irregular areas between them or a large number of thin slit-like passages. For these

meters the limit of linearity must be determined by experiment.

Capillary tube

Capillary bundle

/

Fig. 18 Laminar flowmeters.

The basic relation for flow through capillary tubes as first shown by Hagen and

Poiseuille and now known as the Hagen-Poiseuille law for the units used herein is:

45
Q

. c(gp Q (21ZEZ

} (42)

where C is the coefficient which must be determined by calibration. The effects of inlet

and exit losses, and in the case of coiled capillaries, the effects of curvilinear flow

are included in C. In meters where D is difficult or impossible to determine, the product

of the first three terms may be determined through calibration.

tl 7.3. Variable Area Meters

Another form of meter which can be considered as a head meter is the variable area

meter. In these meters the cross-sectional area of the measuring aperture is caused to vary as a function of flowrate while the differential pressure or head producing the flow

remains essentially constant. Generally, the variation of area with flowrate in this type

of meter is automatic and the position of the area changing element is an indication of

flowrate.

The most common meter of this type consists of a tapered tube and float. The tapered

tube is vertical with its inside diameter increasing with elevation. The direction of

flow is vertical upward through the tube. The float which is free to move vertically re- sponds to the flow until it finds a position where the weight of the float less the buoyant

force of the liquid is equal to the upward force produced by the flow of the liquid past

the float. A schematic drawing of a typical variable area meter is shown in Fig. 19.

ft

Thin disc

float

Fig. 19 Variable area flowmeter.

The parameters which determine the characteristics of such a meter are density and

shape of the float, variation of the annular area (between float and tube), elevation of

the float, and density and viscosity of the flowing liquid. It has been shown that a float with a thin disc, like that shown in Fig. 19, is much more independent of liquid

viscosity than cylindrical and spherical floats. The tapered tubes are generally of a constant taper angle and the annular area is kept less than the cross-sectional area of

46
the float so that the flowrate is nearly linear with elevation.

Glass is the most commonly used material for the manufacture of the tapered tubes.

The floats are guided either with a central rod or with flutes integral with the glass

walls. Other materials are sometimes used for the tubes but these require auxiliary

means for the indication of the float position. In that case a magnetic coupling and

follower can be used. Remote indication and recording are sometimes effected with the

float altering the inductance or capacitance of an electrical circuit.

Equations for the volume and mass rates of flow shown below are derived in [1]:

1/2

Q

= (a*-l) (iScAF) (43)

P

m = (<* 2 -D (2gcAFp) 1/2

(44)

Where a - D/Df

A = tt/4 D

2 F = Buoyed weight of the float, lbf

Of themselves, these equations cannot predict the flow through variable area meters; for

that, calibrations of individual meters are required. However, the equations do show the

variation of flow with float and liquid densities. For accurate liquid metering over a range of liquid densities and viscosities, Shafer, et al. [10], have shown that calibration

at several float positions, with four or five liquids having physical properties covering

the required ranges of densities and viscosities, eliminates the necessity for calibration

with each liquid of interest.

7.4. Turbine Flowmeters

The turbine-type flowmeter is perhaps the most widely used flowmeter utilizing a generated electrical signal which is proportional to fluid flow. Flowmeters of this type

have found considerable application in the aircraft industry because of such advantages

as small size, fast response, accuracy, simplicity and adaptability for remote indication.

The turbine flowmeter consists of a flow tube in which is mounted a bladed turbine

rotor, supported by one or more bearings, together with means of generating an electrical

signal having a frequency proportional to turbine angular velocity. The turbine rotor is

designed in an effort to produce speed of rotation directly proportional to volume rate of

flow. This has been approached for turbine flowmeters in the larger sizes and for moderate

sized meters with liquids of low kinematic viscosities. In an average application a typical turbine flowmeter can be expected to meter volume rate of flow to accuracies of

one-half to one percent over flow ranges varying from 3 to 1 up to 20 to 1 and even higher

depending upon meter design, size and the viscosity of the liquid.

There are several types of electrical signal-producing devices; some of the principal

types are as follows:

a. Inductance system: The routing element of the turbine carries permanent magnets

embedded in the hub or blade tips. The magnetic flux from the moving magnets induces a voltage within a pickup coil located close to the magnetic field.

b. Variable reluctance system: A fixed permanent magnet is centered with a pickup

coil which is located in close proximity to the rotating turbine rotor. Variation

in the magnetic flux results from the passage of each blade of the highly permeable magnetic rotor material.

c. RF type of pickoff : An oscillator applies a high frequency carrier signal to a coil in the pickoff assembly. The rotor blades pass through the field generated

47
by the coil and modulate the carrier signal by shunting action on the field shape.

d. Photoelectric system: A beam of light is interrupted by blades of the rotor

so that a pulsed signal output is developed.

e. Magnetic reed switch system: The contacts of a reed switch are opened and closed

by magnets embedded in the rotor or in the rotating part of the turbine meter.

The switch action causes a constant input to be interrupted so that a pulsed

signal output is produced.

With all of these systems the frequency of the pulses generated becomes a measure of

volumetric flowrate and the total number of pulses measures total volume. Either of these

measurements requires that turbine flowmeter output frequencies be linear with flowrate;

that is, each pulse be equal to the same unit volume of fluid measure, regardless of frequency. Manufacturers of turbine flowmeters have claimed linearity witnin ± 0.5 percent of

flowrate over ranges as great as 75 to 1 for liquids having viscosities of about 1 centistoke. Range of linearity depends primarily on meter size and liquid viscosity. In

general, lower flowrates, smaller meter sizes and higher velocities all tend to decrease

the range of linear operation.

Turbine meter calibration data can be presented in a number of ways. A plot of frequency versus volumetric flowrate can be used as shown in Fig. 20a* Ideally these data

should plot in a straight line passing through zero. However, only the most linear of

meters approach the straight line and all meters will show some viscosity dependency. Note

that as viscosity increases the frequency output decreases as a result of viscous drag.

A more practical method of showing the data, one which more directly displays the

linearity or lack of linearity of the meter is to plot the reciprocal of the slope of the curve shown in Fig. 20a versus frequency. Thus this plot shown in Fig. 20b is a curve of

the frequency divided by the volumetric flowrate versus frequency. This parameter is

termed the "flow coefficient" or "K factor" and for most aircraft work has the units of

cycles or pulses per gallon. The ideal plot of "K" versus frequency is a straight horizontal line.

Another method is a plot of the relation shown by Shafer [11]

f/Q = 4 (f/v)

If this relation is correct a plot of f/Q which is the "K factor" versus f/v (the frequency divided by the kinematic viscosity of the fluid) will give a smooth curve, the

shape of which is determined by the function <j>, throughout the working range of the meter

calibrated. The curves in Fig. 20c especially at the low rates of flow show that no single

form of the function <f> describes the performance of a turbine meter. However, curves of

this form are useful in describing the effects of flowrate and viscosity upon the perform- ance of turbine meters even though significant deviations are obtained.

There are many methods suitable for the calibration of turbine flowmeters; a more com- plete discussion of the methods is treated in Chapter 6. A partial listing of these

methods include

1. Gravimetric Methods

a. Static

b

.

Dynamic

2. Volumetric Methods

3. Running Start and Stop

4. Standing Start and Stop

48
l.OQ

0.8

3

C

0.6

o 0.4

cd 0.2

2G0

(0 o

(0

oo

90

85

80

x 10 3

400 600 '800 1000 1100

Pulses/Sec, f

v 1.74 centistokes

(b)

200

cd o

CO

0)

CO rH3

Pi

90

85 —

80

200 400 600 800

f/v

Fig. 20 Turbine flowmeter characteristics

49

400 600 800 1000 1200

Pulses/Sec, f

1000 1200
The fast response of turbine flowmeters makes possible the use of the Standing

Start-and-Stop Method of calibration. In this procedure there is no flow through the

transducer prior to the beginning and at the end of the calibration run. The procedure

requires that at least 95 percent of the total delivery be at the desired flow. In this

method solenoid valves are synchronized with an electronic counter.

Turbine flowmeters are also conveniently calibrated with mechanical displacement

devices called "ballistic" or "piston" calibrators as described in Section 6.3.

For a more complete treatment of turbine flowmeters [9] and [12] are recommended

reading.

7.5. Positive Displacement Meters

Positive displacement meters measure the volume rate of liquid flow by repeatedly

filling a container. The total volume of liquid flowing through the meter in a given

time is the product of the volume of the container and the number of fillings. The num- ber of fillings is registered by a counter geared to indicate the quantity in gallons,

cubic feet or barrels.

Displacement meters of a great many configurations are available. The more common

types include the reciprocating piston, ring piston meters, sliding and rotating vane

meters, nutating disk meters and gear and lobed impeller meters. The latter are probably

the most used and will be briefly discussed to indicate the principle of displacement

meters.

These meters measure the liquid flow by using a small pressure differential to rotate the pair of elements. These two elements seal the inlet from the outlet thereby

developing the pressure differential. The principle of operation is illustrated in

Fig. 21.

Diagram 1 Diagram 2 Diagram 3

Fig. 21 Positive displacement flowmeter.

When the elements are positioned as in diagram 1, the torque developed on element A due to the differential pressure causes it to rotate and drive element B which has balanced

torques. When A reaches the position shown in diagram 3 its torque is balanced; however,

the torque developed in B causes B to rotate and drive A. The alternate driving action

provides a smooth continuous rotation. If the elements are not geared as shown, they are geared externally with equal gears to keep the rotors in correct relationship to each

other.

As the elements rotate they trap liquid in the crescent shaped gaps of the flowmeter

The total quantity of flow for one revolution of the elements is equal to four times the

volume °f the crescent shaped gap. Thus the volume rate of flow is proportional to the

50
rotational speed of the elements. The clearances between the rotors and the case are

very small so that effective capillary sealing between inlet and outlet is maintained.

The pressure differential between inlet and outlet is kept small through the use of

low friction precision bearings on the rotor shafts. Despite these two features there

is some residual leakage (slip) past the rotors. Slip is defined as the difference

between the quantity which actually passes through the meter and the quantity displaced

by the measuring chambers.

Although it is possible to calculate the volume of the measuring chamber, so as to determine displacement per revolution, this is seldom done. The usual practice to determine the capacity of the meter is by "proving" or calibrating the meter. In proving the

meter

:

1. The meter should be calibrated with the liquid it is to meter or one having

nearly the same properties.

2. The meter should be calibrated over the range of flowrates at which it will be

used

.

3. The liquid should be free of entrained air and vapor.

4. The meter should be calibrated at about the same temperature and pressure at which it will be used.

These meters can be calibrated in essentially the same manner as turbine- type flowmeters. If the meter is equipped with an electrical signal generating device, a counter,

gated with the weigh beam of a dynamic gravimetric calibrator or. with the position sensors

of a volumetric calibrator, is used to count the total pulses. If the meter contains

only a volume register the calibration can be performed by timing the passage of a given

registered volume and simultaneously making a flowrate determination with either a gravimetric or volumetric calibrator.

7.6. Magnetic Flowmeters

The basic principle of the magnetic flowmeter is similar to that of an electric

generator. According to Faraday's law the passage of an electrical conductor through

a magnetic field induces an electromotive force across the conductor in a direction

normal to both the magnetic field and the direction of the conductor. In a magnetic

flowmeter the conductor is the flowing liquid within a pipe or tube of insulating

material. Such a meter is shown schematically in Fig. 22. The direction of liquid

flow is into the plane of the paper and normal to the magnetic field produced by the

permanent magnet. The induced voltage e is detected with insulated electrodes placed

diametrically in the tube and also normal to the magnetic field.

The magnet may be a permanent magnet, a DC electromagnet or an AC electromagnet;

consequently, there are both AC and DC magnetic flowmeters. The flow tube must be non- magnetic material such as glass, plastic, aluminum, brass or stainless steel. Metallic

tubes must be lined with an insulating material to prevent the metal tube from shortcircuiting the path of the induced emf.

From Faraday's law of induction it can be shown that

o q =

4 J- B (45)

where q = rate of flow, in^/s

D = Diameter of tube, inches

6 = Distance between electrodes which may be less than D, inches

e = Induced emf, volts

B = Flux density, webers/in^.

51
Fig. 22 Electromagnetic flowmeter.

The flux density produced in a DC magnetic flowmeter is constant and the volume rate of

flow is proportional to the induced emf. In the case of AC magnetic flowmeters neither

the induced emf nor the flux density is constant. However, the volume rate of flow is

proportional to e/B, therefore with AC magnetic flowmeters it is necessary to measure

this ratio.

Since the magnetic flowmeter is responsive to velocity and therefore a volume rate

flowmeter, its indications are essentially independent of liquid density and viscosity.

It has been said that these meters are not sensitive to changes in velocity profile or helical flow. Another decided advantage of this type of meter is its unobstructed flow

path; therefore, it is not subject to abrasion and has a low overall pressure loss.

A conductivity of 5 micromhos/cm will meet the threshold conductivity for most

magnetic flowmeters. This value of conductivity includes a great many industrial liquids.

Most petroleum products have conductivities below this limit; hence they cannot be metered in this type of meter.

One manufacturer claims an accuracy of flowrate vs indication of ± 0.5 percent of

maximum flow. This of course is after a calibration of the meter.

7.7. Magnetic Resonance Flowmeters

The nuclear magnetic resonance (NMR) flowmeter is another flowmeter utilizing magnetic principles. In this case it is the liquid which becomes magnetized. Most liquids

will become slightly magnetized when passed through a magnetic field. Radio frequency

energy is then used to amplify (resonate) the magnetic effect and produce strong signals

from the weakly magnetized liquids. In order to measure flow, the magnetization is disturbed in small portions of the liquid stream and the passage of these tagged portions

are detected downstream. Each detected tag represents a known volume of liquid and is

totalized on a counter.

Nuclear magnetic resonance can be stimulated in all nuclei having magnetic dipole

moment. For flowmeter applications the hydrogen nucleus (the proton) and the fluorine

52
nucleus have been of the greatest interest for two reasons. First, the magnetic resonance of the proton is stronger than that of any other commonly occurring nucleus. Secondly, the presence of hydrogen and fluorine nuclei in all water-bearing liquids, hydrocarbons and f luorocarbons permits metering of these liquids with meters based on the NMR

principle. The flowing liquid need not be conducting as in the case of the electromagnetic flowmeters.

A typical NMR flowmeter shown schematically in Fig. 23 consists of:

1. A ceramic liner in which there are no obstructions to flow.

2. Strong permanent magnets and static field coils to magnetize the liquid (to re- arrange the orientation of the dipoles from random to alignment with the magnetic

field).

3. A radio frequency field coil (resonator coil) to resonate the magnetization and

produce usable signals for flow measurement.

4. Modulator coil to modulate the magnetic vector in a small portion of the flow

(to tag the flow)

.

5. Detector coil which detects the passage of the small tagged modulated portion of

flow.

PERMANENT MAGNETOSTOIC HELD

C0\L

L

RESONATOR MODULATOR DETECTOR

C0\L COM C0\L

Fig. 23 Nuclear magnetic resonance flowmeter.

Each tag pulse represents the passage of a known discrete volume of liquid. Total

volume of flow is obtained from totalizing pulses on a counter. Volume rate of flow can

be derived from the frequency of the pulses.

A manufacturer has claimed the same meter factor (pulses per gallon) for different

fluids under similar flow conditions.

7.8. Ultrasonic Flowmeters

Several ultrasonic flow measuring devices utilizing the speed of sound in and the

velocity of the flowing fluid have been developed. One of these described herein is

referred to as a "sing- around velocimeter. " Such a device is shown schematically in

Fig. 24.

53
Receiver 2

/ j > / j j/////jj//j\

\fj/???>>?////}j

Transmitter 1

Receiver 1

^zzzzzzzzzzzz

Transmitter 2

Fig. 24 Ultrasonic flowmeter.

It consists of a straight tube, two identical ultrasonic pulse generators and two identical receivers oriented as shown. The time required for a pulse to travel from transmitter

1 to reveiver 1 is

ti =

c + V cos e

where L = distance between transmitter and receiver

c = velocity of sound in the liquid

V = velocity of the liquid. And the time t£ from transmitter 2 to receiver 2 is

t 2 =

c - V cos

thus , At = t2 - ti 2LV

c 2-v2 cos

and when V is very small as compared to c

At = 2LF

Thus for a given meter V <* Ate' (46)

Similarly it can be shown that V a Af = f^ - f2 where f^ and £2 are tne frequencies

detected on receivers 1 and 2

.

The velocity of the liquid can be determined by measurements of either the frequency

difference of the two detectors (beat frequency) or the difference in transit time. The

instrumentation required for these measurements is beyond the scope of this paper.

Suffice it to say such measurements can be routinely made with good accuracy.

The signal output of meters of this type will be a function of the velocity distribution across the duct. As a consequence, these meters will require calibration within the

pipe system in which they will be used.

54
Acknowledgements

The author is indebted to Mr. Henry Vega of Navy Metrology Engineering Center and

Mr. Francis L. Jones of U.S. Army Metrology and Calibration Center for guidance as to the content required and for their review and suggested improvements to this work.

Figures 8, 10, 17 and 22 are modifications of figures taken from [1]. Figure 21

is taken from Catalog DS-900, Brooks Instrument Division, Emerson Electric Company.

Figure 23 is borrowed from Bulletin MRF-125 of the Badger Meter Manufacturing Company.

References

[1]. Fluid Meters Their Theory and Application, Sixth Ed., edited by H. S. Bean, ASME

(1971).

[2]. P. Hidnert and E. L. Peffer, Density of Solids and Liquids, NBS Circ. 487 (1950).

[3]. ASTM-IP Petroleum Measurement Tables, ASTM (1959).

[4J . ASTM Standard Viscosity-Temperature Charts for Liquid Petroleum Products, ASTM

Designation D341-43 (1949).

[5J. NBS Monograph 55 (1962).

[6]. Test for Kinematic Viscosity, ASTM D445 (1960).

[7]. Measurement of Petroleum Liquid Hydrocarbons by Positive Displacement Meters, API

Standard 1101 (1960).

[8]. M. R. Shafer and F. W. Ruegg, Liquid- Flowmeter Calibration Techniques, Trans.

ASME 80, 1369 (1958).

[9]. Measurement of Liquid Hydrocarbons by Turbine Meter Systems, API Standard 2534

(1970)

.

[10]. M. R. Shafer, et al., Correcting for Density and Viscosity of Incompressible Fluids

in Float-Type Flowmeters, J. Research NBS 47, 227 (1951).

[11]. M. R. Shafer, Performance Characteristics of Turbine Flowmeters, ASME J. Basic

Eng. 84, 471 (1962).

[12]. Recommended Practice Specifications, Installation and Calibration of Turbine Flowmeters, ISA-RP 31.1, Instrument Society of America (1972).

[13]. C. H. Page and P. Vigoureux, The International System of Units (SI), NBS (U.S.)

Spec. Pub. 330 (Apr. 1972).

55
FORMNBS-114A (1-71) 2. Gov't Accession No.

U.S. DEPT. OF COMM. 3. Recipient s Accession No. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-831

4. TITLE AND SUBTITLE

Introduction to Liquid Flow Metering and Calibration of

Liquid Flowmeters

5. Publication Date June 1974

6. Performing Organizat i on Code

7. AUTHOR(S)

Lief O. Olsen

t. Performing Organization

10. Project/Task/Work Unit No. 2130467

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No. DoD/CCG Project 73-85

12. Sponsoring Organization Name and Address

DoD/CCG

13. Type of Report & Period Covered Final FY 1973

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 74-600090

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.)

These notes are intended to serve as an instruction manual for technicians and

engineers engaged in metering liquids and calibrating liquid flowmeters. It is a condensed review of the properties of liquids and the mathematical relations required

in this work. References to more complete sources of properties of liquids, theoretical relations and instructions for metering liquids are. included. Separate chapters

discuss liquids and their properties as they affect flow, the theory of incompressible

flow of liquids and the measurements required in the metering of liquids. One chapter

describes several different apparatus and their use in the calibration of liquid

flowmeters. The last chapter contains brief descriptions of the many types of flowmeters such as differential pressure, positive displacement, electromagnetic and

ultrasonic. It also includes a discussion of the physical principles involved in

their design and use.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Calibration; liquid flow; liquid flowmeters; metering.

18. AVAILABILITY STATEMENT

I I UNLIMITED.

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