AlllOO =nM7Qi4 i PUBLICATIONS

NBSIR 81-2313

Performance Evaluation of A

Typical Energy Monitoring

System for Steam Flow in

Buildings

Building Equipment Division

Center for Building Technology

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

January 1981

Issued July 1981

Prepared for •Q^MBiMii^ngineering Laboratory

Construction Battalion Center

lueneme, CA 93010

,Ij56

81-2313

1981

C.2

AUG 1 8 1981

NBSIR 81-2313

/ PERFORMANCE EVALUATION OF A

TYPICAL ENERGY MONITORING

SYSTEM FOR STEAM FLOW IN

BUILDINGS

David W. Baker

James Y. Kao

David A. Didion

Building Equipment Division

Center for Building Technology

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

January 1981

Issued July 1981

Prepared for Civil Engineering Laboratory

Naval Construction Battalion Center

Port Hueneme, CA 93010

z

o

<

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
f:

i
ABSTRACT

Some important design features and measurement techniques are discussed

for determining energy rates in building systems flowing steam. Emphasis is

on use of differential pressure (AP) type flowmeter systems, where only

AP pressure and temperature instrumentation can receive direct calibration.

The role of systematic and random errors in measurement of building

energy is discussed and an appropriate method is given for estimating the

uncertainty in the energy consumed. An illustrative calculation of uncertainty

in accumulated energy used for a one year period is given from estimated

operating data for one of the NBS laboratory buildings. The uncertainty

is estimated to be 3.2! percent for a two-meter (series) configuration and 4.3

percent for a single meter (high range meter) configuration.

Key Words: Building energy monitoring; energy measurement; steam consumption

measurement; uncertainty estimate.
CONVERSION FACTORS TO METRIC (SI) UNITS

Physical

Quantity Symbol

To Convert

From To Multiply By

Acceleration g ft/s^ m/s^ 3.048 X 10

"^

Density P Ib/ft^ kg/m^ 1.602 X 10^

Energy rate E Btu/hr J/s 2.931 X lO'^

Enthalpy h Btu/lb J/kg 2.326 X 10^

Length L in. m 2.540 X 10

"^

Mass flowrate M Ib/hr kg/

s

1.260 X 10

"^

Pressure P psi Pa 6.895 X 10^

hw

in. H

2

O Pa 2.486 X 10^

Specific heat Btu/(lb- °F) J/(kg.K) 4.187 X 10^

Temperature T °F °C t =(t.-32)/1.8

c f

Viscosity M lb/ (f t* s) Pa • s 1.488

Note: English units are customary in fluid measurements and are used in this paper.

IV
TABLE OF CONTENTS

Page

ABSTRACT

CONVERSION FACTORS TO METRIC (SI) iv

LIST OF FIGURES; LIST OF TABLES vi

I. INTRODUCTION . 1

II. DESIGN CONSIDERATIONS 4

1. Meter Installation 5

2. Differential Pressure Measurement 6

3. Metering System Calibration 8

III. ESTIMATE OF UNCERTAINTY 12

1. Systematic Errors 13

2. Random Errors 16

3. Basic Relations 17

4. Error Analysis Results 23

4.1 Correction Factors 24

4.2 Random Error Analyses ........... 24

4.3 Systematic Error Analyses . . . 30

5. Uncertainty in Energy Usage 32

6. Computer Based Calculations .... 36

IV. PROCEDURE SUMMARY ' 39

References 41

APPENDIXES

A. Correction Factor M/M 42

B. Random Error Calculations 46

C. Systematic Error Calculations 50

D. Table of Nomenclature 54

V
LIST OF FIGURES

Page

Figure 1. Orifice meter installation using two pressure transmitters . . 9

Figure 2. A typical steam metering installation 10

Figure 3. Correction factors M/M for north and south branch flowmeters,

900 to 3200 lb /hr . 27

Figure 4. Correction factors M/M for north and south branch flowmeters,

160 to 900 Ib/hr . . .° 28

Figure 5. Random error in flowrates Mnsand M 29

Figure 6, Systematic error in flowrates M^ and M^ 31

LIST OF TABLES

Table 1. Random error and systematic error assignments for energy

rate E 18

Table 2. Random error, systematic error and correction factor

assignments for flowrate M 19

Table 3. Random error and systematic error assignments for differential

pressure AP 20

Table 4. Operating conditions for steam supplied to Materials Research

Building 25

Table 5. Instrumentation systems for steam measurement 26

Table 6. Uncertainty in accumulated energy usage 34

Table Al . Correction factor M/M 45

o Table Bl. Random error 49

Table Cl. Systematic error 53
I . INTRODUCTION

During the late 1970 's large efforts toward improving energy utilization

in buildings were made in various sectors: industrial, residential, commercial, and governmental. Thus, there is a need for accurately measuring

and monitoring energy consumed. In determining costs or energy savings,

often an estimate of or knowledge of the inaccuracy or uncertainty of the

energy measurement is important. For example, a claimed energy savings

or decrease in consumption of, say, 8.0 percent due to a particular operating

procedure or equipment change may actually be greater or less than this.

The uncertainty is never known exactly. However, from knowledge of the

measuring systems, including instrumentation and data processing equipment

and procedures, an estimate of the uncertainty of the resulting energy

measurements can be made through error analysis. Then a statement such

as "an energy savings of 8.0 + 2.0 percent for the period ..." can be

made with a specified degree of confidence, where +2.0 percent refers to

the overall uncertainty of the measurement process. For completeness,

from a measurement point of view, the kinds of error (systematic and/or

random) making up the uncertainty and their magnitudes would be included.

The most accurate measurements of the energy consumed by building

systems which employ flowing fluids such as steam, hot water or chilled

water result from measurements in which each measuring system is calibrated

directly under conditions the same as those which exist when on line.

This means, for example, steam flowrate would be measured with a metering

system calibrated directly at a separate calibration facility flowing

steam at a specified line temperature and pressure. An alternate method

would involve a transfer reference meter system calibrated directly and

subsequently Installed on line to serve as the calibration reference for

1
the (working) meter system. The uncertainty associated with the direct calibration or transfer systems (which combine the effects of many individual

errors) would serve as the basis for assigning an uncertainty to measurements made with the working meter. However, the direct calibration approach

may require facilities not readily available, or it may be too costly or

time consuming particularly when large numbers of meters are Involved.

Also, the adequate installation of a transfer reference meter system into

the building lines may be impractical or impossible in existing building

systems. In fact, adequate installation of the working meter alone is

often difficult. Installation and measurement of other parameters such

as static pressure, differential pressure, single point temperature,

and differential temperature usually offer less difficulty from a

physical point of view.

Thus, indirect methods may be more practical and useful in estimating the

uncertainty of energy measurements. In one approach using differential pressure

(AP) type flowmeters, the flow characteristics of the (working) meter primary

element (orifice, venturi, or nozzle) are determined from published data for

flow coefficients and fluid expansion factors instead of relying on direct

calibration procedures. All other sensors and transducers (temperature, pressure

and differential pressure transmitters) receive direct calibration. Error

analysis then is used to estimate an overall uncertainty based on individiial

errors due to the meter flow characteristics and transducer and sensor

calibrations. This paper Includes a discussion of this approach as

applied to measurements of energy used at one of the laboratories at the

National Bureau of Standards (NBS) , Gaithersburg, Maryland.

2
In the program at NBS , the energy consumed by the heating, ventilating and

.air conditioning (HVAC) systems is being monitored in the Materials Research

Building for the purpose of evaluating energy savings of various operational

and control techniques for NBS general purpose laboratories. The scope of

the program includes monitoring HVAC equipment status, space temperature,

heat transfer medium temperature and flowrate, and building energy consumption

in the form of steam, chilled water and electricity.

Measurements include building supply steam flowrate, temperature and pressure,

and condensate temperature; building supply chilled water flowrate (CHW flow)

and temperature differential (CHW AT) ; and reheat hot water flowrate (HW flow)

and temperature differential (HW AT) for the heat exchangers. Also, for 3 of 10

air conditioning units (ACU) , the reheat HW flow and AT, the CHW flow and AT,

and the preheater steam supply flowrate and temperature, and condensate temperature

are measured. Data is accumulated and processed through an on-line

general purpose digital computer system. This system is designed for

monitoring energy in all main buildings at NBS, Gaithersburg.

The purpose of this paper is (1) to review some important design

considerations for instrumenting energy monitoring systems with emphasis

on systems that measure steam flow, and (2) to present a method of

estimating the uncertainty of energy usage using an Illustrative

calculation for building steam consumption covering a period of one

year. The number of variables is quite large, and a complete list is

included in Appendix D. The accuracy calculations discussed are

currently not computer based.

This paper is written as an aid or guide for the reader who may

be faced, for the first time, with the task of making similar

measurements. Thus, the nature of uncertainty as applied to this

measurement situation is discussed briefly.

3
II. DESIGN CONSIDERATIONS

Establishing an energy monitoring program will include a statement of purpose

or technical goal, a summary of building design and energy systems, an analysis

of any existing building operating data, and a selection of utility parameters

and energy handling systems to monitor. Once these decisions have been made,

design of the measurement system can begin. Starting with design of

individual measuring systems, this may include instrument and hardware

selection, determination of computational, data capacity and instrumentation

accuracy requirements, and consideration of calibration and operating

procedures. Also, the data collection and processing systems need to be

specified and selected. With a computer-based prototype system, data

requirements and program scope often increase during the program so that it

is wise and more economical to use a system on the "large side."

When an energy monitoring system is to be installed in new construction,

this system should be considered an integral system along with other building

systems. In this way, the location and installation of transducer systems,

signal and power cables, and data processing equipment can all receive adequate

attention. Such specific items as location of temperature sensors and wells,

installation of sufficient straight pipe runs for flowmeters, provisions for

inserting on line calibration transducers such as reference flowmeters, location

of transducer or transmitter units for accurate signal sensing and adequate

accessibility for maintenance and calibration usually all can be satisfied

at reasonable costs. Satisfying all such requirements for systems installed

in existing buildings is often expensive and difficult, or sometimes

virtually impossible.

While the energy monitoring program for the existing Materials Research

Building also includes metering chilled water and hot water flows as listed above,

the fluid considered most difficult to measure is steam flow. Systems were in4
stalled in this building to measure steam building supply flowrate, temperature

and pressure, condensate temperature, and for 3 of 10 ACU's, the preheater

supply flowrate and temperature, and condensate temperature.

This discussion pertains to design considerations for the systems

metering the building supply steam flows. The main points discussed are

the meter installation, differential presiiure (AP) measurement, and metering

system calibration techniques.

1. Meter Installation . Steam at nominal 150 psig and nearly saturated enters

the Materials Research Building through an 8-inch supply line and rises to the

4th floor, where its pressure is reduced to a nominal pressure of 27 psia and

temperature of 290®F, i.e., about 45*F superheat. The steam then branches into

two 5-inch lines.

The best flowmeter locations were in the two 5-inch lines from the standpoint

of required straight pipe runs and accessibility. Initially, two reversedtype pitot-static tubes, one in each line, were installed. These have the

advantages of easy and low cost Installation. However, since installation of

adequate plumbing to accommodate transfer reference flowmeters for in-line

calibrations would be quite expensive and impractical in this building,

and since the flow characteristics of such pitot-static tubes are not well

known, each of these were later replaced by a square edge, thin plate, concentric

orifice. The vortex shedder type meter was also seriously considered at this

time but was not adopted because differential pressure transmitter equipment

already installed with the pitot tubes was also suitable for use with the orifices.

Each orifice plate was installed following ASME recommended procedures [1]*.

These included installation of a suitable flow straightener upstream, and use

* Numerals in brackets refer to references listed at the end of the paper.

5
of recommended minimum lengths of straight pipe upstream and dovmstream

of the orifice. Such "flow pattern" control is very important for accurate

metering. Otherwise, transverse or swirling flow components can adversely

influence the differential pressure sensed at the orifice, resulting in

grossly inaccurate flowrate measurement. The importance of flow pattern

control for accurate metering is sometimes not realized and proper installation

of metering systems in existing building systems is often difficult.

2. Differential Pressure Measurement . In addition to a satisfactory meter

installation, successful metering depends on accurate &P measurement across

the flowmeter. The ingredients are selection of suitable transducer or

transmitter and signal processing equipment, installation of transmitter and

pressure sensing lines using "good practice" techniques, and continued

maintenance of the system including transmitter and instrument calibration

checks on a periodic basis as necessary. The importance of an accurate

AP measurement cannot be overemphasized.

In selecting transmitter equipment, several points need consideration.

Usually, the transmitter with the highest "accuracy" rating compatible with the

application needs and dollars available tends to be selected, but transmitter

characteristics covered under the term "accuracy" vary with the manufacturers.

High stability and low hysteresis are essential characteristics. Also,

performance shift due to changes in ambient temperature and line pressure

level should be as low as possible. Highly linear transmitters may not be

essential when used with digital computer systems since nonlinearities can be

corrected through software. Also, fast d3mamic response may not be important

since operating conditions for building systems tend to change slowly.

The full scale differential pressure should be as large as practical compatible

with allowable building system pressure losses, because accurate measurement of

small differentials of a few inches of water full scale is difficult. Finally,

when the flowrate range is larger than say 3:1, use of more than one transmitter

6
should be considered to avoid severe degradation of performance at part scale

operation for transmitters rated on a full scale basis. For example, a transmitter

rated at 0.5 percent full scale would only need produce a differential pressure

measurement within 5 percent of reading at 10 percent full scale to still

comply with its rated performance characteristic.

The AP transmitters selected for this application for all systems were an

industrial type, using diffused silicon strain gages, namely Honeywell Models

41102 and 41105* with rated "accuracy" of 0.35 percent of adjusted span. In order

to cover the nominal 10:1 flowrate range of 320 to 3200 Ib/hr for building supply

steam flow, two transmitters were used in each meter, one with a full atcale

rating of 105 inches of water at 68“F (in. H^O) , and one of 40 inches full scale

with span adjusted to 8.0 inches. The range of 6.37 to 8,00 inches was used as

an overlap region for transmitter checks under computer program control. One

calibration constant was used for scaling each transmitterj thus no corrections

were made for nonlinearity.

The pressure differential sensed by the transmitter should be exactly the

same as that existing at the orifice meter. The interface between steam vapor

and condensate in each pressure sensing line (leg) must be at the same height.

Typical Installation details for both horizontal and nonhorizontal pipe flow

may be found in [1]. Possible troubles include air or vapor trapped in one or

both legs, variation of the density of liquid in each leg due to temperature

gradients, and leakage from the high pressure to the low pressure leg through

any defective bypass valving. To eliminate air or vapors, a vent valve should be

provided at the highest point and a drain valve at the lowest point in each leg.

Systems need purging prior to calibration and should be checked periodically for

presence of air. To minimize effects of possible density gradients in the condensate in each pressure sensing line (exposed to steam temperature at the

* Trade names are used in this paper as a means of identification. They

do not imply endorsement by the National Bureau of Standards

.

7
pressure tap and to room temperature gradients if unlagged) , good practice

dictates keeping these lines short and close together, nearly horizontal if

possible (horizontal pipe run) , with the lines sloping slightly downward toward the

transmitter. With high range transmitters such as 100 in. H^O, this density

gradient effect should be neglible, but when attempting to measure a few

tenths of an inch of water, this effect may be significant though essentially

"unknowable". The bypass valving for the transmitter should provide a positive

check on bypass leakage. Figures 1 and 2 show a schematic diagram and view of

one metering installation using two differential pressure transmitters. Thus,

installation using good practice techniques is essential and measurements should

be confined to high AP whenever possible.

3. Metering System Calibration . As mentioned, for the most accurate measurements, the metering system including adjacent inlet and exit piping along

with its flow straightener and the differential pressure measuring equipment

should be calibrated directly at a calibration facility with the same fluid

(steam) at the same temperatures, pressures, and flow range as those existing

during application. An alternate would be use of a suitable transfer

reference flowmeter calibrated directly on steam and then installed on line

at the application site. Since these approaches are usually expensive,

time consuming, and often not practical, (lack of available calibration

facilities, or existing building systems piping unsuitable), other approaches

to the calibration problem must be considered. One possibility is the direct

calibration of the metering system using a liquid such as water over the Reynolds

number range of interest (thereby obtaining the coefficient of discharge

for the orifice directly) along with the use of published fluid expansion

factors for steam [1] for calculation of the flowrate. Another possibility,

and the one used in this energy study, is direct calibration of just the

differential pressure and the steam temperature and pressure measuripg systems,

and use of known orifice meter coefficients and fluid expansion factors [1]

8
Figure 1. Orifice meter installation using two pressure

transmitters, one high, one low, and showing

shutoff, vent, drain, and bypass valves

9
ensing

lines

and

valves

10

Figure

2.

A

typical

steam

metering

installation
in determining the flowrate.

Calibration of the AP system should be conducted on site if possible.

The calibration equipment may range in complexity from liquid manometers (vertical,

inclined, or micro type using water or mercury) to laboratory reference type

pressure transducer equipment incorporating a digital readout and automated

computational features. Results given here were obtained with vertical manometers

using water or mercury. In any event, the importance of calibration on site and

the establishment of an ongoing calibration program should be emphasized.

This program should include establishing a definite calibration procedure, and

performing calibrations on a periodic basis to monitor the performance of the

differential pressure systems. Use of the control chart method, giving

transmitter system performance accumulated periodically is recommended.

Several points between zero and full scale AP should be checked. If

building system flow can be stopped momentarily, the calibration procedure

should Include a no flow (zero) AP check.

To summarize, certain design considerations, namely the flowmeter installation,

the differential pressure measurement, and the metering system calibration have

been briefly discussed with reference to systems installed in the current NBS

program. It is felt these are basic topics of interest to others involved in

similar building system energy monitoring programs. Next, a method is presented

for estimating the uncertainty of energy usage for the case here, where the flowmeter primary element (orifice) receives no direct calibration, but only the

meter differential pressure, and the steam pressure and temperature instrumentation

are calibrated directly.

11
III. ESTIMATE OF UNCERTAINTY

A statement of the amount of energy used based on site measurements

may be viewed as incomplete unless it includes an assessment of measurement

uncertainty. In many conservation matters, the level of "threshold" savings

will depend on this uncertainty.

Several introductory statements are needed at this point. The uncertainty

or (in) accuracy is usually considered to include two types of error*: systematic

error and random error. Systematic error introduces bias which cause the measured

value of a quantity under specified conditions to be consistently either too

high or too low with respect to the true value. Random or precision error pertains

to those errors which, when repeated measurements are taken of the same quantity

under specified conditions, result in a scatter about the average of these measurements. In making measurements where the true value of a quantity is unknown,

for example the flowrate in a pipe, observations from two independent measuring

systems connected in series can be used as a means of estimating the uncertainty.

That is, the presence of systematic error can be detected and under certain

conditions, the random error in each system can be estimated. This is essentially

the case when calibrating a flowmeter directly with separate calibration equipment

(calibrator) where the calibrator performance becomes arbitrarily the true value.

Differences between the average values of the calibrator and the flowmeter

performances are then assigned to the flowmeter as a systematic error (correction

factor, calibration factor) and from the scatter of these measurements, a random

error for the flowmeter such as the standard deviation may be estimated. A

random error and a possible systematic error associated with the calibrator

itself may be accounted for separately, but most often these errors are included

in the errors assigned to the instrument being calibrated.

* Error is defined as the difference between the value indicated by a measurement process and the true value.

12
However, the use of independent systems for routine on line measurements,

although most basic and sound, is not usually used in building technology work

simply because successful design, installation, debugging and operation of just one

measurement system is often task enough. Rather, a single measurement system

is used and, when required, the uncertainty estimate is usually based on results

of direct calibration of system components, or on an analysis of effects of different

parameters on their performance using error analysis, or a combination of both.

In this section, the uncertainty in the energy usage through steam

consumption in the Materials Research Building at NBS is considered in

detail, and an illustrative calculation of the uncertainty in the steam consumed

over a period of one year is given. In brief, this consists of determining

the systematic error and the random error for the energy usage, both of

which will be found to be flowrate dependent. Since the steam flowrate

varies with time (hourly, dally, seasonly) , the systematic error is

"weighted" with time to determine an effective overall uncertainty for the

one year period. For a large number of data, as would be collected over

a period such as a year, the effects of random error essentially disappear,

For a single observation, the uncertainty would be expressed in a statement

which includes both the systematic error and the random error.

1. Systematic Errors . These errors, due to bias in the measurement

procedure, are usually composed of both "known" and "unknown" components.

A known systematic error is defined as one in which the magnitude and sign

are known for any given set of operating conditions. Such errors are

removed by correction factors, and henceforth in this report, this type of

error will be termed correction factor . In the steam flowrate measurements

here, correction factors were used to remove bias due to:

13
° the variation of the orifice discharge coefficient C with

Reynolds number

° the variation of fluid expansion factor Y with parameter AP/Py

° the variation of steam density p with P and T, and

° the orifice area expansion (parameter F ) with T.

Si

An unknown systematic error is defined as one in which the magnitude and

sign are unknown, but the bounds in which the error will likely remain is

known or can be estimated. Here henceforth in this report, this type of

error will be termed systematic error . These errors include;

° the deviation from a linear response of the differential pressure

transmitter

° the zero and span shift due to ambient temperature changes

for this instrument

° the estimated error in the calibration procedure for this

instrument

° the uncertainty or tolerance in the discharge coefficient C as assigned by the ASME [1]

° the tolerance in the expansion factor Y assigned by the

ASME [1], and

° the uncertainty in orifice diameter d and pipe diameter D

due to manufacturing tolerances, in lieu of direct measurements

of these quantities for each installation.

The bounds for the transmitter errors are based on ratings as published by

the manufacturer. On this basis then, the purpose of the current calibration

program is to check and monitor transmitter performance against the manufacturer

specifications. However, these transmitter errors could be reduced significantly

or essentially eliminated through an extensive calibration program in which the

performance curve (correction factor) is established and monitored for each

transmitter, provided each transmitter operates in a constant or known

ambient temperature environment. Such may require Individual temperaturecontrolled housings, and all of this may be Impractical unless maximum accuracy

14
is required. Zero and span shifts due to line pressure were found negligible for the

transmitters used. Zero and span shifts due to ambient temperature are independent

effects according to the manufacturer. Errors due to tolerances on C and Y could

be reduced significantly by direct calibration of the orifice meter systems

flowing steam at operating conditions P and T.

When considering a system having several sources of systematic error, a

decision has to be made as to how these errors will be combined. When there

are many independent sources of error, each having equal effect on the error

of the result, linehr addition of the errors may result in an overall systematic

error too pessimistic (too large), and one researcher [3] states "if five

or more of the largest independent error sources have approximately equal

effects on the error in the result, i. e. , if the values of each error are

of about the same magnitude, the laws of chance could reasonably be expected

to apply." This means some errors are expected to be positive, and some negative.

Thus, the two types cancel each other to some extent, and combination of the

errors on an RSS (root-sum-square) basis can be justified. On the other

hand, when there are just two independent sources of error, combination

should be on a linear basis [6]

.

In some cases the method of combination

then may become controversial and arguments may even arise on the use of

linear addition in combining the individual systematic errors; that such

an approach is too conservative or pessimistic. A counter argument is that

while use of the individual error approach may be the only practical method

for a given program, it does suffer from the possibility of omitting unknown

(but important) sources of error. The use of linear addition, therefore,

tends to counteract these possible omissions.

15
In this analysis on the measurement of steam flowrates, the largest

error is nominally about twice the next largest error for each meter* , and

even though there are 4 to 6 independent error sources for each meter, the

smallest error being about 1/lOth the largest error, qualification for

combining errors on an RSS basis as set forth in the criterion above [3]

would be weak. Further, when combining systematic errors from the two meters,

there is only a 50/50 chance that the errors tend to cancel, l.e., when the

error of one meter is "+” , there is equal chance the other may be "+" , or

Therefore, for these reasons, the (likely bounds to the) overall systematic

error will be obtained by linear addition of the individual error sources.

2. Random Errors . Due to the nature of this type error and from propagation

of error theory’, for example [4] , random errors can be combined on an RSS

basis. In this analysis, the random errors included are:

o the differential pressure transmitter hysteresis (repeatability)

o the effects of variation of P and T on steam enthalpy h (through p)

o an allowance for variation in density of the condensate in the

transmitter pressure sensing lines

o the accuracy of the analog/digital (A/D) conversion of the transmitter

output, and

° the accuracy of P, T, and measurements.

Tables 1, 2, and 3 list the correction factors, the systematic errors

and the random errors as assigned to energy rate E, mass flowrate M, and

differential pressure h , respectively^*. It is noted that effects of w variation in steam density p include the following: a correction factor

e^^ for the flowrate; (part of) the random error e^ for the enthalpy; and

a random error e, for the flowrate. The effect of variation of p with P and

P T is much greater on the flowrate than on the enthalpy, and since correction

* Appendix C, Table Cl.

** Systematic errors are denoted by an "s" in the subscript and correction

factors by a "c".

16
factors are applied manually at present rather than being computer based,

considerable time savings result from applying just one correction factor*

i.e., e^^ on the flowrate. The effect of this simplification on the accuracy

of the results is negligible.

Following are the basic energy rate and flowrate equations used in this

analysis, and corresponding equations from error propagation theory for evaluation

of the systematic and random errors.

3. Basic Relations , As stated previously, the steam supply line entering the

Materials Research Building divides into two branches, denoted north (n) and

south (s) with an orifice meter installed in each branch. This steam is slightly

superheated at pressure P and temperature T, measured immediately upstream of the

branch. Condensed steam at temperature T^ returns to the power plant at

atmospheric pressure.

The basic relation for net rate of energy consumed is:

E = (M + M )[h - c (T - 32.0)] (1) n s p c

The basic relation for mass flowrates M and M , written as M is, from

n s

[1] for example:

M = 358.93 C Y d^ F [ph /(I - (2) a w

Applying propagation of error theory to equation 1, the random errors for

individual parameters can be combined on an RSS basis giving a random error

e_ in the energy rate E such that:

17
TABLE 1

Random Error and Systematic Error Assignments for Energy Rate E

Error

Symbol Description Random e^ Systematic e^^

Flowrate, see table 2 X

Variation of enthalpy with

density p, and accuracy of

P 6i T measurement

X

^Tc

Variation of condensate

temperature and accuracy

of Tc measurement

X

^s Flowrate, see table 2 X

18
TABLE 2

Random Error, Systematic Error and Correction Factor Assignments for Flowrate M

Error

Symbol

Correction

Description Random e^^ Systematic e^^ Factor

®Cs Tolerance on discharge X

coefficient C

Tolerance on expansion X

factor Y

®ds Manufacturing tolerance on X

orifice diameter d

®3s Tolerance on beta ratio X

d/D

e

P Variation of density p, and X

accuracy of P & T measurements

^hw

Differential pressure, see X

table 3

^hws

Differential pressure, see X

table 3

®Cc

Variation of C with Reynolds X

number

^Yc

Variation of Y with h /P X

w

®Fc

Thermal expansion of the orifice X

epc

Variation with density p, X

and P & T

19
TABLE 3

Random Error and Systematic Error Assignments for Differential Pressure

Error

Symbol Description Random Systematic

e^g Transducer linearity

e. Transducer ambient temperature A Q zero and span shift, RSS basis

e^j^ Transducer hysteresis

e^^^ Reference calibration transducer

current and reference pressure,

RSS basis

Density variation in sensing

lines, allowance for

e. . Analog/digital conversion,

computer system

X

X

X

20
The flowrates M and M refer to the north and south branches, respectively,

n s From simultaneous values of Mnsand M and corresponding r o values of e., /M and

Mn n

e,, /M , error e„/E can be evaluated. Specific heat c is assumed constant, thus

no term appears. It is shown in Appendix B that for the existing operating

conditions all errors are significant, and that e„ and e„ control at the lowest

Mn Ms

AP values for each transmitter.

From propagation of errors through equation 2, random errors

®Ms^^s

written as become:

IL

M

!pV +

2p/ V2h

W' J

1/2

(4)

In addition to transmitter errors, differential pressure data include

possible errors due to the analog/digital conversion and any differences in

density of the fluid (condensate) in (nonhorizontal) pressure sensing lines.

These errors are assumed random in nature along with the transmitter hysteresis

error, and each is assumed to have equal effect. The random error e, on an hw

RSS basis is then:

ehw (5)

Thus, random error has 3 components, those due to AP transmitter hysteresis

®th* ^lis^log/digital (A/D) conversion error sensing line error

These errors are discussed in detail in Appendix B. Random errors must be expressed on the same basis, i.e., confidence intervals

of the same degree or the same number of standard deviations. For this analysis,

the random error data and assumed correspond to two standard deviations or 2o.

On this basis, the estimated random errors are likely not exceeded 95 percent

of the time.

21
The correction factors e_ , e., , e and e„ listed in table 2 are

Cc Yc pc Fc

applied directly to observed flowrates The systematic errors, not

known explicitly, are estimated in terms of bounds or limits likely not

exceeded. No systematic errors in h and c^ were assumed to exist. Thus

from propagation of errors through equation 1 and combining on a linear

addition basis, the systematic error e^^ in energy rate E is:

'Es (e^ /M ) M ^ (e^ /M ) M

Ms , n n n + Ms.s, s s' M + M

n s M + M

n s

( 6 )

where e„ systematic error in flowrate M

Ms,n n

e^ systematic error in flowrate M . iris 9 S S

Table 2 lists the systematic errors for flowrate M, namely Syg*

^ds’ ^6 s ^hws* propagation of errors through equation 2 and combining on a linear addition basis as discussed above, the systematic error

in flowrate M is:

'Ms

M

^Cs + ^Ys +

^ ^ds + ^hws +

^ ^ ^3s

2 h

w 1 - 3 J

(7)

Systematic error three components, ®ij,g» ®^g» ®j^^g

listed in table 3,

They are, respectively, the transmitter linearity characteristic, the effects

of ambient temperature on transmitter performance and the estimated accuracy

of the calibration process. All are assumed have equal effect on e, and thus

'hws on an linear addition basis:

'hws

+ [*e d" 6 , d- e_

*1 = — Ts As RsJ

( 8 )

22
These errors are discussed In greater detail in Appendix C.

A statement expressing the estimate of the overall uncertainty of the

energy rate can be made using the systematic error and the random error, provided

each error is identified or reported separately. For a conservative estimate:

Maximum Overall Uncertainty = + (e^g/E + e^/E) (9)

For perhaps a more probable, realistic estimate:

Probable Overall Uncertainty = + ( 10 )

The uncertainties above refer to a single observation. The effects of

random error involving many observations are expected to essentially disappear.

This is the case when considering uncertainty in energy consumed over a

sustained period, such as a few months or a year. The systematic error,

however, represents a bias which remains and when it can not be quantified

explicitly, it becomes an estimate of the bounds or limits to the error

which is likely not exceeded.

The operating conditions for steam supplied to the Materials Research

Building are summarized in table 4, and characteristics of the instrumentation

systems used to measure this steam are listed in table 5.

4. Error Analysis Results . In this section, results of error analysis are

presented for steam consumed with the above operating conditions and instrumentation systems. These results include the correction factors as applied to

the steam flowrates, and analysis of the systematic errors and the random

errors. Finally, using a set of typical building steam consumption data,

an estimate of the effective overall uncertainty for energy consumed over

a one year period is given. The uncertainty is found due primarily to the

errors encountered in the differential pressure measurement and the tolerance

or uncertainty in the orifice discharge coefficient.

23
4.1 Correction Factors. Figures 3 and 4 give correction factor M/M for the

o flow range 160 to 3200 Ib/hr, where M is the observed or indicated flowrate

o output from the computer for either flowmeter. These data include correctious for

errors e^^ due to variation in the orifice discharge coefficient, e^^ due to

variation in the steam expansion factor, e^^ due to thermal expansion of the

orifice, and e^^ due to steam density variation. Detailed explanation of these

corrections are given in Appendix A. Thus, mass flowrate becomes

M = (M/M )M and M = (M/M )M n o o,n s o' 0,1

( 11 )

For the operating conditions given, M/M^ varies through a range of about 8 percent.

For maximum utility, steam data p = p(P,T) and corrections e and e should

LfC Y C

be computer based,

4,2 Random Error Analysis . Figure 5 shows results of random error calculations

of ®Ms^^s^ using equation 4, This error is flowrate dependent

through the term e, /2h where h varies as M^ and its minimum value hw w w

depends on e , Error e, includes effects of errors e , , e. and e„, as listed

p hw th A/D SL

in table 3, Error e^ is due to uncertainty in measurements of steam conditions

P and T. Appendix B gives details of these calculations.

By using two AP transmitters, denoted HIGH and LOW, the random error is

confined to a range 0.5 to 2.7 percent of flowrate for the nominal range 160

to 3200 Ib/hr, with smallest error at highest flowrates for each meter. The

improvement in precision by the use of two transmitters is quite evident.

As shown, use of the HIGH transmitter alone over this flow range would result

in random error as high as about 30 percent at the lowest flowrate.

The random error e /E in the energy rate expressed in equation 3 depends

^Mn’ ^Ms’ ^h ^Tc* significant with the e^ values controlling

at the low end of the AP range of each transmitter. As discussed in Appendix

B, e /E will vary upward from about 0.7 to 2 percent for the system presented

E

here.

24
TABLE 4

Operating Conditions for Steam Supplied to Materials Research Building

Temperature T = 290°F, nominal; data collected indicated annual

variation in the range 280 to 299®F

Pressure P = 27.3 psia, nominal; with annual variation in the

range 26.8 to 27.9 psia

Enthalpy h = 1185.0 Btu/lb, nominal; range 1179.8 to 1190.0 Btu/lb

Condensate temperature T^ = 176°F nominal; with annual variation in

the range 173 to 179®F

Net enthalpy Ah = 1041.0 Btu/lb

Steam flowrates M and M , each range 400 to 2900 Ib/hr n s

25
TABLE 5

Instrumentation Systems for Steam Measurement

Flovmieters , one each branch (north, south)

Pipe Size D = 6.057 inches

Orifice plate, square edge, stainless steel

d = 2,359 inches

3 = 0.389

Comer pressure taps

Differential pressure transmitters , manufacturer specifications,

one HIGH AP, one LOW AP transmitter for each meter, sensing lines

connections, see figure 1.

Item HIGH AP LOW AP

Full Scale 105 40 in. H^O

Adjusted Span (^^gp) 105 8 in. H^O

Linearity error (e^^ ) 0.35% 0.35% span

Ambient temperature effect. 1.40%/100®F 1.75%/100®F span

zero and span shift error (e. ) /VS Hysteresis error 0. 10% 0.10% span

Steam Temperature Transmitter, resistance type, 40K ohms at 77

well immersion, range 250-475®F, system accuracy + 1.5®F (e^)

Steam Pressure Transmitter , pressure-to-potentiometer , resistance

type, range 0 to 30 psig, system accuracy +0.25 psi (®p)

Condensate Temperature Transmitter , resistance type, 40K ohms at 77 *F,

well immersion, range 250-475®F, system accuracy + 1,5®F

26
oooooo<o ooocsoomoooo00 o o o o o 00 cs o <Ti o\ 9\ a\ ON • • o o o o o

41Uo

00 o , H/H

27

Observed

Mass

Flowrate M,

Ib/hr
o , H/H

M

4-» COIiH

COCO

•d>w

0)

CO Xio

28

Figure

4,

Correction

factor

M/M

for

north

and

south

branch

flowmeters

160

t8

900

lb

/hr
/M,

Percent

of

rate

Flowrate M, Ib/hr

Figure 5. Random error in flowrates M and M

II 9

29
4.3 Systematic Error Analysis . Figure 6 shows systematic error in the flowrate

n^^n’ ^s s^^^s^

calculated through equation 7. This error is primarily

due to the tolerance in the orifice discharge coefficient and the error in the

differential pressure h^. The components of ate the differential pressure

transmitter linearity e , the transmitter ambient temperature error e. and the

1. S aS

estimated error in the calibration procedure for the transmitter, e . These

KS errors are discussed in detail in Appendix C,

Flowrate error e /M ranges from about 1.6 percent of rate to 36 percent

ITio for HIGH and LOW meter configuration over the nominal flow range 160 to 3200 Ib/hr,

with smallest error at highest flowrates for each meter. At high AP values

for each transmitter, the tolerance in the orifice discharge coefficient

(e^^ = 1.00 percent) and error are both significant, while at low

AP values error dominates. For the HIGH transmitter alone, ®^g/M

ranges very high at low AP. This emphasizes the reason for using more

than one transmitter when flow will vary through a range of more than,

say, 3 to 1. This system with two transmitters is considered weak for

extended use at very low flows, e.g., range 160 to 400 Ib/hr where Ej^s/M > 6.0

percent

.

Systematic error in the energy rate depends entirely on flowrate

errors e ^.nd e^ /M and is calculated through equation 6. Errors in r^s ^ s s rrs ^ xi n enthalpy e^ and condensate temperature e^^ have been assigned as random errors,

and the specific heat of the condensate is assumed constant. Since error

®Es^^ depends on flowrates and M^, this error varies with building

operating conditions as these flows vary independently with loads. Thus, no

single value e^^/E applies. However, an effective or weighted systematic error

can be obtained by analyzing steam flowrate usage data over an extended building

operating period. An illustrative calculation is given in the next section.

30
Percent

of

rate

Flowrate M, Ib/hr

Figure 6. Syateoiatic error in flowrates

31
5. Uncertainty in Energy Usage, The uncertainty in energy usage for a period

of one year has been determined from the systematic error in the energy rates.

Effects of random errors become negligible with such a large number of observations.

Therefore, the uncertainty for accumulated usage can be written as:

A =

s

Je„ .At,

1 Esi 1

SE.At,

1 3- X

(12 )

where E e„„,At, denotes the sum of the systematic errors accumulated over a

1 ESi 1

year and, S E.At^ is the total steam energy used in a year.

' i

^ ^

With random variations in Ah, accumulated error A^ is dependent only on

flowrate M and corresponding systematic error With in terms of

(figure 6) and when At^ applies to both north and south flowmeters, equation 12

applied to the two-branch circuit becomes:

A =

S V

(e^ . /M ,)M , + (e^ Msx,n ni ni M; E (M ^ + M ,)At^ ni sx i/M . sx,s sx (13)

Energy consiimptlon data from actual operation, such as data collected

from energy monitoring and control systems, may be used for the uncertainty

calculations. Calculation details for a set of typical building steam

consumption data using equation 13 are presented in table 6. In this case,

the steam consumption of the building was estimated by using the temperature

bin method [7]. Then, the whole year was divided into eight steam flowrate bins

according to the magnitude of the flowrates. These estimated operating data

are given in terms of steam rates Mnsand M for time Intervals At., the

X accumulated time corresponding to the listed flowrates (columns B, C and E)

.

Also listed for equation 13 are values of the numerator (column G) and the

32
V

denominator (column H) quantities for each At^, and accumulated sums which

apply to the transmitter system used. Error data, columns D and F are from

figure 6. Thus, for the HIGH and LOW meter configuration, the uncertainty

in the energy consumed for the 1 year period is estimated not to exceed

3.2 percent. With a net enthalpy change Ah of 1041 Btu/lb (table 4),

6 9

the total energy consumed was 1041(21.54 x 10 ) = 22.423 x 10 Btu

where 21.54 x 10 is the total pounds steam consumed for the 1 year period.

A statement which includes the uncertainty could then be made such as:

"The steam consumed during the one-year period was (22.423 +

9

0.718) X 10 Btu, where the uncertainty due to systematic error in the

9

measurements is estimated not to exceed 0.718 x 10 Btu and the net effect

of random errors are assumed negligible."

For the single transmitter case using the HIGH meter, the estimated uncertainty is 4.3 percent. While for this set of building data, the improvement

from the two transmitter case is just 1.1 percent, it is noted the LOW transmitter

was is use only about 1/4 of the time. In any event, these data illustrate the

dependence of the uncertainty on both the operating conditions and the measurement systems characteristics.

In particular, the characteristics of the flow measurement system are most

important, where with differential pressure type systems, the AP transmitter

will inevitably be a "crucial link" in the system. When operating conditions

show that flowrates vary but little, then one transmitter should suffice, sized

full scale at or slightly above the operating flowrate. When operating conditions

vary through a large range as in this example, or where a low flowrate occurs

for a large portion of the total operating time, then the two transmitter system

appears feasible. An added advantage to the two transmitter system is that of

system self-checking, i. e. comparison of system LOW and HIGH outputs whenever

AP (k^) is in the overlap region. Whenever the two outputs differ by more than

a predetermined amount, an alarm or diagnostic is output and appropriate action

taken. The present NBS software system includes this feature.

33
TABLE 6

Uncertainty in Accumulated Energy Usage

i ^^i

M , ni ^si, n ^si

M

n

^si, s M

s

(D*C + F*E)B

xlO"^

(C + E)B

xlO"^

hrs lb /hr % lb /hr % lb*% lb

A B C D E F G H

HIGH and LOW Transmitters

1 1139 449 5.25 438 5.50 5429 1010

2 350 449 5.25 588 3,45 1535 363

3 526 711 2.75 588 3.45 2096 683

4 350 1252 3.65 588 3.45 2309 644

5 5169 1252 3.65 1369 3.25 46619 13548

6 613 1881 2.28 1855 2.32 5267 2290

7 438 2266 1.95 2312 1.93 3890 2005

8 175 2868 1.72 2830 1.73 1720 997

8760 68865 21540

Uncertainty A^ = ]EG/SH = 68865/21540 = 3.2%

HIGH Transmitter

1 1139 449 18.8 438 19.9 19542 1010

2 350 449 18.8 588 10.1 6695 363

3 526 711 8.1 588 10.1 6153 683

4 350 1252 3.65 588 10.1 3678 644

5 5169 1252 3.65 1369 3.25 46619 13548

6 613 1881 2.28 1855 2.32 5267 2290

7 438 2266 1.95 2312 1,93 3890 2005

8 175 2868 1.72 2830 1.73 1720 997

8760 93564 21540

Uncertainty A^ = 93564/21540 = 4.3%

34
The question can be raised concerning practical ways to improve the uncertainty. The first priority item would be the reduction of the systematic

error due to the pressure transmitter systems. Such would require periodic

calibration of the transmitters resulting in a linearity error "curve" as a

function of AP, or the equivalent, for each transmitter. Such data would then

be applied as a correction factor for h^. In addition, adequate control

of transmitter ambient temperature environment would be needed. Also, it

would be necessary to demonstrate that each transmitter is not significantly

influenced by other parameters such as line pressure level and that hysteresis

(repeatability) is truly a random characteristic. Assuming such a program

results in a systematic error 0.50 percent or less, (now considered

a tolerance) then error from equation 7 would not exceed 1.5 percent,

where = 0.25 percent, = 1.00 percent, = 0.25 percent,

and e, and e„ have been accounted for. With e., /M < 1.5 percent, then from

equations 6 and 13, the uncertainty in the accumulated energy A^ ^ 1.5

percent

.

The second priority item concerns the present tolerances: orifice discharge

coefficient e and fluid expansion factor e . These tolerances could be

^ S 1 s reduced significantly by direct calibration of the orifice meter system

on steam at line conditions P and T. In this case, e^^ approaches zero and

e^^ becomes dependent on the performance of the calibrator facility. Allowing

0.5 percent for the uncertainty in the calibrator performance, the uncertainties

e^^/M and A^ then might be reduced by a factor of two, to, say, 0.75 percent.

Alternately, the orifice meter could be calibrated on a liquid such as water.

The tolerance e^^ = 0.25 percent would still be needed, and allowing 0.25

percent for the uncertainty in the liquid calibrator performance, the

uncertainties and A^ could be reduced significantly, to, say, 0.75

percent. While these estimates include some speculation, it is true that

extending the calibration program will improve system performance most at

low energy rates and least at the higher rates.

35
The requirements of individual programs would dictate whether the first

or both of the two items above would be practical or not. It will only be

noted here that where calibration facilities flowing steam are unavailable,

there appears not much choice other than use of (uncalibrated) meters such as

the orifice, venturi or flow nozzle because the flow characteristics of these

are considered known when the installation conditions are met. Also, the use

of suitable condensate meters is appropriate should the application allow

this. Where a suitable calibrator facility flowing steam is available, other

type flowmeters such as the more recently developed vortex shedder type

should be considered. The big advantage is that its output, an electrical

pulse whose frequency varies directly with volume flowrate, is relatively

easy to measure. However, for best accuracy the vortex meter must be

calibrated directly on a calibrator flowing steam.

6. Computer Based Calculations . In the foregoing, the uncertainty was noted

to be dependent on both building operating conditions and performance of the

measuring systems. Because of this, the estimate of uncertainty involves

handling a number of variables and large amounts of data. Its calculation

becomes lengthy and complex compared to the basic calculations of flowrate

and energy flow. Therefore, it is concluded that when such estimates are

needed on a routine basis, they should best be computer based. The following

considers calculations pertaining to the previous example for use with a

computer based system where the correction factors (e„ , e^ , e„ and e )

are applied and the uncertainty calculations of the energy measurement are

made through software. Assume equations 1 and 2 or the equivalent are being

calculated and that computer based measurements of h^, P, T and T^ are being

made. Also, where two transmitters are used for h , one HIGH and one LOW,

w

it is advisable that check comparison measurements of h^ are made on line

* in the overlap region. Calculate in the following priority:

36
A. Steam density p, under program control from current measurements of

P and T, for use in equations 1 and 2, This involves storing data in

tabular form for p = p(P,T) for the range of interest and use of linear

interpolation, or use of ai appropriate equation to calculate p directly.

Note a check on apparent ste im condition could be made. For exampld,

for current P, test for T > T , where T is the saturation temperature

s s corresponding to P, A diagnostic should be output to indicate ste2un

in the saturation or wet condition.

B. Steam expansion factor Y, under program control from current measurements of h^ and P, for use in equation 2. This involves storing data

in tabular form for Y = Y (AP/Py) for the range of interest of AP/Py

and use of interpolation, or calculating Y from an appropriate equation

as given, for example, in [1],

C. Orifice coefficient of discharge C, under computer control, for use in

equation 2. This involves a calculation of a Reynolds Number (Re),

storage of data for each orifice meter in tabular form for C = C(Re)

and use of interpolation to determine C. Alternately, an appropriate

equation C = C(Re) may be used. Since Re varies with M, a preliminary

calculation of M through equation 2 using a nominal value of C will be

necessary. In turbulent flow, C normally varies slowly with Re, so

that one preliminary calculation of Re will usually suffice. Otherwise,

iterative techniques should be used.

With C, Y and p determined, calculate current values of flowrate M and energy

rate E, and store and output as appropriate for the instrumentation calibration

program and calculation of the uncertainty.

37
D, Read and store under program control all periodic calibration data

for h^, P, T and working instrumentation and manually operated

reference calibration instrumentation. This requires (manual) input

and processing capability for the reference calibration data. Thus,

records of calibration results and checks of instrument performance

would be computer based,

E, Calculation of uncertainty in accumulated energy usage for a

predetermined period of time using equations 12 or 13 or equivalent.

This requires (for equation 13), storage of all flowrates M^, corresponding

At. and calculation of e.. The period of time ZAt. should include

1 Msx 1

sufficient observations so that random error effects approach zero.

Alternately, a current value of A could be calculated, starting at a

s designated time, by updating equation 13 at each current observation.

This would eliminate storage of data.

F, Calculation of overall uncertainty on a single observation basis,

expressing in the form of equation 9 or 10. This requires calculation

of e_ and e„. Es E

38
IV. PROCEDURE SUMMARY

The following summarizes a suggested procedure for estimating the uncertainty or (in) accuracy in the energy measurements for similar cases flowing

steam or other compressible fluids where only the AP, P and T instrumentation

can receive direct calibration.

1. The system design, installation and usage should conform to good metering

techniques. This includes: metering runs which have adequate lengths of straight

pipe and use of flow straighteners; AP sensing lines horizontal and short

with low slopes and no air traps, adequate and convenient provisions for venting and

draining, and positive isolation between the high and low pressure legs; and

periodic calibrations of the AP, P and T systems.

2. Specify sources of possible error. Classify as correction factors,

systematic, or random (precision) errors. Systematic errors are those nonrandom errors of unknown sign and magnitude and for which only the bounds or

limits are known or can be estimated.

3. Decide which correction factors will be significant and quantify these.

For manual corrections, data such as those of figures 3 and 4 can be

calculated. For computer based corrections, software performing calculations

such as outlined in Section III 6, A, B and C above can be developed.

4. Calculate bounds for the systematic error in the energy rate,

once the method of combining Individual errors has been decided. Express error

as function of flowrate, as in figure 6. Such data can then be used for estimates

of the uncertainty for accumulated energy usage through applications of equation

12 or equivalent, either through manual or computer based calculations as outlined in Section III 6, E.

39
5. Calculate random error in the energy rate, combining individual errors

on an RSS basis. Express error as function of flowrate, as in figure 5. These

data are used in any statement of uncertainty as given in equations 9 and 10 for

single observations of the energy rate.

40
REFERENCES

1. ASME; Fluid Meters, Their Application and Theory, American Society of

Mechanical Engineers, Sixth Edition, 1971

2. L.K. Spink, Principles and Practice of Flow Meter Engineering, The Foxboro

Company, Ninth Edition, 1967

3. B.T. Amberg, "Practice and Procedures of Error Calculations," Instrument

Society of America, Symposium on Flow, Volume 1, Part 3, 1974, Pg. 1267-1284

4. Robert P. Benedict, Fundamentals of Temperature, Pressure, and Flow Measurements, John Wiley and Sons, Second Edition, 1977

5. Conversations with representatives of the steel industry, Mr. Bill Franz,

U.S. Steel Corporation October 11, 1978 and Mr. John Holleron, Republic

Steel Corporation, October 11, 1978

6. H.H. Ku, "Notes on the Use of Propagation of Error Formulas," J. of Basic

Research, National Bureau of Standards, Volume 70C, No. 4, October-December

,

1966

7. ASHRAE Handbook and Product Directory, Systems Volume, Chapter 43, Energy

Estimating Methods, American Society of Heating, Refrigerating and

Air-Conditioning Engineers, Inc., 1980.

41
APPENDIX A

CORRECTION FACTOR M/M

o

Actual steam flovnrate M is obtained through correction of observed steam

flowrate M^ by the computer fjrom the equation;

M = (M/M )M (11)

o o

M/M is a correction factor which removes bias due to;

o

® the variation in the orifice discharge coefficient C with

Reynolds Number (e ) Cc

® the variation of fluid expansion factor Y with parameter AP/PY (©y^)

® the variation of steam density p with P and T

° the orifice expansion with T

This correction factor M/M was obtained as follows:

o

1. The observed flowrate measured by the computer is

M = K(A/999)^/^ = K(h /h

o w w^ ’

^ scaling constant, differential pressure transmitter

^wsp span, AP transmitter

A observed count, A/D coverter output, and

999 full scale count, A/D converter

For the HIGH transmitter, K = 3200 lb /hr and h = 105 in. H„0. and for

wsp 2

the LOW transmitter, K = 883.3 lb /hr and h = 8.00 in. H„0. Thus

wsp 2

l/TT

M = 3200(h /105) , 8.00 < h <105 in. H^O, and

o w — V — 2

*

M = 883o3(h /8.00)^^^, 0.26 < h < 8.00 in. H_0 o w — ^ _ 2

42
For both meters;

M„ = 312.29

The A/D converter was assumed linear with a random error of + 1 count,

expressed as

2 1/2 4 1/2

Since M = 358.93 CYd Fa (ph ) /(1~3 ) , (equation 2), a correction factor

w

M/Mo can be expressed as

2 1/2

M/M = 358.93 C Y (2.359)^(1.004) (p) ^ /0. 9885 , or

° 312.29

M/Mo = 6.4963 C Y (p)^^^, where (A2)

d = 2,359 inches

Fa = 1,004, and

3 = 0.389

The discharge coefficient C is based on data for comer taps given in [2]

4 1/2 wherein C = F^K^(1 - 3 ) ; = 0.6098 for 3 = 0.389 (page 144) and F^ is a

function of pipe Reynolds Number as given in Figures B2317, B2318 (page 239).

Pipe Reynolds Number is 48M^/3600TrDy , where y is the absolute viscosity of

— steam at nominal P and T (9.5 x 10 lb/ft*s at 27 psia and 290®F) . Effects of

variation of p and y within the operating range on C (thru R^^) were less than

0,1 percent and were neglected.

4 The expansion factor is calculated from [1], wherein Y = 1 - (0.41 + 0.35 3 ) (AP/Py)

,

equation II-III-7 (page 208), and where P = 27,0 psia, AP = (0.361)h

w

psi, and y

= 1.3, the specific heat ratio for steam. Effects of variation of P

within in the operating range on Y were less than 0.2 percent and were neglected.

The orifice expansion factor F = 1,004 is based on data from [1], page 156,

a for stainless steel with T = 290°F. Effects of variation of T within the operating

range on F were negligible.

Si

43
Table A1 gives data for correction factor M/M as a function of M and

o o summarizes calculation for equation A2. These data are basis for figures 3 and 4.

Note these figures are useful for manual correction of M^ values output from the

computer through the relation M s (M/M^)M^. However, for computer based calculations,

flowrate M would be calculated directly through equation 2 using current values

for p, Y and C and a procedure as outlined in Section III, 6 above. Note, equation

A2 above includes effects of correction factors e^ , e„ , e„ and e and data Cc Yc Fc pc

for these factors are not given separately.

44
M

o lb /hr

3200

2419

1767

1249

883

765 625

442

312 221

159

TABLE A1

CORRECTION FACTOR M/M

o

Orifice meter : D = 2.359 in. , 3=0. 389, comer taps

h c Y n

.

Ib/ft^ ^

V <

in. H^O X lO"^ — — 0.0580 .0600 .0620 .0640 .0660

M/M -

' ”o

105.00 236 0.6013 0.9549 0.898 0.914 0.929 0.944 0.958

60.00 178 .6017 .9742 .917 .933 .948 .963 .978 32.00 130 .6021 .9862 .929 .945 .961 .976 .991 16.00 92 .6026 .9931 .936 .952 .968 .983 1.000

8.00 65 .6033 .9966 .941 .957 .973 .988 1.004

6.00 56 .6037 .9975 .942 .958 .974 .989 1.005

4.00 46 .6040 .9983 .943 .960 .975 .991 1.006

2.00 33 .6051 .9990 .946 .962 .978 .993 1.009

1.00 23 .6061 .9996 .948 .964 .980 .996 1.011

.50 16 .6068 1.000 .949 .966 .981 .997 1.013

.26 12 .6078 1.000 .951 .967 .983 .999 1.014

I
APPENDIX B

RANDOM ERROR CALCULATIONS

Calculations and data are given for random error in the flowrate

as expressed already in equations 4 and 5

%

M

^2 , v2nl/2 I'e \ /e \

_P + / hw\

2p 2hw J

(4)

where

'hw - [^th) (®a/d) (®sl)

_

1/2

(5)

Error e^ is the error in the steam density p due to uncertainties

in the measurements of P and T. These uncertainties are 0.25 psi and

1.5 ®f respectively. Assuming that the density varies with P/(T + 460)

as for an ideal gas, these uncertainties can be combined on an RSS basis

to get

(Bl)

With nominal P = 27 psia and T = 290°F, p = 0.0615 Ib/ft and

e /p = 0,00947, where e„ = 0,25 psi and e„ = 1.5®F.

p P T

Random error e, has 3 components as expressed in equation 5; those hw

due to AP transmitter hysteresis analog/digital conversion and

sensing line error e . Transmitter hysteresis is assigned as a random

error because the hysteresis normally depends on the direction of approach

(h increasing, or h decreasing), and this direction is assumed to occur in

w w

a random fashion. Transmitter hysteresis is rated on an adjusted span basis

as listed in table 5, 0.10 percent of span. Thus, e^j^ = 0,105 in. H

2

O for

the HIGH, and e , = 0.008 in. H^O for the LOW transmitter. Error e. was

assumed to correspond to +1 count (+1 digit) where 999 counts corresponds to

the span of h . Thus, e. ._ = 0.105 and 0.008 in. H.O for the HIGH and LOW

w A/D 2

46
transmitters, respectively. Error e was based on an assumed density

difference not exceeding that due to 1**F temperature difference in the condensate in two pressure sensing lines of height 5 feet. Since AP = pgh, eg^ =

A(AP)/p g = (Ap )h /p , where the change in AP due to the density difference

C C oIj c is A(AP) = (Ap )gh . For water at ambient temperature and h„ = 60 inches,

C oLf e„^ = 0.008 in. H_0 where Ap corresponds to a AT of 1®F. Admittedly, this

SL Z c assigned value for e is quite arbitrary, and in fact its value is ’’unknowable".

oLi However, it is important to minimize its effect by keeping the two sensing

lines short, nearly horizontal, and close together. Under these

conditions, it is believed the effect will likely not exceed the assigned value.

With the above values for e , , e. and e„, , error e, becomes 0.149

th A/D SL* hw

in. H

2

O for the HIGH meter and 0,0139 in. H

2

O for the LOW meter.

The above errors as expressed in equation 4 are tabulated in table Bl,

which gives as a function of M^. These data are plotted on figure 5.

They show that for the combined HIGH/LOW meter configuration, the dominating

error is e except at lowest AP’s for each transmitter, where error e^^^ dominates.

For the HIGH meter alone, the precision error is dominated by error e , = 0»105

th

in. H„0 (through e, ) for low flowrates up to about 1000 Ib/hr. Therefore, when

2 hw

the steam flowrate is essentially constant, the differential pressure transmitter

should be selected to operate near full scale. If the steam rate varies through

a large range, say greater than 3:1, and depending upon relative operating

periods at low flowrates, consideration should be given to use of 2 transmitters,

one active at high AP and one at low AP.

With reference to equation 3, consider random errors in the energy rate E.

Errors e^, e^^ and errors in M (e^^) are all significant. Variations in operating

conditions P and T, and T^ are assumed random, and for the range of P and T

(26.8 to 27.9 psia and 280 to 299“F, table 4), the corresponding range in h is

+5 Btu/lb, Thus, for Ah = 1185 - 144 = 1041 Btu/lb, e /Ah = 0.5 percent. For

47
the range of T from 173 to 179 °F, e„ = +3*F and e„ (c )/Ah = 3(1)/1041 =

c ic — ic p

0.3 percent. Errors in P, T and measurements will have minor effect on

e^^ and are neglected. Referring to figure 4, when = 700

Ib/hr for example, =0.5 percent and + Mg) ~ ®Ms^ ^^n ^s^

”

2

0.25 percent. Substituting these data into equation 3, e„/M = r(.25) +

1/2 “

(.25)^ + (.5)^ + (. 3)^1 = 0.68 percent. Larger values of Sj^/M will result

in less effect of e^ and Therefore, random error e^/E will range upward

from about 0.7 percent. Error e^ will control at lowest AP values for each

transmitter.

48
TABLE B1

RANDOM ERROR

HIGH METER

e ^ = 0.105 in. H.O

th 2

®A/D

= -“5

LOW METER

e ^ = 0.008 in. H„0

th 2

®sl' •°°®

M

w

p 2p

'th 'A/D

2h

w

2h

w

'SL

2h

w

hw

2h

w

^M

M

Ib/hr in. H^O percent

HIGH METER

3200 105.00 0.47 0.05 0.05 0.0 0.07 0.48

2208 50.00 o47 .11 .11 .01 .15 .49

1562 25.00 .47 .21 .21 .02 .30 .56

1082 12.00 .47 .44 .44 .03 .62 .78

788 6.37 o47 .82 .82 .06 1.17 1.26

718 5.28 .47 .99 .99 .08 1.41 1.49

541 3.00 .47 1.75 1.75 .13 2.48 2.53

383 1.50 .47 3.50 3.50 .27 4.96 4.98

271 0.75 .47 7.00 7.00 .53 9.97 9.92

160 .26 .47 20.2 20.2 1.54 28.6 28.6

LOW METER

883 8.00 0.47 0.05 0.05 0.05 0.09 0.48

788 6.37 .47 .06 .06 .06 .11 .48

718 5.28 .47 .08 .08 .08 .14 .49

541 3.00 .47 .13 .13 .13 .23 .52

383 1.50 .47 .27 .27 .27 .46 .66

271 0.75 .47 .53 .53 .53 .93 1.04

160 .26 .47 1.54 1.54 1.54 2.67 2.71

49
APPENDIX C

SYSTEMATIC ERROR CALCULATIONS

Calculations and data are given for systematic error

/M , eu /M ) as expressed previously in equations

s,n n Ms,s s

in the flowrate

7 and 8

fws

M

®Cs + \s+ ^^ds + ^hws +

2h

w 1 - e J

'hws

e_ + e * + e-n Ts As Rs

(7)

( 8 )

where

[^^Asjsp

[®Rs,i

®As,zJ

®Rs,rp]

(Cl)

(C2)

Data for the individual errors and systematic error for the HIGH and

LOW transmitters are listed in table Cl. These data are basis for the systematic

error curves shown on figure 6. Following are further explanation and illustrative

calculations for these data.

In equation 7, errors e^^ and e^^ are tolerances on the orifice discharge

coefficient C and fluid expansion factor Y, respectively, as given in [1]. At

low values of h (<8 in. HO)

,

e„ can be neglected since fluid expansion

w 2 Ys

(density change) through the orifice is practically nil. Thus, e„ becomes zero.

Ys

Errors e, and e. are errors assigned due to manufacturing tolerances,

ds ps

since direct measurements of the orifices and pipe were not made. The tolerance

in the orifice diameter is 0.0005 inch and 2e, /d = 2(0.0005)/2.359 = 0.042

ds

percent. This term is neglected in equation 7. With regard to e , a tolerance

pS

of 1.5 percent on pipe ID was assigned as a reasonable estimate after conversations

with representatives of the steel industry [5], although ASTM standards A-53 and

A-120 allow larger variations o Thus, e„ /D = 0.015 and with 3 = d/D

Ds

50
(C3)

3 4

Since e /D»e, /d, e„ = 0.015. Substituting this into term 23 e, /(I - 3 ) Ds ds ps ps

of equation 7 yields 0.07 percent, where 3 = 0.389. This quantity is neglected

in this analysis.

In equation 8, error e^^ is the AP transmitter linearity characteristic

expressed on an adjusted span basis. Both transmitters are rated 0.35 percent

of span and the (limits for bounds of the) systematic error e^^ was assumed

not exceeding this figure. For the HIGH meter, 105 in. H

2

O span, e^^ = 0.368

in. HO. For the LOW meter, 8 in. HO span, e = 0.028 in. HO.

Z, ^ X s z Error e (equation Cl) gives the transmitter ambient temperature character-

/VS istics with the two components claimed independent by the manufacturer. They

are the span shift e^^ expressed on an adjusted span basis, and the zero

shift e expressed on a maximum (upper range limit, URL) span basis. Table 5

/VS ^ z lists data for these characteristics. The transmitters operated in an environment

75+15°F and e^^ data are given on this basis. Using the data from table 5,

for the HIGH meter:

e = 0.21% span = 0.221 in. H_0

/VS y ^

e = .12% URL = .126 in. H O

/VS ^ z z.

and for the LOW meter:

e = 0.26% span = 0.021 in. H»0

/VS ^ Sp 2a

^As z ~ .23% URL = .090 in. H

2

O

In the above, the range is 15®F, where it is assumed the transmitters are

calibrated in an environment at or near 75 °F.

Error e^^^ estimated for the calibration process has two independent components

the error in the transmitted current measurement e„ , and the error in the

Rs ,1

applied reference pressure e . For the AP transmitters, current output I Ks , rp

varies from 4.00 to 20.00 mA for the AP range zero to adjusted span h , i. e.

51
I - 4.00 + 16.00 h /h mA

w wsp (C4)

The error B in the current measurement for the digital multimeter employed is

B = 0.0005 (I) + 0.0001 mA (C5)

where 0,001 corresponds to + 1 l.s.d. (least significant digit). Scaling B

to units of AP, e^^^ ^ becomes (h^^^) (B/16) in. H

2

O and substituting equations

C4 and C5

,

'Rs ^ = + [o.003(h /16) + 0.0005 (h )1 in H^O,i — L

'^sp wj 2

(C6)

For both transmitters, error e is based on vertical manometry techniques

I Ks , rp

using water or mercury, with AP assumed read to + 0.5 mm, i.e., e = + 0.028

Rs , rp

in. H^O, Some improvement could be realized through use of inclined or micromanometry methods (LOW transmitter), reading to perhaps 0.005 in. H

2

O or better.

52
f'w X TABLE Cl

SYSTEMATIC ERROR e^/M

HIGH METER LOW METER

span h > vsp

105 in. H

2 O span h • wsp

8.00 in. H^O

®Ts 0.368 in. H^O e„ «= Ts 0.028 in. H

2 O

e. . “ A6,sp 0.221

As, sp 0.021

As,z 0.126

®As, z 0.090

®Rs 0.028 + (h/16)

WS (0.003) + 0.0005h , V

both transmitters

M

0

Ib/hr

hw

In n n

®Ts 2hw

^As.sd

2hw

®Ai.z

2hV

®Rs

2hw

®hw

2hw

e_£

C ®YY

®MsM

>

2 < HIGH METER

3200 105.00 0.18 0.11 0.06 0.05 0.40 1.00 0.25 1.65

2208 50.00 .37 .22 .13 .07 .79 1.00 0.25 2.04

1562 25.00 .74 .44 .25 .12 1.55 1,00 0.25 2.80

1082 12.00 1.53 .92 .53 .22 3.20 1.00 0.25 4.45

788 6.37 2.89 1.73 .99 .40 6.01 1.00 0.0 7,01

718 5.28 3.48 2.09 1.19 0.47 7.23 1.00 0.0 8.23

541 3.00 6.13 3.68 2.10 .83 12.7 1.00 0,0 13.7

383 1.50 12.3 7.37 4.20 1.67 25.5 1.00 0.0 26,5

271 0.75 25.5 14.7 8.40 3.33 51.9 2.00 0.0 53.9

160 .26 70.8 42.5 24.2 9.23 147.0 2.00 0.0 149.0

LOW METER

883 8.00 0.18 0.13 0.56 0.21 1.08 1.00 0.0 2.08

788 6.37 .22 .16 .71 .26 1.35 1.00 0.0 2.35

718 5.28 .27 .20 .85 .30 1.62 1.00 0.0 2.62

541 3.00 .47 .35 1.50 .52 2.84 1.00 0.0 3.84

383 1.50 .93 .70 3.00 1.00 5.63 1.00 0.0 6.63

271 0.75 1.87 1.40 6.00 1.99 11.3 2.00 0.0 13.3

160 .26 5.38 4.04 17.3 5.70 32.4 2.00 0.0 34.4

53
APPENDIX D

TABLE OF NOMENCLATURE

Symbol Description Dimensions

A Observed count, output from differential pressure A/D converter —

A

s Uncertainty in accumulated energy usage, bounds to the systematic

error —

C Coefficient of discharge for the orifice meter —

c P

Specific heat of condensate Btu/db-^F)

D Pipe inside diameter in.

d Diameter of orifice in.

E Net energy rate Btu/hr

^A/D Random error, analog/digital conversions, computer system in. H^O

®As Systematic error, AP transmitter, ambient temperature zero and

span shift error

in. H^O

e

.

As ,sp

Systematic error, AP transmitter, ambient temperature, span shift

component

in. H^O

^As,z Systematic error, AP transmitter, ambient temperature, zero shift

component

in. H^O

^Cc Correction factor, orifice discharge coefficient —

^Cs Systematic error (tolerance), assigned to discharge coefficient C —

^ds Systematic error, manufacturing tolerance for d

in.

^Ds Systematic error, manufacturing tolerance for D in.

^E

Random error in energy rate E Btu/hr

^Es Systematic error in energy rate E Btu/hr

®Fc Correction factor, orifice thermal expansion —

Random error in enthalpy h Btu/lb

®hw Random error in differential pressure hw in. H^O

^hws Systematic error in hw in. H^O

*Note; All quantities denoted in. H„0 refer to inches of water at 68°F.

Symbol "lb" refers to pound iMss.

54
Symbol Description Dimensions

®Mn

Random error in flowrate Mn lb /hr

®Ms

Random error in flowrate Ms lb /hr

^s,n Systematic error in flowrate Mn Ib/hr

®Ms,s Systematic error in flowrate Ms lb /hr

Random error in supply steam pressure P psi

®Rs Systematic error in applied reference pressure and transmitter

current measurement

in. H2O

^Rs,i Systematic error, due to AP transmitter current measurement in. H2O

®Rs , rp

Systematic error, AP transmitter reference pressure in. H^O

®SL Random error, allowance for possible density variations in

pressure sensing lines

in. H^O

Random error in supply steam temperature T

°F

®Tc

Random error in condensate temperature Tc °F

^th

Random error, AP transmitter hysteresis in. H^O

^Ts Systematic error, AP transmitter linearity in. H2O

^Yc

Correction factor, steam expansion —

^Ys Systematic error (tolerance) assigned to expansion factor Y —

®

3s Systematic error in 3 ra.tiq due to manufacturing tolerance

in d and D —

e Random error in density p due to uncertainty in P and T measure^ Ib/ft^

P ment

e pcCorrection factors, steam density variation with P and T —

F Area factor for thermal expansion of the orifice

a

g Local acceleration of gravity ft/s^

h Enthalpy of supply steam Btu/lb

hw

Differential pressure in. H2O

h

wsp

Differential pressure transmitter adjusted span value in. H2O

K Scaling constant, differential pressure transmitter lb /hr

M

n

Mass flowrate, north branch meter lb /hr

M Observed or indicated mass flowrate lb /hr

o 55
Symbol Description Dimensions

M

s Mass flowrate, south branch meter lb /hr

P Pressure of supply steam psia

5) Pipe Reynolds number = 48M/3600itDp —

T Temperature of supply steam

T

c Temperature of condensate

T

s Steam saturation temperature at pressure P *F

Y Steam expansion factor —

3 Ratio d/D —

Y specific heat ratio, c /c = 1.3 for steam

p V —

Ah Net enthalpy change = h - c^(T^ - 32.0) Btu/lb

AP Differential pressure psi

P Density Ib/ft^

Density of condensate in AP sensing legs Ib/ft^

y Absolute viscosity of steam Ib/ft.s

56
NBS-n4A (REV. a-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

4.

TITLE AND SUBTITLE

1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date REPORT NO.

NBSIR 81-2313 July 1981

PERFORMANCE EVALUATION OF A TYPICAL ENERGY MONITORING SYSTEM FOR STEAM FLOW IN

BUILDINGS

5. AUTHOR(S)

David W. Baker, James Y. Kao and David A. Didion

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s) 7. Contract/Grant No. national bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, State. ZIP)

National Bureau of Standards U.S. Navy Civil Engineering Laboratory

Washington, D.C. 20234 Port Hueneme, CA 93010

10.

SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200~word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here)

Some important design features and measurement techniques are discussed for

determining energy rates in building systems flowing steam. Emphasis is on use of differential pressure (AP) type flowmeter systems where AP,

pressure and temperature instrumentation only can receive direct calibration.

The role of systematic and random type errors in measurement of building

energy is discussed and an appropriate method is given for estimating the

uncertainty in the energy consumed. An example calculation of uncertainty

in accumulated energy used for a 1 year period is given from typical

operating data for one of the NBS laboratory buildings. The uncertainty

is estimated to be 3.2% for a two-meter (series) configuration and 4.3% for

a single meter (high range meter) configuration.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Building energy monitoring; energy measurement; steam consumption measurement;

uncertainty estimate.

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

[X~| Unlimited

1 1 For Official Distribution. Do Not Release to NTIS oO

r 1 Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402. 15. Price

Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $8.00

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