Annex A: Examples on estimation of uncertainty in

airflow measurement

Introduction

A.2 Example one test facility

A.2. I Definition of the measurement process

This annex contains three examples of fluid flowmeasurement uncertainty analysis. The first dealswith airflow measurement for an entire facility (withseveral test stands) over a long period. It also appliesto a single test with a single set of instruments. Thesecond example demonstrates how comparative de-velopment tests can reduce the uncertainty of thefirst example. The third example illustrates a liquidflow measurement.

A.1 General

Airflow measurements in gas turbine engine systemsare generally made with one of three types offlowmeters: venturis, nozzles and orifices. Selection ofthe specific type of flowmeter to use for a givenapplication is contingent upon a tradeoff betweenmeasurement accuracy requirements, allowable pres-sure drop and fabrication complexity and cost.

Flowmeters may be further classified into two catego-ries: subsonic flow and critical flow. With a criticalflowmeter, in which sonic velocity is maintained atthe flowmeter throat, mass flowrate is a function onlyof the upstream gas properties. With a subsonicflowmeter, where the throat Mach number is lessthan sonic, mass flowrate is a function of bothupstream and downstream gas properties.

Equations for the ideal mass flowrate through noz-zles, venturies and orifices are derived from thecontinuity equation:

W = paV

In using the continuity equation as a basis for idealflow equation derivations, it is normal practice toassume conservation of mass and energy and one-dimensional isentropic flow. Expressions for idealflow will not yield actual flow since actual conditionsalways deviate from ideal. An empirically determinedcorrection factor, the discharge coefficient (C) is usedto adjust ideal to actual flow:

C Wac~/Wjdea~

What is the airflow measurement capability of a givenindustrial or government test facility? This questionmight relate to a guarantee in a product specificationor a research contract. Note that this question impliesthat many test stands, sets of instrumentation andcalibrations over a long period of time should beconsidered.

The same general uncertainty model is applied in thesecond example to a single stand process, the compar-ative test.

These examples will provide, step by step, the entireprocess of calculating the uncertainty of the airflowparameter. The first step is to understand the definedmeasurement process and then identify the source ofevery possible error. For each measurement, calibra-tion errors will be discussed first, then data acquisi-tion errors, data reduction errors, and finally, propa-gation of these errors to the calculated parameter.

Figure 14 depicts a critical venturi flowmeter installedin the inlet ducting upstream of a turbine engineunder test for this example.

When a venturi flowmeter is operated at criticalpressure ratios, i.e., (P2/P1) is a minimum, theflowrate through the venturi is a function of theupstream conditions only and may be calculated from

d2 PW = ~~CFa(P~

A.2.2 Measurementerror sources

(41)

(39) Each of the variables in equation 41 must be carefullyconsidered to determine how and to what extenterrors in the determination of the variable affect thecalculated parameter. A relatively large error in somewill affect the final answer very little, whereas smallerrors in others have a large effect. Particular careshould be taken to identify measurements that influ-ence the fluid flow parameters in more than one way.

In equation (41), upstream pressure and temperature(P1 and T1) are of primary concern. Error sources foreach of these measurements are: (1) calibration, (2)

(40) data acquisition and (3) data reduction.

27

026O~

A.2.2. 1 Figure 15 illustrates a typical calibrationhierarchy. Associated with each comparison in thecalibration hierarchy is a possible pair of elementalerrors, a systematic error limit and an experimentalstandard deviation. Table 7 lists all of the elementalerrors. Note that these elemental errors are not

cumulative, e.g., B21 is not a function of B11. Thesystematic error limits should be based on interlabo-ratory tests if available, otherwise, the judgment ofthe best experts must be used. The experimentalstandard deviations are calculated from calibrationhistory data banks.

Figure 14 Schematic of sonic nozzle flowmeter installation upstream of a turbine engine

Standards Laboratory

1MeasurementStation

Flow

Sonic Nozzle Throat

Plenum

LabyrinthSeal

Belimouth

Calibration

Calibration

Calibration

Interlaboratory Standard

Transfer Standard

Working Standard

Measurement Instrument

Figure 15 Typical calibration hierarchy

Calibration

28

Table 7 Calibration hierarchy error sources Data acquisition error sources for pressure measure-

The experimental standardacquisitionprocess is

S2 = ~ S~2S~2S~2+S~2+S~2+S~2

68.953~ 48.270~+ 10 60 )~77

CalibrationSystematicerror, P0

Experimentalstandard

deviation, P0

Degreesof

freedomSL - ILS

ILS-TS

TS - WS

WS - MI

B11 6&953

B21 = 68.953

B31 68.953

B41 124.

S11 13.787

S21 13.787

S31 13.787

S41 36.541

v11 10

v21 = 15

v31 20

v41 = 30

ment are listed in table 8.

Table 8 Pressure transducer data acquisitionerror sources

Error sourceSystematicerror, P0

Experimentalstandard

Deviation, P0

Degreesof

freedomExcitationVoltage

B12 = 68.953 S12= 34.481 v12 = 40

ElectricalSimulation

B92 68.953 S22 = 34.481 v22 90

SignalConditioning

B32 68.953 S32 = 34.481 v32 = 200

RecordingDevice

B42 = 68.953 S42 = 34.481 v~= 10

PressureTransducer

B52 68.953 S52 = 48.270 v52 100

EnvironmentalEffects

B62 = 68.953 ~62 68.953 v62 = 10

Probe Errors B72 117.223 S72 = 48.270 V72 = 60

The experimental standard deviation for the calibra-tion process is the root-sum-square of the elementalsample standard deviations, i.e.,

S1 = \/Sll+ S21+ S11+ S41

= ~Ji~?~872+ 13.7872 + 13.7872 + 36.5412

43.65 Pa (42)

Degrees of freedom associated with S are calculatedfrom the Welch-Satterthwaite formula as follows:

(S~1S~1S~1S~1)2vl= / ~4 Q4 Q4 Q4I ~1i ~21 ~31 ~~4j

+ + V

11V

21V

31V

41

(13.787~+ 13.787~+ 13.7872 + 36.541~)~ = ( 13.787 13.787 13.787 36.541

10 15 + 20 30(43)

The systematic error for the calibration process is theroot-sum-square of the elemental systematic errorlimits, i.e.,

B1 = ~1B~B~1B~~1 (44)

(45)

deviation for the data

S2 [34.481~~ 34.4812 + 34.4812 + 34.4812 + 48.2702

68.9532 + 48.2702 11/2

= 119.039 P0 (46)

(S~3+S~S~2S~S2S~5~)2= / S~ S~ S32 S42 S52 S02 S32

12 + 22 32 42 + p52 ~/52 + V~

(34.481~34.481234.481234.4812 + 48270~+ 689532 + 48.2702)2

// 34.481 34.481 34.481 34.481~ 48.270/ t, 40 + 90 200 10 100

(47)

= V68.953 68.953 68.953 124.117

172.2 P.

29

0244),

The systematic error limit for the data acquisitionprocess is x

B2 = [68.9532 + 68.9532 + 68.9532 + 68.9532 }l/2

+ 68.9532 + 68.9532 + 117.2232

The systematic error limit for the data reductionprocess is

B3 = ~jB~3B23

B3 = ~/68.9532 + 6.8942

= 205.6 ~a (48)

or

= 69.297 P5

S9 = ~JS~+ S~+ S~

(50)

(51)

Table 9 lists data reduction error sources.

Table 9 Pressure measurement data reductionerror sources

Error source Systematicerror, P0

Experimental Degreesstandard of

deviation, ~a LfreedomCurve Fit

ComputerResolution

B13 = 68.953

B.,3 6.894

S13 = 0

S23 = 0

v~3

v23

The experimental standard deviation for the datareduction process is

S3 = .,JS~3+S~3

= 0.0(49)

= V43.651 92 ~ 119.0392 + 0.02

= 126.790 Pa (52)

Degrees of freedom associated with the experimentalstandard deviation are determined as follows:

v~= (S~1+S~1S~1+S~1S~S~2~S3~+S42+ 5~,S62

+ S~2+S~,S~3)

/ S~~l S~ ~ S~1 S~2 ~ S~2 542/ (__+ +_ +~ V42

+V11

V21

V31

V.51

V12

V22

V3~

S2

S2 S~2 S~3 S~3~+ +V

52V

62V

72V

13V

23I (53)

A computer operates on raw pressure measurementdata to perform the conversion to engineering units.Errors in this process are called data reduction errorsand stem from curve fits and computer resolution.

Computer resolution is the source of a small elemen-tal error. Some of the smallest computers used inexperimental test applications have six digits resolu-tion. The resolution error is then plus or minus one in106. Even though this error is probably negligible,consideration should be given to rounding off andtruncating errors. Rounding-off results in a randomerror. Truncating always results in a systematic error(assumed in this example.)

The experimental sample standard deviation forpressure measurement then is

= [S~1+S~1+S~1S~1S~2S~2S~2

+ S~2~S~S~2S~2+S~52]h/2

30

A.2.2.2 The calibration hierarchy for temperaturemeasurements is similar to that for pressure measure-ments. Figure 16 depicts a typical temperature mea-surement hierarchy. As in the pressure calibrationhierarchy, each comparison in the temperature calib-ration hierarchy may produce elemental systematic

and random errors. Table 10 lists temperature calib-ration hierarchy elementalerrors.

Table 10 Temperature calibration hierarchy ele-mental errors

CalibrationSystematicerror, K

Experimentalstandard

deviation, K

Degreesof

freedomSL - ILS

ILS-TS

TS - WS

WS - MI

B11 0.056

B21 = 0.278

B31 = 0.333

B41 = 0.378

0.002

~21 = 0.028

~31 = 0.028

S41 = 0.039

2

V21

= 10

= 15

v41 30

The calibration hierarchy experimental standard de-viation is calculated as

SI = VS~2S~1+S~1+S~

Degrees of freedom associated with S1 are(S~1S~1S~1-t-S~1)

2V

1= ~

4

(!!V11

+ V21

V31

V41

/

(0.002~0.0282 0.0282+ 0.0392)2 1 0.002 0.028~ 0.028k 0.039~

2 10 15 + 30

The calibration hierarchy systematic error limit is

or

(S~-i-S~+S~)2VP_f S~ S~ S~

~-~- ;;;;-+ -~--

(43.651 92 + 119.0392 + 0.02)2 1 43.651 92 119.0392 O.O~

Is\ 54. 77 +~-

96 therefore t9~= 2. (54)

The systematic error limit for the pressure measure-ment is

B~= [B~1B~1B~1B~1B~2B~+B~2

+ B~2+B~2+B~2B~9B~3B~3]L~2

or

B9 = ~B~+ B~+ B~

B9 = y172.2462

+ 205.593~+ 69.2972

= 277.018 Pa (56)

Uncertainty for the pressure measurement is

U~9= (B9 + t95 S9), U9~= ~JB~+ (t~S9)2

U~= (277.018 + 2 x 126.790)

= 530.598 P U95 = 375.6 PV a (57)

= V0.0022 + 0.0282 + o.o2S2 + 0.0392

= 0.056 K.

= 53 > 30, therefore t95 = 2.

(58)

(59)

(60)

(61)

B1 = ,5/B~1B~1B~1+~2

= ~j0.0562+ 0.2782 + 03332 + 0.3782

= 0.578 ~}(

31

A reference temperature monitoring system willprovide an excellent source of data for evaluatingboth data acquisition and reduction temperaturerandom errors.

Figure 17 depicts a typical setup for measuringtemperature with Chromel-Alumel thermocouples.

Standards Laboratory

Interlaboratory Standard

Transfer Standard

Working Standard

Measurement Instrument

Figure 16 Temperature measurement calibration hierarchy

Ii

If several calibrated thermocouples are utilized tomonitor the temperature of an ice point bath, statisti-cally useful data can be recorded each time measure-ment data are recorded. Assuming that those

thermocouple data are recorded and reduced toengineering units by processes identical to thoseemployed for test temperature measurements, astockpile of data will be gathered, from which dataacquisition and reduction errors may be estimated.

Calibration

Calibration

Calibration

Calibration

Cr Cu

r -II I

IceTO Point

BathL___i

L.i~J

Uniform TemperatureReference

Figure 17 Typical thermocouple channel

32

7) Computer resolution error

For the purpose of illustration, suppose N calibrated ture data if the temperature of the ice bath isChromel-Alumel thermocouples are employed to continuously measured with a working standard suchmonitor the ice bath temperature of a temperature as a calibrated mercury-in-glass thermometer. Theremeasuring system similar to that depicted by figure the systematic error limit is the largest observed17. If each time measurement data are recorded, difference between X and the temperature indicatedmultiple scan recordings are made for each of the by the working standard acquisition and reductionthermocouples, and if a multiple scan average (X1~)iscalculated for each thermocouple, then the average

process. In this example, it is assumed to be O.56K,i.e.,

(Xi) for all recordings of the jth thermocouple is.

B~= 0.56K (66)

Error sources accounted for by this method are:

x = K1 (62)1) Ice point bath reference random error

2) Reference block temperature random errorwhere K- is the number of multiple scan recordingsfor the thermocouple.

.

3) Recording system resolution error

The grand average (X) is computed for all monitor 4) Recording system electrical noise errorthermocouples as

5) Analog-to-digital conversion error

N ~ X3

6) Chromel-Alumel thermocouple millivoltoutput vs. temperature curve-fit error

x= N (63)

The experimental standard deviation (Si) for the Several errors which are not included in the monitor-data acquisition and reduction processes is then ing system statistics are:

S-= (64) -

= 0.094 K (assumed for this example)These errors are a function of probe design andenvironmental conditions. Detailed treatment ofthese error sources is beyond the scope of this work.

The degrees of freedom associated with S~are The experimental standard deviation for temperature

Nmeasurements in this example is

v~= ~(K1-1) (65)S1 =S~S1+S~ (67)

= 200 (assumed for this example)where

Data acquisition and reduction systematic error urn- S1 = calibration hierarchy experimental stand-its may be evaluated from the same ice bath tempera- ard deviation

E~(X8_X1)2j~1ii

~(K3-1)j~1

33

0/Gil,

S7 = 310.0562 + 0.0942

The degrees of freedom associated with S~are

U~= (0.804 + 2 x 0.11), U95 = + (2 x 0.11)2

When v is less than 30, t95 is determined from a(68) Students t table at the value of v. Since v~is greaterthan 30 here, use t95 = 2.

A.2.2.3 There are catalogs of discharge coefficientsfor a variety of venturis, nozzles and orifices. Cata-loged values are the result of a large number of actualcalibrations over a period of many years. Detailedengineering comparisons must be exercised to ensurethat the flowmeter conforms to one of the groupstested before using the tabulated values for dischargecoefficients and error tolerances.

where

B, =

B1 = calibration hierarchy systematic errorlimits

B1 = data acquisition and reduction system-atic error limits

Bc = conduction error systematic error limits(negligible in this example)

BR = radiation error systematic error limits(negligible in this example)

B~ = recovery factor systematic error limits(negligible in this example)

B, = V0.5782 + 0.562

69 To minimize the uncertainty in the discharge coeffi-cient, it should be calibrated using primary standards

in a recognized laboratory. Such a calibration willdetermine a value of Aeff = Ca and the associatedsystematic error limit and experimental standarddeviation.

When an independent flowmeter is used to determineflowrates during a calibration for C,~dimensionalerrors are effectively calibrated out. However, when Cis calculated or taken from a standard reference,errors in the measurement of pipe and throat diame-ters will be reflected as systematic errors in the flowmeasurement.

Dimensional errors in large venturis, nozzles andorifices may be negligible. For example, an error0.001 inch in the throat diameter of a 5 inch criticalflow nozzle will result in a 0.04% systematic error inairflow. However, these errors can be significant atlarge diameter ratios.

A.2.2.4 Non-ideal gas behavior and changes in gascomposition are accounted for by selection of theproper values for compressibility factor (Z), molecularweight (M) and ratio of specific heats (y) for thespecific gas flow being measured.

S1 = data acquisition and reduction experi-mental standard deviation

= 0.11 ~}(

Uncertainty for the temperature measurement is

U, (B,t~5S,)

= (B, 4- t95 S,), U95 = + (t95 S,)2

= 1.02K, 0.83K (S~S~)2

V7 / ~4 ~4

I -ii ~-2+

V1 V2

(0.0562 + 0.0942)2 I 0.056k Q944

53 + 200

= 250 therefore t95 = 2

Systematic error limits for the measurements are

(70)

= 0.804K

34

When values of y and Z are evaluated at the properpressure and temperature conditions, airflow errorsresulting from errors in y and Z will be negligible.

For the specific case of airflow measurement, themain factor contributing to variation of compositionis the moisture content of the air. Though small, theeffect of a change in air density due to water vapor onairflow measurement should be evaluated in everymeasurement process.

A.2.2.5 The thermal expansion correction factor(Fa) corrects for changes in throat area caused bychanges in flowmeter temperature.

For steels, a 17~Kflowmeter temperature difference,between the time of a test and the time of calibration,will introduce an airflow error of 0.06% if no correc-tion is made. If flowmeter skin temperature isdetermined to within 3Kand the correction factorapplied, the resulting error in airflow will be negligi-ble.

A.2.3 Propagation of error to airflow

For an example of propagation of errors in airflowmeasurement using a critical-flow venturi, consider aventuri having a throat diameter of 0.554 meters

operating with dry air at an upstream total pressureof 88 126~aand an upstream total temperature of2659K.

Equation (71) is the flow equation to be analyzed:

icd2 * P1W = ~---CF5p7r~-

____ y+1( 2 ~71 (ygM(p \y+11 ~ZR (71)

Assume, for this example, that the theoretical dis-charge coefficient (C) has been determined to be0.995. Further assume that the thermal expansioncorrection factor (Fa) and the compressibility factor(Z) are equal to 1.0. Table 11 lists nominal values,systematic error limits, sample standard deviationsand degrees of freedom for each error source in theabove equation. (To illustrate the uncertainty meth-odology, we will assume a sample standard deviationof 0.000 5 in addition to a systematic error of 0.003.)

Note that, in table II, airflow errors resulting fromerrors in Fa* Z, k, g, M and R are considerednegligible.

Table 11 Airflow measurement error sources

Errorsource

V

UnitsNominal

value

Systematicerrorlimit

Experime~ta1standarddeviation

Degreesof

freedom,V

UncertaintyU~

P1 P5 88 126 217.02 126.79 96 530.60

T1 K 265.9 0.8 0.11 250 1.02

d m 0.554 2.54X105 2.54X10~ 100 7.62X105

C 0.995 0.003 O000 5 0.003

~a 1.0

z 1.0

y 1.401

g

M kg/kg-mole 28.95

H J/K-kg-mole 8.3 14

35

0260,

From equation (71), airflow is calculated as

w = 3.142 (0554)2 x 0.995 x 1.0

= 52.39 kg/sec.

Taylors series expansion of equation (71) with theassumptions indicated yields equations (72) and (73)from which the flow measurement experimentalstandard deviation and systematic error limits arecalculated.

s~= w s,l (~)2(~L)2(s)2(~5)211 126.790 \2 ( 0.11

= 52.39 L ~. 88 126 1 + k 2 x 265.9 12 2 1/21 0.000 5 \ 1 2 x 0.000 025

~k 995 1 --s 0.554

which results in an overall degrees of freedom> 30,and, therefore, a value oft95 of 2.0.

Total airflow uncertainty is then,

U99 = (B,~+ t95 Sw), U95 = 31B~+ (t95 S~)2

U~ = [0.241 6 + 2 x 0.078 7]

= 0.40 kg/sec

= 0.8%

= 52.39 ~(0.001 4)2 + (0.000 2)2 + (~~05o3)2~O.oo0~j2

= 0.078 7 kg/sec

B~= w~(~)+(4~)(4~)(4t)11 277.02 \2 f 0.804 \2 f 0.003B,, = o239Lk88 126 ) ~ 531.8 1 ~

1 0.000 05~\ 0.554 1 J

= 52.39 ~/(0.003 1)2 + (0.001 5)2 + (0.003 Q)2 + (0.000 09)2

0.241 6 kg/seg

By using the Welch-Satterthwaite formula, the de-grees of freedom for the combined experimentalstandard deviation is determined from

A.3 Example two comparative test

A.3. 1 Definition of the measurement process

The objective of a comparative test is to determinewith the smallest measurement uncertainty the neteffect of a design change, such as a new part. The firsttest is performed with the standard or baseline

(73) configuration. A second test, identical to the firstexcept that the design change is substituted in thebaseline configuration, is then carried out. Thedifference between the measurement results of thetwo tests is an indication of the effect of the designchange.

FI~W ClawL~~S91j ~~-ST1) +vV* = 4 4( aw ~ ~ I a~v

~~p;- Pu ~ ~ TjJ

2 229W \ I ow

aSd; ~-~--sc4 4OW \

~ ~cVd vC

As long as we only consider the difference or neteffect between the two tests, all the fixed, constant,systematic errors will cancel out. The measurementuncertainty is composed of random errors only.

For example, assume we are testing the effect on thegasfiow of a centrifugal compressor from a change tothe inlet inducer. At constant inlet and discharge

4(

2 \0.4012.40 12.401 ) 1 1.401 x 28.95 \ 88 1268314 )x~___

2 /5T )2 + ( 2Sd 2 2 12

~

5T1 ~4 I 2S4 \4 ( __~c_i4(~)+( ~ c~P

Vp1

VT, Vd 1

C

(74)

(72)U95 = 0.29 kg/sec

= 0.55%(75)

36

conditions, and constant rotational speed, will the gasflow increase? If we test the compressor with the oldand new inducers and take the difference in measuredairflow as our defined measurement process, weobtain the smallest uncertainty. All the systematicerrors cancel. Note that, although the comparativetest provides an accurate net effect, the absolute value(gasfiow with the new inducer) is not determined o~ifcalculated, as in example one, it will be inflated by thesystematic errors. Also, the small uncertainty of thecomparative test can be significantly reduced byrepeating it several times.

A.3.2 Measurement error sources

(see equation (65))

A.3.2.3 The test result is the difference in flowbetween two tests.

= WI W2

All errors result from random errors in data acquisi-tion and data reduction. Systematic errors are effec-tively zero. Random error values are identical to thosein example one, except that calibration random errorsbecome systematic errors and, hence, effectively zero.

= (BA,,, + 2SAW)

= (0 + 2SAW)

= 31(BAW)2 (2SAW)2

= 31o2 + (2SA~)2

A.3.2. 1 Comparative tests shall use the same testfacility and instrumentation for each test. All calibra-tion errors are systematic and cancel out in taking thedifference between the test results.

B1 = 0

S1 = 0, Sc = 0

SP = S2

= 2SA~ = 2SAW

UAW~ = 2S~\/~ U~,5= 2S~1J~

S 52 39 { ( 119.037 \2 1 0.094 \2= 88 126 ) ~2x265.9 )/ 0.0005 )2( 0.00005 \,11~2 V

~ 0.995 0.554 J JS,, = 0.076 2 kg/sec SAW = 0.107 8 kg/sec

UA,,~=0.215 5 kg/sec

= 0.41%

= 0.215 5 kg/sec

= 0.41%

= 119.039 Pa (see equation (47)) (see equation (75))

VP = V9

= 77

St = S1

= 0.094K

(see equation (48))

(see equation (64))

37

A.3.2A Note that the differences shown in table 12are entirely due to differences in the measurementprocess definitions. The same fluid flow measurementsystem might be used in both examples. The compar-ative test has the smallest measurement uncertainty,but this uncertainty value does not apply to themeasurement of absolute level of fluid flow, only tothe difference.

V, V1

= 200

SAW 31S~1(i)2S~2=s,,31~

and

A.3.2.2

0260,

1. 2, 3. m Observation points

b1, b2, b3,. . b~ Breadth (metres) of segment associated with the observation pointV d1, d2, d3,. . d~.g Depth of water (metres) at the observation point

Dashed lines Boundary of segments: one heavily outlined

If x and y are respectively horizontal and vertical coordinates of all the points in the cross-section, and A is its total area, then the precise mathematical expression for ~ the truevolumetric flowrate (discharge) across the area, can be written as

Figure 18 Definition sketch of velocity-area method of discharge measurement (midsection method)

L~,4 1)5

lnrt!aIpoint

T~T~

Explanation

38