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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~
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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
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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
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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
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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
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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
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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
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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