Periodic testing of equipment is a common situa-tion where measurement uncertainty must be con-sidered. For this example, a pump is considered to perform consistently if the supply pressure measured at 100% of rated flow is consistent with prior test results. The pump design data is sented in Table 10-5.1-1 and the test data is pre-sented in Table 10-5.1-2 After each test, it is neces-sary to evaluate the test results. As part of this effort the effect of measurement uncertainty must be considered. For the first test in Table 10-5.1-2 the available information is limited since it was a factory test. (See Figs. 10-5.1-1 and 10-5.1-2.)
The conclusions that can be drawn from Fig.
10-5.1-3 are as follows:
(a) The pump is operating consistently when compared to the factory test results since the uncer-tainty bands overlap.
(b) The pump is operating better than the mini-mum required design condition. The confidence for this conclusion is better than 95%.
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The performance of a hydraulic system can be evaluated using Bernoulli’s equation.
H pP
+ v
2
2g+ z where
Hp total head, m Pp system pressure, Pa
pfluid density, kg/m3 vp fluid velocity, m/s
gp gravitational constant, 9.81 m/s2 z1pelevation, m
The effective head produced at a specified pres-sure is a meapres-sure of pump performance. The effective head in terms of the measured test param-eters and physical property data is
⌬P p P2+ 8Q2
2d42
+ (z2− z1)g
where
d2pinside pipe diameter, m
⌬Pp pressure change between taps 1 and 2, Pa P2ppressure at tap 2, Pa
Qp fluid flow rate, m3/s
z2pelevation of pressure tap 2, m
z1pelevation of tap 1 or free-surface eleva-tion, m
The density can be estimated numerically using the relationship [6],
p766.17 + 1.80396 TK−3.4589 T2K
1000 (10-5.1)
where
TKpTC + 273.15 p the absolute temperature and TCis the fluid temperature in Celsius Estimates of the density, , using eq. (10-5.1) are reported to have a systematic standard uncer-tainty of 0.587 kg/m3. The random error in the curve fit was judged to be negligible and so the random uncertainty is set to zero.
During testing it is not feasible to operate exactly at a specified operating condition. For a pump test, the applied flow might be slightly different for each test. This will result in a slight change in the resultant differential pressure. The variation
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Table 10-5.1-1 Pump Design Data (TCⴝ 20°C)
Flow Differential Pressure
m3/s gpm Percent Rated Flow kPa psi
0.000 0 0 827 119.9
0.126 2000 100 689 99.9
0.189 3000 150 552 80.1
Table 10-5.1-2 Summary of Test Results
Field Tests Factory Test
[Note (1)] A B C D E
Raw Data
Flow, m3/s 0.126 0.125 0.126 0.130 0.123 0.129
P exit, Pa . . . 840000 845000 820000 836000 841000
d exit, m . . . 0.254 0.254 0.254 0.254 0.254
z exit, m . . . 2.40 2.40 2.40 2.40 2.40
z inlet, m . . . 15.00 15.00 15.00 15.00 15.00
T, °C 20.15 19.71 20.01 20.31 20.78 21.10
Resultants
, kg/m3 997.7 997.8 997.7 997.7 997.6 997.5
⌬P, Pa 712000 718000 724800 707300 710200 726400
⌬P, psi 103.3 104.1 105.1 102.6 103.0 105.4
NOTE:
(1) The factory test data only provides resultant information.
Fig. 10-5.1-1 Installed Arrangement
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Fig. 10-5.1-2 Pump Design Curve With Factory and Field Test Data Shown
Fig. 10-5.1-3 Comparison of Test Results With Independent Control Conditions
in flow may be handled by normalizing the test results. This is accomplished by adding an addi-tional random term. A normalization coefficient can be estimated from the factory test data in Table 10-5.1-1. A best fit correlation of the data is
⌬P p 827,000 − 376,000Q − 5,710,000Q2 (10-5.2) The normalizing coefficient is the slope of eq.
(10.5.2) at the specified test conditions.
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⌬PNpb(QN− Q) +⌬P (10-5.3) where
⌬PNpexpected pressure change based on the nominal (specified) test conditions, Pa
⌬Pp measured pressure change, Pa
QNpnominal (specified) test flow rate, m3/s Qp measured test flow rate, m3/s
bp slope of eq. (10-5.2), Pa · s/m3
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The slope of eq. (10-5.2) is
∂⌬PN
∂Q
冨
QNpb p −376,000 − 11,420,000 QN
At the 100% rated flow condition (see Table 10-5.1-1), QN p 0.126 m3/s, and the value for the
10-5.2 Comparison With Independent Control The field test data for the pump can be compared with the factory test results or the minimum rated pressure output. For this type of evaluation, the uncertainties are independent and a simple com-parison of the test results with the benchmark value is adequate. The uncertainty is calculated using the method in subsections 7-1 through 7-4.
The partial derivatives necessary to estimate the sensitivity coefficients for this problem are
Symbol,
Xi Formulas for Absolute Sensitivity
P2 ∂⌬P
The temperature sensitivity coefficient is com-puted using the chain rule:
∂⌬PN
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The uncertainty for each test can be calculated as shown in Table 10-5.2-1. Table 10-5.2-2 shows the nominal value, and the systematic, random, and total uncertainties for ⌬P. The uncertainties for each test are presented in Table 10-5.2-3. The results are plotted in Fig. 10-5.2.
10-5.3 Comparative Uncertainties
Correlation of terms is an important consider-ation in comparison testing [23, 24] where two different operating conditions or constructions are being compared by use of a ratio
p ⌿alt
⌿control (10-5.4)
where
pratio of two resultants
⌿altpalternate (or variable) resultant
⌿controlpresultant used for the baseline or control
For the pump test C, with test A considered the control test, eq. (10-5.4) becomes
Ap707 kPa
718 kPap0.985
The test results for the comparative analysis are summarized in Table 10-5.2-3.
The uncertainty for the comparative analysis can be computed using the method from subsection 8-1. The partial derivatives for eq. (10-5.4) are
∂
These may be combined with the partial deriva-tives presented earlier to estimate the sensitivity coefficients for the comparative analysis.
P2
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Table 10-5.2-1 Uncertainty Propagation for Comparison With Independent Control
Independent Parameters
Uncertainty Contribution of
Parameter Information Parameters to the Result
(in Parameter Units) (in Result Unit Squared)
Absolute Absolute
Absolute Absolute Systematic Random
Systematic Random Standard Standard
Standard Standard Absolute Uncertainty Uncertainty Nominal Uncertainty, Uncertainty, Sensitivity, Contribution, Contribution,
Symbol Description Units Value bX
i sX
i i (bX
ii)2 (sX
ii)2
Q Flow m3 0.125 3.0ⴛ10−3 1.0ⴛ10−3 1.86ⴛ106 3.11ⴛ107 3.46ⴛ106
P2 Exit pressure Pa 840.0ⴛ103 3500 3000 1.0 1.23ⴛ107 9.0ⴛ106
d2 Exit diameter m 0.254 1.0ⴛ10−3 0 −47810 2.29ⴛ103 0
z2 Exit elevation m 2.40 0.0125 0 9786 1.50ⴛ104 0
z1 Inlet elevation m 15.00 0.0125 0 −9786 1.50ⴛ104 0
Tc Fluid temperature °C 19.71 0.25 0.10 26.76 44.8 7.16
Fluid density kg/m3 997.8 0.6 0 −120.5 5.23ⴛ103 0
b Correlation Pa·s/m3 −1.82ⴛ106 5000 0 1.0ⴛ10−3 25.0 0
coefficient
Table 10-5.2-2 Summary: Uncertainties in Absolute Terms
Combined
Absolute Absolute Standard
Systematic Random Uncertainty Total
Standard Standard of the Absolute
Calculated Uncertainty, Uncertainty, Result, Uncertainty,
Symbol Description Units Value bR sR uR UR,95
⌬P Differential pressure Pa 718,000 6,590 3,530 7,476 14,951
Table 10-5.2-3 Summary of Results for Each Test
⌬P, bX, sX, U⌬P,95,
Test kPa kPa kPa kPa b s U,95
Factory Test 712 . . . . . . . . . 0.9916 0.0091 0.0049 0.0207
A 718 6.5 4 15 1.0000 0.0048 0.0069 0.0168
B 725 6.5 4 15 1.0097 0.0049 0.0070 0.0170
C 707 6.5 4 15 0.9847 0.0049 0.0069 0.0169
D 710 6.5 4 15 0.9889 0.0048 0.0069 0.0168
E 726 6.5 4 15 1.0111 0.0049 0.0070 0.0170
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Fig. 10-5.2 Comparison of Test Results Using the Initial Field Test as the Control
The uncertainty of the comparison ratio,, when all of the systematic standard uncertainty terms are correlated and the corresponding sensitivity coefficients are equal (i.e., i,altp i,control) is zero.
For eq. (10-5.4) the systematic standard uncertainty summation based on eq. (8-1.2).