**Appendix C. Uncertainty Analysis of Flow Rate Measurements by Means of the Pressure-Time Method**

Standards [4,6] specify the requirements that must be met so that the uncertainty of the flow rate measurement obtained using the pressure-time method is in the range of ±1.5% (2.3%) according to [4] and ±1.0% according to [6]. However, a way to calculate this uncertainty is not provided. The algorithm for estimating this uncertainty was the subject of only few available papers [36,37] but the presented algorithms do not comply with the applicable principles of expressing measurement uncertainty, presented in [19].

Below is a method for estimating the uncertainty of flow rate measurement under the considered conditions. The method is currently used by the authors of this contribution and complies with the recommendations presented in [19]. To present it, a simplified formulation of Equation (2) is introduced in the following form:

$$Q\_0 = \frac{1}{\rho F} (\Delta p\_{\text{ff}} + \Delta p\_{dm} + P\_{rm}) \left(t\_f - t\_0\right) + Q\_f \tag{A21}$$

where Δ*pm*, Δ*pdm*, and *Prm* are the values of Δ*p*, Δ*pd* and *Pr,* respectively, after averaging over the time interval from *t*<sup>0</sup> to *tf*.

Treating all the constituent quantities (components) in the above dependence as uncorrelated with each other, the value of the relative standard total uncertainty δ*c*(*Q*0) can be calculated from the formula resulting from the law of uncertainty propagation:

$$\delta\varsigma(Q\_0) = \sqrt{\delta^2(\rho) + \delta^2(F) + \delta^2(\Delta p\_m) + \delta^2(\Delta p\_{dm}) + \delta^2(P\_{rm}) + \delta^2(t\_f - t\_0) + \delta^2(Q\_f)}\tag{A.22}$$

The largest uncertainty component is related to the measurement and recording of the pressure difference. In the measurement procedure used by the authors of this work, the initially recorded pressure difference signal Δ*p*(*ti*) is numerically corrected taking into account characteristic of signal between limits *tf* and *t*ff as well as the flow rate at final conditions (*Qf*) and the *Cr* coefficient of frictional resistance characterizing the pipeline between measuring cross-sections. All measurement results of differential pressure values Δ*p*(*ti*) are corrected according to the formula:

$$
\Delta p(t\_i)\_{correction} = \Delta p(t\_i) - \left(\frac{1}{N\_f} \sum\_{t\_f}^{t\_{ff}} \Delta p(t\_i) - C\_r Q\_f \left| Q\_f \right| - \Delta p\_{df} \right) \tag{A23}
$$

where the second component on the right is the average value calculated from the recorded signal Δ*p*(*ti*) in the time interval (*tf,t*ff), i.e., in the phase of suppression of free pressure oscillations after the flow is cut off, *Nf* is the number of recorded values of Δ*p*(*ti*) in the time interval (*tf, t*ff), and Δ*pdf* means the difference of dynamic pressures in the final steady state conditions (the method of calculating the difference of dynamic pressures is analogous to the calculation of the average difference of dynamic pressures—see the further part of the Appendix).

The *Cr* factor is determined from the formula (A23) on the basis of the measured pressure difference Δ*p*0*correction* = *Pr*<sup>0</sup> + Δ*pd*<sup>0</sup> caused by friction losses in the pipeline measuring section and dynamic pressure difference in the initial steady flow conditions, i.e., immediately before the closing of the flow shutoff device. Thus, the value of *Pr*<sup>0</sup> is calculated as the average of the measured pressure difference (after correction) in the time interval (*t*00, *t*0):

$$P\_{r0} = \Delta p\_0 - \Delta p\_{d0} = \frac{1}{N\_0} \sum\_{t=0}^{t\_0} \Delta p(t\_i)\_{correction} - \Delta p\_{d0} \tag{A24}$$

where *N*<sup>0</sup> is the number of recorded values of Δ*p*(*ti*) in the time interval from *ti* = *t*<sup>00</sup> to *ti* = *t*0, and Δ*pd*<sup>0</sup> means the difference of dynamic pressures in the initial steady.

The method of correction according to formula (A23) allows us to get rid of the most important part of uncertainty arising from the exact determination of the "zero" pressure differential transducer. The residual uncertainty associated with it is estimated when analyzing the effect of *tf* limit on the uncertainty value. It should be emphasized that the correction applied takes place in the iterative process of calculating the *Q*<sup>0</sup> value.

Therefore, the mean pressure difference Δ*pm* is calculated from the measured and corrected values of Δ*p*(*t*)*correction* using the formula:

$$
\Delta p\_m = \frac{1}{N} \sum\_{t\_0}^{t\_f} \Delta p(t\_i)\_{correction} \tag{A25}
$$

where *N* is the number of recorded values of Δ*p*(*ti*) in the time interval from *ti* = *t*<sup>0</sup> to *ti* = *tf*.

The absolute standard uncertainty of type B measurement of pressure difference Δ*p*, resulting from the classes of transducers used, was determined as follows:

$$
\Delta \mu\_{kB}(\Delta p\_m) = \frac{K\_{\Delta p} \cdot \Delta p\_{\text{range}}}{\sqrt{3}} \tag{A26}
$$

After considering the pressure difference transducer class *K*Δ*<sup>p</sup>* = 0.075% and its range Δ*prange* = ±500 kPa (1 MPa), this uncertainty was: *ukB*(Δ*pm*) = 0.43 kPa.

To record Δ*pm*, a computer data acquisition system with a measurement card with an absolute accuracy of 0.55 mV was used. In order to determine the measurement uncertainty of the water level resulting from the used measurement card, the scaling of the level transducer should be taken into account (in the case under consideration the full width of the transducer measuring range corresponded to a 3.5 V voltage change). The resulting standard uncertainty of level measurement can be determined by the formula:

$$
\Delta u\_{rB}(\Delta p\_m) = \frac{\Delta\_{DAQ}}{\sqrt{3}} \frac{\Delta p\_{m-mng\epsilon}}{U\_{\Delta p\_{m-mng\epsilon}}} \cong \frac{0.00055 \cdot 1000}{\sqrt{3} \cdot 3.5} \approx 0.09 \text{ kPa} \tag{A27}
$$

In connection with the above, the total standard uncertainty *u*(Δ*pm*), calculated from the formula:

$$
\Delta u(\Delta p\_m) = \sqrt{\mu\_{k\text{B}}^2 (\Delta p\_m) + \mu\_{r\text{B}}^2 (\Delta p\_m)}\tag{A28}
$$

was not worse than:

*u*(Δ*pm*) = 0.44 kPa

After referring these uncertainty values to the average pressure difference increases caused by the inertia forces after flow cut-off during the measurement, i.e.,

$$
\Delta p\_{m-i
u\_{rm}} = \left(\Delta p\_m + \Delta p\_{dm} + P\_{rm}\right) \tag{A29}
$$

the relative standard uncertainty δ(Δ*pm*) is determined, which, together with other uncertainty components, has been presented in the uncertainty balance table Table A2. This uncertainty is approximately 0.36% and 0.43% for turbine and pump mode of operation, respectively.

In addition to the *Pr*<sup>0</sup> value resulting from the measurement and calculations, the values of friction pressure drop *Pr* during flow cut off are calculated according to the relationship (A24) in the time interval (*t*0, *tf*). For this range, the average pressure drop *Prm* can be calculated from the formula:

$$P\_{\rm rm} = \frac{C\_r}{N\_0} \sum\_{t\_0}^{t\_f} Q(t\_i) \left| Q(t\_i) \right| - C\_r Q\_f \left| Q\_f \right| - \Delta p\_{df} \tag{A30}$$

where *N* is the number of calculated *Q*(*ti*) values in the range (*t*0, *tf*). The values of the second and third components to the right of the above dependence are negligibly small, so it can be neglected when estimating their uncertainty.

The standard uncertainty type B resulting from the calculation of the *Prm* value was estimated from the formula:

$$
\mu\_B(P\_{rm}) = \mu(P\_{rm}) = \frac{\delta\_{P\_{rm}} P\_{rm}}{\sqrt{3}} \tag{A31}
$$

where δ*Prm* is the average, relative difference in friction losses calculated using the quasi-stationary model (friction coefficient depending on the *Re* number) and the stationary model (constant friction coefficient)—the δ*Prm* value was adopted according to approximately parabolic dependence of this difference on the flow rate proposed in monograph [11]: δ*Prm* = δ*Prmax*/3 = ~0.025/3 = 0.0083. It is worth emphasizing here that for calculating the flow rate, δ*Prm* value was not used to correct friction loss calculations, i.e., the calculations were carried out assuming a constant *Cr* factor, not dependent on *Re*.

The effect of other factors on uncertainty *u*(*Prm*), e.g., unsteadiness of flow, was omitted as irrelevant from the practical point of view. References [38,39] indicate that dissipation of mechanical energy during flow deceleration (taking place when the pressure-time method is applied) is only slightly less than that obtained from the quasi-steady hypothesis. It is the opposite to accelerating flow where energy dissipation is much larger than according the quasi-steady modeling. Some unsteady friction loss models in the closed conduits use these features [40]. These models have been confirmed experimentally—there is a high conformity between experimental and numerical results of the water hammer course [30]. With reference to the pressure-time method, the above assessment is confirmed by [27–29].

Finally, after the referring the *u*(*Prm*) to the value of Δ*pm-inertia*, the relative standard uncertainty associated with the calculation of *Pr*, for the highest value of flow rate measured is presented in the uncertainty balance table Table A2.

The uncertainty of calculating the dynamic pressure difference between the pipeline measuring cross-sections, *u*(Δ*pdm*) was estimated as follows. The average dynamic pressure difference, Δ*pdm*, in the time interval (*t*0, *tf*) was calculated from the formula:

$$
\Delta p\_{dm} = \frac{1}{2} \left( \frac{\alpha\_B \rho}{A\_B^2} - \frac{\alpha\_A \rho}{A\_A^2} \right) \frac{1}{N} \sum\_{t\_0}^{t\_f} \left[ Q(t\_i) \right]^2 \tag{A32}
$$

in which *N* denotes the number of calculated *Q*(*ti*) values in the interval (*t*0, *tf*), and *AA* with *AB* are the cross-sectional areas of the upper and lower pipeline measuring cross-sections, and α*<sup>A</sup>* and α*B*—Coriolis coefficients.

In the considered case, it was assumed that *AA* = ∞ and the effect of calculating Δ*pm* on the uncertainty of flow measurement results from changes in the Coriolis coefficient in the lower measuring cross-section of the pipeline. In calculations α*<sup>B</sup>* = 1.05 was taken as the average value of the Coriolis coefficient for fully developed turbulent flow in the pipeline determined within the limits from 1.04 to 1.06 [23]. On this basis, the standard uncertainty type B resulting from the calculation of the Δ*pdm* value was calculated using the following formula:

$$
\Delta u\_B(\Delta p\_{dm}) = \frac{0.01 \Delta p\_{dm}}{\sqrt{3}}\tag{A33}
$$

For the cases with the highest values of measured flow rates, the values of standard uncertainty *u*(Δ*pdm*) determined in this way was 0.21 kPa and 0.16 kPa for turbine and pump mode of operation, respectively.

Table A2 of the uncertainty balance lists the relative standard uncertainty associated with the calculation of Δ*pdm*, after relating *u*(Δ*pdm*) to the value of Δ*pm-inertia* for the highest values of measured flow rates in the turbine and pump mode of operation of the tested machine.

The time accuracy of the computer acquisition system measuring the pressure difference signal *p*(*ti*) was omitted as having no impact on the standard uncertainty type B regarding the measurement of the time interval from *t*<sup>0</sup> to *tf*. It can be calculated using the following formula:

$$
\mu\_B(\Delta t) = \mu\_B(t\_f - t\_0) = \frac{\Delta\_{tDAQ}(t\_f - t\_0)}{\sqrt{3}} \tag{A34}
$$

where <sup>Δ</sup>*tDAQ* <sup>=</sup> <sup>50</sup> <sup>×</sup> <sup>10</sup>−<sup>6</sup> is the time accuracy of measurement card used in the data acquisition system including its resolution.

The value of *uB*(*t*) is about 0.0007 s and 0.0005 s for turbine and pump mode of operation, respectively.

For the flow rate measurements, the time interval (*t*0*, tf*) during turbine mode of operation was not longer than *T* = ~25 s, and during pump mode of operation *T* = ~20 s, using a sampling frequency of 200 Hz. Table A2 of the uncertainty balance lists the relative standard uncertainty associated with the measurement of the time interval (*t*0, *tf*).

The method of determining the *tf* time limit, i.e., the upper limit of integration in the pressure-time method, was presented in [21]—an earlier author's publication. This method significantly influences the uncertainty of measuring *Q*<sup>0</sup> in cases where the free pressure oscillations after the closing of the shut-off device have relatively high amplitudes compared to the average Δ*pm* values. The value of *tf* should be selected at the top of the peak or the bottom of the valley of free oscillations of pressure differences, with its exact determination taking place in the calculation program. It is recommended to choose the limit *tf* from the first clear peak or valley of free oscillations in order to minimize the impact of these oscillations on the measurement result *Q*0. Pulsations superimposed on these oscillations, which are random in nature, have been included in the estimation of uncertainty type A. For this reason, a series of calculations of *Q*0*<sup>i</sup>* values for several values of time *tfi*, selected in close proximity of the original value of *tf* selected in accordance with the above principle, was carried out in the range covering only one valley and one peak visible in the measured quick-change pressure difference signal (pressure difference pulsation). It should be emphasized that it is not advisable to significantly shift the *tfi* value from the tops of peaks and bottom of valleys of free differential pressure oscillations. The obtained *Q*0*<sup>i</sup>* calculation results were subjected to statistical analysis, i.e., the average *Q*0*<sup>m</sup>* value and standard uncertainty were calculated using the formula:

$$
\mu\_{tfA}(Q\_0) = k \sqrt{\frac{1}{n(n-1)} \sum\_{i=1}^{n} \left(Q\_{0i} - Q\_{0m}\right)^2} \tag{A35}
$$

where *k* is the extension coefficients calculated from the Student's *t*-distribution for the confidence level *p* = 68.2% and the number of degrees of freedom (*n* − 1), *n*—number of calculated *Q*0*<sup>i</sup>* values.

After relating the *utfA*(*Q*0) values determined in the above described manner to the measured flow rate *Q*0, the relative standard uncertainties δ*tfA* did not exceed δ*tfA*(*Q*0) = 0.08% for turbine mode of operation and δ*tfA*(*Q*0) = 0.1% for pump mode of operation.

The uncertainty δ(ρ) results from the variability of water density with pressure change and from the accuracy of its determination for given temperature and average absolute pressure in the pipeline occurring during tests. This uncertainty is very small; therefore, it was omitted in calculating the uncertainty of flow rate measurement.

The standard uncertainty δ(*F*) for determining the geometric factor *F* results from the accuracy of measuring the length of individual pipeline segments (*Li*) and the area of their internal cross-sections (*Ai*) and from the accuracy of the correction of the *F* factor using CFD calculations. The uncertainty of determining the *F* factor based on the available post-completion documentation of the pipeline, positively verified by direct measurement of *Li* and *Ai*, was not worse than:

$$
\delta(F\_{\text{дем}}) = 0.15\% \tag{A36}
$$

Due to the fact that the uncertainty of the *F* factor correction introduced reaches about 0.75%, the uncertainty of this correction based on CFD calculations is of the same order assuming even 20% accuracy of CFD calculations, and as a result we get standard uncertainty:

$$\delta(F) = \sqrt{\delta^2(F\_{\otimes \text{om}}) + \delta^2(F\_{\text{CFD}})} \approx 0.21\% \tag{A37}$$

The flow rate under final conditions, being that the leakage through the closed wicket gates of the pump-turbine, *Qf*, was measured in a separate way. For this purpose, under closed wicket gate conditions, pressure changes in the pipeline were recorded when closing the shut-off valve characterizing with very high tightness. On this basis, also using the pressure-time method, *Qf* values were determined. For turbine mode of operation, it was equal *Qf* = ~0.14 m3/s, while for pump mode of operation *Qf* = ~0.18 m3/s, which is about 0.7% in relation to the minimum flow rate values for turbine and about 0.65% for pump flow direction. No detailed analysis of the *Qf* uncertainty was carried out, but it was assumed with a large excess that it is not worse than 10%, which gives uncertainty:


The uncertainty resulting from the iterative algorithm for calculating the flow rate is δ(*Q*0*iter*) = 0.1%. This is due to the condition used for ending the calculations, which assumes that the calculations are finished when two subsequent values *Q*0*iter*-1 and *Q*0*iter* do not differ by more than 0.1%.

The balance of the estimated uncertainty of *Q* measurement using the pressure-time method is presented in Table A2.

**Table A2.** Summary results of calculations of uncertainty of flow rate results measured using the pressure-time method.

