2.3.1. Basic Information

The pressure-time method is based on the relationship between flow rate at steady state conditions and pressure-time change occurring in the pipeline during cutting off the flow [7,8]. The value of *Q*<sup>0</sup> indicating the discharge at initial liquid flow conditions is calculated using the definite integral over a time interval in which the flow varies from initial conditions to conditions after the flow is completely shut off [4,6,11]:

$$Q\_0 = \frac{1}{\rho F} \int\_{t\_0}^{t\_f} (\Delta p(t) + \Delta p\_d(t) + \Delta P\_r(t)) \mathrm{d}t + Q\_f \tag{2}$$

where:

ρ is the density of a liquid,

*t*<sup>0</sup> and *tf* are the initial and final time-limits of integration, respectively,

*Qf* is the discharge under final steady-state conditions (after complete closing of the shut-off device) due to the leakage through the closed shut-off device,

Δ*p* is the difference in pressures measured between the pipeline measuring cross-sections *B-B* and *A-A*, which geometrical centers are at level *zA* and *zB*, respectively (Figure 4):

$$
\Delta p = p\_B + \rho g z\_B - p\_A - \rho g z\_A \tag{3}
$$

Δ*pd* is the difference in dynamic pressures between the pipeline measuring cross-sections with area of each section equal *AA* and *AB*:

$$
\Delta p\_d = \alpha\_2 \frac{\rho Q^2}{2A\_B^2} - \alpha\_1 \frac{\rho Q^2}{2A\_A^2} \tag{4}
$$

where:

α1, α<sup>2</sup> are the kinetic energy correction factors for *A-A* and *B-B* sections (the value of the kinetic energy correction factor for fully developed turbulent flow in the pipeline, dependent on *Re* number is within the limits from 1.03 to 1.11 [20,21]);

Δ*Pr* is the pressure loss caused by hydraulic resistance in pipeline between the measurement cross-sections—quantity calculated as proportional to the square of flow rate (accounting for its direction):

$$
\Delta P\_r = \mathbb{C}\_r \cdot \mathbb{Q}|Q| \tag{5}
$$

**Figure 4.** Scheme of the penstock measuring section with markings.

One of the most important parameters in Formula (2) is the *F* factor. Its value depends on the geometry of the pipeline flow system between the pressure measurement cross-sections. The following formula can be used to calculate the *F* factor in case of the pipeline segment with length *L* and *j* sub-segments with different sizes:

$$F = \int\_0^L \frac{\mathbf{dx}}{A(\mathbf{x})} = \sum\_{j=1}^{j=J} \frac{\Delta \mathbf{x}\_j}{A\_j}, \quad \text{with } \sum\_{j=1}^{j=J} \Delta \mathbf{x}\_j = L \tag{6}$$

where Δ*xj* and *Aj* indicate the length and internal cross-sectional area of the *j*-th sub-segment, respectively. As shown in Equation (2), the pressure loss, Δ*Pr*, representing hydraulic resistance and the dynamic pressure difference, Δ*pd*, should be separated from the pressure difference measured between the pipeline measurement cross-sections, Δ*p*. In total, the integral expression of Equation (2) defines the pressure difference resulting from the inertia force of the mass of liquid retained in the pipeline measuring section (segment). The values of Δ*Pr* and Δ*pd* can be calculated with good accuracy using their dependence on the square of the flow rate in the forms written in Equation (4) and (5).

Measurements made using the pressure-time method, as was the case concerning the volumetric gauging method, were carried out for both flow directions through a reversible machine equipped with Francis type runner. Measuring flow rate in the pump direction requires appropriate modifications of the pressure-time method to the calculation procedures described in the standards, which were postulated by the authors in earlier publications [11,22,23] and which resulted with formula in Equation (5) (introducing term *<sup>Q</sup>*|*Q*<sup>|</sup> instead of *<sup>Q</sup>*2).

A comprehensive discussion of some problems related to the computational procedures in the pressure-time method is provided in standards [4,6] as well as in monograph [11]. A description of some important problems related to the use of the pressure-time method for measuring flow rate in hydropower plants can also be found in publications [22–29]. Calculation of friction losses according to the quasi-stationary hypothesis is consistent with the conclusions presented in [30]. It was proved that the modelling of unsteady friction losses has little effect on the course of water hammer in its initial time-phase that is taken into account in the pressure-time method. Nevertheless, it should be emphasized that including the transient nature of friction losses into the calculation method, under certain circumstances, may improve predictions of the pressure-time method as described in [27–29].

Several variants of the pressure-time method are used in practice. They differ mainly in methods of measuring the pressure differences between pipeline measurement cross-sections. In the considered case, the pressure-time method was used in the variant based on measuring the pressure changes at the cross-section of the pump-turbine spiral case outlet/inlet and relating these changes to the pressure exerted by the water column from the head water reservoir. This variant requires the determination of the geometric factor *F* accounting the entire penstock of the tested machine, starting from the inlet section and ending with the outlet/inlet cross-section of the spiral case.

The recommendations of the standards [4–6] allow the use of the *F* factor for straight-axis measuring pipelines of variable diameter (according to the Formula (6), taking into account their geometry). However, in the case of more complex changes in the geometry occurring in the measuring section of the pipeline (changes in the shape of the flow section, changes in the direction of the pipeline axis or branches), there is a need to take into account the influence of these changes on the flow conditions.

Irregular parts (components) of the penstock cause flow disturbances in the form of non-uniform water velocity distribution. This should be taken into account in order to ensure better accuracy of discharge measurement. In the considered case, except for the straight pipe sections with constant internal diameters, the penstock has three elbows (two vertical and one horizontal), a number of short conical sections connecting pipes of different diameters, and two short branches, where one branch remained closed during the tests. In addition, the square cross-section as well as transition section from square to the circular cross-section in the highest part of the penstock had to be taken into consideration. In the previously published work [24], authors presented the procedure, based on CFD, used for correction of *F*-factor calculated in case of penstocks with elbows. The assumption of equal kinetic energy resulting from the simulated and the uniform water flow velocity distributions in the same flow parts of the penstock was the main, except mass conservation law, theoretical basis for this procedure. In this work, using CFD, an extended procedure was developed and applied to correct the value of the *F*-factor for the above-mentioned irregular components of the penstock under consideration. The procedure is presented in detail in Appendix A. The selected results of CFD calculations and the *F*-factor correction for the studied case are presented later in this paper.
