**Appendix A. Procedure for Calculating Equivalent Geometrical F Factor in The Pressure-Time Method for Pipelines with Irregular Shape Sections of the on the Basis of CFD Analysis**

The determination of the geometrical *F*-factor from Equation (3) is fully acceptable for straight measuring sections of pipelines where there are no flow irregularities. This equation does not take into account changes in the flow velocity profiles in irregularly shaped pipeline elements, such as elbows, bifurcations, cones, pipe inlets, etc. Therefore, the authors of this paper recommend a special calculation procedure to consider the effect of these irregular shaped flow elements on the pressure-time measurement results.

The procedure shown below is an extension of the procedure for the curved pipe sections published in [24].


**Figure A1.** A pipe elbow with marked computational space.


$$
\varepsilon\_{k\Box\Diamond} = \varepsilon\_{\text{kri}}; \quad \rho = \text{const} \tag{A1}
$$

$$c\_{\rm kCFDi} = \frac{1}{\dot{m}} \iint\_{A\_i} \frac{1}{2} V\_i^2 [\rho V\_i dA] = \frac{\rho}{2\dot{m}} \iint\_{A\_i} V\_i^3 dA; \quad \dot{m} = \rho V\_{\rm ai} A\_i \tag{A2}$$

$$c\_{kni} = \frac{1}{2}V\_{ai}^2 = \frac{1}{2in} \rho A\_i V\_{ai}^3 \tag{A3}$$

$$V\_{ai} = \left[\frac{\iint\_{A\_i} \left(V\_j^3 dA\right)}{A\_i}\right]^{1/3} \tag{A4}$$

where *Vi* is the flow velocity axial component—the component perpendicular to the *i*-th numerical cross-section.

*Step 5:* Computation of the equivalent cross-sectional area, *Aei*, for each numerical cross-section (*i* = 1, 2, ..., *I*) using the continuity equation *Qj* = *Vai Aei* = *const*:

$$A\_{ei} = \frac{Q\_j}{V\_{ai}}\tag{A5}$$

*Step 6* Computation of coordinates of flow velocity centers in each *i*-th numerical cross-section, *i* = 1, 2, ..., *I*:

$$\mathbf{x}\_{\mathrm{Ci}} = \frac{\iint\_{A\_i} \mathbf{x} \, V(\mathbf{x}, \mathbf{y}, \mathbf{z}) dA}{V\_{\mathrm{oi}} A\_{\mathrm{ci}}}; \; y\_{\mathrm{Ci}} = \frac{\iint\_{A\_i} \mathbf{y} \, V(\mathbf{x}, \mathbf{y}, \mathbf{z}) dA}{V\_{\mathrm{oi}} A\_{\mathrm{ci}}}; \; z\_{\mathrm{Ci}} = \frac{\iint\_{A\_i} \mathbf{z} \, V(\mathbf{x}, \mathbf{y}, \mathbf{z}) dA}{V\_{\mathrm{oi}} A\_{\mathrm{ci}}} \tag{A6}$$

*Step 7:* For the considered flow rate *Qj* through the analyzed pipe element, computing the equivalent factor *FeQj* from the formula:

$$F\_{cQ\_j} = \sum\_{i=1}^{I-1} \frac{l\_{i \to i+1}}{0.5(A\_{ci} + A\_{ci+1})} \tag{A7}$$

where *li*→*i*+<sup>1</sup> denotes the distance between the resultant velocity centers for computational sections *i* and *i* + 1, *Aei* and *Ae i*+1—equivalent areas of computational cross-sections *i* and *I* + 1, respectively.

The above computation should be performed for several discharge values (*Qj*, *j* = 1, 2, ..., *m*) from the whole scope of its variation (*Q*min < *Qj* ≤ *Q*max). The average value of equivalent factor, *Fe*, can be calculated from the relationship:

$$F\_{\varepsilon} = \frac{1}{m} \sum\_{i=1}^{m} F\_{\varepsilon Q\_{j}} \tag{A8}$$

In the above procedure, it was assumed that the changes in velocity profiles are the same under steady and transient flow conditions. This assumption is correct for cases where the flow shut devices are not closed very quickly when using the pressure-time method. Practically, such cases occur in all hydraulic machines, due to the need to protect their flow systems against the destructive effects of the water hammer phenomenon.

Taking the equivalent value of *Fe* instead of the value *F* calculated directly from the geometry of pipeline sections it is possible to increase the pressure-time method accuracy in cases when pipelines have irregular flow elements.
