## 2 Computational Modeling
### Entropy Production Theory  
To numerically determine the indicators of the pump stage being studied, it is necessary to calculate the losses in the individual elements. The entropy generation theory is applied for this purpose. This theory has been used to analyze losses in centrifugal pumps in Refs. [19; 20; 21; 22; 23]. In Refs. [24; 25], the authors demonstrate the implementation of entropy production calculation for turbulent shear flows in CFD.  
In Ref. [23], the authors suggest calculating the power of the energy losses in a pump by applying the entropy generation theory and the absolute temperature, using the following equation:  
\[\mathrm{P}_{\mathrm{gen}}=\mathrm{T}\big{(}\Delta\mathrm{S}_{\mathrm{gen,V}}+ \Delta\mathrm{S}_{\mathrm{gen,T}}+\Delta\mathrm{S}_{\mathrm{gen,W}}\big{)}\ (\mathrm{W}) \tag{2}\]  
Figure 2: Boundary conditions.  
where \(\mathrm{T}=\mathrm{const},\ \left(\mathrm{K}\right)\) is the absolute temperature of the working fluid (water), which is assumed to remain constant in this case; \(\mathrm{\Delta S_{gen,V}}\) is the total entropy generation due to viscous diffusion; \(\mathrm{\Delta S_{gen,T}}\) is the total entropy generation due to velocity fluctuation; \(\mathrm{\Delta S_{gen,W}}\) is total entropy generation caused by the wall effect. The individual terms of Equation (9) are calculated as follows [23]:  
\[\mathrm{\Delta S_{gen,V}}=\int\limits_{V}\mathrm{S_{vd}dV},\ \left(\mathrm{WK^{-1}}\right) \tag{3}\]  
\[\mathrm{\Delta S_{gen,T}}=\int\limits_{V}\mathrm{S_{rd}dV},\ \left(\mathrm{WK^{-1}}\right) \tag{4}\]  
\[\mathrm{\Delta S_{gen,W}}=\int\limits_{a}\mathrm{S_{wd}dA},\ \left(\mathrm{WK^{-1}}\right) \tag{5}\]  
The rate of entropy generation due to viscous diffusion can be calculated as:  
\[\begin{array}{l}\mathrm{S_{V}}=\frac{2\mu}{\mathrm{T}}\left[\left(\frac{ \partial\overline{u}}{\partial k}\right)^{2}+\left(\frac{\partial\overline{v} }{\partial\overline{y}}\right)^{2}+\left(\frac{\partial\overline{w}}{\partial \overline{z}}\right)^{2}\right]+\frac{\mu}{\mathrm{T}}\left[\left(\frac{ \partial\overline{v}}{\partial k}+\frac{\partial\overline{v}}{\partial \overline{y}}\right)^{2}+\right.\\ \left.+\left(\frac{\partial\overline{w}}{\partial k}+\frac{\partial\overline{ u}}{\partial\overline{z}}\right)^{2}+\left(\frac{\partial\overline{v}}{\partial \overline{z}}+\frac{\partial\overline{w}}{\partial\overline{y}}\right)^{2} \right]\end{array} \tag{6}\]  
where \(\mu,\ \left(\mathrm{Pa\cdot s}\right)\) is the fluid dynamic viscosity; \(\overline{u},\overline{v}\) and \(\overline{w}\) are Reynolds averaged velocity components in a Cartesian coordinate system.  
The entropy generation rate due to velocity fluctuation can be determined by applying the following equation [23]:  
\[\mathrm{S_{T}}=\beta\frac{\rho\omega\kappa}{\mathrm{T}} \tag{7}\]  
where \(\beta=0.09\) is a constant; \(\omega,\ \left(\mathrm{s}^{-1}\right)\) is the turbulent eddy frequency; \(\mathrm{k},\ \left(\mathrm{m}^{2}/\mathrm{s}^{2}\right)\) is the turbulent kinetic energy.  
The rate of entropy generation, caused by the wall effect, can be calculated as follows [23]:  
\[\mathrm{S_{W}}=\frac{\overset{\rightarrow}{\tau}\cdot\overset{\rightarrow}{ \tau}}{\mathrm{T}}=\frac{\tau_{x}u+\tau_{y}v+\tau_{z}w}{\mathrm{T}} \tag{8}\]  
where \(\overset{\rightarrow}{\tau},\ \mathrm{Pa}\) is the shear stress on the wall; \(\overset{\rightarrow}{\upsilon},\ \left[\mathrm{m}/\mathrm{s}\right]\) is the velocity of the first grid near the wall. The parameters used in the estimation of \(\mathrm{S_{V}},\ \mathrm{S_{T}}\) and \(\mathrm{S_{W}}\) can be determined after solving URANS using the SST model. The determination of losses in the form of head can be performed by applying the following equation:  
\[\mathrm{h_{v}}=\frac{\mathrm{P_{gen}}}{\mathrm{\rho gQ}}\left(\frac{\mathrm{J }}{\mathrm{N}}\equiv\mathrm{m}\right) \tag{9}\]