## 2 Numerical Simulation Method
### Entropy Production Theory  
Entropy is a parameter that represents the state of an object. During pump operation, mechanical energy dissipation occurs in the flow field due to fluid viscosity and Reynolds stress, and the lost mechanical energy is converted into internal energy, which leads to an increase in entropy. Thus, local entropy production can effectively characterize the spatial distribution of mechanical energy losses and provide a better physical understanding about loss evaluation. The entropy transport equation is defined as follows [31]:  
\[\rho(\frac{\partial S}{\partial t}+v_{1}\frac{\partial S}{\partial x}+v_{2} \frac{\partial S}{\partial y}+v_{3}\frac{\partial S}{\partial z})=div(\frac{ \rightarrow}{T})+\frac{\Phi}{T}+\frac{\Phi_{\mathbf{0}}}{T^{2}} \tag{5}\]  
where \(S\) is specific entropy. \(T\) represents absolute temperature.\(v_{1}\), \(v_{2}\) and \(v_{3}\) are velocity components in the Cartesian coordinate direction: \(x\), \(y\), \(z\). \(div(\frac{\rightarrow}{T})\) is the reversible heat transfer term. \(\frac{\Phi_{\mathbf{0}}}{T^{2}}\) and \(\frac{\Phi}{T}\) are entropy produced by heat transfer and dissipation, respectively.  
Since the governing equation is URANS, entropy transport equation should be time-averaged as follows [18,19]:  
\[\rho(\frac{\partial\overline{S}}{\partial t}+\overline{v_{1}}\frac{\partial \overline{S}}{\partial x}+\overline{v_{2}}\frac{\partial\overline{S}}{ \partial y}+\overline{v_{3}}\frac{\partial\overline{S}}{\partial z})=\overline {div(\frac{\rightarrow}{T})}-\rho(\frac{\partial\overline{v^{\prime}_{1}S^{ \prime}}}{\partial x}+\frac{\partial\overline{v^{\prime}_{2}S^{\prime}}}{ \partial y}+\frac{\partial\overline{v^{\prime}_{3}S^{\prime}}}{\partial z})+ \overline{\left(\frac{\Phi}{T}\right)}+\overline{\left(\frac{\Phi_{\mathbf{0 }}}{T^{2}}\right)} \tag{6}\]  
where \(\overline{v_{1}}\), \(\overline{v_{2}}\), and \(\overline{v_{3}}\) are time-averaged velocity components. \(v^{\prime}_{1}\), \(v^{\prime}_{2}\), and \(v^{\prime}_{3}\) are velocity fluctuation components. \(\overline{S}\) and \(S^{\prime}\) represent time-average specific entropy and specific entropy fluctuation. \(\overline{T}\) is time-average absolute temperature.  
In this study, the working medium inside the pump was set as constant temperature and incompressible flow fluid [27, 28, 29]. Therefore, the time-averaged entropy production in the pump only results from \(\overline{\left(\frac{\Phi}{T}\right)}\) which can be calculated as follows [18,19]:  
\[\overline{\left(\frac{\Phi}{T}\right)}=\frac{\Phi_{\overline{\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline \  
\[P_{\text{PRO,D}^{\prime}}=\int\limits_{V}\Phi_{\text{D}^{\prime}}dV \tag{11}\]  
To summarize, the total dissipation rate \(\Phi_{\text{T}}\) and total power dissipation \(P_{\text{total}}\) can be calculated as follows:  
\[\Phi_{\text{T}}=\Phi_{\overline{\text{D}}}+\Phi_{\text{D}^{\prime}} \tag{12}\]  
\[P_{\text{total}}=P_{\text{PRO,\overline{\text{D}}}}+P_{\text{PRO,D}^{\prime}} \tag{13}\]