## 3 Methods
### CFD Model  
#### 3.2.1 Mathematical Model  
The continuity, momentum conservation and energy equations applicable to this study are listed and solved based on the finite volume method. The Reynolds numbers in Table 2 are all greater than 4000, indicating a turbulent flow at the cold water inlet. Due to the coexistence of laminar and turbulent flows within the storage tank, both laminar and turbulent models were tested in this study. The analysis revealed that in the central region of the tank, both models could simulate the flow field characteristics effectively. However, near the inlet and outlet, the computational results of the two models showed significant differences, with the turbulent model providing more accurate and predictive results for the turbulent effects in these areas. This observation is consistent with the description in Reference [30]. The Realizable \(k-\epsilon\) model is an enhancement of the standard \(k-\epsilon\) model, incorporating the constraints of turbulent stresses for better physical consistency. It exhibits excellent numerical stability even in complex geometries and under high strain rate conditions. Additionally, it offers better convergence when computing turbulence, has acceptable computational costs, and is widely used in engineering applications. Figure 6: Verification of time-step irrelevance. In order to make the calculation results more realistic, the Realizable \(k-\varepsilon\) turbulence model was selected in the calculation settings [31]. The governing equations for this study are as follows:  
Energy equation:  
\[\frac{\partial_{\rho}}{\partial_{t}}+\
abla(\rho v)=0 \tag{3}\]  
Considering water as an incompressible fluid, the equations are:  
\[\
abla v=0 \tag{4}\]  
Momentum equation:  
\[\rho\frac{\partial_{v}}{\partial_{t}}+(\rho v\cdot\
abla)v=-\
abla p+\
abla \cdot\tau+\rho f_{(v)} \tag{5}\]  
\[ \begin{array}{l}\frac{\partial u}{\partial t}+\frac{\partial uu}{\partial a} +\frac{\partial uu}{\partial b}+\frac{\partial uu}{\partial t}=-\rho^{-1}\frac {\partial p}{\partial a}+\varepsilon(\frac{\partial u}{\partial a}+\frac{ \partial^{2}u}{\partial\beta^{2}}+\frac{\partial^{2}u}{\partial\beta^{2}})\\ \frac{\partial v}{\partial t}+\frac{\partial uv}{\partial a}+\frac{\partial uv }{\partial b}+\frac{\partial uv}{\partial b}+\frac{\partial uv}{\partial b} =-\rho^{-1}\frac{\partial p}{\partial b}+\varepsilon(\frac{\partial^{2}v}{ \partial a^{2}}+\frac{\partial^{2}v}{\partial\beta^{2}}+\frac{\partial^{2}v }{\partial\beta^{2}})+g\chi(T_{i}-T_{in})\\ \frac{\partial uv}{\partial t}+\frac{\partial uv}{\partial a}+\frac{\partial uv }{\partial\beta}+\frac{\partial uv}{\partial b}=-\rho^{-1}\frac{\partial p}{ \partial b}+\varepsilon(\frac{\partial^{2}v}{\partial a^{2}}+\frac{\partial^{ 2}v}{\partial\beta^{2}}+\frac{\partial^{2}w}{\partial\beta^{2}})\end{array}  \tag{6}\]  
Energy equation:  
\[\lambda\left(\frac{\partial^{2}T}{\partial a}+\frac{\partial^{2}T}{\partial \beta}+\frac{\partial^{2}T}{\partial\theta}\right)=C_{P}\rho\left[\frac{ \partial_{T}}{\partial_{t}}+\
abla(vT)\right] \tag{7}\]  
In this study, the SIMPLE algorithm was selected for pressure-velocity coupling in the solution process. The SIMPLE algorithm achieves excellent numerical convergence by solving the velocity and pressure fields in a step-by-step manner and iteratively correcting them to satisfy the continuity equation. Its pressure correction principle enhances the physical consistency of the computational results. Combining the SIMPLE algorithm with the aforementioned Realizable \(k-\varepsilon\) model ensures the proper resolution of incompressible fluid flow, providing more accurate computational results. This combination strikes a favorable balance between convergence, physical consistency, and computational cost [31].