## 2 Numerical Simulation Modeling of Plate-Fin Heat Exchanger
### Porous Media Model  
Due to the difference of several orders of magnitude between the geometric size of the fin and the overall size of the heat exchanger, if the overall flow and heat transfer \begin{table}
\begin{tabular}{c c} \hline \hline
**Water Physical Properties** & **Calculation Formula** \\ \hline Density/ (kg x m\({}^{-3}\)) & \(\rho_{2}\ =\ 1000.7-0.0852l-0.0034l^{2}\) \\ Constant pressure heat & \(\text{c}_{22}\ =\ 4184.4-0.6964t+1.036\times 10^{-2}t^{2}\) \\ \(\text{ratio}/\left(\text{J}\times\text{kg}^{-1}\times\text{K}^{-1}\right)\) & \(\lambda_{2}\ =\ 0.5980+1.373\times 10^{-3}t-5.333\times 10^{-6}t^{2}\) \\ Thermal conductivity/ (W x m\({}^{-1}\times\text{K}^{-1}\)) & \(\mu_{1}\ =\ 0.00178-4.607\times 10^{-5}t+5.285\times 10^{-7}-2.156\times 10^{-9}\) \\ Dynamic viscosity/(Pa x) & \(\mu_{1}\ =\ 0.00178-4.607\times 10^{-5}t+5.285\times 10^{-7}-2.156\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Water parameters. Figure 3: The solid line is 90-degree fin, and the dashed line is 75-degree fin. Figure 2: Fin model. performance of the heat exchanger is to be calculated by numerical simulation on the premise of retaining the geometric shape of the whole fin, the number of meshes will exceed 1 billion. The existing computer is difficult to meet its huge computational volume. Therefore, in this study, the porous media model is used to replace the fin region with complex and dense structure. [PERSON] et al. [15] first to applied the porous media model to the overall study of heat exchangers, where the fluid heat transfer and pressure drop through the fin region were approximated with the porous effect. This approximation sacrifices the details of the flow heat transfer in the simplified region, but allows the overall performance of the heat exchanger to be studied by virtue of a smaller computational volume. The porous media distribution resistance model is [16; 25]:  
\[S_{i}\ =\ -\Delta P_{i}\ =\ -(a_{1}u_{i}+a_{2}|u_{i}|u_{i}) \tag{1}\]  
where \(S_{i}\) is the \(i\)-directional (\(x,y,z\)) momentum source term, \(\Delta P_{i}\) is the fluid pressure drop per unit length, \(a_{1}\), \(a_{2}\) are constants, \(u_{i}\) is the \(i\)-directional fluid rate, and the resistance of the fins to the fluid is expressed on the momentum equation through the source term \(S_{i}\). The resistance of the fins in each direction is not uniform and all are calculated using this model. Ultimately, the anisotropic porous media model is obtained. In the working process of heat exchanger, there is a certain temperature difference between the fluid and the solid. Therefore, the porous medium model is a non-thermal equilibrium model, and this model uses different energy conservation equations for the fluid and the solid, respectively [26], where the fluid energy equation is:  
\[\gamma\frac{\partial(c_{f}\rho_{f}T_{f})}{\partial t}+\
abla\cdot\left(\rho _{f}V_{f}c_{f}T_{f}\right)\ =\ \
abla\cdot\left(k_{f,eff}\
abla T_{f}\right)+h_{sf}A_{sf}\Big{(}T_{s}-T_{f} \Big{)} \tag{2}\]  
where \(\gamma\) is the porosity of the medium, \(c_{f}\) is the fluid specific heat capacity at constant pressure, \(\rho_{f}\) is the fluid density, \(T_{f}\) is the fluid temperature, \(V_{f}\) is the fluid rate, \(k_{f,eff}\) is the fluid equivalent thermal conductivity, \(h_{sf}\) is the fluid-solid heat transfer coefficient, \(A_{sf}\) is the specific surface area, and \(T_{s}\) is the solid temperature. The solid energy equation is:  
\[(1-\gamma)\rho_{s}c_{s}\frac{\partial(T_{s})}{\partial t}\ =\ \
abla\cdot\left(k_{s,eff}\
abla T_{s}\right)-h_{sf}A_{sf}\Big{(}T_{s}-T_{f} \Big{)} \tag{3}\]  
where \(c_{s}\) is the constant pressure specific heat capacity of the solid, \(\rho_{s}\) is the density of the solid, \(k_{s,eff}\) is the equivalent thermal conductivity of the solid. Using Equation (1), the inertia resistance coefficient and viscous resistance coefficient of the corresponding porous media with different angle fins can be calculated by the contrast coefficient method.