## 2 Mathematical Model
### Entropy Generation Analysis
#### 2.1.1 Heat Transfer  
The heat transfer behavior of the core is summarized by the classical \(\varepsilon\)-NTU method. The parameter \(\varepsilon\) is the effectiveness of the heat exchanger. The effectiveness \(\varepsilon\) of the cross-flow heat exchanger with both fluids unmixed is shown in Equation (2) [32].  
\[\varepsilon=1-\exp\left\{\frac{NTU^{0.22}}{c_{\text{R}}}[\exp(-c_{\text{R}}NTU ^{0.78})-1]\right\} \tag{2}\]  
where \(c_{\text{R}}\) and the number of transfer units (\(NTU\)) are defined as follows:  
\[c_{\text{R}}=\frac{\min[(\dot{m}c_{\text{p}})_{\text{f}}\left(\dot{m}c_{\text{ p}}\right)_{\text{e}}]}{\max[(\dot{m}c_{\text{p}})_{\text{f}}\left(\dot{m}c_{ \text{p}}\right)_{\text{e}}]} \tag{3}\]  
\[NTU=\frac{KA_{\text{total}}}{c} \tag{4}\]  
\[c=\min[(\dot{m}c_{\text{p}})_{\text{f}^{\prime}}\left(\dot{m}c_{\text{p}}\right) _{\text{e}}] \tag{5}\]  
The parameter \(A_{\text{total}}\) is the total heat transfer area. It is twice as large as the total area of the corrugated triangular plates in the heat exchanger. The parameter \(A\) is the total area of the corrugated triangular plates, which is calculated as follows:  
\[A=A_{\text{cyc}}\cdot\frac{L_{\text{f}}}{2H\cdot\tan\frac{\theta}{2}}\cdot \frac{L_{\text{e}}}{2H\cdot\tan\frac{\theta}{2}}\cdot\frac{L_{\text{h}}}{2H+2 \delta_{\text{plate}}} \tag{6}\]  
The parameter \(A_{\text{cyc}}\) is the surface area of a unitary cell of the core, and \(L_{\text{f}}\), \(L_{\text{e}}\) and \(L_{\text{h}}\) are the fresh air channel length, exhaust air channel length and height of the heat exchanger, respectively. In addition, the \(H\), \(\theta\) and \(\delta_{\text{plate}}\) are channel height, apex angle and thickness of plate, respectively.  
The total heat transfer coefficient \(K\) in Equation (4) is expressed in Equation (7):  
\[K=\frac{1}{\frac{1}{h_{\text{f}}}+\frac{\delta_{\text{plate}}}{\lambda_{\text{ plate}}}+\frac{1}{h_{\text{e}}}} \tag{7}\]  
where \(\lambda_{\text{plate}}\) is thermal conductivity of the plate. The parameters \(h_{\text{f}}\) and \(h_{\text{e}}\) are convective heat transfer coefficients of the fresh air and exhaust air, respectively. The convective heat transfer coefficient \(h\) is defined by Equation (8):  
\[h=\frac{\lambda_{\text{a}}Nu}{D_{\text{h}}} \tag{8}\]The parameter \(Nu\) is the Nusselt number. [PERSON] [33] developed a correlation of the corrugated triangular ducts under uniform heat flux boundary conditions.  
\[Nu=0.274 Re^{0.569}Pr^{0.333} \tag{9}\]  
The Colburn factor \(j\), is a heat transfer performance index of the heat exchanger, which is defined in Equation (10).  
\[j=\frac{Nu}{Re\cdot Pr^{1/3}} \tag{10}\]  
The heat transfer rate is expressed in Equation (11):  
\[Q=\varepsilon c(T_{\rm fi}-T_{\rm ei}) \tag{11}\]  
In addition, the heat balance between the two streams is calculated as follows:  
\[\left\{\begin{array}{l}Q=\left(\dot{m}c_{\rm p}\right)_{\rm f}(T_{\rm fi}-T_ {\rm fo})\\ Q=\left(\dot{m}c_{\rm p}\right)_{\rm e}(T_{\rm eo}-T_{\rm ei})\end{array}\right. \tag{12}\]  
Then, the outlet temperatures of two sides are expressed as follows:  
\[T_{\rm fo}=T_{\rm fi}-\frac{\varepsilon c(T_{\rm fi}-T_{\rm ei})}{\left(\dot{ m}c_{\rm p}\right)_{\rm f}}\]  
\[T_{\rm eo}=T_{\rm ei}+\frac{\varepsilon c(T_{\rm fi}-T_{\rm ei})}{\left(mc_{ \rm p}\right)_{\rm e}} \tag{13}\]