## 2 Formulation of the Problem
### Entropy Generation Analysis  
The entropy equations can be written in the following form:  
\[s_{f}^{\prime\prime\prime}=\frac{k_{lnf}}{T_{0}^{2}}\big{(}\
abla T_{fp}\big{)} ^{2}+\frac{\mu_{lnf}}{T_{0}\,K(y)}\left(V\cdot V\right)+\frac{\mu_{lnf}}{T_{0} }\big{(}\tau_{ij}:\
abla V\big{)}+\frac{\sigma_{lnf}}{T_{0}}\big{|}V\wedge B \big{|}^{2}+\frac{h\big{(}T_{p}-T_{fp}\big{)}^{2}}{T_{0}^{2}} \tag{21}\]  
\[s_{p}^{{}^{\prime\prime\prime}}=\frac{k_{p}}{T_{0}^{2}}(\
abla T_{p})^{2}+ \frac{h\big{(}T_{p}-T_{fp}\big{)}^{2}}{T_{0}^{2}} \tag{22}\]  
where \(T_{0}=\frac{T_{b}+T_{c}}{2}\), using the dimensionless variables and the characteristics of entropy  
\[S^{\prime\prime\prime}0=\frac{k_{f}(\Delta T)^{2}}{L^{2}T_{0}^{2}}),\ S^{ \prime\prime\prime}0_{p}=\frac{k_{p}(\Delta T)^{2}}{L^{2}T_{0}^{2}} \tag{23}\]In the above equation, \(\Theta=\frac{\mu_{f}T_{0}}{K_{0}}\Big{(}\frac{\alpha_{f}}{H\Delta T}\Big{)}^{2}\) is the ratio of the irreversibility distribution. In addition, the local and average Bejan number (Be) is expressed as:  
Using the dimensionless quantities, the entropy generation is given as:  
\[S^{\prime\prime\prime}{}_{f}=\frac{k_{lnf}}{k_{f}}\Bigg{[}\bigg{(}\frac{ \partial\theta_{fp}}{\partial X}\bigg{)}^{2}+\bigg{(}\frac{\partial\theta_{fp} }{\partial Y}\bigg{)}^{2}\Bigg{]}+\Theta\frac{\mu_{lnf}}{\mu_{f}}\Bigg{[}\frac{ 1}{Da(Y)}\Big{(}U_{p}^{2}+V_{p}^{2}\Big{)}+2\bigg{(}\frac{\partial U_{p}}{ \partial X}\bigg{)}^{2}+2\bigg{(}\frac{\partial V_{p}}{\partial Y}\bigg{)}^{2} +\bigg{(}\frac{\partial U_{p}}{\partial Y}+\frac{\partial V_{p}}{\partial X} \bigg{)}^{2}\Bigg{]} \tag{24}\]  
\[S^{\prime\prime\prime}{}_{p}=\Bigg{[}\bigg{(}\frac{\partial\theta_{p}}{ \partial X}\bigg{)}^{2}+\bigg{(}\frac{\partial\theta_{p}}{\partial Y}\bigg{)}^ {2}\Bigg{]}+Kr(1-\varepsilon)H\Big{(}\theta_{p}-\theta_{fp}\Big{)}^{2} \tag{25}\]  
The total dimensionless entropy generation is the sum of these components ([PERSON] et al. [46]), and it is written as:  
\[S^{\prime\prime\prime}{}_{total}=S^{\prime\prime\prime}{}_{f}+S^{\prime\prime \prime}{}_{p} \tag{26}\]  
The entropy generation due to the entire domain can be obtained by integrating the previous component over the entire domain as:  
\[S_{T}=\int\limits_{\Omega}S^{\prime\prime\prime}{}_{T}\,d\Omega \tag{27}\]  
\[S_{f}=\int\limits_{\Omega}S^{\prime\prime\prime}{}_{f}\,d\Omega \tag{28}\]  
\[S_{T_{S}}=\int\limits_{\Omega}S^{\prime\prime\prime}{}_{p}\,d\Omega \tag{29}\]  
\[S_{total}\ =\int\limits_{\Omega}S^{\prime\prime\prime}{}_{total}\,d\Omega \tag{30}\]  
The local Be is the ratio between the local entropy generation due to the transfer of heat (sum of the entropy generation due to the heat transfer for the fluid phase and for the solid phase) and the entropy generation due to the fluid friction; thus, it can be expressed as:  
\[Be=\frac{k_{lnf}}{k_{f}}\Bigg{[}\bigg{(}\frac{\partial\theta_{fp}}{\partial X }\bigg{)}^{2}+\bigg{(}\frac{\partial\theta_{fp}}{\partial Y}\bigg{)}^{2} \Bigg{]}+\bigg{[}\bigg{(}\frac{\partial\theta_{p}}{\partial X}\bigg{)}^{2}+ \bigg{(}\frac{\partial\theta_{p}}{\partial Y}\bigg{)}^{2}\bigg{]}+H\Big{(} \theta_{p}-\theta_{fp}\Big{)}^{2}(\varepsilon+Kr(1-\varepsilon))}{S_{total}} \tag{31}\]  
Here, it should be mentioned that the ratio of the irreversibility distribution \(\Theta\) is fixed at \(10^{-4}\) for all the computations and this value is selected according to [PERSON] et al. [47].