## 4 Results and Discussion
### Semi-Empirical Correlation  
Figure 16 illustrates the energy-balance schematic diagram for the air flow heated via the heat sink in this issue. The inlet air flow has the uniform temperature (\(T_{i}\)) and velocity (\(u_{i}\)). According to Figure 10, the average velocity of air flow into windward side (\(u_{w,k}\), _i.e._, the average velocity of flow-in air flow) of each heat sink decreases along the downward stream. In other words, a part of flow-in air continuously results in leakage air at each downward heat sink to join the bypass flow. Therefore, the energy balance for the flow-in air heated via each heat sink can be expressed as follows:  
\[\rho\cdot u_{i}\cdot\left(\frac{U_{w,k}+U_{w,k+1}}{2}\right)\cdot H\cdot L\cdot C _{p}\cdot\left(T_{o,k}-T_{i,k}\right)=Q_{w,k}^{\ast} \tag{15}\]  
\[T_{b,k}=\frac{T_{o,k}+T_{i,k}}{2}\,,\;U_{w,k}=\frac{u_{w,k}}{u_{i}} \tag{16}\]  
where \(T_{i,k}\) and \(T_{o,k}\) are the average air temperatures at the inlet and outlet faces of the \(k\)th heat sink, respectively; \(T_{b,k}\) is the bulk mean temperature of the air through the \(k\)th heat sink; and \(Q_{w,k}^{\ast}\) is the heat transferred to the air flow through the \(k\)th heat sink. Because that the leakage-flow effect, the mean velocity of air flow through the \(k\)th heat sink is expressed as \(u(U_{w,k}+U_{w,k+1})/2\). Besides, the \(Q_{w,k}^{\ast}\) is less than the total heat (\(Q_{w,k}\)) dissipated from the \(k\)th heat sink since that the heat carried by the leakage air does not join to result in the \(T_{o,k}\).  
Equations (17) and (18) are the definitions of the average Nusselt number (\(Nu_{i,k}\) and \(Nu_{b,k}\)) of the \(k\)th heat sink separately based on \(T_{i,1}\) (_i.e._, \(T_{i}\)) and \(T_{b,k}\). Substitute Equations (17) and (18) into Equation (15), the relationship between \(Nu_{i,k}\) and \(Nu_{b,k}\) can be written as Equation (19).  
\[Nu_{i,k}=\frac{Q_{w,k}\cdot H}{(T_{w,k}-T_{i,1})\cdot k_{f}\cdot L^{2}} \tag{17}\]  
\[Nu_{b,k}=\frac{Q_{w,k}\cdot H}{(T_{w,k}-T_{b,k})\cdot k_{f}\cdot L^{2}} \tag{18}\]  
Figure 16: Energy-balance schematic diagram for the air flow heated via the heat sink.  
\[{Nu_{i,k}}^{-1}={Nu_{b,k}}^{-1}+\sum_{1}^{k}\frac{{Q_{w,k}}^{*}}{Q_{w,k}}\cdot \frac{L}{2\,\text{Re}(\frac{U_{w,k}-U_{w,k+1}}{2})\cdot\text{Pr}\cdot H} \tag{19}\]  
Finally, a semi-empirical correlation (see Equation (20)) can be completed to be a closed form by combined with Equations (7) and (8) suggested by [PERSON] _et al._[24], the present numerical data of \(U_{w,k}\) plotted in Figure 10, and the assumption of \({Q_{w,k}}^{*}/Q_{w,k}=U_{w,k+1}/U_{w,k}\). For the present numerical simulation, \(C_{1}\) and \(C_{2}\) are 0.363 and 0.542, respectively (see Equation (7)).  
\[\begin{split}{Nu_{b,k}}=&\Bigg{(}\frac{h_{\beta} \cdot d}{k_{f}}\Bigg{)}\cdot\Bigg{(}\frac{H}{d}\Bigg{)}\cdot\Bigg{(}\frac{a_{ \beta}\cdot L^{2}\cdot H}{L^{2}}\Bigg{)}\\ =&\Bigg{[}\varepsilon\cdot C_{1}\Bigg{(}\text{Re} \cdot\frac{U_{w,k}+U_{w,k+1}}{2}\cdot\frac{d}{H}\cdot\frac{S_{T}}{S_{T}-d} \Bigg{)}^{C2}\Bigg{]}\cdot\Bigg{(}\frac{H}{d}\Bigg{)}\cdot\Big{(}a_{\beta} \cdot H\Big{)}\\ &\frac{{Q_{w,k}}^{*}}{Q_{w,k}}=\frac{U_{w,k+1}}{U_{w,k}}\end{split} \tag{20}\]  
The comparison results of the present numerical Nusselt number with the semi-empirical predictions and the experimental data for Sample 3 and Sample 5 heat-sink arrays with \(Re=500\) are plotted in Figure 17. The predicted \(Nu\) values generally agree with those numerical data. In addition, the numerical data and the predictions of Equation (20) also agree with the typical experimental results, demonstrating the correctness of the present study.