## 2 Optimal Design Methodology
### Governing Equation  
The LFHX is one of the heat exchangers in charge of heat exchange in the cooling system and is a thermal management component that cools the temperature of the coolant through heat exchange with outside air to control the vehicle temperature. The LFHX must be used by selecting the type of heat exchanger that is designed to meet the heat transfer requirements within the constraints. It was intended to measure the outlet temperature utilizing the coolant, air flow rate, and inlet temperature as parameters based on the heat exchange theory, the effectiveness-NTU method, in order to correctly understand the heat exchange characteristics. When the type and dimensions of the heat exchanger are specified, the effectiveness-NTU method calculates the heat transfer and outlet temperature for a particular mass flow rate and inlet temperature. In other words, it is used when the heat transfer area is known and the outlet temperature needs to be derived, and the heat transfer performance of a specified heat exchanger needs to be determined, or whether the heat exchanger can work. Effectiveness is defined as the ratio of the actual heat transfer rate/maximum possible heat transfer rate and is based on a dimensionless coefficient called heat transfer effectiveness. The heat exchange characteristics of the LFHX can be determined by calculating the effectiveness '\(\epsilon\)' based on the flow rate and temperature of the cooling water and outside air. Figure 3 is a flowchart of the procedure for obtaining the heat transfer rate, which is one of the performance indicators of the heat exchanger, and the heat transfer rate is finally obtained using the effectiveness-NTU method. In this study,  
Figure 2: Flowchart of genetic algorithm. the temperature of the cooling water is determined according to the temperature of the outside air by using the effectiveness-NTU method based on the geometry of the louver fin, and the pressure is calculated by taking into account the pressure drop and head loss in the LFHX inner tube. The heat transfer coefficient of the cooling water (\(h_{w}\)) can be calculated as in the following equation [19; 20]:  
\[h_{w}=\frac{N_{u}\times k_{w}}{D_{h}} \tag{1}\]  
\[N_{u}=0.332\,Re_{w}{}^{0.5}Pr_{w}{}^{0.333} \tag{2}\]  
\[D_{hw}=\frac{2(T_{l}\times 2T_{t})(T_{d}-2T_{t})}{(T_{l}+T_{d}-2T_{t})} \tag{3}\]  
\[Re_{w}=\frac{\left(\dot{m}\times D_{hw}\right)}{\mu_{w}\{2(T_{l}\times 2T_{t})(T_{ d}-2T_{t})N_{T}\}} \tag{4}\]  
where \(N_{u}\), \(D_{hw}\), and \(Re_{w}\) are the Nusselt number, hydraulic diameter, and Reynolds number of the coolant, respectively. The heat transfer coefficient of air (\(h_{a}\)) is calculated as  
\[h_{a}=j\;G\;C_{pa}\;Pr^{-2/3} \tag{5}\]  
The Colburn factor (\(j\)) is given by [21]  
\[j=\left( \begin{array}{cc}Re_{air}^{-0.49}&\times\left(\frac{L_{x}}{90} \right)^{0.27}\times\left(\frac{Fp}{Lp}\right)^{-0.14}\times\left(\frac{Fl}{ Lp}\right)^{-0.29}\times\left(\frac{T_{d}}{Lp}\right)^{-0.23}\\ &\times\left(\frac{Ll}{Lp}\right)^{0.68}\times\left(\frac{Fl+T_{t}}{Lp}\right) ^{-0.28}\times\left(\frac{F}{Lp}\right)^{-0.05}\end{array} \right) \tag{6}\]  
In this case, \(L_{a}\) is defined in the range between \(15^{\circ}\) and \(30^{\circ}\), which implies that, as the angle of the louver pin increases, the heat transfer area increases and the air flows between the louvers in the low Reynolds number region; thus, the angle must be appropriately increased. However, in the high Reynolds number region, air flows between the louvers even if the angle is small; therefore, it must be chosen appropriately, considering the friction coefficient and pressure drop according to the increase in angle [22; 23; 24]. Figure 3: Calculation flowchart of performance for heat exchanger. The Prandtl number of air (\(Pr\)) is given by  
\[Pr=\frac{C_{pa}\ \mu_{a}}{k_{a}} \tag{7}\]  
where \(C_{pa}\), \(\mu_{a}\), and \(k_{a}\) are the specific heat capacity, dynamic viscosity, and thermal conductivity of air, respectively.