## 2 Numerical Methodology
### Model Assumptions and Governing Equations
By solving a single heat transfer equation with effective material properties for both phases, the apparent heat capacity formula provides implicit capturing of the phase transition interface. The latent heat is taken into account by modifying the heat capacity [36]. This method has been used to simulate the solid-liquid phase change processes in many studies and has been proven to be effective [37; 38; 39]. It is assumed that the phase transition occurs in a temperature interval between \(T_{lower}\) (\(T_{lower}=T_{m}-\Delta T2\)) and \(T_{upper}\) (\(T_{upper}=T_{m}+\Delta T2\)). In the temperature interval, the phase state is expressed by a smooth function, \(\theta\), which represents the liquid fraction and is equivalent to 0 before \(T_{lower}\) and to 1 after \(T_{upper}\). The expressions of the density (\(\rho\)) and the specific enthalpy (\(H^{th}\)) are as follows. \[\rho=(1-\theta)\rho_{1}+\theta\rho_{2} \tag{4}\]  
\[H^{th}=\frac{1}{\rho}\Big{(}(1-\theta)\rho_{1}H_{1}^{th}+\theta\rho_{2}H_{2}^{ th}\Big{)} \tag{5}\]  
where, subscripts 1 and 2 indicate the material phase before or after transition, respectively. Differentiating with respect to temperature and after some equation transformations, the specific heat capacity is as follows:  
\[C_{p}=\frac{\partial H^{th}}{\partial T}=\frac{1}{\rho}\big{(}(1-\theta)\rho_{ 1}C_{p,1}+\theta\rho_{2}C_{p,2}\big{)}+\Big{(}H_{2}^{th}-H_{1}^{th}\Big{)} \frac{d\alpha_{m}}{dT}=C_{eq}+C_{L} \tag{6}\]  
where, \(C_{eq}\) represents the equivalent heat capacity and \(C_{L}\) represents the distribution of latent heat (\(L\)), \(\alpha_{m}\) represents the mass fraction and is defined by  
\[\alpha_{m}=\frac{1}{2}\frac{\theta\rho_{2}-(1-\theta)\rho_{1}}{(1-\theta)\rho_ {1}+\theta\rho_{2}} \tag{7}\]The \(C_{L}\) is approximated by  
\[C_{L}(T)=L\frac{d\alpha_{m}}{dT} \tag{8}\]  
The total heat released per unit volume during the phase change coincides with \(L\) as follows. \[\int_{T_{lower}}^{T_{upper}}C_{L}(T)dT=L\int_{T_{lower}}^{T_{upper}}\frac{d \alpha_{m}}{dT}dT=L \tag{9}\]  
Therefore, the apparent heat capacity (\(C_{p}\)) applied in the heat equation is expressed by  
Figure 1:0  
The effective thermal conductivity and the effective density are as follows. Figure 1:1  
Figure 1:2  
The metallic PCM used in this work was a 49 Bi/21 In/18 Pb/12 Sn alloy whose mean temperature of the phase change (\(T_{m}\)) was 58 \({}^{\circ}\)C, and its thermophysical properties are shown in Figure 1:3  
According to the symmetry assumption and the selection of calculation region, the geometric central planes were defined as symmetrical boundaries. The coolant inlet temperature was 35 \({}^{\circ}\)C, and the inlet volumetric flow rate (\(Q_{V}\)) of the single jet was 0.32 L/min. The outlet gauge pressure was 0. The boundary heat source was set for the area where the 3 \(\times\) 3.5 mm\({}^{2}\) chip was located, and the heat flux (\(q^{\prime}\)) was 150 W/cm\({}^{2}\) for the simulation cases of constant heat flux. \(q^{\prime\prime}\) was set to be 1.2 W/cm\({}^{2}\) for the rest area around the chip. For the heat transfer interface, the other boundaries were defined as adiabatic boundaries, and the initial temperature of all areas was 35 \({}^{\circ}\)C.