## 2 Methodology
### The Second Law Efficiency Analysis of the Minichannel Heat Sink Flow  
The second law efficiency evaluates the quality of the thermodynamic process performed by a heat sink. It does so by considering the irreversibility of the cooling process. The primary sources of irreversibility are the temperature drop in the heat that is transported and the friction losses in the working fluid. One method to determine the magnitudes of these irreversibilities is evaluating the steady flow exergy balance [55], using the inlet flow temperature and the outflow pressure of the working fluid as the reference dead state. Figure 3 shows the control-volume two-dimensional schematic of a heat sink with a steady through flow and a constant heat flux from the bottom. The exergy destruction rate \((\dot{E}_{xd})\) from the exergy balance through the dashed line control volume is [7]:  
\[\dot{E}_{xd}=-\iint_{base}\biggl{(}1-\frac{T_{r}}{T}\biggr{)}\,\dot{q}_{i}n_{ i}\,dA-\iint_{intlet}\rho_{nf}\psi u_{i}n_{i}\,dA-\iint_{outlet}\rho_{nf}\psi u _{i}n_{i}\,dA \tag{28}\]  
where \(T_{r}\) is the dead state reference temperature, which is equal to the inlet temperature \(T_{in}=303.15\) K, \(T\) is the temperature at the control volume surface, \(\dot{q}_{i}\) is the heat flux vector, which is defined positive outwards from the control volume, \(n_{i}\) is the control volume boundary normal unit vector, which is positive outwards, \(u_{i}\) is the working fluid velocity, and \(\psi\) is the flow exergy. The latter is defined as  
\[\psi=(h_{0}-h_{0r})-T_{r}(s-s_{r}) \tag{29}\]  
where \(h_{0}\), \(h_{0r}\), \(s\), and \(s_{r}\) are the specific stagnation enthalpy, the specific stagnation enthalpy at \(T_{r}\), the specific entropy, and the specific entropy at \(T_{r}\), respectively. The contribution to the flow exergy owing to gravity is disregarded, as the inlet and the outlet of the minichannel are horizontally aligned [8]. The constant property incompressible flow assumption in the single-phase and two-phase mixture models enables restating Equation (29) as  
\[\psi=Cp_{nf}\,\left(T_{0}+\frac{{u_{i}}^{2}}{2 Cp_{nf}}-T_{0r}-\frac{{u_{i}}^ {2}}{2 Cp_{nf}}-T_{r}\ln\left(\frac{T}{T_{r}}\right)\right) \tag{30}\]  
where \(Cp_{nf}\) is evaluated from Equation (17) using \(\phi=1\%\) for all simulations.  
Figure 3: The control-volume two-dimensional schematic of a minichannel heat sink. The heat flux direction is indicated by the up-pointing arrow.  
The rate of exergy expended \((\dot{E}_{x})\) through the minichannel cooling process is the sum of the total reversible work rate potential of the heat supplied through the base, as identified in [PERSON], et al. [35], plus the reversible work rate potential of the coolant supplied to the inlet, as identified in [PERSON], et al. [7]. This gives  
\[\dot{E}_{x}=-\iint_{base}\left(1-\frac{T_{r}}{T}\right)\,\dot{q}_{i}n_{i}\,dA- \iint_{intlet}\rho_{nf}\psi u_{i}n_{i}\,dA \tag{31}\]  
In Equations (28) and (31), \(\rho_{nf}\) is the density of the working fluid evaluated by Equation (16) with \(\phi=1\%\).  
By combining the exergy destruction rate and the rate of exergy expended, the second law efficiency of the minichannel heat sink is determined as [8].  
\[\eta_{II}=\left(1-\,\frac{\dot{E}_{xd}}{\dot{E}_{x}}\right) \tag{32}\]