## 2 Materials and Methods
### Analyzed Parameters  
This section presents formulas used to calculate the values of parameters characterizing the quality of liquid flow in microchannels and the quality of heat exchange between a solid and a liquid.  
The first parameter is the difference in the mean pressure at the inlet and outlet of the microchannel (pressure drop):  
\[\Delta p=\left.p\right|_{y=0}-\left.p\right|_{z=l}=p_{0}-\left.p\right|_{z=l} \tag{16}\]  
where the average outlet pressure is:  
\[\left.p\right|_{z=l}=\frac{1}{A_{f}}\iint\limits_{A_{f}}p\;dxdy\right|_{z=l} \tag{17}\]  
and \(A_{f}=a\times c\) is the microchannel cross-section. The power needed to force the flow of a given intensity is:  
\[P=p_{0}v_{0}A_{f}-\iint\limits_{A_{f}}pv\;dxdy\right|_{z=l} \tag{18}\]  
The value of the pressure drop in the microchannels is determined by the Darcy-Weisbach equation:  
\[\Delta p=f\frac{l}{D_{h}}\frac{\rho_{f}v_{0}^{2}}{2} \tag{19}\]  
where for a square section the hydraulic diameter is \(D_{h}=a\). Rearranging Equation (19) gives the Darcy friction factor \(f\):  
\[f=\frac{2a\Delta p}{\rho_{f}v_{0}^{2}l} \tag{20}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Property** & **A** & **B** & **C** & **D** & **E** \\ \hline Viscosity, Pa-s & \(-51.964\) & \(3670.6\) & \(5.7331\) & \(-5.3495\times 10^{-29}\) & 10 \\ Density, kmol/m\({}^{3}\) & \(5.459\) & \(0.30542\) & \(647.13\) & \(0.081\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: The coefficients values for properties temperature correlations [34].  
The Darcy friction coefficient is a metric of flow resistance. It offers insights that can be leveraged to optimize flow systems, thereby minimizing pressure losses and enhancing the overall efficiency of equipment.  
The heat flux carried away by the fluid results from the difference between the outgoing and supplied streams:  
\[\dot{Q}=\iint\limits_{A_{f}}\rho_{f}v_{z}c_{pf}T_{f}\;dxdy\Bigg{|}_{z=l}-\left( \rho_{f}c_{pf}\right)_{z=0}v_{0}T_{f0}A \tag{21}\]  
This heat flux is the most important parameter that describes the cooling effect of the microchannel.  
The volumetric average fluid temperature is:  
\[\overline{T}_{f}=\frac{1}{V_{f}}\iint\limits_{V_{f}}T_{f}(x,y,z)dV \tag{22}\]  
while the average temperature of the surface separating the liquid and the solid is:  
\[\overline{T}_{sl}=\frac{1}{A_{sl}}\iint\limits_{A_{sl}}T_{s}(x,y,z)dA \tag{23}\]  
The heat flux received from the cooled element, equal to the heat flux carried away by the fluid, is described by the Newton equation:  
\[\dot{Q}=\alpha A\Delta T \tag{24}\]  
where \(\alpha\) is the average heat transfer coefficient. This coefficient can be calculated by assuming that the temperature difference concerns the average values:  
\[\Delta T=\overline{T}_{sl}-\overline{T}_{f} \tag{25}\]  
and the solid-liquid interface of a simple microchannel is taken as the heat transfer surface (reference surface):  
\[A=(a+2h)l \tag{26}\]  
In this case, the average heat transfer coefficient \(\alpha\) is:  
\[\alpha=\frac{\dot{Q}}{(a+2h)l\Big{(}\overline{T}_{sl}-\overline{T}_{f}\Big{)}} \tag{27}\]  
Based on knowledge of the average heat transfer coefficient, the average value of the Nusselt number can be calculated:  
\[\text{Nu}=\frac{\alpha D_{h}}{\lambda_{f}} \tag{28}\]  
The Nusselt number describes the quality of convective heat transfer in a dimensionless way. It is used in practice in the design of all types of heat exchangers.  
To assess the quality of operation of microchannels of various shapes, working under different flow conditions and in different temperature ranges, the quality factor can be introduced, which is the ratio of the total heat transfer rate during cooling to the power needed to force the flow:  
\[\eta=\frac{\dot{Q}}{P} \tag{29}\]