## 4 Numerical Simulation Results on Heat Transfer Characteristics
### Analytical Heat Transfer Equation  
In heat transfer, the Nusselt number Nu is a dimensionless number describing the strength of convective heat transfer [20]. Through numerical simulation, Nu suitable for this model is analyzed in depth, and an accurate fitting formula of Nusselt number (Nu) is established, which can provide a reliable prediction for engineering applications and theoretical research. The Prandtl number (Pr) is a dimensionless number in fluid mechanics and heat transfer, which is used to describe the important parameters of momentum diffusivity and thermal diffusivity of different concentrations of ethylene glycol. According to different flow rates, multiple sets of Reynolds number (Re) are calculated, and then the  
Figure 16: Temperature distribution of 30% concentration ethylene glycol solution. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & \multicolumn{4}{c}{**Heat Flow Density (W/cm\({}^{2}\))**} \\ \hline numerical value & 125 & 300 & 500 & 1000 \\ Maximum temperature of the T/R module K & 312.7 & 338.7 & 358.2 & 445.9 \\ Minimum temperature of the T/R module K & 302.5 & 316.3 & 331.8 & 370.2 \\ Maximum temperature of cold plate K & 304.6 & 320.9 & 339.5 & 386.6 \\ Minimum temperature of cold plate K & 293.8 & 295.3 & 296.3 & 299.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Temperature of T/R components at different heat flow densities. Nusselt number (Nu) is calculated by the Nusselt number definition (Nu). The relationship between the Nusselt number (Nu), Reynolds number (Re), and Prandtl number (Pr) is fitted in the origin. Table 7 shows the different flow velocities randomly selected in the model and the Reynolds number (Nu) and Nusselt number (Nu) at this time [21]. The formulas used in the calculation are the Reynolds number definition formula, the Nusselt number definition formula, and the Prandtl number definition formula, which are Equation (11), Equation (12), and Equation (13), respectively. The Reynolds number is defined as follows:  
\[\text{Re}=\frac{\rho\text{u}L}{\mu} \tag{11}\]  
In the equation,  
\(\rho\)--Density, kg/m\({}^{3}\);  
u--Flow rate, m/s. The average velocity in the cold plate pipe is used here;  
\(L\)--Characteristic length, m. The characteristic length is calculated by the size of the cooling pipe;  
\(\mu\)--Kinematic viscosity, Pa-s. The definition of the Nusselt number is shown in Equation (12):  
\[N\text{u}=\frac{\text{h}L}{\text{k}} \tag{12}\]  
In the equation,  
\(h\)--Convective heat transfer coefficient, W/(m\({}^{2}\)-K). The convective heat transfer coefficient is calculated by the heat flux and its average temperature difference, which is the average convective coefficient of the cold plate; \begin{table}
\begin{tabular}{c c c} \hline \hline
**Flow Velocity** & **Re** & **Nu** \\
**kg/s** & & \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 7: Reynolds number and Nusselt number under different flow rates. \(L\)--Characteristic Length, m. The characteristic length is calculated by the size of the cooling pipe;  
\(k\)--Thermal Conductivity, W/(m\(\cdot\)K). The definition formula of the Prandtl number is shown in Equation (13):  
\[\mathrm{Pr}=\frac{\gamma}{\alpha} \tag{13}\]  
In the equation,  
\(\gamma\)--Kinematic viscosity, m\({}^{2}\)/s;  
\(\alpha\)--Thermal diffusivity, m\({}^{2}\)/s. The formula of the convective heat transfer coefficient is shown in Equation (14):  
\[\mathrm{h}=\frac{\mathrm{q}}{T_{\mathrm{s}}-T_{\mathrm{f}}} \tag{14}\]  
In the equation,  
\(q\) --Heat flux density, W/m\({}^{2}\). Here is the heat flux density set in the numerical simulation;  
\(T_{\mathrm{s}}\)--Solid surface temperature, K;  
\(T_{\mathrm{f}}\)--Mainstream temperature of the fluid, K. Here is the average temperature of the coolant. The fitted Nusselt number formula is shown in Equation (15):  
\[N\mathrm{u}=139.0643\mathrm{Re}^{-0.1589}\bullet\mathrm{Pr}^{-0.6061} \tag{15}\]  
The condition of this formula is \(3000<Re<9500\), \(15<Pr<50\).