# The Thermal Analysis and Heat Dissipation Structure Optimization of a Propeller Driver System
## 2 Thermal Analysis of Propeller Driver  
### Numerical Model Description  
The propeller driver device and its simulation model are shown in Figure 1. It is made up of a base, a rectangular heat sink, two capacitors, and four driver PCB layers. The rectangular heat sink has a total size of 80 mm \(\times\) 27 mm \(\times\) 150 mm. The cylindrical capacitors have a height of 35 mm and a radius of 9 mm. The dimensions and power of each driver board are shown in Table 1. The total power of this driver is 10.6 W. The power of the 4 th PCB layer, including U4, is 10 W. That means that most of the heat generated in this work comes from the 4 th PCB layer. Heat conduction within a solid and heat convection between layers and surroundings can be used to categorize the heat transfer manners of multi-layer plate [1]. Since there is little space between the layers, heat is primarily transferred by heat conduction within the driver. The conduction processes can be described with the Fourier heat conduction law. For the steady-state heat transfer with an internal heat source, the heat conduction differential equation for a three-dimensional system can be expressed as in Equation (1) [30]. \[\frac{\partial}{\partial x}(\lambda\frac{\partial t}{\partial x})+\frac{ \partial}{\partial y}(\lambda\frac{\partial t}{\partial y})+\frac{\partial}{ \partial z}(\lambda\frac{\partial t}{\partial z})+\dot{\Phi}=0 \tag{1}\]  
The heat convection process can be expressed with [PERSON]'s cooling law as Equation (2) [31]. \[Q=hA\Delta t \tag{2}\]  
In the formula, \(\Delta t\) is the temperature difference between the solid wall and the fluid. In the thermal analysis software, three major conservation equations are mainly used to calculate and solve the model, namely the mass conservation equation, momentum conservation equation, and energy conservation equation. The governing equations of fluid flow are shown in Equation (3) [31]. \[\frac{\partial p}{\partial\tau}+divp\overset{\rightarrow}{\psi}\Phi=div\Gamma grad \Phi+s \tag{3}\]  
In Cartesian coordinates, the continuity equation is given in Equation (4) [31]. \[\frac{\partial p}{\partial\tau}+\frac{\partial(\rho u)}{\partial x}+\frac{ \partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0 \tag{4}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Layer Name**} & \multirow{2}{*}{**Size (mm)**} & **Main Heating** & \multirow{2}{*}{\
\begin{tabular}{c} **Power of the** \\ **Component** \\ \end{tabular} } & **Main Heating** & \multirow{2}{*}{
\begin{tabular}{c} **Main Heating** \\ **Component** \\ \end{tabular} } & **Main Heating** \\  & & & **Component (W)** & & **Component** \\ \hline
1 st PCB layer & 36 \(\times\) 30 \(\times\) 0.8 & U1 & 0.15 & 1 \\
2 nd PCB layer & 36 \(\times\) 30 \(\times\) 0.8 & U2 & 0.1 & 1 \\
3 rd PCB layer & 36 \(\times\) 33 \(\times\) 0.4 & U3 & 0.35 & 1 \\
4 th PCB layer & 36 \(\times\) 33 \(\times\) 0.4 & U4 & 1 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The main parameters of the driver layer. Figure 1: (a) Driver device physical drawing. (b) Driver device model. In the Cartesian coordinate system, the momentum conservation equations in each sub-direction are shown in Equations (5)-(7) [31].