## 2 Materials and Methods
### Determining Convection Coefficients and Simulation Boundary Conditions  
Determining the temperature distribution in a semiconductor diode case is possible after using the FEM method. This method consists of dividing the analyzed section of the case into a finite number of smaller solids of a specific shape. The nodes are located at the vertices of the resulting solids. Temperature information is stored at the node. The temperature values at certain nodes (e.g., those located on the section of the diode case that has been split) are set as a boundary condition. Knowing the heat flow equation that the software uses, the thermal properties (e.g., thermal resistance) of the material of the case fragment being analyzed, and the distances between nodes, the temperature distribution in the case can be determined. The Solidworks 2020 SP05 software (Dassault Systemes, Velizy-Villacoublay, France) was used in the research carried out. The software allowed the necessary simulation work to be carried out through the use of FEM. The created model was subdivided into tetrahedral finite elements. The number of these elements can be specified by the user. Note that the more complex the model is, the more finite elements and nodes it will contain. Consequently, the greater the number of nodes (and finite elements), the more time is needed to calculate the values at the nodes and obtain the temperature distribution sought [37]. Figure 5: Placement of markers made with Velvet Coating 811-21 paint and numbers given to each marker. The equation that describes the transient heat conduction in a three-dimensional element can be written as follows (1) [38]:  
\[dc\frac{\partial T(x,y,z,t)}{\partial t}=(k_{x}\frac{\partial^{2}T}{\partial x^{2 }}+k_{y}\frac{\partial^{2}T}{\partial y^{2}}+k_{z}\frac{\partial^{2}T}{ \partial z^{2}}) \tag{1}\]  
where \(c\) is the specific heat; \(d\) is the density (kg/m\({}^{3}\)); \(k_{x}\), \(k_{y}\), and \(k_{z}\) (W/(m\(\cdot\)K)) are the thermal conductivities in the \(x\), \(y\), and \(z\) directions, respectively; \(T\) (K) is the temperature; and \(t\) (s) is the time. In a steady state, when the heat flow in a single direction in a homogeneous environment is considered, Equation (2) can be used instead of Equation (1) [39]:  
\[J=-k\frac{\partial T}{\partial x} \tag{2}\]  
where \(J\) is a radiative heat flux (W\(\cdot\)m\({}^{-2}\)). After separating the variables contained in Equation (2), the time constant can be obtained from the boundary conditions with Equation (3) [39]:  
\[ \begin{array}{c}for\ x=0\ \rightarrow\ T=T_{1}\\ for\ x=x_{e}\ \rightarrow\ T=T_{2}\end{array}  \tag{3}\]  
where \(x_{e}\) is the end point of the analyzed heat flow path (m), \(T_{1}\) is the temperature at the starting point of the analyzed heat flow path (K), and \(T_{2}\) is the temperature at the end point of the analyzed heat flow path (K). After separating the variables, integrating both sides and determining the time constant, Equation (2) takes the form of Equation (4) [39]:  
\[T_{1}-T_{2}=\frac{P_{c}}{S\cdot K}x_{k} \tag{4}\]  
where \(P_{c}\) is the total power, which was applied to the wall (W), and S is the area of the flat wall, which was penetrated by \(J\). Using Equation (4), the temperature difference between two points can be determined. In practice, the temperature distribution obtained in one line (1D model) might not provide enough data. For this reason, a 3D model is used to obtain more information. In each tetrahedral element (into which the semiconductor diode case is divided), the temperature field is interpolated based on the temperature at the nodes of that element and the linear shape function, Equation (5) [40]:  
\[T(x,y,z,t)=\sum_{i=1}^{4}H_{i}\cdot T_{i}(t) \tag{5}\]  
where \(T_{i}(t)\) is the nodal temperature at node \(i\), and \(H_{i}\) is the linear shape function. Determining the correct temperature distribution in the semiconductor diode case requires the determination of the radiation coefficient \(h_{r}\).