# Study on Thermoelectric Conversion and Conjugate Heat Transfer for PCBA by Finite Element Analysis
## 3 Theories and Research Methods
### Numerical Simulation Theory  
#### 3.4.1 Fluid Dynamics Theory  
The numerical simulation of the temperature field of the LED module in a tightly closed space is carried out, and the problem of heat transfer between the gas and the solid in a hermetic space must be considered. The heat transfer mode is called conjugate heat transfer. Therefore, the study uses the Fluent module in ANSYS Workbench for numerical simulation of the LED temperature field. In today's fluid analysis, most of the Finite Volume Method is used to solve the problem. The finite volume method is to divide the volume of the simulated object and generate nodes in the volume. The boundary conditions set in the numerical simulation are used to integrate the governing equations and discretize them to solve the numerical results. The governing equations used in the study include discrete equations such as continuous, momentum, and energy equations, and equations such as turbulence and radiation are added.  
Proposed by Fluent, there are five calculation models for the radiation module:  
1. Discrete Ordinates Model (DO)
2. Discrete Transfer Radiation Model (DTRM)
3. P1 Radiation Model (P1)
4. Rosseland Model
5. Surface-to-surface (S2S)  
The Discrete Transfer Radiation Model's computational model is relatively simple, and one can increase the numerical accuracy by increasing the number of rays during the calculation. However, all the faces of the model are diffuse reflections, which also causes the reflection of the radiation to be more complicated than the incidence, and also causes the CPU's resource consumption to increase relatively when solving a large number of rays. The P1 Radiation Model consumes less computational resources than DTRM when calculating thermal radiation conduction. It is relatively stable when considering large optical depths, such as high-temperature combustion models, but it magnifies radiation transmission when computing local heat sources and heat sinks. Surface-to-surface is ideal for solving problems in a closed space without the medium transmitting radiation, but not for the thermal radiation problems that the medium is involved in. The Discrete Ordinates Model (DO) mode is applicable to all wavelength range radiation problems in analysis and can analyze Surface-to-surface (S2S) problems without medium participation and can also solve the problem of heat radiation transmission involving medium, so it will take longer in the numerical solution time.  
When the radiation reaches the same order of heat convection and heat conduction or significant heat transfer occurs, the effect of heat radiation must be considered. When considering radiative heat transfer, attention must be paid to the radiation intensity, \(I(r,s)\) in the radiation transfer equation has directionality and spatiality. When radiative heat transfer, the radiation intensity has the following mechanism:  
1. Local absorption
2. Out-scattering
3. Local emission
4. In-scattering  
The radiation equation used in this study is the DO Equation module. The DO module is suitable for radiation problems in various optical depth ranges. This module can solve the radiation problem of the S2S non-medium closed region and can also solve the model with medium participation. The numerical theory equation derivation of DO Equation is performed below:  
\[\
abla.\left(I\left(\overrightarrow{r},\overrightarrow{s}\right) \overrightarrow{s}\right)\ +\ (\alpha+\sigma_{s})I\left(\overrightarrow{r}, \overrightarrow{s}\right)\ =\ an^{2}\frac{\sigma T^{4}}{\pi}\ +\ \frac{\sigma_{s}}{4\pi}\int\limits_{0}^{4x}I\left( \overrightarrow{r},\overrightarrow{s}\right)\Phi\left(\overrightarrow{s} \cdot\overrightarrow{s}\right)\mathrm{d}\Omega^{\prime} \tag{5}\]  
where  
\(\alpha\)= Gas absorption coefficient  
\(\sigma_{s}\)= Gas scattering coefficient  
\(I\)= Emission intensity  
\(\overrightarrow{r}\)= Position vector  
\(\overrightarrow{s}\)= Scattering direction vector