*2.4. Governing Equations*

The thermoelectric module works on the conjugate physics of thermal and electrical effects; hence, the coupled equations related to heat transfer (temperature), and the electric field has to be solved in order to analyze the thermoelectrical characteristics of the thermoelectric modules [4,29].

The thermoelectric coupling equations are presented as:

$$
\nabla \cdot \left( p \cdot \overrightarrow{J} \right) - \nabla \cdot \left( k \cdot \nabla T \right) = \overrightarrow{J} \cdot \overrightarrow{E} \tag{1}
$$

$$
\nabla \cdot (\sigma \cdot \mathbf{a} \cdot \nabla T) + \nabla \cdot (\sigma \cdot \nabla \varpi) = 0 \tag{2}
$$

The term → *J* . → *E* in Equation (1) represents volumetric joule heating.

For all configurations of the thermoelectric module, the thermoelectric coupling equations were solved in the thermal electric solver of ANSYS 19.1 software using the Galerkin finite element method and the corresponding boundary conditions in order to predict their thermoelectric behavior. The dimensionless displacement-strain relation needed to be solved in order to deal with the coupled thermal stress effect [7,30]:

$$
\overline{\varepsilon}\_{\text{xx}} = \frac{\overline{\partial \overline{u}}}{\overline{\partial \overline{x}}} , \qquad \qquad \overline{\varepsilon}\_{yy} = \frac{\overline{\partial \overline{v}}}{\overline{\partial \overline{y}}} , \qquad \qquad \overline{\varepsilon}\_{zz} = \frac{\overline{\partial \overline{w}}}{\overline{\partial \overline{z}}} \tag{3}
$$

$$
\overline{\varepsilon}\_{xy} = 0.5 \left( \frac{\partial \overline{u}}{\partial \overline{y}} + \frac{\partial \overline{v}}{\partial \overline{x}} \right) \qquad \overline{\varepsilon}\_{yz} = 0.5 \left( \frac{\partial \overline{w}}{\partial \overline{y}} + \frac{\partial \overline{v}}{\partial \overline{z}} \right) \qquad \overline{\varepsilon}\_{zx} = 0.5 \left( \frac{\partial \overline{w}}{\partial \overline{x}} + \frac{\partial \overline{u}}{\partial \overline{z}} \right) \tag{4}
$$

*Symmetry* **2020**, *12*, 786

The exact dimensionless relationship between the stress and the strain is presented in terms of the non-dimensional Jacobian matrix, which is deducted from Equations (3) and (4) using Newton's method [7,30].

$$
\begin{bmatrix}
\overline{\sigma}\_{xx} \\
\overline{\sigma}\_{yy} \\
\overline{\sigma}\_{zz} \\
\overline{\sigma}\_{yz} \\
\overline{\sigma}\_{xy}
\end{bmatrix} = \frac{E}{(1+v)(1-2v)} \times \begin{bmatrix}
1-v & v & v & 0 & 0 & 0 \\
v & 1-v & v & 0 & 0 & 0 \\
v & v & 1-v & 0 & 0 & 0 \\
0 & 0 & 0 & 1-2v & 0 & 0 \\
0 & 0 & 0 & 0 & 1-2v & 0 \\
0 & 0 & 0 & 0 & 0 & 1-2v
\end{bmatrix} \begin{bmatrix}
\overline{\varepsilon}\_{xx} \\
\overline{\varepsilon}\_{yy} \\
\overline{\varepsilon}\_{zz} \\
\overline{\varepsilon}\_{yz} \\
\overline{\varepsilon}\_{zx} \\
0 \\
0
\end{bmatrix} - \begin{bmatrix}
1 \\
1 \\
1 \\
0 \\
1 - 2v
\end{bmatrix} \tag{5}
$$

The non-symmetrical Jacobian matrix was solved in the static structure solver of ANSYS 19.1 software for all configurations of the thermoelectric module using the corresponding boundary conditions in order to predict the stress and strain generated within them.
