6. Concluding remarks

Another important aspect to be considered for the proper operation of Peltier cells is the maximum temperature inside the thermoelements since an overheating produces thermal stresses that could break the semiconductors from a mechanical point of view. For that matter, Figure 8 shows the maximum and minimum temperature inside the thermoelements for two

As observed, for <sup>I</sup> <sup>¼</sup> 3 (A) the maximum temperature becomes approximately 80<sup>∘</sup> C due to the increasing of the Joule heating that depends on the prescribed intensity. To sum up, the present numerical tool can be used for a proper design and optimization of cooling devices from both

The last example deals with the minimization of thermoelements. Currently, there exists a trend towards minimization of thermoelectric devices for two reasons: decreasing power

Figure 8. Minimum and maximum temperature inside thermoelements vs. temperature at the cold end for I ¼ 1:7 (left)

Figure 9. Voltage drop (left) and extracted heat (right) vs. the thermoelement length.

C.

operating conditions: <sup>I</sup> <sup>¼</sup> <sup>1</sup>:7 and <sup>I</sup> <sup>¼</sup> 3 (A), at Th <sup>¼</sup> <sup>50</sup><sup>∘</sup>

thermoelectric and thermomechanic interactions.

consumption and reducing the size of the devices.

5.3. Miniaturization of thermoelements

284 Bringing Thermoelectricity into Reality

and <sup>I</sup> <sup>¼</sup> 3 (A) (right), Th <sup>¼</sup> <sup>50</sup><sup>∘</sup> C.

This chapter has presented a thermodynamically consistent formulation to obtain the governing equations of thermoelectricity, from a theoretical point of view. Then, a nonlinear numerical formulation within the finite element method is developed. The nonlinearities emerge from the presence of Joule heating and the temperature-dependence of the material properties and they are numerically solved by the Newton-Raphson algorithm. Notice that this material nonlinearity directly allows the inclusion of the Thomson effect. Furthermore, the formulation is dynamic and monolithic; the former feature is solved by backward finite differences and the latter is carried out by defining a coupled assembled matrix that increases the accuracy of the formulation. Finally, several examples are reported to show the capabilities of the numerical formulation that can be used as a "virtual laboratory." In particular, hrefinements and residual convergence tests are conducted to validate the codification. Then, comparisons between analytical and numerical solutions for cooling thermoelectric cells are reported in order to highlight the advantages of the simulations against the simple onedimensional analytical solutions. In conclusion, the use of the present numerical tool could be applied for a proper design and optimization of thermoelectric devices. For instance, this tool was used to optimize the shape of a pulsed thermoelectric in [25].
