Thermoelectric quadrupole of a leg to model transient state

The transient heat transfer in a thermoelectric leg is modelled. The Thomson effect is into account, whereas it is often neglected in other studies. Thanks to the expressions of the temperature and the heat flux expressed in the Laplace domain, a semi-analytical technique, the so-called quadrupole method is performed. No mesh is required. The whole matrix formulation is given for different initial conditions. It will allow, for instance, to evaluate easily the efficiency or the coefficient of performance of the thermoelement or use it as inverse method, for instance, to perform estimations.