Influence of the linear distribution of the charge-carrier density along the arm of a thermocouple on the regime of its operation

Abstract

The problem on the temperature field in an inhomogeneous one-dimensional arm of a thermocouple has been solved by numerical methods with regard for both the distributed Peltier effect and the Thomson effect. It was assumed that the charge carriers are nondegenerate. The temperature field of the thermocouple was optimized for the regimes of maximum temperature drop and maximum refrigerating capacity. An optimum range of change in the charge-carrier density gradient has been determined.The widespread use of thermoelectric temperature transducers in practice poses the problem of increasing their efficiency. This can be done first of all by increasing the thermoelectric Q of these devices. The values of this parameter attained to date in practice are far from the theoretically predicted limit [1]. As is known, a thermocouple with an arm with properties changing along its length has a higher thermoelectric Q than a thermocouple with a homogeneous arm [2]. An investigation of thermocouples with arms with properties changing along their length, as a limiting case of a complex thermocouple, [3] has shown that the thermoelectric Q of a thermocouple increases with increase in its conductivity and decrease in the thermal e.m.f. from the hot to the cold end of the arm. A distinctly different approach to the solution of this problem has been proposed in [4]. This approach is based on solution of the boundary problem on steady-state heat conduction. In this case, the optimum distribution of an impurity along the length of an arm is determined by solving the variational problem with the use of the Pontryagin maximum principle. However, in the case where this formalism is used, the problem should be substantially simplified. For example, in [4] it was assumed that the thermal e.m.f., the heat conduction, and the electrical conduction depend only slightly on the temperature and the Thomson effect can be disregarded, which significantly diminishes the usefulness of the results obtained. Moreover, only one regime was considered in this work -the regime of maximum temperature drop, while the regime of maximum refrigerating capacity is the most interesting from the practical standpoint. In the present work, we made an effort to solve the boundary problem of [4] with allowance for the temperature dependence of the kinetic coefficients and for the Thomson effect. Since the problem is nonlinear in this case, we numerically solved the boundary problem and numerically optimized the solution obtained. The regimes of maximum temperature drop and maximum refrigerating capacity were considered.To determine the role of the Thomson effect, we solved both problems with and without regard for this effect. The last-mentioned case was considered earlier in [5]. The temperature field of the adiabatically isolated inhomogeneous one-dimensional arm of an unloaded thermocouple is defined, without regard for the Thomson effect, by the steady-state heat-conduction equation [4] d dx