Stationary Temperature Distribution in an Electrically Heated Conductor

Abstract

The thermodynamic theory of irreversible processes as developed by Onsager, de Groot, and Callen is used to derive in a straightforward way the partial differential equation for the stationary temperature distribution in an electrically heated, chemically inhomogeneous conductor. It is shown that the form of this differential equation given in 1900 by Diesselhorst (for homogeneous media) in terms of the electrical potential gradient $\mathbf{\nabla }\mathrm{·\kappa }\mathbf{\nabla }\mathrm{T+\sigma \tau }\mathbf{\nabla }\mathrm{T·}\mathbf{\nabla }\mathrm{\phi +\sigma (}\mathbf{\nabla }{\mathrm{\phi )}}^2\text{=0}$ , is incorrect. Diesselhorst's equation reads $\mathbf{\nabla }\phi$ in which κ is the thermal conductivity (for zero electrical current), T the temperature, σ the isothermal electrical conductivity, and τ the Thomson coefficient. The correct form of the equation, for the special case of a chemically homogeneous conductor, is $\mathbf{\nabla }\mathrm{·\kappa }\mathbf{\nabla }{\text{T+(1/σ)J}}^2\mathrm{-\tau }\mathbf{J}·\mathbf{\nabla }\text{T=0}$ , where J is the electrical current density. The correct form can be obtained from Diesselhorst's equation by substitution of the ``isothermal Ohm's law'' $\mathbf{J}\mathrm{=-\sigma }\mathbf{\nabla }\phi$ , which, however, is not valid in a nonisothermal medium. The apparent difficulty is resolved by the method of Onsager‐de Groot‐Callen, and it is shown that the correct differential equation expressed in terms of electrical potential is much more complicated than the form given by Diesselhorst.