Quasi-linear electrical potentials in steady-state Joule heating

Abstract

It is shown that, when account is taken of the temperature dependence of the electrical conductivity of a medium in steady-state conditions, the electrical potential obeys a quasi-linear, second-order partial differential equation. The equation is shown to be easily solved by means of a generalized Kirchhoff transformation, giving coordinate-free solutions in terms of functions which obey Laplace's equation. Steady-state temperatures resulting from the potential are shown to be significantly influenced by the quasi-linear potential when compared with the expected form which assumes the potential to satisfy Laplace's equation.

Nonlinear effects of variable conductivities in thermistor-related problems: an explicit example

“…This equation is of the form which was shown in Young (1987) and Tenti & Chamberland (1986) to have solution obtainable in terms of a function, x, which does satisfy Laplace's equation and is related to xjr through…”

“…) From a physical point of view, on the other hand, a temperature-dependent electrical conductivity indicates in (1.3) that the potential <P will not satisfy Laplace's equation within the conductor, and it is clearly important to understand quantitatively whether or not the nonlinearity leads to effects that cannot be neglected in practical applications. Light in this direction was shed by an explicit class of solutions derived by Young (1986Young ( , 1987 and further discussed by Tenti (1986) and Tenti & Chamberland (1986) in which the interior temperature (whose elevations are above some reference 7^) was regarded as being a function of the potential within the conducting region. Physically, the basis for such a solution is simply in the assumption that the spatially varying electrical potential is the cause of the heating.…”

Existence and regularity of weak solutions for a thermoelectric model

“…This simplified model was used to analyse the Joule heating of electrically conducting media; see [30][31][32] and the references therein. See also [7,10] and the references therein for the use of this model in the thermistor problem with a current limiting device.…”

Existence and Regularity of Weak Solutions for a Thermoelectric Model

“…This simplified model was used to analyze the Joule heating of electrically conducting media, see [30,31,32] and the references therein. See also [10,7] and the references therein for the use of this model in the thermistor problem with a current limiting device.…”