Heat conduction in a symmetric body subjected to a current flow of symmetric input and output

Steady heat conduction in symmetrical electro-thermal problems is analyzed under the influence of a steady direct current passing through symmetrical regions of the boundary. In the present approach, solution is obtained by dividing the temperature field of the electro-thermal problem into two fields-one is related to the heat conduction problem without Joule heating and the other corresponds to a symmetric temperature field related to Joule heating induced by current supply. A Joule heating residue vector is introduced in the present analysis, which is expressed as a summation of the real heat flux and a vector representing the effect of Joule heating. It is shown that the Joule heating residue vector of symmetrical electro-thermal problem is related to the gradient of the temperature field associated with the problem without Joule heating. Moreover, the results of the present analysis show that, even if the Joule heating residue vector assumes a non-zero value, the temperature along the symmetric axis/plane remains constant when the applied current field is accompanied with antisymmetric heat fluxes on the boundary.