An unconstrained mathematical description for conduction heat transfer problems with linear temperature-dependent thermal conductivity

•A systematic procedure for describing heat transfer with temperature dependent conductivity.•Existence and uniqueness of the Kirchhoff transformation always ensured.•An unconstrained mathematical representation that always ensures physical sense.•A general closed form formula for the inverse of the Kirchhoff transformation.•An associate convex and coercive functional which admits one and only one minimum.

This work presents a systematic modeling for conduction heat transfer problems in which the thermal conductivity is assumed a linear function of the temperature. In general, the mathematical descriptions arising from a linear relationship between thermal conductivity and temperature give rise to more than one solution, some of them without physical sense. In this work a convenient mathematical representation is proposed, avoiding physically inadmissible solutions.A conduction heat transfer problem in which the thermal conductivity decreases linearly with the temperature in a given interval is considered in this work. A physically equivalent alternative form, valid for any absolute temperature is proposed, giving rise to an unrestricted mathematical modeling and circumventing the need of a posterior choice for establishing the solution with physical meaning.An equivalent minimum principle for the problem is presented, showing that the extremum of a proposed functional corresponds to the solution of the problem.