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.