Linearization of Sparrow and Koopman’s Method for Heat Transfer with Variable Conductivity

It has been shown that the heat conduction equation could be made linear by choosing the integrated conductivity, instead of the temperature, as the unknown; unfortunately, the boundary conditions for this variable, at the interface between solid and coolant, become nonlinear. We propose a linear representation of these conditions, based on the notion of an effective coolant temperature, which removes this difficulty; the accuracy of such a representation is demonstrated.