Semi-analytical solution to steady-state and transient heat transfer problem with variable conductivity properties of the domain

Abstract

Introduction. Present paper studies the case of heat transfer problem where the conductivity property of the domain is not a constant value but varies depending on its temperature. In particular, a conductivity that linearly dependent onthe temperature is reviewed. A semi-analytical formulation is developed and proposed to solve steady-state and transient types of these kinds of heat transfer problems.
 Materials and methods. The domain is mathematically modeled by using finite element method where non-linearity property of the problem has been incorporated. For time-dependent (transient) case, the temporal solution has been achieved by analytical methods wherefore all necessary formulation has been derived. Non-linear equations have been solved by both Newton and Picard methods.
 Results. In order to proof check, the proposed calculations a wall with three layers has been calculated by using the proposed transient problem formulation. The derivative boundary conditions at the faces of the wall are given as the functions of time. Non-linearity has been solved by two different methods that gave the identical results. Having solved the non-linearity by these two methods, allowed to compare the efficiency of Newton method versus Picard method. Both methods reached the solution with the same number of iterations. The paper proposes the algorithms for solving the problems. The authors have used MatLab environment to implement those algorithms.
 Conclusions. Proposed formulation solves coupled heat transfer problems in multi-layered exterior walls. Multi-layered calculation ability allows to cover all non-homogeneous cases of the computation domain. The formulation solves the heat loss problems through the multi-layered walls with due and reliable accuracy.