COMPARATIVE STUDY OF DIFFERENT EIGENFUNCTION BASED APPROACHES FOR 1D MULTILAYER HEAT CONDUCTION PROBLEM WITH TIME DEPENDENT BOUNDARY CONDITIONS
The separation of variables (SOV) is a widely used method to obtain analytical solution of multilayer heat conduction problem with homogenous BCs and no sources. This method is also applicable for problems with time independent BCs and sources after a small modification in the solution procedure. But the problem with homogeneous BCs and time dependent source cannot be solved by SOV. The similar problem with time dependent BCs (either homogeneous or inhomogeneous) in finite domain can be solved analytically mainly by two eigenfunction based approaches finite integral transform (FIT) and eigenfunction orthogonal eigenfunction expansion method (OEEM). However, the problems with inhomogeneous BCs cannot be solved in a straightforward manner. The FIT is relatively quiet straightforward to implement and it is shown in the present work this approach leads to inconsistent solutions at the boundaries. To avoid this discrepancy, OEEM approach is used to separate out an auxiliary function from the temperature in order to homogenize the BCs. This leads to a modified source term (either original governing equation has source or not). In this paper, it is shown that auxiliary functions have to satisfy a set of conditions and are not uniquely defined. The auxiliary functions have two available formulations. In the first formulation it is considered that these functions are defined for each layer of the composite. To evaluate these auxiliary functions, it is proposed that auxiliary function for each layer goes under "pseudo steady state" condition (Laplacian of a time and space varying function is identically zero). This proposal helps to reduce one term from each modified source term. In the second formulation, it is considered that these functions are zero in all layers except the two extreme layers (innermost and outermost). It is obvious that these different approaches for multilayer heat conduction problem with time dependent BCs lead to a correct and unique solution. However, there is quite a significant difference in the complexity and computational cost of evaluating each term of the series solution for approaches discussed in this work. Moreover, number of terms required for a reasonably converge solution also vary significantly with approach used. A detailed comparison of different approaches in regard to complexity of the implementation and computational cost is also made.