In eigenfunction based solutions for heat conduction problems, the inhomogeneous boundary
conditions (BCs) and the eigenfunctions (corresponding to homogeneous BCs) are incompatible
with each other at the boundaries. Therefore, homogenization of BCs is essential to
avoid the abovementioned incompatibility. However, homogenization is usually quite
tedious especially for time dependent BCs. The process becomes even more difficult
for multi-dimensional multi-layer problems. These difficulties come from the fact
that, in general, an auxiliary function (quasi-static part of temperature) needs to
be subtracted from the temperature of each layer of the composite. This treatment
yields a set of conditions on auxiliary functions. In this work it is demonstrated
that these auxiliary functions are not unique and subsequently, two formulations of
auxiliary function are proposed here for cylindrical polar (r, theta) coordinates.
The second formulation leads to relatively simple computational procedure. The computational
ease of the proposed methodology arises from the fact that auxiliary functions are
defined only in the innermost and the outermost layers of the composite in this approach.
The solutions of a few illustrative examples are obtained without homogenization as
well as with homogenization (using two distinct approaches). The results without homogenization
of BCs exhibits large mismatch near the boundaries, while such mismatch goes way in
case of homogenized BCs. Moreover, the novel (second) approach proposed here for homogenization
is easy to implement and efficient compared to the first formulation approach. (C)
2018 Published by Elsevier Ltd.
