Homogenization of time dependent boundary conditions for multi-layer heat conduction problem in cylindrical polar coordinates

Abstract

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, θ) 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.