Analytical solution of time-dependent multilayer heat conduction problems for nuclear applications

Analytical solutions for one-dimensional time dependent multilayer heat conduction problems were developed several decades ago. Mathematical theory for such problems in more than one dimensions was also developed during that time. Several of these methods were based on separation of variable and finite integral transform. However, the application of these methods was hindered by the fact that the eigenvalue problems, which are essential for this methodology are difficult to solve. Moreover, in two and three dimensional Cartesian coordinates these eigenvalues were imaginary rendering their solutions even more difficult. It has been recently shown that similar problems in two dimensional cylindrical and spherical coordinates do not have imaginary eigenvalues. It is also helpful that the softwares which are capable of analytical manipulations are now ubiquitous. This paper discusses the methodology as well as possible application in nuclear reactors of analytical solutions of two-dimensional multilayer heat conduction in spherical and cylindrical coordinates.

1. INTRODUCTION

Multilayer components are of significant interest due to the added advantage of combining different thermo-physical properties. These components find a wide range of applications in various automotive, space, chemical, civil and nuclear industries. Time dependent temperature distribution in such components, with the presence of sources (with all three types of boundary conditions) can be obtained analytically or numerically. Though not always available, exact analytical solutions are desirable since: (1) better insight can be gained through the mathematical form of an analytical solution compared to a discrete numerical solution; and (2) these analytical solutions can be used as benchmark or reference results to verify numerical algorithms and codes. There are several approaches for solving such problems [1]–[3]. In this paper, only solutions obtained with separation of variables are discussed. Series Solutions of one-dimensional problems, using separation of variables, were obtained several decades ago [4]. However, since computation of eigenvalues needed for the solution, is difficult, these solution were of little use for obtaining temperature distribution. Only recently with advances in computational capabilities such distributions have been obtained. For the multidimensional cases a solution and generalized coordinates has been given by Yener and Ozisik (1974) [5]. In theory this solution is applicable to all types of multiregion (including multilayer) problems. However, practical implementation is not trivially straightforward. It has been found that in multilayer multidimensional Cartesian coordinates eigenvalues of the problem may be imaginary making the solution of the characteristic equation very difficult, if not impossible. In contrast it has been recently shown that in cylindrical and 2D and 3D spherical coordinates similar problems have real eigenvalues. In the conventional nuclear reactors, heat conduction in the fuel rods is through several layers and is also asymmetric. Moreover, in pebble bed reactor, which is a new design proposed for the reactors, similar multilayer heat conduction problem exists in spherical coordinates. In this paper recently developed analytical solution in multilayer cylindrical and spherical coordinates and its applicability to the nuclear engineering problems is discussed. Development and application of such ID problems is also discussed.