Analytical Solution to Transient Asymmetric Heat Conduction in a Multilayer Annulus

Abstract

In this paper, we present an analytical double-series solution for the time-dependent asymmetric heat conduction in a multilayer annulus. In general, analytical solutions in multidimensional Cartesian or cylindrical ͑r,z͒ coordinates suffer from existence of imaginary eigenvalues and thus may lead to numerical difficulties in computing analytical solution. In contrast, the proposed analytical solution in polar coordinates (2D cylindrical) is "free" from such imaginary eigenvalues. Real eigenvalues are obtained by virtue of precluded explicit dependence of transverse (radial) eigenvalues on those in the other direction.

An exact analytical solution for two-dimensional, unsteady, multilayer heat conduction in spherical coordinates

“…The same is also true for multilayer heat conduction problems in cylindrical polar coordinate system [21]. Consequently, absence of explicit dependence leads to a complete solution which does not have imaginary radial eigenvalues, and thus k imp are real [22][23].…”

“…Moreover, the following relationship between k imp and k 1mp must hold for all values of t [6,11,[21][22][23] for T i ðr; l; tÞ to be continuous at the layer interfaces,…”

“…Though useful, analytical solutions to the problem of multidimensional transient heat conduction in several multilayer geometries are still lacking. Such solutions are only available for the parallel slabs [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and are, very recently, developed for cylindrical geometries composed of layers [19][20][21][22][23]. Here, we report the first analytical solution to twodimensional (r-h spherical coordinates), time-dependent heat conduction problem in concentric layers.…”

“…Analytical solutions in the cylindrical coordinate system were restricted to the r-z coordinates [19,20]. Only recently, series solutions based on the Separation of variables method in polar coordinates are reported for both pie-slice [21] and annular [22] geometries. These solutions are valid for any combinations of the time-independent, inhomogeneous boundary conditions at the inner and outer radii of the domain.…”

Untitled

“…The difficulty is due to the algebraic nature of such problems, and the fact that certain transcendental equations must be solved. Explicit formulations of the transient heat conduction problem in polar coordinates with multiple annular layers are given in [15], [21]. The resulting equations were solved numerically.…”