Exact solution of the heat conduction equation in eccentric spherical annuli

• Published in: International journal of thermal sciences 2013-06, Vol.68, p.158-172
• Main Authors: Yılmazer, Ayhan, Kocar, Cemil
• Format: Article
• Language: English
• Subjects: Bispherical coordinates; Computational fluid dynamics; Conduction; Eccentric spheres; Eccentrics; Exact sciences and technology; Exact solutions; Fundamental areas of phenomenology (including applications); Green's function; Green's functions; Heat conduction; Heat transfer; Mathematical analysis; Mathematical models; Physics
• ISSN: 1290-0729; 1778-4166
• DOI: 10.1016/j.ijthermalsci.2013.01.015

In this study, an analytical solution to the heat conduction equation in an annulus between eccentric spheres with isothermal boundaries and with heat generation is obtained using Green's function method. Deriving Green's function in bispherical coordinates for eccentric spherical annuli, an exact solution of heat equation for eccentric spheres with constant surface temperatures is expressed in terms of Green's function and source distribution. The solution is general and can easily be applicable to any space dependent heat source. Results are presented as temperature distributions and local Nusselt numbers for sources having practical and theoretical importance: uniform source, impulse source, shell source. Analytical results are compared with the results of the Computational Fluids Dynamics (CFD) solver Fluent and perfect agreement is observed. ► Conduction equation for eccentric spherical annuli is analytically solved. ► The Green's function method in bispherical coordinates is used. ► The method can easily be applicable to any space dependent heat source. ► Numerical results are presented for uniform source, impulse source, and shell source. ► Rate of convergence is very high due to hyperbolic nature of Green's function.