Simple Algebraic Treatment of Unsteady Heat Conduction in Solid Spheres with Heat Convection Exchange to Fluids Covering the Entire Time Domain

Abstract

Accurate estimates of temperature histories in solid bodies of regular shape (slabs, cylinders and spheres) exposed to cold fluids have been traditionally done by the method of separation of variables evaluating infinite Fourier series. The infinite Fourier series correspond to the exact, analytic solution of the unidirectional heat conduction equation in various coordinate systems. The central goal of this technical paper is to bypass this traditional procedure involving infinite Fourier series. The idea is to predict the three most important temperatures (mean, surface and center) in a solid sphere by implementing a new 1-D composite lumped analysis, which constitutes a natural extension of the standard 1-D lumped analysis. The computational methodology to be proposed is effortless and brings to the table a handful of compact algebraic equations for the mean, surface and center temperatures that cover the complete gamma of the controlling Biot numbers (0 < Bi < 100) in the entire time domain, i.e., at all dimensionless times or Fourier numbers (0 < τ < ∞).