The Sparrow–Galerkin solution of radiation exchange and transition to finite element

Abstract

This paper presents a historical perspective on the development of the Galerkin-type and finite-element calculations of radiation exchange between surfaces. It is shown that the formulation of the variational solution of radiation exchange introduced by Sparrow in the 1950s is similar to that used by Galerkin to solve differential equations. The extension of Galerkins method to the integral equations of the Fredholm type yields the formulations developed by Sparrow using the variational method. For this reason, the Galerkin-type solutions of radiation exchange will be referred to as the Sparrow–Galerkin method. The Sparrow–Galerkin solution technique, in most cases, provides highly accurate results. However, for a certain set of parameters, the series solutions may not converge. This leads to formulation of a finite-element technique as the discretized form of the Sparrow–Galerkin method evolves into the finite-element method.