Two-dimensional steady-state temperature distributions and heat flows in square corners

Abstract

The Schwartz-Christoffel transformation is applied to the case of symmetrical and asymmetrical square corners, with solid walls and isothermal surfaces. The steady-state patterns of temperature distribution and of heat flow lines are shown. The distribution of heat flow intensity along the inside surfaces is illustrated for a range of wall thickness ratios. The Finite Element Method is applied to obtain the components of the maximum intensity occurring at the point where the two inside surfaces meet, and the results are tabulated for a range of wall thickness ratios. The total rate of heat flow for various cases is compared with that obtained from an analytical expression derived by Carslaw and Jaeger. ( Conduction of Heat in Solids, 2nd edn, p. 454). This is an approximate expression, asymptotically correct at points distant from the corner, but involving some error near to the corner. A simple graphical procedure is demonstrated, which now permits the total heat flow to be calculated at any distance from the inside corner. Additionally, a method is shown for finding the distance from the corner at which the analytical equation applies within a given error.