Corrected quasi one-dimensional heat conduction equation for the analysis of straight fins of uniform profile

Abstract

Justification of the quasi one-dimensionality assumption in the analysis of fins of various shapes has been traditionally based on the criterion that fins have large slenderness ratios, i.e., large length and small thickness. This simplistic approach is based on a mere geometric consideration that obviates the implications of the two-dimensional heat paths in the axial and transversal directions that occurs in fins. Beginning with the formal differential formulation of the two-dimensional heat conduction equation for straight fins of uniform profile and the appropriate boundary conditions, the central objective of the present paper is to develop a systematic mathematical procedure that revolves around a new corrected quasi one-dimensional heat conduction equation that is more physically sound. The step-by-step mathematical development depends on two controlling parameters: (1) a thermo-geometric parameter, i.e., the transverse Biot number based on the half-thickness, and (2) a geometric parameter, i.e., the slenderness ratio. The computed two-dimensional heat transfer rates clearly demonstrate that the corrected quasi one-dimensional heat conduction equation captures the two-dimensional heat paths flawlessly and as a direct result is better than the standard quasi one-dimensional heat conduction equation. The discrepancy between the corrected quasi one-dimensional and the standard quasi one-dimensional heat transfer rates is of the order of 10%.