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Technical Brief

THE LUMPED CAPACITANCE MODEL FOR UNSTEADY HEAT CONDUCTION IN REGULAR SOLID BODIES WITH NATURAL CONVECTION TO NEARBY FLUIDS ENGAGES THE NONLINEAR BERNOULLI EQUATION

[+] Author and Article Information
Antonio Campo

Department of Mechanical Engineering, The University of Texas at San Antonio, San Antonio, TX 78249
campanto@yahoo.com

1Corresponding author.

ASME doi:10.1115/1.4038539 History: Received April 11, 2017; Revised September 14, 2017

Abstract

For the analysis of unsteady heat conduction in solid bodies comprising heat exchange by forced convection to neighboring fluids there are two feasible models: 1) the differential model and 2) the lumped capacitance model. With regards to the latter, the suited lumped heat equation is linear. In addition, the lumped Biot number criterion stipulates that < 0.1, where the mean convective coefficient is affected by the imposed fluid velocity. Conversely, when the heat exchange happens by natural convection, the pertinent lumped heat equation is nonlinear because the mean convective coefficient is dependent upon the instantaneous mean temperature in the solid body. Normally, the nonlinear lumped heat equation must be solved with a numerical procedure, such as the fourth order Runge-Kutta method. In addition, the lumped Biot number criterion engages a variable mean convective coefficient: which implies that the lumped Biot number criterion < 0.1 must be modified to < 0.1. For the case of cooling, stands for the maximum mean convective coefficient at the initial temperature Tin and initial time t = 0. Interestingly, by way of a temperature transformation the nonlinear lumped heat equation can be channeled through a nonlinear Bernoulli equation, which admits an exact, analytic solution. Thereby, the exact, analytic solution gives way to the mean temperature distributions T (t) for a class of regular solid bodies: vertical plate, vertical cylinder, horizontal cylinder and sphere, which are constricted to < 0.1.

Copyright (c) 2017 by ASME
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