Analytical Modeling of Natural Convection in Concentric Spherical Enclosures

A modeling procedure is developed for natural convection heat transfer from an isothermal heated sphere located at the center of an isothermal, cooled, sphericalshaped enclosure. The model is based on the linear superposition of conduction and convection solutions, where the convective component is determined based on a combination of two limiting cases, laminar boundary layer convection and transition flow convection. The model is validated using experimental and numerical data from the literature, including data from an experimental test program performed by the authors in a previous publication for four different enclosure configurations, 1 5 do di 4 8, over the range of Rayleigh number 10 Ra Ai 1 107. The model accurately predicts the experimental data with an RMS difference of 2 4 % and is in good agreement with the other data from the literature. Nomenclature A = area; m2 C = coefficient Ccs = Raithby and Hollands16 coefficient d = diameter; m F Pr = Prandtl number function g = gravitational acceleration; m s2 GL = body gravity function k = thermal conductivity; W mK ke = effective thermal conductivity, kNuL S L ; W mK L = general characteristic length; m † Research Assistant Professor ‡ Fellow AIAA, Distinguished Professor Emeritus Associate Professor, Director MHTL Copyright c 2004 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. n = combination parameter NuL = Nusselt number, QL kAi∆T Pr = Prandtl number, ν α Q = total heat transfer rate; W r = radius, m R = thermal resistance, Ti To Q; oC W RaL = Rayleigh number, gβ Ti To L3 να S = conduction shape factor; m S L = dimensionless shape factor, SL Ai T = temperature; oC Tb = bulk fluid temperature; oC Greek Symbols α = thermal diffusivity; m2 s β = thermal expansion coefficient; 1 K φ = dimensionless bulk temperature ν = kinematic viscosity; m2 s Subscripts b = bulk fluid bl = boundary layer flow conv = convection tr = transition flow i = inner body o = outer body LB = lower bound UB = upper bound