An Analysis of Laminar Forced Convection Between Concentric Spheres

Abstract

A numerical solution for predicting the behavior of laminar flow and heat transfer between concentric spheres is developed. Axial symmetry is assumed. The Navier-Stokes equations and energy equation are simplified to parabolic form and solved using finite-difference methods. Hydrodynamic and energy equations are uncoupled, which allows the hydrodynamic problem to be solved independently of the heat-transfer problem. Velocity and temperature are calculated in terms of the two spatial coordinates. Solutions depend on radius ratio of the concentric spheres, Reynolds number of the flow, Prandtl number, initial conditions of temperature and velocity, temperature distribution along the spherical surfaces, and azimuthal position of the start of the flow. The effect on flow and heat transfer of these variables, except surface temperature distribution, is evaluated. While the computer solution is not restricted to isothermal spheres, this is the only case treated. Velocity profiles, pressure distribution, flow losses, and heat-transfer coefficients are determined for a variety of situations. Local and average Nusselt numbers are computed, and a correlation is developed for mean Nusselt number on the inner surface as a function of Reynolds number, Prandtl number, and radius ratio. Flow separation is predicted by the analysis. Separation is a function of Reynolds number, radius ratio, and azimuthal location of the initial state. Separation was observed at the outer surface as well as from the inner surface under some conditions. In cases where separation occurred, the solution was valid only to the point of separation.