Convective heat transfer and temperature stratification in a sphere completely filled with a liquid, with a given heat flux

Abstract

A number of articles have been devoted to the theoretical and experimental investigation of natural convection in spherical vessels completely filled with a liquid [1–6]. Analytical solutions are known, obtained by the expansion of the sought function in series in powers of the Rayleigh number (see, for example, [1]), valid for very small values of this number. A numerical solution of the nonlinear Boussinesq equations can be used to obtain solutions with larger Rayleigh numbers, but the existing data for spherical regions [2, 3] embrace a relatively narrow range of Rayleigh numbers. The experimental data with a given heat flux, published in [4–6], were obtained with relatively large Rayleigh numbers (Ra*=109−1011) and Prandtl numbers (P= 3−1500). Data on the characteristics of convection in spherical vessels are still not very numerous and, in a number of cases, contradictory. This relates, in particular, to the boundaries of unsteady-state conditions. The present article, continuing [7–9], expounds a method and gives the results of a calculation of convection in a sphere with a thinwalled shell, in a range of Rayleigh and Fourier numbers embracing the principal conditions of unsteady-state laminar convection with a given heat flux.