A one-dimensional model that describes diffusion and advection process has been developed. The approximate solution to the nonlinear equations has been obtained analytically with an iterative method. A two-dimensional numerical solution with computational fluid dynamics (CFD) software is also derived for a two-dimensional crucible around which the convective and diffusive flows are more precisely presented. The results showed that as long as both heat and oxygen diffusion processes are considered at a time, the one-dimensional solution could provide as accurate a result as that obtained by using the two-dimensional solution evaluated in terms of temperature, but not in terms of the oxygen content. The weight and the geometry of the soot bed significantly affect the distribution of temperature and oxygen content that warrants the use of two-dimensional model because a two-dimensional character such as a buoyant upward flow from the relatively hot bed is predominant. (C) 2010 Elsevier B.V. All rights reserved.