Treatment of radiative transfer in glass melts: validity of Rosseland and P-1 approximations

Abstract

The Rosseland (or diffusion) approximation for the radiative transfer in glass melts has been widely used in the numerical simulation of flow and heat transfer in glass furnaces and forming processes. Its validity is not well confirmed, however and it is necessary to verify to which limit if is applicable. A simple but realistic problem of steadily cooled flowing glass layer is analysed using two Rosseland approximations, one with prescribed boundary temperature and another with Deissler jump boundary condition, and also using the P-1 approximation and an exact solution scheme. The three approximate methods fire compared to the exact solution and their validities are scrutinised. When the depth of the glass melt is greater than about 0.5 m, the Rosseland approximation with prescribed boundary temperature is shown to be sufficiently accurate except near the free surface. The Deissler's jump boundary condition cures the inaccuracy near the free surface and the heat flux is better predicted than the Rosseland approximation; however, deep in the melt it results in greater error in temperature than the former. When the melt depth is small the shape of temperature profile is not correctly predicted by the Rosseland approximations. Even the qualitative shape may be different from reality. Significant improvement of accuracy is available by the P-1 approximation even when the glass melt is as shallow as several centimetres. For all depths of the glass melt the P-1 approximation gives better results than the Rosseland and the Deissler approximations. When the depth of glass melt is greater than about 10 cm the P-1 approximation gives almost the same result as the exact solution. Upgrading the radiation treatment to P-I method is highly recommended not only for shallow glass melts bur also for deep ones when the exact temperature profile near the boundary is important as is frequently such the case.