Lumped models of gas bubbles in thermal gradients

The motion of spherical gas bubbles in isothermal and nonisothermal glassmelts is analyzed both analytically and numerically by means of lumped models. The analytical studies provide expressions for the bubble radius and location as functions of time in the absence of mass transfer and have been obtained using a linear temperature gradient, a linear dependence of the surface tension on the temperature, and average values for the dynamic viscosity and/or surface tension. Three flow regimes have been analyzed: the buoyant the thermocapillary, and the mired buoyant-thermocapillary regimes. Numerical solutions to both the bubble radius and location have been obtained using the local values of both the dynamic viscosity and surface tension in the absence of mass transfer. These numerical results indicate that the bubble radius at refining is about 8% of the bubble's initial radius, while the bubble velocity increases as the initial bubble radius, mean temperature, and thermal gradient are increased. Bubbles in zero-gravity environments and without mass transfer move slowly, and the bubble radius and velocity increase as the initial bubble radius and temperature gradient are increased, but they decrease as the glassmelt mean temperature is decreased. Numerical studies of gas bubbles with mass transfer in nonisothermal glassmelts indicate that, for a bubble containing only oxygen initially, nitrogen, carbon dioxide, and water vapor diffuse from the glassmelt to the bubble, whereas the oxygen diffuses from the bubble to the glassmelt. The time required by the bubble to be dissolved increases as the initial bubble radius, mean temperature, and thermal gradient are increased. (C) 1997 by Elsevier Science Inc.