Dynamics of bubble population undergoing mass transfer and coalescence in a glass forming liquid
Résumé
During the melting of raw materials, bubbles are created due to the trapping of atmospheric gases
and the decomposition of mineral species used as batch components. Recent experiments achieved by
Pereira et al. [1] on glass cullet melting in a small crucible confirm this assumption. The bubble dynamics
can be rescaled by the residence time of a bubble in the crucible directly linked to the temperature via
the dynamical viscosity. A first attempt to model the dynamics of bubble density by taking into account
the bubble rising and mass transfer between the two phases failed. Cable [2] underlined that bubble
coalescence can play a role in the dynamics at the early times of the process.
To rationalize the influence of mass transfer and coalescence, a population balance equation is derived
to study the dynamics of bubbles in a glass forming liquid. According to the nomenclature of Hulburt
and Katz [3], internal coordinates are the radius, and the Ng-1 molar fractions of the Ng gas species
dissolved either physically or chemically in the glass forming liquid. The coalescence kernel is determined
by taking into account the velocity difference of the bubble rising in the crucible. A collision efficiency
based on the dynamics of film drainage between two bubbles is also derived according to Guémas et al.
[4]. The temporal behaviours of the dissolved gas are also included in the theoretical model.
A Direct Quadrature Method of Moments (DQMOM) [5] is used to solve numerically the population
balance equation. In this method, the size distribution function is decomposed over N discrete Dirac
distributions characterised by its weight with and its abscissa. The description
of the 2N first moments allows to determine 2N balance equations.This system
is coupled to the balance equations of Ng-1 molar fractions in each discrete distribution and to the
Ng molar concentrations of gas species dissolved in the liquid. The whole system is numerically solved
using a Runge-Kutta method at the fourth order. Initially, the bubble size is distributed according to a
log-normal distribution.
The numerical results are compared to Experimental data showing that coalescence is important at short
time mainly due to the fact that the bubble population is dense. In a such case, the coalescence efficiency is relevant. When the bubble population decreases enough, coalescence events are scarce. The
dynamics of the bubble is limited to the removing from the free surface of the crucible.
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