Sand Dissolution
All solids surviving primary batch melting have to dissolve in the viscous rough melt by slow diffusion processes under comparatively low driving chemical forces. This is one of the reasons why, even today, long dwell times are required for the fusion process. By mass, the sand represents the major part of solids that have to dissolve in this way. The process suffers from an especially unfavorable feature (Figure 5): the decrease of the silica concentration from the sand grain to the melt phase represents a strong chemical gradient that causes the grain to be surrounded by a seam of melt with a high viscosity and a low basicity. This gradient affects not only mass transport but also the solubility of gases, which generally decreases with decreasing basicity (cf. Chapter 5.7). Thus, gases dissolved in the rough melt tend to form bubbles around a dissolving sand grain.
Figure 5 Schematic view of a dissolving sand grain; the grain is surrounded by a solid reaction layer (e.g. tridymite) followed by a liquid high‐viscosity diffusion seam with decreasing SiO2 concentration, hence decreasing acidity, from inside to outside; gas bubbles – mostly O2 – precipitate at the interface solid/liquid; upon complete dissolution of the sand grain, a bubble cluster remains in the melt.
In addition, temperature‐induced reduction of ferric iron takes place as described by the reaction
(2)
describing how firm [Fe3+O4] oxygen complexes give rise to the weak [Fe2+O6] complexes formed by ferrous iron. The equilibrium constant of the reaction is given by
(3)
so that, at constant redox state, tiny oxygen bubbles emerge at the boundary of the dissolving grain. Any dissolving sand grain leaves behind it a cluster of small bubbles, removal of these bubbles makes sense only if their generation is over. This is one of the reasons why sand dissolution and the fining process need to take place in separate parts of the furnace.
In summary, successful sand dissolution is a prerequisite for successful fining. Even apparently small differences in the grain‐size distributions of sands have a big impact in this respect. This statement will be demonstrated for two different sands. Let us assume that a spherical sand grain with radius r dissolves according to Jander's kinetics:
(4)
Figure 6 Grain‐size distributions of two different glass‐grade sand qualities as determined with sieves of increasing mesh width.
Here, α(r,t) denotes the turnover, with 0 ≤ α(r,t) ≤ 1 and D a diffusion coefficient. The grain‐size distribution is mathematically represented by a log‐normal distribution, the differential form of which reads
(5)
where r50 is the median radius of the particle size distribution and σ = ½ · ln(r84/r16) is the standard deviation denoting the width of the distribution; 16 and 84% by mass of the sand are contained in the fraction smaller than r16 and r84, respectively. The values of r50 and σ are determined by an evaluation of the sieve analysis (Figure 6). Both sand qualities have an identical median d50 = 2·r50 = 180 μm, but different σ. An ensemble of grains with a size distribution q(r) then dissolves according to the equation
(6)
where 0 ≤ A(t) ≤ 1 denotes the reaction turnover of the entire ensemble. The results for the two selected sand qualities upon isothermal dissolution are shown in Figure 7 as obtained with the solution of Eq. (18) given in the Appendix. At first sight, both kinds of sands dissolve in about the same manner. But on closer inspection (see inset), the difference does become large toward the very end of the process since Sand 2 needs significantly many more hours than Sand 1 to reach a 99.9% dissolution level, which is crucial for glass quality.
Figure 7 Dissolution turnover of the two sands of Figure 6 as a function of process time for isothermal diffusion with D = 1·10−13 m2/s. Inset: magnification of the results for nearly complete dissolution.
5 Fining, Refining, Homogenization
5.1 Physical Fining
As noted above, the ideal onset of fining takes place when sand dissolution is complete. Physically, fining relies on two simultaneous processes, namely bubble removal by buoyancy and coalescence of small bubbles to form larger ones. The latter is driven by the release of energy associated with the excess internal pressure of a bubble relative to ambient. As given by Laplace's formula, this excess pressure is ΔP = 2σ/r for a bubble of radius r with a surface tension σ so that the energy gained amounts to about 3.5·σ·r when two bubbles of identical size merge. As for the buoyancy velocity v0 of a single bubble in a melt of viscosity η, it is given by a modification of Stokes' law for dispersed phases with mobile boundaries known as Hadamard's law:
(7)
where g is the gravitation constant and ∆ρ the density difference between the melt and bubble.
For a melt with a volume fraction ϕ of bubbles, the effective viscosity becomes
