Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
alt="Bar chart depicts the grain-size distributions of two different glass-grade sand qualities as determined with sieves of increasing mesh width."/>
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
Figure 8 Rising velocity vSLIP of bubble swarms in a melt at a viscosity of 150 dPa·s as a function of bubble radius r and volume fraction ϕ of bubbles.
(8)
where ϕmax = 0.64 is the maximum value of ϕ as given by random close spherical packing. But the density decrease caused by the presence of bubbles, which is proportional to 1 − ϕ/ϕmax, must also be taken into account. The rising velocity vSLIP of an individual bubble within a bubble swarm of volume fraction ϕ thus is
(9)
The situation is illustrated in Figure 8 for a viscosity of 150 dPa·s, which is that of a typical float glass melt near 1400 °C. Up to a volume fraction of 0.4, bubbles bigger than 0.5 mm in radius safely escape during the available process time, whereas those smaller than 0.1 mm hardly reach any noticeable rising velocity. They rather rest relative to the environment. An especially critical situation occurs when the volume fraction approaches the limit ϕmax. In this case, bubbles of any size become stagnant so that a foam forms on top of the melt in the fining area as observed in a glass of beer. Hence, this problem calls for utmost care in the design of the chemical part of the fining process, and especially of the amount of fining agent used.
5.2 Chemical Fining
As indicated by old glass specimens, bubbles cannot be completely eliminated with only physical fining. In a somewhat paradoxical way, better results are achieved if additional bubbles are produced within the melt at a sufficiently high, yet not too high, volume fraction to coalesce with the bubbles formed or entrapped during melting. The process is known as chemical fining as it involves reactions with gas‐releasing substances.
For reasons of cost, chemical compatibility, and effectiveness, the most widely used agent is sodium sulphate (Na2SO4). By experience, 4 kg of Na2SO4 are added per ton of produced glass. During the early stages of batch melting, the sulfate dissolves in the melt. Under oxidizing conditions, it decomposes at 1400–1450 °C according to the reaction
(10)
where the braces {−} denote the state “dissolved in the melt.” Under reducing conditions, sodium sulphate reacts with the Na2S formed during primary batch melting as follows:(11)
The latter reaction already occurs at temperatures slightly below 1400 °C.
Oxygen fining is an alternative option. The agent typically used is Sb2O3; it is added to the batch in amounts of 3–5 kg per 1000 kg of sand, in combination with a four‐ to eightfold amount of NaNO3 [5]. At the moderately low temperatures of primary batch melting, Sb2O3 converts to {Sb2O5} provided that a sufficiently high oxygen partial pressure in the batch is established (Figure