Читать книгу Encyclopedia of Glass Science, Technology, History, and Culture - Группа авторов - Страница 81

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 v_SLIP 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 ϕ_max = 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/ϕ_max, must also be taken into account. The rising velocity v_SLIP 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 ϕ_max. 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.