Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
rel="nofollow" href="#u3408c313-7fae-51e4-9be4-450c464d9a9a">Chapter 4.1).
Glass flow is forced to a certain extent by the introduction and melting of batch as well as by draining through the throat of the furnace. However, additional forces significantly affect flow patterns. Density changes caused by temperature variations give rise to buoyancy forces, which significantly affect flow patterns in the glass melt. These are accounted for through a body force, which is the last term on the right side of each momentum Eqs. (5)–(7). It is convenient and typical for one of the coordinate directions (e.g. the z‐direction) to be aligned with the direction of gravity (or at least opposite to it), so that the body force in its respective momentum equation is represented as
(16)
where ρ(T) is the local density evaluated at the local temperature and gi represents gravitational acceleration in coordinate direction i (e.g. gz = −9.806 m/s2).
Although the glass is heated from above, which usually results in a stable, vertical temperature gradient, freshly melted material from the batch blanket is relatively cool and dense so that it provides a significant driving force for recirculation in the melt. These buoyancy‐driven recirculation velocities can be an order of magnitude larger than those resulting from the forward flow of glass associated with the melter pull. Furthermore, lateral temperature gradients along sidewalls and electrodes provide additional density variations that alter the flow structure. Accounting for flow‐inducing density variations thus is essential in the glass melt.
Bubblers also induce significant recirculation of glass caused by forced convection. Buoyancy forces acting on the bubbles cause them to rise, and in doing so, they drag glass upward along with them (Chapter 1.3). With sufficient multiphase modeling techniques, it is possible to track explicitly the flow of both glass and bubbles, but one commonly treats the effects of the glass bubbles more abstractly by applying a momentum source to the appropriate component of the momentum equation in the columnar region associated with each bubbler. That is, the source term will be augmented by a calculated force per unit volume based upon either Stokes' law or a modified version of it [14].
Another means of affecting glass flow is with mechanical stirrers. These can be accounted for in several ways including appropriately scaled volumetric‐source terms or through basic boundary conditions where the motion of a wetted wall is prescribed.
Other walls, such as the sidewalls, floor, and electrode surfaces, are simply treated as nonslip boundaries where the fluid velocity is zero. Along the top surface, the glass interacts with the batch, foam and possibly the combustion fumes. Because of extreme density differences, the influences of foam and combustion fumes on the glass velocities are often assumed to be negligible. The interface between the batch and glass, however, represents a greater challenge, because the momentum exchange between these two zones is not considered negligible, and the interface itself can be difficult to define precisely. Moreover, the batch–glass interface is where freshly melted glass enters the glass domain from the batch layer. Commercial codes treat this interface in different ways. Because it is beyond our scope to cover the details, we will just note that this topic is an area of needed, ongoing development.
Equation (8) governs some of the energy transport in the glass and is the basis for which temperature distributions are determined. Sometimes an enthalpy formulation is used in place of Eq. (8) to couple intrinsically the batch and glass zones with a single equation governing energy transport, in which case temperatures are determined from enthalpy through an appropriate thermodynamic equation of state. Energy is also transported into and through the glass by electrical dissipation and thermal radiation. Joule dissipation is determined from the solution of equation (F) in Table 2. The rate of conversion of electrical energy to thermal energy is represented by the following:
(17)
where ∇E is the gradient in electric potential,
Thermal radiation is usually accounted for with the aforementioned Rosseland approximation. But a more detailed accounting of thermal radiation transport is possible with methods such as discrete ordinates, which can, for example, be used to resolve spectral characteristics. Other means are available [3].
The combustion zone above the glass is modeled with the same basic governing equations for momentum and energy conservation, but their application is different for a variety of reasons. Furthermore, transport equations for individual species and thermodynamic state relationships must be applied to account for the reaction of fuel and oxidizer and the creation of products of combustion.
Since the combustion zone is turbulent, all diffusion coefficients are replaced with effective values that account for diffusive‐like transport. Hence, it is common to include the k and ε equations (D and E in Table 2), from which μt is determined. Another difference involves radiation for which the assumption of optically thick media required by the Rosseland approximation is not valid. Use of discrete ordinates in combustion zones is common. The absorption coefficients required of the DOM depend on species concentrations, especially CO2 and H2O, which must be determined from a combustion model that accounts for chemical reactions (i.e. the creation and destruction of molecular species) and the transport of the related species.
Through radiative and convective transport, the combustion gases heat virtually all surfaces, including the walls of the superstructure, the top of the batch layer, the foam, and the glass. Furthermore, these surfaces exchange heat through radiation, which is intrinsically included with a discrete ordinates model. Owing to nonlinearities and to the strong coupling between the various zones and between the various transport equations within a zone, a robust, iterative solver is required to converge on a solution. Typically, iterations are performed until conservation laws are satisfied to within 0.1%, whereas adjustments to URCs are sometimes required to improve convergence.
A model of a glass melting furnace must account for transport not only in the glass and combustion zones, but also within the batch, foam, and walls. Whereas all of these zones must obey the same basic laws of physics, their dissimilar material characteristics require different mathematical treatments. Perhaps the easiest to consider are the walls and other solid objects. The energy Eq. (C) (in Table 2) is applied without the advection term since velocities in the walls are zero. Equation (F) is in addition applied to account for electrical current and Joule heating with the assumption that the electric potential is uniform within an electrode, since its electrical conductivity is orders of magnitude larger than that of any other material.
The batch and foam require additional considerations. Considering first the foam, there are many questions to ask. Where does it exist? How thick is it? Does it absorb radiation from the combustion zone and crown, or does it transmit such radiation? What is the gaseous species within the liquid glass membrane? How large are the foam cells? All of these and other factors will affect transport so that choosing a modeling method presents a significant challenge.
One way to deal with foam is to invoke several simplifying