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Within a complete furnace model (Figure 5), the common zones included are the glass melt, a batch cover and a foam layer both floating on the top surface of the glass, the hot combustion zone above the glass/batch/foam surface, and the wall zones enclosing the glass and combustion zones. Most of these zones are coupled to each other through exchange of mass, momentum, and energy. The governing equations, boundary conditions, and sources applied to each zone depend on the physical phenomena which occur within each of them, as well as the manner in which they are coupled to the others.

Glass flow in the melter is laminar so that Eqs. (1) and (5–7) apply. To account for the exponential dependence of viscosity upon temperature in a manner, the empirical Fulcher law is often used,

(15)

where F₁, F₂, and F₃ depend on the specific glass composition (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 g_i represents gravitational acceleration in coordinate direction i (e.g. g_z = −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, is the current flux density, and k_e is the temperature‐dependent electrical conductivity of glass. The Joule dissipation calculated in this way is included in the source term, S_T, in the energy Eq. (8).

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 μ_t 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 assumptions allowing adjustments based on foam conditions, without requiring detailed information regarding its phenomenological behavior. For example, foam can be treated as a layer of material that acts to impede heat transfer between combustion and glass zones, but ignores advection transport within it. A relatively small number of parameters can be used to characterize the thermal behavior of the foam, which can be adjusted, within reason, to render a well‐tuned model.

A similar set of questions arises when considering the batch. Whereas the foam is a two‐phase mixture of liquid membranes enclosing gas cells, the batch is a multiphase mixture of solid particles, with interstitial gas and liquid, whose proportions depend on temperature. Unlike in the foam, advective energy transport within the batch zone cannot be ignored without large compromises such that, therefore, the velocity field within the batch layer must be computed. A common way to accomplish it is to treat the batch as a pseudo‐fluid with a characteristic viscosity that depends on temperature. The batch is assumed to float on top of the glass and to provide an inflow of melt whenever the temperature at its interface with the glass achieves or exceeds a specified temperature where melting occurs. In this way, the batch zone is treated as another fluid zone governed by equations similar to those of the glass. In addition to providing an inlet flow of melt to the glass zone, the batch zone also produces a small amount of gases into the combustion zone because of the chemical reactions that occur upon melting.

The model described in the preceding paragraphs is an example of a powerful tool constructed from well‐defined assumptions and mathematical abstractions. It is summarized in Table 3, which indicates for each zone of the furnace the governing equations and interactions with other zones. The required boundary conditions and other physical parameters needed to specify the operating conditions are summarized in Table 4 where a “coupled” condition indicates an internal boundary condition between two zones where the field variable and associated flux are forced to be the same. In addition, many numerical parameters must be specified in such a way as to bring about a converged solution that satisfies the various conservation principles. Modeling procedures and material properties for glass are discussed in greater detail elsewhere [15].

Table 3 Interacting zones of a complete glass melting‐furnace model.

			Zone couplings
Zone	Equations (Table 2)	Radiation treatment	Glass	Batch	Foam	Walls	Combustion
Glass	A,B,C,F	Rosseland
Batch	A,B,C,F	Surface emissivity	Mass, momentum, energy, electric current
Foam	C	Surface emissivity and transparency	Energy	Energy
Walls	C,F	Surface emissivity	Energy, electric current	Energy, electric current	Energy
Combustion	A,B,C,D,E,G	DOM	Energy	Mass, energy	Energy	Energy
			Glass	Batch	Foam	Walls	Combustion