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 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 |
4.2.3 Post‐processing
