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It is usually the case that μt ≫ μ and μeff ≈ μt. Likewise, similar substitutions are made for the diffusion coefficients of other transport equations.
3.4 Radiative Heat Transfer
Heat transfer by conduction and convection is accounted for by Eq. (8) (or C in Table 2), which is also compatible with radiative heat transfer applied as a boundary condition on an opaque surface. However, glass and other media (e.g. combustion gases) are commonly semitransparent in glass processes, that is, they emit, absorb, and scatter IR radiation volumetrically.
Radiative heat transfer significantly differs from transport by advection and diffusion so that it cannot be mathematically described by an equation of the form of Eq. (9). It is instead governed by an integrodifferential equation, known as the radiative transfer equation (RTE) [3],
where
As for the first term on the left side of Eq. (14), it represents the change in beam intensity per unit length in beam direction
Radiation is a directional phenomenon and is in addition spectral in nature in that its intensity in principle depends on the wavelength of the IR beam. When spectral variations can be assumed to have negligible effects, Eq. (14) is written for a medium that is said to be gray. Extending the RTE to include spectral effects is straightforward [3], but not presented here.
Note that, Eq. (14) only accounts for IR intensity without determining directly temperatures within a material. Integrated over all directions, however, the net effects of absorption and emission are added to the source term ST in the energy Eq. (8), thereby affecting local temperatures. There are several methods to account for the directional nature of the RTE. Referring to texts on radiation heat transfer for the details of their derivation [3], we will discuss some of them in Section 4.
3.5 Discretization Methods, Solution Algorithms, and Model Specifications
3.5.1 Finite Element and Control Volume Formulations
Several kinds of computational algorithms exist to solve for the field variable in Eq. (9). One category is known as the finite element method (FEM), where the field variable is assumed to have a functional form or shape over discrete portions of the problem domain. In finite element, the governing equations are multiplied by a weight function and then integrated over an element. The weight function can have various forms. As an example, with the Galerkin method, the weight function is the shape function itself. Another category is known as the CV method, where the problem domain is divided instead into a multitude of small volume elements, each characterized by a single, representative value for each relevant field variable. The conservation laws and fluxes are enforced on each CV, where transport or exchange across adjoining boundaries are determined with finite‐difference estimates (usually, a truncated Taylor Series expansion based on unknown or estimated adjacent CV values) of the various derivatives in the governing equation. Whereas both of these numerical methods involve discretizing the problem domain into a multitude of elements or volumes that appear to be virtually the same, they are different as explained in detail in [7, 8]. Generally, more mathematics are involved with the FEM whereas the CV method, dealing with fluxes, can easily be associated with representations giving a physical significance to the problem.
3.5.2 Physical and Numerical Specifications
Once geometrical definition and meshing are done, simulation models are set up with various input specifications (i.e. Steps 3 and 4 in Table 1). Those associated with Step 3 are physical; that is, they describe the physical characteristics of the process, which in turn affect the degree to which field variables at neighboring grid cells (or mesh nodes) influence each other. Essential are material specifications and associated properties, such as viscosity (for fluids), thermal conductivity, density, specific heat, etc. Many software products have material properties defined for commonly used materials, which must be complemented for materials of interest not dealt with in the existing material database. Various boundary conditions such as flow rates, temperatures, and other state parameters at all flow inlets are other physical specifications to be included along with wall boundaries conditions, which could be, for example, a prescribed temperature, heat flux, or temperature‐dependent heat flux.
Every portion of the boundary of the simulation domain requires a boundary condition for each transport equation considered. These conditions can be in the form of a prescribed field variable (e.g. temperature T), prescribed flux by definition proportional to the gradient of the field variable
