Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
are absorption and scattering coefficients, respectively; n is the index of refraction;
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
In Table 1, model set up specifications for Step 4 are related to the numerical methodologies employed to render a solution. Examples include so‐called under‐relaxation coefficients (URCs), which are used to stabilize the evolution of iterative calculation procedures required to solve nonlinear problems. These coefficients are very important since many factors cause virtually all glass‐process simulations to be nonlinear.
URCs have values between 0 and 1, where 1 represents no under‐relaxation and 0 does not allow the estimated field variable to change from one iteration to the next. In general, larger URC values thus allow for more rapid convergence, but divergence will occur instead if a URC is too high. Conversely, small values of URCs tend to be more robust but require many more iterations to satisfy convergence criteria. It remains a bit of an art to specify URCs, especially because optimal values can very much depend on other numerical specifications.
Additional specifications can include the manner in which advection terms are discretized, whether velocity components are solved consecutively as scalar components or coupled to one another, along with pressure, or if energy and radiation equations are solved in a coupled or uncoupled manner. Choosing these options can depend on the capabilities of the computer used as algorithms that couple equations require larger amounts of memory.
As just noted, prescribing numerical parameters is an art so that experience is required for an analyst to become efficient and develop realistic expectations. Nevertheless, many commercial software providers offer recommended or default values to begin a simulation. Most numerical parameters will not affect the converged solution, but only the time required to obtain the solution. However, some numerical schemes will provide more accurate results for a given mesh than others although their differences should become imperceptible with sufficient grid refinement. For example, a second‐order upwind differencing scheme for advection terms will produce