Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
property solution serves as the initial estimate for the more accurate, variable property simulation. It is recommended to begin a simulation project erring on the side of simplicity, and then to add complexity in subsequent simulations.
It is also the case that the physical modeling decisions will affect the geometrical and meshing procedures. For example, a perforated plate through which a fluid flows could require extremely detailed meshing (Figure 2); or the effects of the hole pattern could be accounted for with a much less refined mesh using a more abstract method, in which the screen is characterized by a permeability which relates velocity and pressure drop. The decision on how to proceed will depend on the objectives of the study.
Post‐processing (Step 5) is also very important, as it requires the analyst to execute good judgment on the results obtained. Before extracting information and insights from the results, the analyst must scrutinize the calculated field variables, as well as checking for balanced conservation laws; that is, checking for sufficient numerical convergence must be performed. Finally, management of simulation data (i.e. electronic model files) should not be overlooked since it determines the efficiency with which the simulation process is executed.
Figure 1 Examples of burner block geometry. (a) Accurate representation with poor mesh quality; (b) slight modification with improved mesh quality.
Figure 2 Example of a screen with small‐scale features requiring a high mesh density to be resolved.
3 Fundamental Phenomena, Governing Equations, and Simulation Tools
3.1 Glass as a Continuum
The basic principles of engineering science are applied in CFD simulations. These involve fluid mechanics and usually various other phenomena that are typically considered to fall within the category of thermal sciences. A brief overview is provided, but for complete development, interested readers not yet familiar with the details are referred to various textbooks (e.g. [1–3]). Physical phenomena specific to glass processes must be accounted for within the framework of three fundamental principles and will be reviewed later with respect to a few chosen examples. Readers are also directed to a volume edited by Krause and Loch [4] for a collection of excellent examples of numerical simulations applied to glass processes.
Forming the foundation on which CFD models are constructed, the fundamental principles of classical physics account for conservation of mass, momentum, and energy. Conservation of momentum follows from Newton's three laws of motion, whereas energy conservation is of course the first law of thermodynamics. Contrary to what is done for the very small systems simulated in theoretical studies (Chapters 2.8 and 2.9), it is impractical to account for the motion or energy level of each individual atom or structural entity at the scale relevant to industrial processes. Instead, it is recognized that, for length scales of engineering practicality, substances can be characterized with intensive properties (i.e. per unit volume or mass). Because it describes the mass per unit volume of a particular substance, density is a simple example of such a property that is independent of the size of the system. This abstraction allows substances to be treated as a continuum and allows for powerful mathematical models to be constructed.
3.2 Transport by Advection and Diffusion
Referring again to well‐recognized texts [1, 2], we will remind that the principle of conservation of mass, applied to an infinitesimally small control volume (CV), is mathematically expressed as
(1)
where ρ is the density and
(2)
whose left‐hand side represents the time rate of change of momentum (i.e. mass times acceleration), and the right the forces acting upon the fluid by adjacent fluid particles (i.e. the divergence of the stress tensor,
An important requirement for mathematical modeling of fluid flows is to relate internal fluid stresses to characteristics of the fluid's motion. Such a relationship, which depends on the substance, is termed a constitutive relationship. There are many types of material behavior, requiring different constitutive models, but the most commonly used model relates shear stresses to strain rate (i.e. velocity gradient) in a linear manner. Fluids to which this linear relationship applies are known as Newtonian fluids. For example, a shear stress component τxy for a Newtonian fluid is characterized with
(3)
where μ is the dynamic viscosity and u is the x‐direction component of the velocity vector. For a Newtonian fluid, viscosity is independent of the strain rate. As long as it is homogeneous, glass is a Newtonian fluid although its viscosity is a very strong function of both composition and temperature (Chapter 4.1).
When the Newtonian constitutive model is substituted into the more general momentum Eq. (2) and the fluid is assumed to be incompressible, the result is the well‐known Navier–Stokes equation
(4)
where P is the fluid pressure. Not all fluids behave according to Eq. (3). Such non‐Newtonian fluids must be mathematically modeled with different constitutive relations substituted into Eq. (2) [5, 6].
Note that Eq. (4) is a vector equation, although it can be decomposed into vector component (scalar) equations, which is often done for the numerical application. In Cartesian coordinates, these component