specifications can be the root cause of the frustrating experience of trying to resolve inconsistencies between simulation results and measured data and/or expectations.
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 less “false diffusion” than a first‐order upwind scheme [7].
4 Simulations in Glass Manufacturing Processes: A Few Examples
4.1 Fundamental Studies
As already stated, the importance of a particular phenomenon can be explored by numerical simulations in such a way that the mathematical treatment and/or its numerical implementation can be examined to assess the best way to account for the physical effects of interest. A good example of this approach was the early one‐dimensional study of Glicksman [9] of various physical effects on fiber formation. He formulated his model by manipulating the conservation equations (A)–(C) in Table 2, where glass velocity, filament diameter, and temperature were assumed to vary only in the axial direction of the fiber draw. This relatively simple model was very helpful in understanding the relative roles of glass viscosity and surface tension on fiber‐forming dynamics, as well as the influences of radiative and convective cooling.
As computational resources grew, more sophisticated models were developed to explore further fiber‐forming dynamics from 2‐D axisymmetric models with free surface boundary conditions [10, 11]. Both steady‐state and transient simulations were performed and revealed the onset of unstable forming conditions, which could lead to poor product quality and/or reduced process efficiency. A view of the deformed finite element mesh, representing fiber attenuation as it is drawn, is shown in Figure 3, whereas the excellent agreement found between numerical and experimental results of the fiber attenuation (Figure 4) illustrated the reliability of the method.
Figure 3 Deformed finite element mesh during a simulation of the drawing of a glass fiber.
Source:International Congress on Glass[10]
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Figure 4 Fiber radius attenuation: comparison of numerical and experimental results for an extension ratio of 19 000.
Source:International Congress on Glass[10]
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Another study examined the manner in which radiation within a semitransparent glass is considered [12, 13]. It was performed in the context of a simplified glass furnace geometry. Because radiative energy is both absorbed and emitted volumetrically, this study examined two methods for accounting the radiative transport with equations (A)–(C) in Table 2. One method is the computationally convenient Rosseland approximation [3], in which one accounts for radiative transport by appropriately adjusting the thermal conductivity of glass; the other employs the discrete ordinates method (DOM) [3] to solve independently the radiative‐transport Eq. (14), the results of which are then coupled to the energy Eq. (8) through source terms. The DOM requires significantly more computational effort and, thus, longer run times. For large models with millions of mesh cells/elements, the difference in run times can be significantly important. For many problems in glass processing, the Rosseland approximation will yield sufficiently accurate results but some situations require a more detailed accounting of the radiative transport. For example, if the refractory‐wall temperatures are of interest to assess wear rates, then a DOM might be a better choice. Also, in forming operations, length scales associated with the forming apparatus may be significantly smaller than those for which the Rosseland approximation is valid. A modeling simulation aimed at assessing such fundamental matters is sometimes an appropriate ancillary simulation to perform. Both methods were investigated and compared in [12].
4.2 Glass Melting Furnace
4.2.1 Models
Relatively small simulation models provide a means to understand the behavior of an isolated part or function of a glass process, or to assess numerical treatments. However, many problems require the mutual interactions of several parts or processes to be considered simultaneously. A good example is a glass melting furnace, in which there exist several flow regimes, multimode heat transfer, physical and chemical reactions, and other related phenomena.
Modeling a glass melting tank provides a means to estimate the effects of many things contemplated by a glass maker. Simulations made with a properly constructed model can, for instance, assist in prescribing or changing the profiles of combustion burners, E‐boost power zones, or bubbler flow rates. Changes in pull rate, insulation thickness or type, added wind cooling to outside walls, and surface treatments to alter emissivity of crown materials are just a few other examples of what can be considered with a furnace model.
The effects of various changes can be measured in different ways, too. The overall rate of energy consumption
