Читать книгу Encyclopedia of Glass Science, Technology, History, and Culture - Группа авторов - Страница 158

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].