Mathematical simulation in glass technology

This book, entitled Mathematical Simulation in Glass Technology, is one of a series reporting on research and development activities on products and processes conducted by the Schott Group. The scientifically founded development of new products and technical pro cesses has traditionally been of vital importance to Schott and has always been performed on a scale determined by the prospects for application of our special glasses. Since the reconstruction of the Schott Glaswerke in Mainz, the scale has increased enormously. The range of expert knowledge required could never have been supplied by Schott alone. It is also a tradition in our company to cultivate collaboration with customers, universities, and research institutes. Publications in numerous technical journals, which since 1969 we have edited to a regular schedule as Forschungsberichte - "research reports" - describe the results of these cooperations. They contain up-to-date infor mation on various topics for the expert but are not suited as survey material for those whose standpoint is more remote. This is the point where we would like to place our series, to stimulate the exchange of thoughts, so that we can consider from different points of view the possibilities offered by those incredibly versatile materials, glass and glass ceramics. We would like to share the knowledge won through our research and development at Schott in cooperation with the users of our materials with scientists and engineers, interested customers and friends, and with the employees of our firm.

Horst LochRuud G.C. BeerkensWolf S. KuhnHenry Eisermann and Ulrich Lange and Horst Loch and GUnter WeidmannUlrich Lange and Horst LochNorbert Siedow and Horst Loch and Sandro ManservisiMatthias Brinkmann and Norbert SiedowMatthias Brinkmann and Norbert Siedow and Thomas KorbMatthias Brinkmann and Thomas KorbUlrich LangeGünter Weidmann and Kurt Holtberg and Henry EisermannUlrich LangeUlrich LangeThoralf Johansson and Ulrich FotheringhamThoralf JohanssonPeter Thomas and Clemens KunischClemens KunischVolker Seibert and Andreas BensbergKurt Nattermann and Christoph HermannsKurt Nattermann and Harald Krummel and Ludwig FrankVolker SeibertSeth Nzahumunyurwa and Hans-Jürgen Hoffmann and Hans Oechsner and Joachim Disam

1 Overview	p. 1
1.1 Introduction	p. 1
1.2 Systematics and Boundary Conditions of This Book	p. 3
1.3 Some Important 3D Continuum Equations	p. 6
References	p. 15
2 Melting and Fining	p. 17
2.1 Modeling of the Melting Process in Industrial Glass Furnaces	p. 17
2.1.1 Application of Process Simulation Models for Glass Furnaces	p. 18
2.1.2 Modeling of Heat Transfer and Convection Flows in Glass-Melting Tanks	p. 23
2.1.3 Sand-Grain Dissolution, Behavior of Gas Bubbles in Glass Melts, and Glass-Quality Index	p. 47
2.1.4 Models for Evaporation and Superstructure Refractory Attack by Vapors	p. 61
2.1.5 Dynamic Modeling	p. 71
2.1.6 Concluding Remarks	p. 72
2.2 Mathematical Modeling of Batch Melting in Glass Tanks	p. 73
2.2.1 Motivation and Requirements on Batch Modeling	p. 74
2.2.2 Survey of Batch Melting	p. 76
2.2.3 Theoretical Basis of Batch Modeling	p. 93
2.2.4 Key Values and Non-Dimensional Numbers	p. 105
2.2.5 Batch Models	p. 110
2.3 High-Frequency Melting of Glass in Crucibles Frank-Thomas Lentes	p. 126
2.3.1 Basics of Electrodynamics	p. 126
2.3.2 Mathematical Formulation of the Simulation Model	p. 129
2.3.3 Simulation Results	p. 134
2.3.4 Conclusion and Outlook	p. 136
2.4 Model-Based Glass Melter Control Ton Backx	p. 137
2.4.1 Model Concepts	p. 138
2.4.2 Model-Predictive Control	p. 142
2.4.3 Extensions of the MPC Technology	p. 148
2.4.4 Application of MPC in the Glass Industry	p. 150
References	p. 155
3 Homogenizing and Conditioning	p. 165
3.1 The Intensity of Mixing Processes	p. 165
3.1.1 Description and Quantification of Mixing Processes	p. 165
3.1.2 Flows and Particle Paths in Stirrers	p. 169
3.1.3 Statistics of Residence Time and Dispersion	p. 173
3.1.4 Deformation of Infinitesimal Test Bodies Along Particle Paths	p. 176
3.1.5 Deformation Statistics	p. 187
3.1.6 Example: a Simple Paddle Stirrer	p. 189
3.1.7 Outlook	p. 192
3.2 Instabilities and Stabilization of Glass Pipe Flows	p. 193
3.2.1 Stationary Temperature and Pressure Profiles in the Pipe	p. 193
3.2.2 A Stability Phenomenon	p. 197
3.2.3 Appendix: Derivation of Several Equations	p. 205
3.3 Shape Optimization of Flanges	p. 208
3.3.1 General Shape Optimization: Continuously Varying Thicknesses and Contours	p. 209
3.3.2 Finite-Dimensional Shape Optimization: the 3-Ring/Spoke Flange	p. 230
References	p. 237
4 Shaping at Low Viscosities	p. 239
4.1 Heat Transfer Between Glass and Mold During Hot Forming	p. 239
4.1.1 Heat Transfer Coefficient Between Glass and Mold	p. 241
4.1.2 Physics and Mathematics of the Heat Transfer	p. 245
4.1.3 Sample Computations	p. 252
4.1.4 Radiative Contributions to the Heat Transfer	p. 255
4.1.5 Laboratory Experiments	p. 259
4.2 Remote Spectral Temperature Profile Sensing	p. 262
4.2.1 Thermal Radiation in Hot Glass	p. 263
4.2.2 The Inverse Problem of Spectral Temperature Sensing	p. 266
4.2.3 Sample Computations	p. 273
4.2.4 Laboratory Experiment	p. 275
4.2.5 Spectral Imaging of Hot Glass	p. 279
4.3 Heat Transfer During Casting Experiments	p. 286
4.3.1 Experimental Set Up	p. 287
4.3.2 Comparison Between "Exact" Modeling and Measurement	p. 289
4.3.3 Alternative Modeling Using the Active Thermal Conductivity	p. 290
4.4 Thin-Layer Flows of Glass	p. 293
4.4.1 Example of a Thin-Layer Model	p. 294
4.4.2 Simplified Energy Balance	p. 298
4.4.3 Validation of the Model	p. 300
4.4.4 Fiber- and Tube-Drawing Models	p. 302
4.4.5 More Comprehensive Thin-Layer Flow Models	p. 305
4.5 Pressing of Drinking-Glass Stems	p. 306
4.5.1 Model 1: Finite-Element Modeling	p. 309
4.5.2 Model 2: Analytical Modeling	p. 310
4.5.3 Comparison of Model 1 and Model 2	p. 316
4.6 The Use of Remeshing Methods in Pressing Simulations	p. 317
4.6.1 Some Technical Aspects of the Method	p. 319
4.6.2 Example: Pressing of a Tumbler	p. 320
4.6.3 Example: Pressing of an "Axisymmetric TV Screen"	p. 321
4.7 Chill Ripples in Pressing and Casting Processes	p. 326
4.7.1 A Simple Casting Process	p. 328
4.7.2 A Model for Kluge's Experimental Set-Up	p. 330
References	p. 335
5	p. 339
5.1 Temperature-Dependent Elasticity in Reshaping Simulations	p. 339
5.1.1 Model	p. 339
5.1.2 Simulation Results	p. 341
5.2 Sagging and Pressing of Glass Sheets	p. 343
5.2.1 Model and Boundary Conditions	p. 344
5.2.2 Results of the Model Computations	p. 345
5.3 Calibration of Glass Tubes ThoralfJohansson	p. 349
5.3.1 Model Description	p. 349
5.3.2 Results of the Model Computations	p. 350
6 Thermal Treatment	p. 359
6.1 Verification of Relaxation Models	p. 359
6.1.1 Mathematical Models	p. 359
6.1.2 Experiments in the Lehr	p. 362
6.1.3 Simulation	p. 363
6.1.4 Measuring Stress and Compaction	p. 363
6.1.5 Results	p. 363
6.2 Stresses and Crack Growth in Continuously Formed Slabs	p. 367
6.2.1 Cooling a Continuous Strip	p. 369
6.2.2 Crack Growth	p. 370
6.2.3 Modified Temperature Program in Order to Avoid Cracking	p. 371
6.2.4 Cutting the Strip into Slabs	p. 372
6.3 Thermal Tempering of Drinking Glasses	p. 374
6.3.1 Principles of Thermal Tempering	p. 375
6.3.2 Results for Spatially Inhomogeneous Quenching	p. 376
6.3.3 Realization of a Quenching Process	p. 378
7 Post-Processing by Laser Cutting	p. 381
7.1 Rough Estimation of Process Parameters	p. 381
7.1.1 Stress Levels	p. 381
7.1.2 Laser-Beam Profiling	p. 382
7.1.3 Selection of Laser	p. 384
7.2 Numerical Analysis of Cutting Processes	p. 385
7.2.1 Calculation of Temperature Distributions	p. 386
7.2.2 Calculation of Stress Distributions	p. 391
7.2.3 Condition for Cut Elongation	p. 396
7.2.4 Calculation of Stress Intensities for Laser Cutting	p. 399
7.3 Practical Realization	p. 404
7.4 Appendix: Fundamentals of Fracture Mechanics	p. 408
7.4.1 Fracture Mechanics for Brittle Solids	p. 408
7.4.2 FEA Calculation of Stress-Intensity Factors	p. 410
7.4.3 Prediction of the Crack Path	p. 411
8 Glass Products Under Mechanical and Thermal Loads	p. 413
8.1 Strength Optimization of Airbag Igniters	p. 413
8.1.1 FEA for Axial-Symmetric Models	p. 413
8.1.2 FEA of 3D Models	p. 420
8.1.3 Pull-Out Tests	p. 423
8.1.4 Push-Out Tests	p. 432
8.1.5 Pressure Tests	p. 435
8.1.6 Appendix: Statistical Procedure	p. 437
8.2 Stiffness and Weight Optimization of a Reticle Stage for Optical Lithography	p. 438
8.2.1 Requirements for a (9 × 9)" Reticle Stage	p. 439
8.2.2 Design of a Prototype	p. 440
8.2.3 FEM Optimization Without Additional Masses	p. 442
8.2.4 FEM Analysis With Additional Masses	p. 444
References	p. 446
9 Simulation and Test of the Spinning Process Applied to Platinum Metals	p. 449
9.1 Necessity to Shape Materials	p. 449
9.2 Qualitative Description of the Spinning Process	p. 449
9.3 Essential Assumptions for the Modeling of the Spinning Process	p. 451
9.4 General Relations for the Model of the Spinning Process	p. 453
9.5 Approximations	p. 455
9.5.1 First Approximation: Quasi-Homogeneous Deformation	p. 455
9.5.2 Second Approximation: Linearly Decreasing Deformation	p. 458
9.6 A Practical Example for the First and Second Approximations	p. 460
9.7 Experimental Observations and Discussion	p. 464
References	p. 465
List of Contributors	p. 467
Sources of Figures and Tables	p. 471
Index	p. 473

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