This is a book about mathematical modelling. It focuses on the modelling of the preparation of materials. Materials are important, of course, in an economic sense: the "goods" of goods-and-services are made of materials. This provides a strong incentive to produce good materials and to improve existing materials. Mathematical modelling can help in this regard. Without a doubt, modelling a materials processing operation is not strictly necessary. Materials synthesis and fabrication processes certainly existed before the invention of mathematics and computers, and well before the combined use of…mehr
This is a book about mathematical modelling. It focuses on the modelling of the preparation of materials. Materials are important, of course, in an economic sense: the "goods" of goods-and-services are made of materials. This provides a strong incentive to produce good materials and to improve existing materials. Mathematical modelling can help in this regard. Without a doubt, modelling a materials processing operation is not strictly necessary. Materials synthesis and fabrication processes certainly existed before the invention of mathematics and computers, and well before the combined use of mathematics and computers. Modelling can, however, be of assistance--if done properly--and if used properly. The mathematical modelling described in this book is, at its root, a rather formal, structured way of thinking about materials synthesis and fabrication processes. It requires looking at a process as a whole. It requires considering everything that is or might be important. It requires translating the details of a given physical process into one or more mathematical equations. It requires knowing how to simplify the equations without over-simplifying them.
1. Balance Equations.- 1. Balance equations: An informal description.- 2. Conserved quantities for the balance equations.- 3. Balance equations for a single-phase material.- 4. Balance equations for a multiphase material.- II. Constitutive Relationships.- 5. Constitutive relationships: Near-equilibrium thermodynamics.- 6. Expressions for a single-phase material.- 7. Internal geometry & topology of multiphase materials.- 8. Expressions for the system in a multiphase material.- 9. Expressions for the solid phase in a multiphase material.- 10. Expressions for the liquid phase in a multiphase material.- 11. Expressions for the gas phase in a multiphase material.- III. Practical Considerations.- 12. Interlude: Use of chapters 1-11.- 13. Sources of material property values.- 14. Numerical solution: Some finite difference methods.- 15. Numerical solution: Some finite element methods.- IV. Implementation.- 16. Example: Glass refining.- 17. Example: Fiber spinning.- 18. Example: Alloy solidification.- 19. Example: Microwave heating of ceramics.- 20. Example: Combustion synthesis of refractory materials.- 21. Example: Binder removal from shaped powder compacts.- 22. Example: Flow of a solid-liquid suspension.- Appendices.- Appendix A: Details of Mathematical Quantities and Operations Used in the Text.- A1 Simple Algebra.- A2 Coordinate Systems.- A2.1 Essential Features of a Coordinate System.- A2.2 Transformations From One Coordinate System to Another.- A2.3 Particular Coordinate Systems.- A3 Algebra.- A3.1 Representation of Scalars, Vectors & Tensors.- A3.2 Algebraic Operations.- A3.3 Special Vectors and Tensors.- A3.4 Uses of Certain Special Second-Order Tensors.- A3.5 Useful Relationships Among Algebraic Operations.- A4 Spatial Derivatives and Differentials.- A4.1 SpatialDerivatives.- A4.2 Useful Relationships Involving Spatial Derivatives.- A4.3 Spatial Derivatives in Particular Coordinate Systems.- A4.4 Differentials & Regions of Infinitesimal Extent.- A5 Time Derivatives.- A5.1 Motion, Paths & Moving "Objects".- A5.2 Time Derivatives in Three Dimensions - General Case.- A5.3 Time Derivatives in Three Dimensions - Specific Cases.- A5.4 Time Derivatives Involving a Surface.- A6 Integrals.- A7 Derivatives of Integrals.- A7.1 Spatial Derivatives of Volume & Surface Integrals.- A7.2 Time-Derivatives of Volume Integrals.- A7.3 Time-Derivatives of Surface Integrals.- A8 Other Mathematical Details.- A8.1 Evaluation of Gaussian Integrals.- A8.2 Orthogonal Functions.- Appendix B: Units and Dimensions.- B1 Fundamental and Derived Units.- B2 Unit Conversions.- B3 Dimensional Analysis.- B3.1 Dimensional Analysis of the Balance Equations.- B3.2 Dimensional Analysis of Coupled Transport Processes.- B4 Non-dimensionalization.- Appendix C: Correlations for Heat and Mass Transfer Coefficients.- C1 Heat Transfer Coefficients.- C1.1 Natural Convection Involving "Inside" Flow.- C1.2 Forced Convection Involving "Inside" Flow.- C1.3 Natural Convection Involving "Outside" Flow.- C1.4 Forced Convection Involving "Outside" Flow.- C2 Mass Transfer Coefficients.- C2.1 Mass Transfer Involving "Inside" Flow.- C2.2 Mass Transfer Involving "Outside" Flow.- Glossary of Symbols Used.
1. Balance Equations.- 1. Balance equations: An informal description.- 2. Conserved quantities for the balance equations.- 3. Balance equations for a single-phase material.- 4. Balance equations for a multiphase material.- II. Constitutive Relationships.- 5. Constitutive relationships: Near-equilibrium thermodynamics.- 6. Expressions for a single-phase material.- 7. Internal geometry & topology of multiphase materials.- 8. Expressions for the system in a multiphase material.- 9. Expressions for the solid phase in a multiphase material.- 10. Expressions for the liquid phase in a multiphase material.- 11. Expressions for the gas phase in a multiphase material.- III. Practical Considerations.- 12. Interlude: Use of chapters 1-11.- 13. Sources of material property values.- 14. Numerical solution: Some finite difference methods.- 15. Numerical solution: Some finite element methods.- IV. Implementation.- 16. Example: Glass refining.- 17. Example: Fiber spinning.- 18. Example: Alloy solidification.- 19. Example: Microwave heating of ceramics.- 20. Example: Combustion synthesis of refractory materials.- 21. Example: Binder removal from shaped powder compacts.- 22. Example: Flow of a solid-liquid suspension.- Appendices.- Appendix A: Details of Mathematical Quantities and Operations Used in the Text.- A1 Simple Algebra.- A2 Coordinate Systems.- A2.1 Essential Features of a Coordinate System.- A2.2 Transformations From One Coordinate System to Another.- A2.3 Particular Coordinate Systems.- A3 Algebra.- A3.1 Representation of Scalars, Vectors & Tensors.- A3.2 Algebraic Operations.- A3.3 Special Vectors and Tensors.- A3.4 Uses of Certain Special Second-Order Tensors.- A3.5 Useful Relationships Among Algebraic Operations.- A4 Spatial Derivatives and Differentials.- A4.1 SpatialDerivatives.- A4.2 Useful Relationships Involving Spatial Derivatives.- A4.3 Spatial Derivatives in Particular Coordinate Systems.- A4.4 Differentials & Regions of Infinitesimal Extent.- A5 Time Derivatives.- A5.1 Motion, Paths & Moving "Objects".- A5.2 Time Derivatives in Three Dimensions - General Case.- A5.3 Time Derivatives in Three Dimensions - Specific Cases.- A5.4 Time Derivatives Involving a Surface.- A6 Integrals.- A7 Derivatives of Integrals.- A7.1 Spatial Derivatives of Volume & Surface Integrals.- A7.2 Time-Derivatives of Volume Integrals.- A7.3 Time-Derivatives of Surface Integrals.- A8 Other Mathematical Details.- A8.1 Evaluation of Gaussian Integrals.- A8.2 Orthogonal Functions.- Appendix B: Units and Dimensions.- B1 Fundamental and Derived Units.- B2 Unit Conversions.- B3 Dimensional Analysis.- B3.1 Dimensional Analysis of the Balance Equations.- B3.2 Dimensional Analysis of Coupled Transport Processes.- B4 Non-dimensionalization.- Appendix C: Correlations for Heat and Mass Transfer Coefficients.- C1 Heat Transfer Coefficients.- C1.1 Natural Convection Involving "Inside" Flow.- C1.2 Forced Convection Involving "Inside" Flow.- C1.3 Natural Convection Involving "Outside" Flow.- C1.4 Forced Convection Involving "Outside" Flow.- C2 Mass Transfer Coefficients.- C2.1 Mass Transfer Involving "Inside" Flow.- C2.2 Mass Transfer Involving "Outside" Flow.- Glossary of Symbols Used.
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