Mathematical Modeling (eBook)

Cover 1
CONTENTS 8
PREFACE 18
PART I: METHOD AND MANNER 24
Chapter 1. What is Mathematical Modeling? 26
A Very Simple Example 26
Review of the Simplest Example 31
The Simplest Distributed Model 32
The General Balance Equations for Distributed Systems 33
Respecting Uniformity 38
Extensive and Intensive Quantities 41
General Observations on Forming the Model 43
Chapter 2. Manipulation of Models 48
Getting Rid of Unnecessary Equations 49
The Reduction of the Equations to Dimensionless Form 51
An Alternative Method of Reduction 53
Scaling 56
Shape Factors 59
A Priori Estimates 62
Scaling and Partial Solution in Linear Systems 63
Chapter 3. Solving the Equations 68
Getting a Feel for the Solution 68
Special Forms 72
Getting the Most from Calculations 74
Asymptotics and Perturbations 82
Moments and Generating Functions 87
Observing Conditions 90
Chapter 4. Presenting the Model and Its Behavior 98
The Phase Plane 99
Oscillations in Three Dimensions 110
Forced Oscillations and the Stroboscopic Phase Plane 111
Chapter 5. Maxims for Modelers 116
Chapter 6. Style 118
Literary Style 119
Genre 121
Plagiarism and Attribution 124
Publish or Perish 125
PART II: MATTER 128
Chapter 7. Dispersion in Flow 130
A. On the Dispersion of a Solute in a Fluid Flowing through a Tube 132
1. Introduction 132
2. The General Equations of Diffusion and Flow in a Straight Tube 133
3. The Tube of Circular Cross-Section 134
4. Some Special Initial Distributions of Solute 138
5. The General Case 139
6. Turbulent Flow in a Tube of Circular Cross-Section 141
7. Viscous Flow in a Tube of Arbitrary Cross-Section 141
References 143
B. On the Dispersion of a Solute by Diffusion, Convection, and Exchange between Phases 144
1. Introduction 144
2. Dispersion in Coaxial Cylindrical Annuli 145
3. Certain Special Cases 151
4. Dispersion in Coaxial Streams of Arbitrary Cross-Section 152
5. Application to the Theory of Chromatography 156
6. Application to a Simplified Theory of Distillation 156
References 158
C. On the Dispersion of Linear Kinematic Waves 159
1. Introduction 159
2. The Dispersion of a Flood Wave 160
3. General Theorems 163
4. A Kinematic Temperature Wave 163
5. The Ultimate Form of a Kinematic Wave 167
References 169
Chapter 8. Formal Kinetics 170
D. Prolegomena to the Rational Analysis of Systems of Chemical Reactions 172
1. Introduction 172
2. The Representation of Molecular Species and Reactions between Them 173
3. The Representation and Calculus of Composition Changes 177
4. Equilibrium in Systems of Reactions 180
5. Kinetics of Reactions 182
6. Reaction Mechanisms 184
7. Entropy Production 188
8. Discussion 189
Nomenclature 190
References 191
E. Prolegomena to the Rational Analysis of Systems of Chemical Reactions II. Some Addenda 193
1. Introduction 193
2. The Uniqueness of Equilibrium under Adiabatic Conditions 194
3. The Consistency of Certain Kinetic and Equilibrium Expressions 195
4. Reaction Mechanisms and Exact Sequences 197
5. Of Chemical Kinetics in General 200
References 202
F. Modelling Cubic Autocatalysis by Successive Bimolecular Steps 203
1. Introduction 203
2. Kinetic Schemes and Mass-Balance Equations 204
3. Behaviour of First-Order Correction to Autocatalator: Stationary-States and Limit Cycles 206
4. Comparison of First-Order Equations with Full, Three-Variable Model 209
Conclusions 210
References 211
G. Reactions in Continuous Mixtures 212
Introduction 212
General Formulation for a Single Index 214
Parallel Reaction in a Doubly Distributed Continuum 217
Examples 218
Generalized Background Kinetics 222
Discrete Distributions 224
Distributions of k(x) 225
Asymptotic Behavior 227
Sequential Parallel Reactions 228
Mechanisms 230
Literature Cited 232
H. Reaction of a Continuous Mixture in a Bubbling Fluidized Bed 234
Introduction 234
Gamma Distributions 236
Application of the Gamma Distribution 237
A General Theorem for Simple, Linear Reactor Models 238
Application to a Model of the Bubbling Fluidized Bed 238
The Damköhler Number 241
The Fluid Bed with Astarita's Uniform Kinetics 243
Nomenclature 244
References 246
Chapter 9. Statics and Dynamics of Chemical Reactors 248
I. Some Common Features of Periodically Forced Reacting Systems 250
Introduction 250
Numerical Methods 252
Forced Dynamic Phenomena 263
Conclusions 271
References 272
J. *Yet Who Would Have Thought the Old Man to Have Had So Much Blood in Him?"„Reflections on the Multiplicity of Steady States of the Stirred Tank Reactor 275
Introduction 275
The System 277
Discussion I: Butterfly Points 284
Discussion II: Maximum Multiplicity 293
Conclusions 301
Notation 303
References 303
K. Autonomous Bifurcations of a Simple Biomolecular Surface-Reaction Model 305
1. Introduction 305
2. Surface Reaction Model 306
3. Bifurcation Analysis 309
4. The Stability of the Steady States 317
5. Hopf Bifurcations 321
6. Homoclinic Bifurcations 325
7. Discussion 326
References 328
L. Forced Oscillations of a Self-Oscillating Bimolecular Surface Reaction Model 330
1. Introduction 330
2. Surface Reaction Model 332
3. Mathematical and Numerical Framework 334
4. Excitation Diagram 337
5. Discussion 350
References 354
Chapter 10. Mass and Heat Transfer 358
M. An Example of the Relation between Discrete and Continuous Models 360
The Geometry of the Hexaga 360
Heat Transfer 362
The Discrete Model 364
The Continuous Model 365
Two Lemmas 366
Equivalence of the Models in the Limit e.0 367
N. A General Theory of Anisotropic Membranes 368
Introduction 368
Exponential Dependence 369
Designing for Maximum Anisotropy 373
Application 376
Other Configurations 378
Nomenclature 380
References 381
Chapter 11. Modeling in General 382
O. Of Chemical Engineering and the Liberal Arts: An Inaugural for the Olaf Hougen Visiting Professorship: October 3, 1979 384
P. Two Eyes Are Better Than One: Some Refledons on the Importance of Having More Than One Viewpoint in Mathematical Modelling and Other Disciplines 397
References 422
Q. Reflections on Keats' Equation 423
Keats' Equation 423
A Model of Algal Growth 431
Notation 436
References 437
R. Chemical Engineering Greetings 438
PART III: MISCELLANEA 440
Acknowledgments: An Autobiographical Appendix with Asides 442
Early Education, 1935–1943 443
Canford, 1943–1946 444
Billingham, 1946–1948 445
Edinburgh, 1948–1950 446
Billingham, 1950–1955 449
Minnesota, 1955–1956 453
Edinburgh, 1956–1958 455
Minneapolis, 1958–1964 456
Cambridge, 1964–1965 459
Aside on Formal Chemical Kinetics, 1963–1995 461
Minneapolis, 1965–1971 465
Cambridge, 1971–1972 466
Minneapolis, 1972–1974 467
Minneapolis, 1974–1996 470
Minnesota and Sabbaticals, 1978–1996 471
BIBLIOGRAPHY 478
Books 478
Edited Books 479
Chapters in Books Edited by Others 479
Journal Papers 479
INDEX OF GRADUATE STUDENTS AND CO-AUTHORS 490
SUBJECT INDEX TO THE PAPERS IN THE BIBLIOGRAPHY 492
INDEX 496