1 Elements of Thermodynamically Constrained Averaging
Theory ..................................................... 1
1.1 Overview ................................................... 1
1.2 Identification of Scales for Modeling ...................... 4
1.2.1 Length Scales ....................................... 4
1.2.2 Time Scales ........................................ 11
1.3 Thermodynamically Constrained Averaging Theory Approach ... 13
1.3.1 Overview ........................................... 13
1.3.2 Microscale Conservation Principles ................. 13
1.3.3 Microscale Thermodynamics .......................... 15
1.3.4 Microscale Equilibrium Conditions .................. 16
1.3.5 Averaging Theorems ................................. 18
1.3.6 Larger-scale Conservation Principles ............... 19
1.3.7 Larger-scale Thermodynamics ........................ 20
1.3.8 Evolution Equations ................................ 22
1.3.9 Constrained Entropy Inequality ..................... 22
1.3.10 Simplified Entropy Inequality ...................... 24
1.3.11 Closure Relations .................................. 25
1.3.12 Closed Models ...................................... 26
1.3.13 Subscale Modeling and Applications ................. 27
1.4 Summary ................................................... 28
Exercises ................................................. 29
References ................................................ 29
2 Microscale Conservation Principles ........................ 37
2.1 Overview .................................................. 37
2.2 General Conservation and Balance Principles ............... 39
2.2.1 General Qualitative Formulation .................... 39
2.2.2 General Quantitative Formulation ................... 40
2.2.3 Species-based Quantitative Formulation ............. 42
2.3 Conservation and Balance Principles for a Phase ........... 44
2.3.1 General Microscale Point Forms ..................... 44
2.3.2 Specific Conservation and Balance Principles ....... 47
2.4 Conservation and Balance Principles for an Interface ...... 56
2.4.1 General Microscale Point Form ...................... 57
2.4.2 Specific Conservation and Balance Principles ....... 61
2.5 Conservation and Balance Principles for a Common Curve .... 68
2.5.1 General Microscale Point Form ...................... 69
2.5.2 Specific Conservation and Balance Principles ....... 73
2.6 General Multispecies Formulation for a Common Point ....... 78
2.6.1 General Microscale Point Form ...................... 79
2.7 Summary ................................................... 81
Exercises ................................................. 82
References ................................................ 84
3 Microscale Thermodynamics ................................. 87
3.1 Overview .................................................. 87
3.2 Essence of Equilibrium Thermodynamics ..................... 90
3.3 Fluid-phase Equilibrium Thermodynamics .................... 91
3.3.1 Fundamental and Differential Forms ................. 91
3.3.2 Euler Equation for Internal Energy ................. 96
3.3.3 Gibbs-Duhem Equation for a Fluid ................... 97
3.4 Normalized Internal Energy Formulation .................... 98
3.5 Other Thermodynamic Potentials ........................... 100
3.5.1 Helmholtz Free Energy ............................. 101
3.5.2 Enthalpy .......................................... 103
3.5.3 Gibbs Free Energy ................................. 105
3.5.4 Comments on Energy Potentials ..................... 107
3.6 Relation between  P and V .............................. 108
3.7 Solid-phase Equilibrium Thermodynamics ................... 110
3.8 Interface and Common Curve Equilibrium Thermodynamics .... 114
3.8.1 Interface Thermodynamics .......................... 115
3.8.2 Common Curve Thermodynamics ....................... 117
3.9 Microscale Multiphase System Notation .................... 118
3.10 Partial Mass Quantities .................................. 120
3.10.1 Fluid Phase ....................................... 120
3.10.2 Solid Phase ....................................... 122
3.10.3 Interface ......................................... 123
3.10.4 Common Curve ...................................... 124
3.11 Classical Irreversible Thermodynamics (CIT) .............. 124
3.12 Other Thermodynamic Theories ............................. 128
3.12.1 Rational Thermodynamics (RT) ...................... 128
3.12.2 Extended Irreversible Thermodynamics (EIT) ........ 129
3.12.3 Theory of Internal Variables (TIV) ................ 129
3.13 Summary .................................................. 130
Exercises ................................................ 131
References ............................................... 132
4 Microscale Equilibrium Conditions ........................ 135
4.1 Overview ................................................. 135
4.2 Components of Variational Analysis ....................... 136
4.3 Variation of Microscale Quantities ....................... 137
4.3.1 Variation of Green's Strain Tensor ................ 137
4.4 Variation of Energy Integrals ............................ 139
4.4.1 Analysis of the Integral of as:8(Cs/js) ........... 144
4.4.2 General Condition of Equilibrium .................. 147
4.5 Single-fluid-phase Porous Medium System .................. 148
4.5.1 Equilibrium Conditions ........................... 151
4.6 Two-fluid-phase Porous Medium System ..................... 152
4.6.1 Identification of Index Sets ...................... 153
4.6.2 Identification of Simpler Equilibrium Conditions .. 154
4.6.3 Equilibrium Variational Analysis .................. 155
4.6.4 Additional Equilibrium Conditions ................. 160
4.7 Summary .................................................. 163
Exercises ................................................ 163
References ............................................... 164
5 Microscale Closure for a Fluid Phase ..................... 167
5.1 Overview ................................................. 167
5.2 System Definition ........................................ 169
5.3 Conservation and Thermodynamic Equations ................. 170
5.3.1 Entropy Inequality ................................ 171
5.3.2 Conservation Equations ............................ 172
5.3.3 Thermodynamic Relations ........................... 172
5.4 Constrained Entropy Inequality ........................... 173
5.4.1 Introduction of Constraints ....................... 173
5.4.2 Selection of Lagrange Multipliers ................. 175
5.4.3 Reduction to the CEI .............................. 178
5.5 Simplified Entropy Inequality ............................ 181
5.5.1 Introduction of Approximations .................... 182
5.5.2 Consideration of Equilibrium Conditions ........... 183
5.6 Closure Relations ........................................ 184
5.6.1 Count of Variables ................................ 184
5.6.2 Diffusive Flux Rearrangement ...................... 186
5.7 Special Cases ............................................ 186
5.7.1 Single-species Phase .............................. 187
5.7.2 Single-species, Isothermal Phase .................. 187
5.8 Conjugate Force-flux Closure ............................. 187
5.8.1 Stress Tensor ..................................... 188
5.8.2 Diffusion Vector .................................. 191
5.8.3 Non-advective Heat Flux ........................... 192
5.8.4 Chemical Reaction ................................. 193
5.9 Cross-coupled Closure .................................... 193
5.10 Summary .................................................. 196
Exercises ................................................ 197
References ............................................... 198
6 Macroscale Conservation Principles ....................... 201
6.1 Overview ................................................. 201
6.2 Averaging Conventions and Notation ....................... 203
6.2.1 Intrinsic Averages ................................ 205
6.2.2 Mass Averages ..................................... 205
6.2.3 Unique Averages ................................... 206
6.2.4 Examples of Averaging Notation .................... 206
6.3 Averaging Theorems ....................................... 210
6.3.1 Averaging Theorems for Phases ..................... 211
6.3.2 Averaging Theorems for Interfaces ................. 214
6.3.3 Averaging Theorems for Common Curves .............. 216
6.3.4 Averaging Theorem for Common Points ............... 217
6.4 Application of Averaging Process ......................... 217
6.5 Macroscale Principles for a Phase ........................ 218
6.5.1 Conservation of Mass .............................. 218
6.5.2 Conservation of Momentum .......................... 225
6.5.3 Conservation of Energy ............................ 230
6.5.4 Balance of Entropy ................................ 235
6.5.5 Body Force Potential .............................. 238
6.6 On the Forms of Macroscale Equations ..................... 240
6.7 Macroscale Principles for an Interface ................... 242
6.7.1 Example: Conservation of Species Mass ............. 242
6.7.2 Comment on Interface Equations .................... 245
6.8 Macroscale Principles for a Common Curve ................. 246
6.8.1 Example: Conservation of Common Curve Momentum .... 247
6.8.2 Comment on Common Curve Equations ................. 249
6.9 Mixed Forms of Macroscale Equations ................. 250
6.9.1 Species-based Equations ........................... 250
6.9.2 Entity-based Energy and Entropy ................... 251
6.9.3 Entity-based Momentum, Energy, and Entropy ........ 254
6.10 Internal Energy Equation ................................. 256
6.10.1 Species-and Entity-based Equations ................ 256
6.10.2 Mixed Formulation with Species Conservation ....... 257
6.11 Summary .................................................. 258
Exercises ................................................ 260
References ............................................... 260
7 Macroscale Thermodynamics ................................ 263
7.1 Overview ................................................. 263
7.2 Macroscale Euler Equations ............................... 264
7.2.1 Fluid Phase ....................................... 264
7.2.2 Solid Phase ....................................... 267
7.2.3 Interface and Common Curve ........................ 268
7.3 Macroscale Energy Differentials .......................... 269
7.4 Fluid Energy Dynamics .................................... 271
7.4.1 Fluid Species Energy .............................. 272
7.4.2 Fluid Species Potential Energy .................... 275
7.4.3 Fluid-phase Energy ................................ 276
7.4.4 Fluid-phase Potential Energy ...................... 277
7.5 Solid-phase Energy Dynamics .............................. 278
7.5.1 Solid Species Energy .............................. 278
7.5.2 Solid Species Potential Energy .................... 284
7.5.3 Solid-phase Energy ................................ 284
7.5.4 Solid-phase Potential Energy ...................... 285
7.6 Interface Energy Dynamics ................................ 285
7.6.1 Interface Species Energy .......................... 286
7.6.2 Interface Species Potential Energy ................ 289
7.6.3 Interface-entity Energy ........................... 289
7.6.4 Interface-entity Potential Energy ................. 290
7.7 Common Curve Energy Dynamics ............................. 291
7.7.1 Common Curve Species Energy ....................... 291
7.7.2 Common Curve Species Potential Energy ............. 292
7.7.3 Common Curve Entity Energy ........................ 292
7.7.4 Common Curve Entity Potential Energy .............. 293
7.8 Equilibrium Conditions ................................... 293
7.8.1 Two-phase Equilibrium Conditions .................. 294
7.8.2 Three-phase Equilibrium Conditions ................ 296
7.9 Summary .................................................. 299
Exercises ................................................ 299
References ............................................... 300
8 Evolution Equations ...................................... 301
8.1 Overview ................................................. 301
8.2 Derivation of Evolution Equations ........................ 303
8.2.1 General Expression ................................ 303
8.3 Single-fluid-phase Flow .................................. 305
8.3.1 Phases ............................................ 305
8.3.2 Interface ......................................... 306
8.4 Single-fluid-phase Flow Geometric Relations .............. 306
8.5 Two-fluid-phase Flow ..................................... 308
8.5.1 Solid Phase ....................................... 309
8.5.2 Fluid Phases ...................................... 310
8.5.3 Fluid-solid Interfaces ............................ 311
8.5.4 Fluid-fluid Interface ............................. 312
8.5.5 Common Curve ...................................... 314
8.6 Two-fluid-phase Flow Geometric Relations ................. 316
8.6.1 Solid Phase and Fluid-solid Interfaces ............ 316
8.6.2 Fluid-fluid Interface Evolution ................... 317
8.6.3 Common Curve Evolution ............................ 320
8.7 Average Normal Velocities ................................ 321
8.7.1 Fluid-fluid Interface Velocity Approximation ...... 322
8.7.2 Common Curve Velocity Approximation ............... 323
8.8 Summary .................................................. 324
Exercises ................................................ 325
References ............................................... 326
9 Single-fluid-phase Flow .................................. 327
9.1 Overview ................................................. 327
9.2 Single-phase-flow System Definition ...................... 329
9.3 Conservation and Thermodynamic Equations ................. 332
9.3.1 Entropy Inequality ................................ 332
9.3.2 Conservation Equations ............................ 333
9.3.3 Thermodynamic Relations ........................... 334
9.4 Constrained Entropy Inequality ........................... 335
9.4.1 Augmented Entropy Inequality ...................... 335
9.4.2 Selection of Lagrange Multipliers ................. 336
9.4.3 Elimination of Time Derivatives ................... 339
9.4.4 Manipulation Insights ............................. 341
9.4.5 Formulation of the CEI ............................ 344
9.5 Simplified Entropy Inequality ............................ 348
9.5.1 The Need for Approximations ....................... 349
9.5.2 Elimination of Terms .............................. 349
9.5.3 Approximation of Averages ......................... 351
9.5.4 General SEI ....................................... 354
9.6 Example Restricted Application ........................... 355
9.6.1 Statement of Secondary Restriction ................ 356
9.6.2 Count of Variables ................................ 357
9.6.3 Reduction in Number of Variables .................. 358
9.6.4 Conjugate Force-flux Closure ...................... 360
9.6.5 Closed Conservation Equation Set .................. 362
9.7 Model of Fluid and Elastic Solid ......................... 363
9.7.1 Compressible Elastic Solid with Small
Deformation ....................................... 364
9.7.2 Passive Solid Phase ............................... 367
9.8 Summary .................................................. 370
Exercises ................................................ 370
References ............................................... 371
10 Single-fluid-phase Species Transport ..................... 373
10.1 Overview ................................................. 373
10.2 System Definition by Primary Restrictions ................ 375
10.3 Constrained Entropy Inequality ........................... 377
10.3.1 Augmented Entropy Inequality ...................... 377
10.3.2 Determination of Lagrange Multipliers ............. 380
10.3.3 Formulation of the CEI ............................ 384
10.4 Simplified Entropy Inequality ............................ 389
10.4.1 Elimination of Small Terms ........................ 390
10.4.2 Breaking of Averages .............................. 392
10.4.3 General SEI ....................................... 394
10.5 SEI for Application to Non-isothermal Species Transport .. 396
10.5.1 Imposition of Secondary Restrictions .............. 396
10.6 Isothermal Transport with No Interphase Mass Exchange .... 400
10.6.1 Simplification of SEI ............................. 400
10.6.2 Count of Equations and Variables .................. 401
10.7 Isothermal Transport with Interphase Mass Exchange ....... 402
10.7.1 Simplification of the SEI ......................... 402
10.7.2 Additional Variables and Constraints .............. 403
10.8 Unitemperature, Nonisothermal Transport .................. 404
10.8.1 Simplification of the SEI for a Single
Temperature ....................................... 404
10.8.2 Accounting for Additional Variables ............... 405
10.9 Multi-temperature Species Transport ...................... 407
10.9.1 SEI for the Multi-temperature Case ................ 407
10.9.2 Treatment of Additional Variables ................. 408
10.10 Example of Conjugate Force-flux Closure ................. 409
10.10.1 Closure Relations ................................ 410
10.10.2 Closed Set of Conservation Equations ............. 414
10.11 Summary ................................................. 418
Exercises ................................................ 419
References ............................................... 420
11 Two-phase Flow ........................................... 421
11.1 Overview ................................................. 421
11.2 Primary Restrictions ..................................... 423
11.3 Constrained Entropy Inequality Statement ................. 424
11.3.1 Entropy Inequality ................................ 424
11.3.2 Augmented Entropy Inequality ...................... 424
11.3.3 Selection of Lagrange Multipliers ................. 425
11.3.4 Expanded CEI ...................................... 425
11.4 Simplified Entropy Inequality ............................ 429
11.4.1 Required SEI Approximations ....................... 430
11.4.2 Basic SEI Approximations .......................... 431
11.4.3 Complex SEI Approximations ........................ 435
11.4.4 Product Breaking SEI Approximations ............... 442
11.4.5 General SEI ....................................... 445
11.5 Example Application ...................................... 448
11.5.1 Selection of Secondary Restrictions ............... 448
11.5.2 Identification of Variables ....................... 450
11.5.3 Specification of Closure Conditions ............... 450
11.5.4 Closed Set of Conservation Equations .............. 454
11.6 Simplified Momentum Equations for Example Model .......... 456
11.7 Immobile Solid ........................................... 459
11.8 Summary .................................................. 461
Exercises ................................................ 462
References ............................................... 463
12 Modeling Approach and Extensions ......................... 465
12.1 Overview ................................................. 465
12.2 Modeling Process ......................................... 466
12.2.1 On Primary Restrictions ........................... 467
12.2.2 On SEI Approximations ............................. 468
12.2.3 On Secondary Restrictions and Closure ............. 469
12.3 Subscale Modeling ........................................ 470
12.3.1 Capillarity Effects ............................... 471
12.3.2 Testing of Approximations ......................... 473
12.4 Macroscale Modeling ...................................... 473
12.4.1 Model Validation .................................. 474
12.4.2 Model Verification ................................ 476
12.5 Extensions of TCAT Models ................................ 477
12.5.1 Model Class Extensions: Equations ................. 477
12.5.2 Mixed-scale Dimensionality: Averaging ............. 479
12.5.3 Multiphysics: Linking of Larger-scale Systems ..... 480
12.5.4 Alternative Thermodynamic Theories ................ 481
12.5.5 Nonlinearities: Closure Relations ................. 482
12.5.6 Applications: Dynamically Coupled Multiscale
Systems ........................................... 483
12.5.7 Subscale Modeling and Applications: Stochastic
Systems ........................................... 483
12.6 Summary .................................................. 484
Exercises ................................................ 485
References ............................................... 485
A Considerations on Calculus of Variations ................. 489
A.l Fundamentals of Variational Approaches ................... 489
A.2 Classical Approach to Volume Integrals ................... 490
A.3 Indicator Functions ...................................... 493
A.3.1 Universal Properties .............................. 494
A.3.2 Integral over a Phase ............................. 495
A.3.3 Integral over an Interface ........................ 495
A.3.4 Integral over a Common Curve ...................... 496
A.4 Variation of a Volume Integral ........................... 497
A.5 Variation of a Surface Integral .......................... 498
A.6 Variation of an Integral over a Curve .................... 501
A.7 Summary .................................................. 505
Exercises ................................................ 506
References ............................................... 506
В Derivations of Averaging Theorems ........................ 509
B.1 Overview ................................................. 509
B.1.1 Naming Convention ................................. 510
B.2 Coordinate Systems ....................................... 510
B.3 Averaging Theorems for Volumes ........................... 512
B.3.1 D[3,(3,0),0] ...................................... 514
B.3.2 G[3,(3,0),0] ...................................... 514
B.3.3 T[3,(3,0),0] ...................................... 515
B.4 Averaging Theorems for Surfaces .......................... 515
B.4.1 D[2,(3,0),0] ...................................... 517
B.4.2 G[2,(3,0),0] ...................................... 519
B.4.3 T[2,(3,0),0] ...................................... 519
B.5 Averaging Theorems for Curves ............................ 521
B.5.1 D[1,(3,0),0] ...................................... 522
B.5.2 G[1,(3,0),0] ...................................... 524
B.5.3 T[1,(3,0),0] ...................................... 524
B.6 Averaging Theorems for Points ............................ 526
B.6.1 T[0,(3,0),0] ...................................... 526
Exercises ................................................ 526
References ............................................... 528
С Constrained Entropy Inequality Derivations ............... 529
C.l CEI for Single-fluid-phase Flow, Eq. (9.43) .............. 529
C.2 CEI for Single-fluid-phase Transport, Eq. (10.14) ........ 541
C.3 CEI for Two-fluid-phase Flow, Eq. (11.5) ................. 555
Index ......................................................... 573
|
