Table of Contents
Preface ..................................................... xxiii
Acknowledgments .............................................. xxix
List of Symbols .............................................. xxxi
Chapter 1
Problems 111; One Variable, 1st Order, 1st Kind Boundary
Condition ................................................. 1
1.1 Introduction
1.1.1 Model and Reality ................................. 1
1.1.2 Verification of Solutions ......................... 4
1.1.3 Use of Dimensionless Variables .................... 4
1.2 Heating of a Solid ........................................ 5
1.2.1 Solution by Separation of Variables ............... 7
1.3 Flow between Two Drums ................................... 10
1.4 Heating of a Fluid in a Stirring Tank .................... 12
1.4.1 Variation of Parameters .......................... 15
1.5 Heated Batch Reactor ..................................... 17
1.5.1 Solution by Laplace Transform .................... 18
1.5.2 Solution by Weighted Residuals ................... 21
1.5.2.1 First Approximation .................... 22
1.5.2.2 Another Alternative .................... 22
1.5.2.3 Second Approximation ................... 25
1.5.2.4 When to Stop ........................... 26
1.6 Reactor with a Time-Controlled Rate ...................... 26
1.6.1 Solution by Picard's Method ...................... 26
1.6.2 Exact Solution ................................... 28
1.7 Pressure in a Resting Fluid .............................. 30
1.7.1 Solution by Separation of Variables .............. 32
1.8 Pressure in Fluid under Rotational Movement .............. 33
1.8.1 Solution by Separation of Variables .............. 34
1.9 Plug-Flow Reactor ........................................ 36
1.9.1 Solution by Separation of Variables .............. 38
1.9.2 Comments ......................................... 40
1.10 Heat Conduction in an Indefinite Wall .................... 41
1.10.1 Solution by Separation of Variables .............. 43
1.11 Plate-and-Cone Viscometer ................................ 43
1.12 Thermocouple ............................................. 48
Exercises ...................................................... 50
References ..................................................... 52
Chapter 2
Problems 112; One Variable, 1st Order, 2nd Kind
Boundary Condition ............................................. 55
2.1 Introduction ............................................. 55
2.2 Heating of a Solid ....................................... 55
2.2.1 Solution by Separation of Variables .............. 56
2.3 Heat Conduction in a Spherical Shell ..................... 57
2.3.1 Solution by Separation of Variables .............. 58
2.4 Batch Reactor ............................................ 60
2.4.1 Solution by Separation of Variables .............. 62
2.4.2 Solution by Laplace Transform .................... 62
2.4.3 Discussion ....................................... 63
2.5 Plug-Flow Reactor ........................................ 64
2.5.1 Solution by Separation of Variables .............. 66
Exercises ...................................................... 68
Reference ...................................................... 69
Chapter 3
Problems 113; One Variable, 1st Order, 3rd Kind
Boundary Condition ............................................. 71
3.1 Introduction ............................................. 71
3.2 Heating of a Solid with Controlled Heat Transfer Rate .... 71
3.2.1 Solution by Separation of Variables .............. 72
3.3 Temperature-Controlled Batch Reactor ..................... 72
3.3.1 Solution by Separable Equation ................... 75
3.3.2 Solution by Laplace Transform .................... 77
3.3.3 Arrhenius Relation ............................... 78
3.3.4 Solution by Weighted Residuals ................... 78
3.3.4.1 Choosing the Form of Approximations .... 79
3.3.4.2 First Approximation .................... 80
3.3.4.3 Collocation Method ..................... 80
3.3.4.4 Second Approximation ................... 81
3.4 Batch Reactor ............................................ 82
3.4.1 Solution by Separation of Variables .............. 83
Exercises ...................................................... 85
Chapter 4
Problems 121; One Variable, 2nd Order, 1st Kind Boundary
Condition ................................................ 87
4.1 Introduction ............................................. 87
4.2 Mass Transfer through a Cylindrical Rod .................. 87
4.3 Mass Transfer in a Rod with Variable Diffusivity ......... 90
4.3.1 Application of Method of Weighted Residuals ...... 93
4.3.1.1 Choice of Trial Functions .............. 93
4.3.1.2 First Approximation .................... 93
4.3.1.3 Second Approximation ................... 96
4.4 Conduction through a Pipe Wall ........................... 99
4.5 Electrically Heated Pipe Wall ........................... 101
4.6 Heat Transfer in a Spherical Shell ...................... 105
4.7 Absorption without Reaction ............................. 107
4.8 Diffusion through a Spherical Shell with Zero-Order
Reaction ................................................ 112
4.9 Absorption with Homogeneous Reaction .................... 114
4.9.1 Case of Low Solubility .......................... 115
4.9.2 Case of High Solubility ......................... 117
4.9.3 Solution by MWR ................................. 118
4.9.4 Variable Global Density or Concentration ........ 120
4.10 Reacting Particle ....................................... 120
4.10.1 Solution for Spherical Particles ................ 125
4.10.1.1 Solution through Power Series ......... 127
4.10.1.2 Solution Using Laplace Transform ...... 130
4.10.1.3 Solution Using Method of Weighted
Residuals ............................. 130
4.10.2 Solution for Cylindrical Particles .............. 132
4.10.3 Solution for Flat Particles ..................... 132
4.10.4 Genera] Irregular Shape ......................... 133
4.11 Heat Transfer through a Reacting Plate .................. 133
Exercises ..................................................... 136
References .................................................... 140
Chapter 5
Problems 122; One Variable, 2nd Order, 2nd Kind Boundary
Condition ............................................... 141
5.1 Introduction ............................................ 141
5.2 Flow on an Inclined Plate ............................... 141
5.3 Flow in an Inclined Tube ................................ 144
5.4 Rectangular Fin ......................................... 148
5.5 Circular Fin ............................................ 154
5.5.1 Solution by Laplace Transform ................... 158
5.6 Film Condensation ....................................... 159
5.6.1 Momentum Conservation ........................... 161
5.6.2 Energy Conservation ............................. 163
5.6.3 Comments ........................................ 165
5.7 Heat Transfer through a Reacting Plate .................. 165
Exercises ..................................................... 168
References .................................................... 169
Chapter 6
Problems 123; One Variable, 2nd Order, 3rd Kind Boundary
Condition ............................................... 171
6.1 Introduction ............................................ 171
6.2 Heat Transfer between a Plate and Fluids ................ 171
6.3 Heat Transfer in a Spherical Shell ...................... 173
6.4 Reacting Particle ....................................... 176
6.4.1 Unreacted-Core Model ............................ 177
6.4.1.1 Comments .............................. 183
6.4.2 Exposed-Core Model .............................. 183
6.5 Heat Transfer between a Reacting Plate and a Fluid ...... 185
Exercises ..................................................... 188
References .................................................... 189
Chapter 7
Problems 211; Two Variables, 1st Order, 1st Kind Boundary
Condition ............................................... 191
7.1 Introduction ............................................ 191
7.2 Pressure in Fluid under Rotational Movement ............. 191
7.2.1 Separable Equations ............................. 192
7.3 Heating a Flowing Liquid ................................ 195
7.3.1 Solution by Laplace Transform ................... 198
7.3.2 Comments ........................................ 199
7.3.3 Complete Dimensionless Form ..................... 201
7.3.4 Considerations on Possible Application of
Similarity ...................................... 202
7.3.5 Considerations on Possible Application of
Method of Weighted Residues ..................... 206
7.3.5.1 First Approximation ................... 207
7.4 Plug-Flow Reactor ....................................... 208
7.4.1 Solution by Laplace Transform ................... 211
7.4.2 Complete Dimensionless Form ..................... 213
7.4.3 Solution by Method of Weighted Residues ......... 217
7.4.3.1 First Approximation ................... 218
Exercises ..................................................... 219
Chapter 8
Problems 212; Two Variables, 1st Order, 2nd Kind Boundary
Condition ............................................... 221
8.1 Introduction ............................................ 221
8.2 Heating of Flowing Liquid ............................... 221
8.2.1 Solution by Laplace Transform ................... 223
8.2.2 Complete Dimensionless Form ..................... 225
8.2.3 Comments on Similarity .......................... 226
8.3 Plug-Flow Reactor ....................................... 228
8.3.1 Solution by Laplace Transform ................... 231
Exercises ..................................................... 233
Chapter 9
Problems 213; Two Variables, 1st Order, 3rd Kind Boundary
Condition ............................................... 235
9.1 Introduction ............................................ 235
9.2 Heating of Flowing Liquid ............................... 235
9.2.1 Solution by Laplace Transform ................... 238
9.3 Dynamic Plug-Flow Reactor ............................... 240
9.3.1 Solution by Laplace Transform ................... 243
Exercises ..................................................... 246
Chapter 10
Problems 221; Two Variables, 2nd Order, 1st Kind Boundary
Condition ............................................... 247
10.1 Introduction ............................................ 247
10.2 Heating an Insulated Rod or a Semi-Infinite Body ........ 248
10.2.1 Solution by Laplace Transform ................... 250
10.2.2 Heat Flux to the Rod ............................ 251
10.2.3 Complete Dimensionless Form ..................... 252
10.2.4 Solution by Similarity .......................... 253
10.3 Sudden Motion of a Plate ................................ 256
10.3.1 Boundary Conditions ............................. 257
10.3.2 Solution by Laplace Transform ................... 258
10.3.3 Solution by Similarity .......................... 259
10.3.4 Shear Stress .................................... 262
10.4 Heating a Flowing Liquid ................................ 262
10.4.1 Application of Laplace Transform ................ 264
10.5 Plug-Flow Reactor ....................................... 268
10.5.1 Solution by Laplace Transform ................... 270
10.6 Temperature Profile in a Rectangular Plate .............. 275
10.6.1 Solution by Separation of Variables ............. 276
10.6.2 Solution by the Method of Weighted Residuals .... 279
10.6.2.1 First Approximation ................... 280
10.6.2.2 Further Approximations ............... 280
10.7 Heating a Liquid Film ................................... 282
10.7.1 Solution by Similarity .......................... 284
10.8 Absorbing Flowing Film .................................. 288
10.8.1 Solution by Similarity .......................... 291
10.8.2 Comments ........................................ 292
Exercises ..................................................... 295
Reference ..................................................... 298
Chapter 11
Problems 222; Two Variables, 2nd Order, 2nd Kind Boundary
Condition ............................................... 299
11.1 Introduction ............................................ 299
11.2 Heating an Insulated Rod or Semi-Infinite Body .......... 299
11.2.1 Solution by Laplace Transform ................... 301
11.2.2 Solution by Separation of Variables ............. 303
11.2.2.1 Comments .............................. 307
11.3 Drying of a Spherical Particle .......................... 309
11.3.1 Solution by Laplace Transform ................... 312
11.4 Heating a Cylinder ...................................... 315
11.4.1 Solution by Laplace Transform ................... 317
11.5 Insulated Rod with Prescribed Initial Temperature
Profile ................................................. 320
11.5.1 Solution by Fourier Transform ................... 322
11.6 Plate-and-Cone Viscometer ............................... 326
11.6.1 Solution by the Method of Variables
Separation ...................................... 327
11.7 Heating a Rectangular Plate ............................. 329
11.7.1 Solution by Separation of Variables ............. 332
Exercises ..................................................... 335
References .................................................... 336
Chapter 12
Problems 223; Two Variables, 2nd Order, 3rd Kind Boundary
Condition ............................................... 339
12.1 Introduction ............................................ 339
12.2 Convective Heating of an Insulated Rod
or Semi-Infinite Body ......................................... 339
12.2.1 Solution by Laplace Transform ................... 342
12.3 Drying of a Spherical Particle .......................... 345
12.3.1 Solution by Laplace Transform ................... 348
12.3.2 Solution by Separation of Variables ............. 350
12.4 Convective Heating of a Cylinder ........................ 354
12.4.1 Solution by Laplace Transform ................... 356
12.4.2 Solution by Separation of Variables ............. 358
12.5 Convective Heating of Insulated Rod with Prescribed
Temperature at One End .................................. 362
12.5.1 Solution by Methods of Weighted Residuals ....... 364
12.5.1.1 First Approximation ................... 365
12.5.1.2 Alternative Form for the
Approximation ......................... 367
12.5.2 Solution by Separation of Variables ............. 371
12.5.3 Comments ........................................ 374
12.6 Convective Heating of a Plate ........................... 375
12.6.1 Solution by Separation of Variables ............. 378
12.7 Convective Heating of a Plate with Prescribed
Temperature Function at One Face ........................ 382
12.7.1 Solution by Separation of Variables ............. 384
12.7.2 Solution by the Method of Weighted Residuals .... 386
12.7.2.1 First Approximation ................... 388
Exercises ..................................................... 389
References .................................................... 391
Chapter 13
Problems 311; Three Variables, 1st Order, 1st Kind Boundary
Condition ............................................... 393
13.1 Introduction ............................................ 393
13.2 Heating a Flowing Liquid ................................ 393
13.2.1 Solution by Laplace Transform ................... 396
13.2.1.1 Completely Dimensionless Variables .... 399
13.2.2 Heating Rate .................................... 401
13.3 Two-Dimensional Reacting Flow ........................... 402
13.3.1 Solution by Laplace Transform ................... 405
13.3.2 Rate of Reactant Consumption .................... 408
Exercises ..................................................... 409
Chapter 14
Problems 312; Three Variables, 1st Order, 2nd Kind Boundary
Condition ............................................... 411
14.1 Introduction ............................................ 411
14.2 Heating a Flowing Liquid ................................ 411
14.2.1 Basic Equations and Boundary Conditions ......... 413
14.2.2 Solution by Laplace Transform ................... 414
14.3 Two-Dimensional Reacting Flow ........................... 418
14.3.1 Solution by Laplace Transform ................... 421
Exercises ..................................................... 424
Chapter 15
Problems 313; Three Variables, 1st Order, 3rd Kind Boundary
Condition ............................................... 427
15.1 Introduction ............................................ 427
15.2 Heating a Flowing Liquid ................................ 427
15.2.1 Solution by Laplace Transform ................... 430
15.3 Two-Dimensional Reacting Flow ........................... 436
15.3.1 Solution by Laplace Transform ................... 439
Exercises ..................................................... 445
Chapter 16
Problems 321; Three Variables, 2nd Order, 1st Kind Boundary
Condition ............................................... 447
16.1 Introduction ............................................ 447
16.2 Temperatures in a Rectangular Plate ..................... 447
16.2.1 Solution by Separation of Variables ............. 449
16.2.2 Example of Application .......................... 451
Exercises ..................................................... 454
Appendix A
Fundamental Equations of Transport Phenomena .................. 459
A.l Introduction ............................................ 459
A.2 Global Continuity ....................................... 460
A.3 Momentum Transfer ....................................... 460
A.4 Energy Transfer ......................................... 463
A.5 Mass Transfer ........................................... 463
A.5.1 Important Correlations on Mass Transfer ......... 465
A.5.1.1 Molar and Mass Concentrations ......... 465
A.5.1.2 Average Molecular Mass ............... 465
A.5.1.3 Sums of Molar and Mass Fractions ...... 465
A.5.1.4 Molar and Mass Fractions .............. 468
A.5.1.5 Mass and Molar Average Velocities ..... 469
A.5.1.6 Binary (A and В Species) Mass and
Molar Fluxes in Relation to Inertial
Coordinates ........................... 469
A.5.1.7 Sums of Mass and Molar Fluxes ......... 469
A.6 Parabolic, Elliptic, and Hyperbolic Partial Differential
Equations ............................................... 469
References .................................................... 470
Appendix В
Fundamental Aspects of Ordinary Differential Equations ........ 471
B.l Introduction ............................................ 471
B.2 First-Order Linear Equations ............................ 471
B.2.1 Separable Equations ............................. 472
B.2.2 Variation of Parameters ......................... 472
B.3 First-Order Nonlinear Equations ......................... 473
B.3.1 Picard's Method ................................. 473
B.3.2 Existence and Uniqueness of Solutions ........... 474
B.4 Second-Order Linear Equations ........................... 474
B.4.1 Second-Order Linear Equation with Constant
Coefficients .................................... 475
B.4.1.1 Homogeneous Linear Second Order ....... 475
B.4.1.2 Nonhomogeneous Linear Second Order .... 477
B.4.1.3 How to Obtain the Particular
Solution for Any Case ................. 479
B.4.2 Euler-Cauchy Equation ........................... 479
B.4.2.1 Different Real Roots .................. 480
B.4.2.2 Double Root ........................... 480
B.4.2.3 Complex Conjugate Roots ............... 480
B.4.3 Existence and Uniqueness ........................ 481
References .................................................... 481
Appendix С
Method of Power Series and Special Functions .................. 483
C.l Introduction ............................................ 483
C.2 Power Series ............................................ 483
C.2.1 An Example ...................................... 484
C.3 Frobenius Method ........................................ 486
C.3.1 Indicial Equation ............................... 486
C.3.2 Example ......................................... 487
C.4 Bessel Equations ........................................ 488
C.4.1 Bessel Functions of the First Kind .............. 489
C.4.2 Bessel Functions of the Second Kind ............. 489
C.4.3 Modified Bessel Functions ....................... 490
C.4.4 Selected Relations .............................. 491
C.5 Legendre Functions ...................................... 492
References .................................................... 492
Appendix D
Laplace Transform ............................................. 493
D.l Introduction ............................................ 493
D.2 Basic Principle ......................................... 493
D.3 A Few Basic Properties .................................. 494
D.3.1 Fundamental Operations .......................... 494
D.3.2 Transforms of Derivatives ............................. 494
D.3.3 Transforms of Integrals ............................... 495
D.3.4 s-Shifting ............................................ 495
D.3.5 t-Shifting ............................................ 495
D.3.6 Impulse Function ...................................... 496
D.3.7 Convolution ........................................... 498
D.3.8 Derivatives of Transforms ............................. 498
D.4 Transform of a Partial Derivative ....................... 499
D.5 Tables of Laplace Transforms ............................ 499
References .................................................... 508
Appendix E
Method of Weighted Residuals .................................. 509
E.l Introduction ............................................ 509
E.2 General Theory .......................................... 510
E.3 Various Methods ......................................... 511
E.3.1 Method of Moments ............................... 511
E.3.2 Collocation ..................................... 512
E.3.3 Subdomain ....................................... 512
E.3.4 Least Squares ................................... 513
E.3.5 Galerkin ........................................ 513
E.3.6 Comparing the Methods ........................... 513
E.4 Selecting the Approximation Order ....................... 514
E.5 Example of Application .................................. 514
E.5.1 First Approximation ............................. 515
E.5.1.1 Method of Moments ..................... 515
E.5.1.2 Collocation Method .................... 516
E.5.1.3 Subdomain Method ...................... 516
E.5.1.4 Least Squares Method .................. 516
E.5.1.5 Galerkin's Method ..................... 516
E.5.2 Second Approximation ............................ 517
E.5.2.1 Collocation ........................... 517
E.5.2.2 Application of Approximation
Criterion ............................. 517
E.5.3 Third Approximation ............................. 518
E.5.3.1 Collocation ........................... 518
References .................................................... 519
Appendix F
Method of Similarity .......................................... 521
F.l Introduction ............................................ 521
F.2 Generalized Method of Similarity ........................ 524
F.2.1 Definitions ..................................... 524
F.2.2 Similarity ...................................... 525
References .................................................... 525
Appendix G
Fourier Series and Method of Separation of Variables .......... 527
G.l Fourier Series .......................................... 527
G.l.l Periodic Functions .............................. 527
G.l.2 Coefficients of Fourier Series .................. 528
G.1.3 Even and Odd Functions .......................... 528
G.l.4 Half-Range Expansions ........................... 529
G.l.5 Fourier Integrals ............................... 530
G.2 Method of Separation of Variables ....................... 531
G.2.1 Procedure ....................................... 531
References ................................................... 531
Appendix H
Fourier Transforms ............................................ 533
H.l Introduction ............................................ 533
H.2 Fourier Cosine and Sine Transforms ...................... 533
H.2.1 Examples of Transforms .......................... 534
H.3 Complex or Exponential Fourier Transform ................ 535
H.4 Existence of Fourier Transforms ......................... 535
H.5 Properties of Fourier Transforms ........................ 536
H.5.1 Linearity of Fourier Transforms ................. 536
H.5.2 Transform of Derivatives ........................ 536
H.5.3 Convolution ..................................... 537
H.6 Generalized Integral Transforms ......................... 537
References .................................................... 538
Appendix I
Generalized Representation by Series .......................... 539
1.1 Introduction ............................................ 539
1.2 Orthogonal Functions .................................... 539
1.3 Orthogonal Series ....................................... 540
Exercises ..................................................... 541
References .................................................... 541
Index ......................................................... 543
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