《Mathematical Methods in Chemical Engineering》

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Chapter 1THE MATHEMATICAL STATEMENT OF THE PROBLEM1

1.1 INTRODUCTION1

1.2 REPRESENTATION OF THE PROBLEM1

1.3 SOLVENT EXTRACTION IN TWO STAGES3

1.4 SOLVENT EXTRACTION IN N STAGES4

1.5 SIMPLE WATER STILL WITH PREHEATED FEED6

1.6 UNSTEADY STATE OPERATION8

1.7 SALT ACCUMULATION IN A STIRRED TANK11

1.8 RADIAL HEAT TRANSFER THROUGH A CYLINDRICAL CONDUCTOR14

1.9 HEATING A CLOSED KETTLE16

1.10 DEPENDENT AND INDEPENDENT VARIABLES,PARAMETERS17

1.11 BOUNDARY CONDITIONS18

1.12 SIGN CONVENTIONS19

1.13 SUMMARY OF THE METHOD OF FORMULATION21

Chapter 2ORDINARY DIFFERENTIAL EQUATIONS23

2.1 INTRODUCTION23

2.2 ORDER AND DEGREE23

2.3 FIRST ORDER DIFFERENTIAL EQUATIONS24

2.4 SECOND ORDER DIFFERENTIAL EQUATIONS33

2.5 LINEAR DIFFERENTIAL EQUATIONS41

2.6 SIMULTANEOUS DIFFERENTIAL EQUATIONS66

2.7 CONCLUSIONS72

Chapter 3SOLUTION BY SERIES74

3.1 INTRODUCTION74

3.2 INFINITE SERIES74

3.3 POWER SERIES79

3.4 SIMPLE SERIES SOLUTIONS86

3.5 METHOD OF FROBENIUS90

3.6 BESSEL'S EQUATION106

3.7 PROPERTIES OF BESSEL FUNCTIONS113

Chapter 4COMPLEX ALGEBRA117

4.1 INTRODUCTION117

4.2 THE COMPLEX NUMBER117

4.3 THE ARGAND DIAGRAM118

4.4 PRINCIPAL VALUES119

4.5 ALGEBRAIC OPERATIONS ON THE ARGAND DIAGRAM120

4.6 CONJUGATE NUMBERS123

4.7 DE MOIVRE'S THEOREM124

4.8 THE nTH ROOTS OF UNITY125

4.9 COMPLEX NUMBER SERIES126

4.10 TRIGONOMETRICAL—EXPONENTIAL IDENTITIES128

4.11 THE COMPLEX VARIABLE128

4.12 DERIVATIVES OF A COMPLEX VARIABLE130

4.13 ANALYTIC FUNCTIONS131

4.14 SINGULARITIES132

4.15 INTEGRATION OF FUNCTIONS OF COMPLEX VARIABLES,AND CAUCHY'S THEOREM137

4.16 LAURENT'S EXPANSION AND THE THEORY OF RESIDUES142

Chapter 5FUNCTIONS AND DEFINITE INTEGRALS149

5.1 INTRODUCTION149

5.2 THE ERROR FUNCTION149

5.3 THE GAMMA FUNCTION151

5.4 THE BETA FUNCTION154

5.5 OTHER TABULATED FUNCTIONS WHICH ARE DEFINED BY INTEGRALS157

5.6 EVALUATION OF DEFINITE INTEGRALS159

Chapter 6THE LAPLACE TRANSFORMATION163

6.1 INTRODUCTION163

6.2 THE LAPLACE TRANSFORM163

6.3 THE INVERSE TRANSFORMATION167

6.4 PROPERTIES OF THE LAPLACE TRANSFORMATION170

6.5 THE STEP FUNCTIONS174

6.6 CONVOLUTION179

6.7 FURTHER ELEMENTARY METHODS OF INVERSION180

6.8 INVERSION OF THE LAPLACE TRANSFORM BY CONTOUR INTEGRATION182

6.9 APPLICATION OF THE LAPLACE TRANSFORM TO AUTOMATIC CONTROL THEORY188

Chapter 7VECTOR ANALYSIS199

7.1 INTRODUCTION199

7.2 TENSORS200

7.3 ADDITION AND SUBTRACTION OF VECTORS203

7.4 MULTIPLICATION OF VECTORS210

7.5 DIFFERENTIATION OF VECTORS216

7.6 HAMILTON'S OPERATOR,？218

7.7 INTEGRATION OF VECTORS AND SCALARS222

7.8 STANDARD IDENTITIES227

7.9 CURVILINEAR COORDINATE SYSTEMS228

7.10 THE EQUATIONS OF FLUID FLOW231

7.11 TRANSPORT OF HEAT,MASS,AND MOMENTUM236

Chapter 8PARTIAL DIFFERENTIATION AND PARTIAL DIFFERENTIAL EQUATIONS238

8.1 INTRODUCTION238

8.2 INTERPRETATION OF PARTIAL DERIVATIVES239

8.3 FORMULATING PARTIAL DIFFERENTIAL EQUATIONS245

8.4 BOUNDARY CONDITIONS252

8.5 PARTICULAR SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS259

8.6 ORTHOGONAL FUNCTIONS269

8.7 METHOD OF SEPARATION OF VARIABLES272

8.8 THE LAPLACE TRANSFORM METHOD290

8.9 OTHER TRANSFORMS302

8.10 CONCLUSIONS306

Chapter 9FINITE DIFFERENCES307

9.1 INTRODUCTION307

9.2 THE DIFFERENCE OPERATOR,△307

9.3 OTHER DIFFERENCE OPERATORS311

9.4 INTERPOLATION315

9.5 FINITE DIFFERENCE EQUATIONS321

9.6 LINEAR FINITE DIFFERENCE EQUATIONS322

9.7 NON-LINEAR FINITE DIFFERENCE EQUATIONS331

9.8 DIFFERENTIAL-DIFFERENCE EQUATIONS338

Chapter 10TREATMENT OF EXPERIMENTAL RESULTS349

10.1 INTRODUCTION349

10.2 GRAPH PAPER349

10.3 THEORETICAL PROPERTIES354

10.4 CONTOUR PLOTS355

10.5 PROPAGATION OF ERRORS356

10.6 CURVE FITTING360

10.7 NUMERICAL INTEGRATION369

Chapter 11NUMERICAL METHODS380

11.1 INTRODUCTION380

11.2 FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS380

11.3 HIGHER ORDER DIFFERENTIAL EQUATIONS(INITIAL VALUE TYPE)385

11.4 HIGHER ORDER DIFFERENTIAL EQUATIONS(BOUNDARY VALUE TYPE)388

11.5 ALGEBRAIC EQUATIONS397

11.6 DIFFERENCE-DIFFERENTIAL EQUATIONS406

11.7 PARTIAL DIFFERENTIAL EQUATIONS409

Chapter 12MATRICES437

12.1 INTRODUCTION437

12.2 THE MATRIX438

12.3 MATRIX ALGEBRA439

12.4 DETERMINANTS OF SQUARE MATRICES AND MATRIX PRODUCTS443

12.5 THE TRANSPOSE OF A MATRIX443

12.6 ADJOINT MATRICES444

12.7 RECIPROCAL OF A SQUARE MATRIX444

12.8 THE RANK AND DEGENERACY OF A MATRIX446

12.9 THE SUB-MATRIX448

12.10 SOLUTION OF LINEAR ALGEBRAIC EQUATIONS448

12.11 MATRIX SERIES449

12.12 DIFFERENTIATION AND INTEGRATION OF MATRICES451

12.13 LAMBDA-MATRICES452

12.14 THE CHARACTERISTIC EQUATION454

12.15 SYLVESTER'S THEOREM457

12.16 TRANSFORMATION OF MATRICES459

12.17 QUADRATIC FORM461

12.18 APPLICATION TO THE SOLUTION OF DIFFERENTIAL EQUATIONS463

12.19 SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS465

12.20 CONCLUSIONS472

Chapter 13OPTIMIZATION473

13.1 INTRODUCTION473

13.2 TYPES OF OPTIMIZATION474

13.3 ANALYTICAL PROCEDURES475

13.4 THE METHOD OF STEEPEST ASCENT483

13.5 THE SEQUENTIAL SIMPLEX METHOD485

13.6 DYNAMIC PROGRAMMING486

Chapter 14COMPUTERS492

14.1 INTRODUCTION492

14.2 PASSIVE ANALOGUE COMPUTERS493

14.3 ACTIVE ANALOGUE COMPUTERS496

14.4 DIGITAL COMPUTERS505

14.5 COMPARISON OF THE USES OF ANALOGUE AND DIGITAL COMPUTERS509

PROBLEMS511

APPENDIX532

SUBJECT INDEX543