Mathematical Modelling and Simulation in Chemical Engineering

Contents......Page 6
Figures......Page 12
1.1 Mathematical Model......Page 20
1.2 Development of Mathematical Model......Page 22
Review Problems......Page 29
Fig. 2.1 Force balance on a spherical particle in a liquid......Page 31
2.3 Vaporization from a Single Droplet in Quiescent Air......Page 32
2.4 Parallel Couette Flow......Page 34
2.5 Plane Poiseuille Flow......Page 35
2.6 Modelling of a Surge Tank......Page 37
2.7 Rapid Thermal Processing......Page 38
2.8 Modelling of a pH Process......Page 39
2.10 Diffusion of Dopants in the Semiconductor Substrate......Page 42
2.11 Modelling of a Transport Reactor......Page 44
2.11.1 Numerical evaluation and discussion......Page 45
Fig. 2.11 Effect of particle diameter on the fractional solid hold up......Page 46
Table 2.1 Values of parameters used for simulation......Page 47
2.12 PDE Model for Tubular Reactor with Axial Dispersion Model......Page 48
3.1 Isothermal Continuously Stirred Tank Reactor (CSTR)......Page 50
Fig. 3.3a Local information of system with 4 equilibrium states......Page 53
3.2 Bioreactor Modelling......Page 54
Fig. 3.5 Schematic of the MAGLEV system......Page 56
3.4 Balancing a Stick on a Finger......Page 57
3.5 Cholette’s Model with Input Multiplicities......Page 58
3.6 Isothermal CSTR with Input Multiplicities......Page 59
Fig. 3.9 Steady state x2 versus u......Page 60
3.8 Weather Forecasting......Page 62
Fig. 3.11 An illustration of convection according to Lorenz’s equation......Page 63
3.9 Population Model......Page 64
Fig. 3.13 Phase plane diagram for the system......Page 66
3.11 Non-isothermal Continuous Stirred Tank Reactor......Page 68
Fig. 3.15 Sustained oscillatory response of CSTR [c versus q]......Page 69
3.14 Nonlinear Phenomena......Page 72
Fig. 3.17 A period doubling sequence for the Rossler equations......Page 74
4.3 Application of Laplace Transform to Solve a Linear PDE......Page 80
4.4.1 Equations with coefficients asymptotic to constants......Page 86
Fig. 4.3 Graphs of tan l a and –kl/h......Page 90
5.7 Finite Difference Scheme Method for Nonlinear BVP ODEs......Page 106
6.1 Axial and Radial Mixing of Catalyst in a Bubble Column Slurry Contactor (BCSC)......Page 112
6.2.1 Model equations and solution......Page 116
Fig. 6.3 Effect of catalyst decay velocity constant on verses behaviour......Page 119
Fig. 6.4 Effect of particle diameter on the particle residence time......Page 120
6.3.1 Dispersion model......Page 121
6.3.2 Application to the experimental data......Page 122
Fig. 6.7 Effect of particle diameter on the mean residence time of the particles......Page 127
Fig. 6.9 (a) Average conversion versus particle dispersion coefficient (b) Effect of particle diameter on the mean residence time of the particles......Page 128
Fig. 6.11 Average conversion versus particle dispersion coefficient. Countercurrent flow......Page 129
6.4.2 Comparison between cocurrent and countercurrent BCSRs......Page 130
Fig. 7.1 Periodic inlet concentration function......Page 142
Fig. 7.3 Increasing trend for model I and resonance behaviour for model II......Page 143
7.2.4 Results and discussions......Page 148
7.3.1 Model equations......Page 149
7.3.2 Periodic operation......Page 151
7.3.3 Results and discussions......Page 152
8.7 The General Parameter Mapping (GPM)......Page 166
9.1 Optimization of a Function of a Single Variable......Page 172
Fig. 9.2 System description for Example 9.1......Page 173
Fig. 9.3 Circuit for Example 9.2......Page 174
9.13 Variable Transformation to Get a Linear Equation......Page 195
Fig. 9.5 y versus x for Example 9.19......Page 196
Fig. 9.6 The response (solid) with b = 0.63 and (- -) with b = 1......Page 200
Fig. 10.1 Output for problem 10.6......Page 212
Fig. 10.2 Output for problem 10.10......Page 215
Fig. 10.3 Solution of the delayed logistic equation......Page 216
Fig. 10.4 Output for problem 10.12 via algebraic equation solver......Page 218
10.4 Parameter Estimation......Page 219
Fig. 10.6 Output for problem 10.21 using Scilab......Page 230
Fig. 11.1 Block diagram of a feedback control system......Page 235
11.3 Transfer Function Models......Page 236
Fig. 11.3 Block diagram of an open loop system for obtaining the process reaction curve......Page 237
11.4 Stability of Closed Loop System......Page 238
11.8 Controller Design by Synthesis Method......Page 246
Fig. 11.6 Block diagram for simple feedback control system......Page 247
Tables......Page 16
Table 1.1 Model classification......Page 23
1.3.1 Nonlinear differential equations......Page 24
1.5 Random Processes......Page 27
4.2.1 Particular integrals: method of variation of parameters......Page 79
4.3.2 Solution of linear ODE......Page 81
Table 6.1 Analytical solutions for XD and XB for ash layer diffusion controlled and chemical reaction controlled steps......Page 131
Appendix......Page 132
9. Optimization Methods......Page 171
9.3 Optimization in Model Reduction......Page 177
Table 11.1 Magnitude and phase angle for simple transfer function models......Page 240
Table 11.2 Equation for kmax and wc for some of the transfer function models......Page 243
Preface......Page 18
1.4 Conservation of Mass/Energy/Momentum......Page 25
1.6 Black Box Model and Grey Box Model......Page 28
2.1 Settling Velocity of a Particle......Page 30
2.5.1 Plane Poiseuille flow with slip......Page 36
2.9 Modelling of a Long Chain Polymerization Reaction......Page 41
2.10.2 Constant total dopant......Page 43
Review Problems......Page 49
3.1.1 Linearization of a nonlinear system......Page 51
3.1.2 Linearization of coupled ODEs......Page 52
3.3 Magnetic Levitation System......Page 55
3.7 Model for Predators and Prey Populations......Page 61
3.9.1 Two competing species......Page 65
3.10 Epidemiological Model......Page 67
3.12 The Belousov–Zhabotinski (Oscillating) Reaction......Page 70
3.13 Breathing Model......Page 71
3.15 One Route to Chaos: Period Doubling......Page 73
Review Problems......Page 75
4.1 Homogeneous ODEs with Constant Coefficients......Page 76
4.2 Non-homogeneous ODE with Constant Coefficients......Page 78
4.4 Series Solution of Variable Coefficient Linear ODE......Page 85
4.4.2 Summary of the series solution method......Page 87
4.5 The Sturm–Liouville Equation......Page 88
4.6 Method of Separation of Variables......Page 89
4.7 Types of Boundary Conditions......Page 91
Review Problems......Page 92
5.1.1 Scalar equation......Page 94
5.1.2 Multivariable equations......Page 95
5.2.1 Development of second-order algorithm......Page 97
5.2.2 Fourth-order algorithms......Page 98
5.2.4 Conversion to state variable form......Page 99
5.2.5 Coupled higher order models......Page 100
5.2.7 Predictor–corrector method......Page 101
5.3 Solution of Second-order Nonlinear BVP ODEs......Page 102
5.4 Coupled Two Point BVP ODEs......Page 103
5.5 Method of Parameter Differentiation......Page 104
5.6 MPD for Simultaneous Equations......Page 105
5.8 Conversion of a PDE into a Set of IVP ODEs......Page 109
Review Problems......Page 110
6.2.2 Hydrodynamic and mass transfer considerations......Page 118
6.4 Non-catalytic Reactions in Flow BCSR......Page 124
6.4.1 Mathematical model and analytical solution......Page 125
Review Problems......Page 138
7.1 Modelling Deactivation Disguised Kinetics......Page 140
7.2.1 Assumptions and model equations......Page 144
7.2.2 Model description......Page 146
7.2.3 Periodic operation......Page 147
Review Problems......Page 153
8.1 Introduction......Page 154
8.2 Calculation of Sensitivity......Page 155
8.3.1 Direct differential method......Page 157
8.3.3 The Green’s function method......Page 158
8.4 Examples......Page 159
8.6 Dependence of the Solution of a Set of Nonlinear Equations on a Parameter......Page 164
8.8 Sensitivity Analysis in Process Control......Page 168
Review Problems......Page 169
9.2 Optimization of a Function of Multi-variables......Page 176
9.4 Optimization in Model Fitting......Page 178
9.5.1 Newton’s method......Page 179
9.5.2 Steepest descent method......Page 180
9.6.1 The method of Lagrange multipliers......Page 181
9.7 The Mathematical Programming......Page 183
9.8 Converting Inequality Constraint to Equality Constraint......Page 185
9.9 Kuhn–Tucker Conditions......Page 187
9.10 Multi-objectives Optimization Problem......Page 189
9.11 Random Search Optimization Techniques......Page 190
9.11.2 Conley method......Page 191
9.11.3 Simulation results......Page 192
9.12 Linear Least Square Curve Fitting......Page 193
9.14 Set-point Weighted PI/PID Controllers......Page 197
Review Problems......Page 203
10.1 MATLAB (MATrix LABoratory)......Page 206
10.2.1 Stiff problems......Page 208
10.3 Matlab Programs with Examples......Page 209
10.5 Constrained Optimization Problem......Page 224
10.6 PDE Solver in Matlab......Page 225
10.7.1 Development of Scilab......Page 227
10.7.3 ODE package in Scilab......Page 228
10.8.2 Fsolve......Page 229
Review Problems......Page 232
11.2 Feedback Control Systems......Page 234
11.5 Design of Controllers for Stable Systems......Page 241
11.6 Design of PI/PID Controllers for Unstable Systems......Page 244
11.7 Control of Unstable Bioreactor......Page 245
Review Problems......Page 251
References......Page 254
Index......Page 262