Applied Parameter Estimation for Chemical Engineers
Hardcover: 462 pages
Publisher: CRC Press; 1st edition (December 15, 2000)
Language: English
IN-10: 082479561X
IN-13: 978-0824795610
Contents
Preface v
1 Introduction 1
2 Formulation of the Parameter Estimation Problem 7
2.1 Structure of the Mathematical Model 7
2.1.1 Algebraic Equation Models 7
2.1.2 Differential Equation Models 11
2.2 The Objective Function 13
2.2.1 Explicit Estimation 14
2.2.1.1 Simple or Unweighted Least Squares (LS) Estimation 15
2.2.1.2 Weighted Least Squares (WLS) Estimation 15
2.2.1.3 Generalized Least Squares (GLS) Estimation 15
2.2.1.4 Maximum Likelihood (ML) Estimation 15
2.2.1.5 The Determinant Criterion 19
2.2.1.6 Incorporation of Prior Information About the Parameters 19
2.2.2 Implicit Estimation 19
2.3 Parameter Estimation Subject to Constraints 22
3 Computation of Parameters in Linear Models - Linear Regression 23
3.1 The Linear Regression Model 23
3.2 The Linear Least Squares Objective Function 26
3.3 Linear Least Squares Estimation 27
3.4 Polynomial Curve Fitting 29
3.5 Statistical Inferences 32
3.5.1 Inference on the Parameters 3 2
3.5.2 Inference on the Expected Response Variables 33
3.6 Solution of Multiple Linear Regression Problems 35
3.6.1 Procedure for Using Microsoft Excel™ for Windows 35
3.6.2 Procedure for Using SigmaPlot™ for Windows 42
3.7 Solution of Multiresponse Linear Regression Problems 46
3.8 Problems on Linear Regression 46
3.8.1 Vapor Pressure Data for Pyridine and Piperidine 46
3.8.2 Vapor Pressure Data for R142b and R152a 47
4.Gauss-Newton Method for Algebraic Models 49
4.1 Formulation of the Problem 49
4.2 The Gauss-Newton Method 50
4.2.1 Bisection Rule 52
4.2.2 Convergence Criteria 52
4.2.3 Formulation of the Solution Steps for the Gauss-Newton Method: Two Consecutive Chemical Reactions 53
4.2.4 Notes on the Gauss-Newton Method 55
4.3 Examples 55
4.3.1 Chemical Kinetics: Catalytic Oxidation of 3-Hexanol 55
4.3.2 Biological Oxygen Demand (BOD) 56
4.3.3 Numerical Example 1 57
4.3.4 Chemical Kinetics: Isomerization of Bicyclo [2,1,1] Hexane 58
4.3.5 Enzyme Kinetics 60
4.3.6 Catalytic Reduction of Nitric Oxide 61
4.3.7 Numerical Example 2 62
4.4 Solutions 64
4.4.1 Numerical Example 1 65
4.4.2 Numerical Example 2 66
5.Other Nonlinear Regression Methods for Algebraic Models 67
5.1 Gradient Minimization Methods 67
5.1.1 Steepest Descent Method 69
5.1.2 Newton's Method 71
5.1.3 Modified Newton's Method 76
5.1.4 Conjugate Gradient Methods 76
5.1.5 Quasi-Newton or Variable Metric or Secant Methods 77
5.2 Direct Search or Derivative Free Methods 78
5.2.1 LJ Optimization Procedure 79
5.2.2 Simplex Method 81
5.3 Exercises 83
6. Gauss-Newton Method for Ordinary Differential Equation (ODE) Models 84
6.1 Formulation of the Problem 84
6.2 The Gauss-Newton Method 85
6.2.1 Gauss-Newton Algorithm for ODE Models 88
6.2.2 Implementation Guidelines for ODE Models 88
6.3 The Gauss-Newton Method - Nonlinear Output Relationship 92
6.4 The Gauss-Newton Method - Systems with Unknown Initial Conditions 93
6.5 Examples 96
6.5.1 A Homogeneous Gas Phase Reaction 96
6.5.2 Pyrolytic Dehydrogenation of Benzene to Diphenyl and Triphenyl 98
6.5.3 Catalytic Hydrogenation of 3 -Hydroxyprop (HPA) to l,3-Propanediol(PD) 102
6.6 Equivalence of Gauss-Newton with Quasilinearization Method 111
6.6.1 The Quasilinearization Method and its Simplification 111
6.6.2 Equivalence to Gauss-Newton Method 114
6.6.3 Nonlinear Output Relationship 114
7 Shortcut Estimation Methods for ODE Models 115
7.1 ODE Models with Linear Dependence on the Parameters 115
7.1.1 Derivative Approach 116
7.1.2 Integral Approach 118
7.2 Generalization to ODE Models with Nonlinear Dependence on the Parameters 119
7.3 Estimation of Apparent Rates in Biological Systems 120
7.3.1 Derivative Approach 122
7.3.2 Integral Approach 123
7.4 Examples 129
7.4.1 Derivative Approach - Pyrolytic Dehydrogenation of Benzene 129
8. Practical Guidelines for Algorithm Implementation 133
8.1 Inspection of the Data 133
8.2 Generation of Initial Guesses 135
8.2.1 Nature and Structure of the Model 135
8.2.2 Asymptotic Behavior of the Model Equations 135
8.2.3 Transformation of the Model Equations 136
8.2.4 Conditionally Linear Systems 138
8.2.5 Direct Search Approach 139
8.3 Overstepping 139
8.3.1 An Optimal Step-Size Policy 140
8.4 Ill-Conditioning of Matrix A and Partial Remedies 141
8.4.1 Pseudoinverse 143
8.4.2 Marquardt's Modification 144
8.4.3 Scaling of Matrix A 145
8.5 Use of "Prior" Information 146
8.6 Selection of Weighting Matrix Q in Least Squares Estimation 147
8.7 Implementation Guidelines for ODE Models 148
8.7.1 Stiff ODE Models 148
8.7.2 Increasing the Region of Convergence 150
8.7.2.1 An Optimal Step-Size Policy 150
8.7.2.2 Use of the Information Index 152
8.7.2.3 Use of Direct Search Methods 155
8.8 Autocorrelation in Dynamic Systems 156
9. Constrained Parameter Estimation 158
9.1 Equality Constraints 158
9.1.1 Lagrange Multipliers 159
9.2 Inequality Constraints 162
9.2.1 Optimum Is Internal Point 162
9.2.1.1 Reparameterization 162
9.2.1.2 Penalty Function 163
9.2.1.3 Bisection Rule 165
9.2.2 The Kuhn-Tucker Conditions 165
10 Gauss-Newton Method for Partial Differential Equation (PDE) Models 167
10.1 Formulation of the Problem 167
10.2 The Gauss-Newton Method for PDE Models 169
10.3 The Gauss-Newton Method for Discretized PDE Models 172
10.3.1 Efficient Computation of the Sensitivity Coefficients 173
11 Statistical Inferences 177
11.1 Inferences on the Parameters 177
11.2 Inferences on the Expected Response Variables 179
11.3 Model Adequacy Tests 182
11.3.1 Single Response Models 182
11.3.2 Multivariate Models 184
12 Design of Experiments 185
12.1 Preliminary Experimental Design 185
12.2 Sequential Experimental Design for Precise Parameter Estimation 187
12.2.1 The Volume Design Criterion 188
12.2.2 The Shape Design Criterion 189
12.2.3 Implementation Steps 190
12.3 Sequential Experimental Design for Model Discrimination 191
12.3.1 The Divergence Design Criterion 192
12.3.2 Model Adequacy Tests for Model Discrimination 193
12.3.3 Implementation Steps for Model Discrimination 195
12.4 Sequential Experimental Design for ODE Systems 196
12.4.1 Selection of Optimal Sampling Interval and Initial State for Precise Parameter Estimation 196
12.4.2 Selection of Optimal Sampling Interval and Initial State for Model Discrimination 200
12.4.3 Determination of Optimal Inputs for Precise Parameter Estimation and Model Discrimination 200
12.5 Examples 202
12.5.1 Consecutive Chemical Reactions 202
12.5.2 Fed-batch Bioreactor 207
12.5.3 Chemostat Growth Kinetics 213
13 Recursive Parameter Estimation 218
13.1 Discrete Input-Output Models 218
13.2 Recursive Least Squares (RLS) 219
13.3 Recursive Extended Least Squares (RELS) 221
13.4 Recursive Generalized Least Squares (RGLS) 223
14 Parameter Estimation in Nonlinear Thermodynamic Models: Cubic Equations of State 226
14.1 Equations of State 226
14.1.1 Cubic Equations of State 227
14.1.2 Estimation of Interaction Parameters 229
14.1.3 Fugacity Expressions Using the Peng-Robinson EoS 230
14.1.4 Fugacity Expressions Using the Trebble-Bishnoi EoS 231
14.2 Parameter Estimation Using Binary VLB Data 231
14.2.1 Maximum Likelihood Parameter and State Estimation 232
14.2.2 Explicit Least Squares Estimation 233
14.2.3 Implicit Maximum Likelihood Parameter Estimation 234
14.2.4 Implicit Least Squares Estimation 236
14.2.5 Constrained Least Squares Estimation 236
14.2.5.1 Simplified Constrained Least Squares Estimation 237
14.2.5.2 A Potential Problem with Sparse or Not Well Distributed Data 238
14.2.5.3 Constrained Gauss-Newton Method for Regression of Binary VLB Data 240
14.2.6 A Systematic Approach for Regression of Binary VLB Data 242
14.2.7 Numerical Results 244
14.2.7.1 The n-Pentane-Acetone System 244
14.2.7.2 The Methane-Acetone System 245
14.2.7.3 The Nitrogen-Ethane System 246
14.2.7.4 The Methane-Methanol System 246
14.2.7.5 The Carbon Dioxide-Methanol System 246
14.2.7.6 The Carbon Dioxide-n-Hexane System 247
14.2.7.7 The Propane-Methanol System 248
14.2.7.8 The Diethylamine-Water System 250
14.3 Parameter Estimation Using the Entire Binary Phase Equilibrium Data 255
14.3.1 The Objective Function 255
14.3.2 Covariance Matrix of the Parameters 257
14.3.3 Numerical Results 258
14.3.3.1 The Hydrogen Sulfide-Water System 258
14.3.3.2 The Methane-n-Hexane System 259
14.4 Parameter Estimation Using Binary Critical Point Data 261
14.4.1 The Objective Function 261
14.4.2 Numerical Results 264
14.5 Problems 266
14.5.1 Data for the Methanol-Isobutane System 266
14.5.2 Data for the Carbon Dioxide-Cyclohexane System 266
15 Parameter Estimation in Nonlinear Thermodynamic Models: Activity Coefficients 268
15.1 Electrolyte Solutions 268
15.1.1 Pitzer's Model Parameters for Aqueous Na2SiO3 Solutions 268
15.1.2 Pitzer's Model Parameters for Aqueous Na2SiO3 - NaOH Solutions 270
15.1.3 Numerical Results 273
15.2 Non-Electrolyte Solutions 274
15.2.1 The Two-Parameter Wilson Model 276
15.2.2 The Three-Parameter NRTL Model 276
15.2.3 The Two-Parameter UNIQUAC Model 277
15.2.4 Parameter Estimation: The Objective Function 278
15.3 Problems 279
15.3.1 Ootic Coefficients for Aqueous Solutions of KC1 Obtained by the Isopiestic Method 279
15.3.2 Ootic Coefficients for Aqueous Solutions of High-Purity NiCl2 280
15.3.3 The Benzene (l)-i-Propyl Alcohol (2) System 281
15.3.4 Vapor-Liquid Equilibria of Coal-Derived Liquids: Binary Systems with Terralin 282
15.3.5 Vapor-Liquid Equilibria of Ethylbenzene (l)-o-Xylene (2) at 26.66 kPa 283
16 Parameter Estimation in Chemical Reaction Kinetic Models 285
16.1 Algebraic Equation Models 285
16.1.1 Chemical Kinetics: Catalytic Oxidation of 3-Hexanol 285
16.1.2 Chemical Kinetics: Isomerization of Bicyclo [2,1,1] Hexane 287
16.1.3 Catalytic Reduction of Nitric Oxide 288
16.2 Problems with Algebraic Models 295
16.2.1 Catalytic Dehydrogenation of sec-butyl Alcohol 295
16.2.2 Oxidation of Propylene 297
16.2.3 Model Reduction Through Parameter Estimation in the s-Domain 300
16.3 Ordinary Differential Equation Models 302
16.3.1 A Homogeneous Gas Phase Reaction 302
16.3.2 Pyrolytic Dehydrogenation of Benzene to Diphenyl and Triphenyl 303
16.3.3 Catalytic Hydrogenation of 3-Hydroxyprop (HPA) to l,3-Propanediol(PD) 307
16.3.4 Gas Hydrate Formation Kinetics 314
16.4 Problems with ODE Models 316
16.4.1 Toluene Hydrogenation 317
16.4.2 Methylester Hydrogenation 318
16.4.3 Catalytic Hydrogenation of 3-Hydroxyprop (HPA) to 1,3-Propanediol (PD) - Nonisothermal Data 320
17 Parameter Estimation in Biochemical Engineering Models 322
17.1 Algebraic Equation Models 322
17.1.1 Biological Oxygen Demand 322
17.1.2 Enzyme Kinetics 323
17.1.3 Determination of Mass Transfer Coefficient (kLa) in a Municipal Wastewater Treatment Plant (with PULSAR aerators) 327
17.1.4 Determination of Monoclonal Antibody Productivity in a Dialyzed Chemostat 330
17.2 Problems with Algebraic Equation Models 338
17.2.1 Effect of Glucose to Glutamine Ratio on MAb Productivity in a Chemostat 338
17.2.2 Enzyme Inhibition Kinetics 340
17.2.3 Determination of kLa in Bubble-free Bioreactors 341
17.3 Ordinary Differential Equation Models 344
17.3.1 Contact Inhibition in Microcarrier Cultures of MRC-5 Cells 344
17.4 Problems with ODE Models 347
17.4.1 Vero Cells Grown on Microcarriers (Contact Inhibition) 347
17.4.2 Effect of Temperature on Insect Cell Growth Kinetics 348
18 Parameter Estimation in Petroleum Engineering 353
18.1 Modeling of Drilling Rate Using Canadian Offshore Well Data 353
18.1.1 Application to Canadian Offshore Well Data 355
18.2 Modeling of Bitumen Oxidation and Cracking Kinetics Using Data from Alberta Oil Sands 358
18.2.1 Two-Component Models 358
18.2.2 Three-Component Models 359
18.2.3 Four-Component Models 362
18.2.4 Results and Discussion 364
18.3 Automatic History Matching in Reservoir Engineering 371
18.3.1 A Fully Implicit, Three Dimensional, Three-Phase Simulator with Automatic History-Matching Capability 371
18.3.2 Application to a Radial Coning Problem (Second SPE Comparative Solution Problem) 373
18.3.2.1 Matching Reservoir Pressure 373
18.3.2.2 Matching Water-Oil Ratio, Gas-Oil Ratio or Bottom Hole Pressure 374
18.3.2.3 Matching All Observed Data 374
18.3.3 A Three-Dimensional, Three-Phase Automatic History-Matching Model: Reliability of Parameter Estimates 376
18.3.3.1 Implementation and Numerical Results 378
18.3.4 Improved Reservoir Characterization Through Automatic History Matching 380
18.3.4.1 Incorporation of Prior Information and Constraints on the Parameters 382
18.3.4.2 Reservoir Characterization Using Automatic History Matching 384
18.3.5 Reliability of Predicted Well Performance Through Automatic History Matching 385
18.3.5.1 Quantification of Risk 388
18.3.5.2 Multiple Reservoir Descriptions 388
18.3.5.3 Case Study-Reliability of a Horizontal Well Performance 389
References 391
Appendix 1 403
A. 1.1 The Trebble-Bishnoi Equation of State 403
A. 1.2 Derivation of the Fugacity Expression 403
A. 1.3 Derivation of the Expression for (91nfj/3xj)TiPjX 405
Appendix 2 410
A.2.1 Listings of Computer Programs 410
A.2.2 Contents of Accompanying CD 411
A.2.3 Computer Program for Example 16.1.2 412
A.2.4 Computer Program for Example 16.3.2
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