Process modeling, simulation, and control for chemical engineers

Preface Introduction 1 Examples of the Role of Process Dynamics and Control Historical Background Perspective Motivation for Studying Process Control General Concepts Laws and Languages of Process Control 1.6.1 Process Control Laws 1.6.2 Languages of Process Control 1 6 7 8 8 11 11 12 Mathematical Models of Chemical Engineering Systems Xxi 2 Fundamentals 2.1 Intreduction 2.1.1 Uses of Mathematical Models 2.1.2 Scope of Coverage 2.1.3 Principles of Formulation 2.2 Fundamental Laws 2.2.1 Continuity Equations 2.2.2 Energy Equation 2.2.3 Equations of Motion 2.2.4 Transport Equations 2.2.5 Equations of State 2.2.6 Equilibrium 2.2.7 Chemical Kinetics Problems 15 15 15 16 16 17 17 2 3 2 7 31 3 2 3 3 3 6 3 8 xi I 4 Xii CONTENTS 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 Part II Examples of Mathematical Models of Chemical Engineering Systems Introduction Series of Isothermal, Constant-Holdup CSTRs CSTRs With Variable Holdups Two Heated Tanks Gas-Phase, Pressurized CSTR Nonisothermal CSTR Single-Component Vaporizer Multicomponent Flash Drum Batch Reactor Reactor With Mass Transfer Ideal Binary Distillation Column Multicomponent Nonideal Distillation Column Batch Distillation With Holdup pH Systems 3.14.1 Equilibrium-Constant Models 3.14.2 Titration-Curve Method Problems Computer Simulation 4 0 40 4 1 43 44 45 46 51 54 57 62 64 70 72 74 74 75 77 4 Numerical Methods 89 4.1 Introduction 8 9 4.2 Computer Programming 90 4.3 Iterative Convergence Methods 9 1 4.3.1 Interval Halving 93 4.3.2 Newton-Raphson Method 96 4.3.3 False Position 100 4.3.4 Explicit Convergence Methods 101 4.35 Wegstein 103 4.3.6 Muller Method 103 4.4 Numerical Integration of Ordinary Differential Equations 105 4.4.1 Explicit Numerical Integration Algorithms 106 4.4.2 Implicit Methods 113 Problems 114 5 Simulation Examples 5.1 Gravity-Flow Tank 5.2 Three CSTRs in Series 5.3 Nonisothermal CSTR 5.4 Binary Distillation Column 5.5 Multicomponent Distillation Column 5.6 Variable Pressure Distillation 5.6.1 Approximate Variable-Pressure Model 5.6.2 Rigorous Variable-Pressure Model 116 116 119 124 129 132 141 141 142 .. . CONTENTS xlu 5.7 Batch Reactor 150 5.8 Ternary Batch Distillation With Holdup 157 Problems 162 Part III Time-Domain Dynamics and Control 6 Time-Domain Dynamics 6.1 Classification and Dethtition 6.2 Linearization and Perturbation Variables 6.2.1 Linearization 6.2.2 Perturbation Variables 6.3 Responses of Simple Linear Systems 6.3.1 First-Order Linear Ordinary Differential Equation 6.3.2 Second-Order Linear ODES With Constant Coefficients 6.3.3 Nth-Order Linear ODES With Constant Coefficients 6.4 Steadystate Techniques Problems 7 Corkentional Control Systems and Hardware 205 7.1 Control Instrumentation 205 7.1.1 Sensors 207 7.1.2 Transmitters 211 7.1.3 Control Valves 213 7.1.4 Analog and Digital Controllers 222 7.1.5 Computing and Logic Devices 226 7.2 Performance of Feedback Controllers 226 7.2.1 Specifications for Closedloop Response 226 7.2.2 Load Performance 227 7.3 Controller Tuning 231 7.3.1 Rules of Thumb 231 7.3.2 On-Line Trial and Error 234 7.3.3 Ziegler-Nichols Method 235 Problems 238 8 Advanced Control Systems 8.1 Ratio Control 8.2 Cascade Control 8.3 Computed Variable Control 8.4 Override Control 8.4.1 Basic System 8.4.2 Reset Windup 8.5 Nonlinear and Adaptive Control 8.5.1 Nonlinear Control 8.5.2 Adaptive Control 8.6 Valve-Position Control 8.7 Feedfonvard Control Concepts 167 167 171 171 175 177 177 182 192 195 198 253 253 255 257 259 259 261 262 262 263 263 265 Xiv C O N T E N T S 8.8 Control System Design Concepts 268 8.8.1 General Guidelines 268 8.8.2 Trade-Offs Between Steadystate Design and Control 273 8.8.3 Plant-Wide Control 274 8.9 Dynamic Matrix Control 281 8.9.1 Review of Least Squares 281 8.9.2 Step-Response Models 284 8.9.3 DMC Algorithm 287 Problems 288 Part IV Laplace-Domain Dynamics and Control 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 10 10.1 10.2 Laplace-Domain Dynamics Laplace-Transformation Fundamentals 9.1.1 Definition 9.1.2 Linearity Property Laplace Transformation of Important Functions 9.2.1 Step Function 9.2.2 Ramp 9.2.3 Sine 9.2.4 Exponential 9.2.5 Exponential Multiplied By Time 9.2.6 Impulse (Dirac Delta Function 6,,,) Inversion of Laplace Transforms Transfer Functions 9.4.1 Multiplication by a Constant 9.4.2 Differentiation With Respect To Time 9.4.3 Integration 9.4.4 Deadtime Examples Properties of Transfer Functions 9.6.1 Physical Realizability 9.6.2 Poles and Zeros 9.6.3 Steadystate Gains Transfer Functions for Feedback Controllers Problems Laplace-Domain Analysis of Conventional Feedback Control Systems Openloop and Closedloop Systems 10.1.1 Openloop Characteristic Equation 10.1.2 Closedloop Characteristic Equation and Closedloop Transfer Function Stability 10.2.1 Routh Stability Criterion 10.2.2 Direct Substitution For Stability Limit 303 303 303 304 304 304 305 306 306 307 307 308 311 312 312 314 315 316 325 325 326 327 329 331 339 340 340 341 345 346 348 CONTENTS xv 10.3 10.4 Performance Specifications 350 10.3.1 Steadystate Performance 350 10.3.2 Dynamic Specifications 351 Root Locus Analysis 353 10.4.1 Definition 353 10.4.2 Construction of Root Locus Curves 357 10.4.3 Examples 363 Problems 367 11 11.1 11.2 11.3 11.4 11.5 Laplace-Domain Analysis of Advanced Control Systems Cascade Control 11.1.1 Series Cascade 11.1.2 Parallel Cascade Feedforward Control 11.2.1 Fundamentals 11.2.2 Linear Feedforward Control 11.2.3 Nonlinear Feedforward Control (Ipenloop Unstable Processes 11.3.1. Simple Systems 11.3.2 Effects of Lags 11.3.3 PD Control 11.3.4 Effects of Reactor Scale-up On Controllability Processes With Inverse Response Model-Based Control 11.51 Minimal Prototype Design 11.52 Internal Model Control Problems Part V Frequency-Domain Dynamics and Control 376 376 377 382 383 383 384 3 8 9 391 392 397 397 398 398 402 402 404 407 1 2 Frequency-Domain Dynamics 12.1 Definition 12.2 Basic Theorem 12.3 Representation 12.3.1 Nyquist Plots 12.3.2 Bode Plots 12.3.3 Nichols Plots 12.4 Frequency-Domain Solution Techniques Problems 415 415 417 420 421 427 440 442 452 1 3 Frequency-Domain Analysis of Closedloop Systems 455 13.1 Nyquist Stability Criterion 456 13.1.1 Proof 456 13.1.2 Examples 460 13.1.3 Representation 468 . xvi coNTENls 13.2 13.3 ii4 Closedloop Specifications in the Frequency Domain 470 13.2.1 Phase Margin 470 \’ 13.2.2 Gain Margin 470 13.2.3 Maximum Closedloop Log Modulus (Ly) 472 Frequency Response of Feedback Controllers 478 13.3.1 Proportional Controller (P) 478 13.3.2 Proportional-Integral Controller (PI) 479 13.3.3 Proportional-Integral-Derivative Controller (PID) 481 Examples 481 13.4.1 Three-CSTR System 481 13.4.2 First-Order Lag With Deadtime 488 13.4.3 Openloop Unstable Processes 490 Problems 493 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 Process Identification 502 Purpose 502 Direct Methods 503 14.2.1 Time-Domain (L Eyeball ” Fitting of Step Test Data 503 14.2.2 Direct Sine-Wave Testing 505 Pulse Testing 507 14.3.1 Calculation of Gu,, From Pulse Test Data 508 14.3.2 Digital Evaluation of Fourier Transformations 512 14.3.3 Practical Tips on Pulse Testing 515 14.3.4 Processes With Integration 516 Step Testing 518 ATV Identification 519 14.51 Autotuning 520 14.52 Approximate Transfer Functions 522 Least-Squares Method 525 State Estimators 529 Relationships Among Time, Laplace, and Frequency Domains 530 14.8.1 Laplace to Frequency Domain 530 14.8.2 Frequency to Laplace Domain 530 14.8.3 Time to Laplace Domain 530 14.8.4 Laplace to Time Domain 530 14.85 Time to Frequency Domain 532 14.8.6 Frequency to Time Domain 532 Problems 533 Part VI Multivariable Processes 1 5 Matrix Properties and State Variables 537 15.1 Matrix Mathematics 537 15.1.1 Matrix Addition 538 15.1.2 Matrix Multiplication 538 comwrs xvii 151.3 Transpose of a Matrix 539 151.4 Matrix Inversion 540 15.2 Matrix Properties 541 15.2.1 Eigenvalues 542 15.212 Canonical Transformation 543 152.3 Singular Values 546 15.3 Representation of Multivariable Processes 548 15.3.1 Transfer Function Matrix 548 15.3.2 State Variables 551 15.4 Openloop and Closedloop Systems 554 15.4.1 Transfer Function Representation 554 15.4.2 State Variable Representation 556 15.5 Computer Programs For Matrix Calculations 559 Problems 561 1 6 Analysis of Multivariable Systems 16.1 Stability 16.1.1 Openloop and Closedloop Characteristic Equations 16.1.2 Multivariable Nyquist Plot 16.1.3 Characteristic Loci Plots 16.1.4 Niederlinski Index * 16.2 Resiliency 16.3 Interaction 16.3.1 Relative Gain Array (Bristol Array) 16.3.2 Inverse Nyquist Array (INA) 16.3.3 Decoupling 16.4 Robustness 16.4.1 Trade-off Between Performance and Robustness 16.4.2 Doyle-Stein Criterion 16.4.3 Skogestad-Morari Method Problems 17 Design of Controllers For Multivariable Processes 17.1 17.2 17.3 17.4 17.5 17.6 17.7 Problem Definition Selection of Controlled Variables 17.2.1 Engineering Judgment 17.2.2 Singular Value Decomposition Selection of Manipulated Variables Elimination of Poor Pairings BLT Tuning Load Rejection Performance Multivariable Controllers 17.7.1 Multivariable DMC 17.7.2 Multivariable IMC Problems 562 562 562 564 568 572 573 575 576 579 581 584 585 585 588 591 594 594 596 596 596 598 598 599 605 606 606 609 611 .. . XVIII CONTENTS Part VII Sampled-Data Control Systems 18 Sampling and z Transforms 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 Introduction 18.1.1 Definition 18.1.2 Occurrence of Sampled-Data Systems in Chemical Engineering Impulse Sampler Basic Sampling Theorem z Transformation 18.4.1 Definition 18.4.2 Derivation of z Transforms of Common Functions 18.4.3 Effect of Deadtime 18.4.4 z-Transform Theorems 18.4.5 Inversion Pulse Transfer Functions Hold Devices Openloop and Closedloop Systems 18.7.1 Openloop Systems 18.7.2 Closedloop Systems -. Discrete Approximation of Continuous Transfer Functions Modified z Transforms 18.9.1 Definition 18.9.2 Modified z Transforms of Common Functions Problems 19 19.1 19.2 Stability Analysis of Sampled-Data Systems 19.3 19.4 Stability in the z Plane Root Locus Design Methods 19.2.1 z-Plane Root Locus Plots 19.2.2 Log-z Plane Root Locus Plots Bilinear Transformation Frequency-Domain Design Techniques 19.4.1 Nyquist Stability Criterion 19.4.2 Rigorous Method 19.4.3 Approximate Method Problems 20 20.1 20.2 20.3 Design of Digital Compensators Physical Realizability Frequency-Domain Effects Minimal-Prototype Design 203.1 Basic Concept 20.3.2 Other Input Types 20.3.3 Load Inputs 20.3.4 Second-Order Processes 20.3.5 Shunta Dual Algorithm 20.3.6 Dahlin Algorithm 615 615 615 615 620 623 626 626 626 629 630 631 636 638 639 639 643 648 651 651 652 655 657 657 660 660 669 672 675 675 676 681 682 685 686 687 689 689 692 694 696 699 7 0 1 CONTENTS Xix 20.4 Sampled-Data Control of Processes With Deadtime 702 20.5 Sampled-Data Control of Openloop Unstable Processes 705 Problems 709 Appendix Instrumentation Hardware 711 Index 721