Optimal Control in Thermal Engineering (eBook)

Preface 6
Contents 8
1 Introduction 17
1.1 Control of Systems 17
1.2 Optimization Classes 19
References 23
Introductory Elements 24
2 Functions Optimization 25
2.1 Weierstrass Theorem 25
2.2 Conditions of Extreme 26
2.2.1 Real Functions of One Variable 26
2.2.2 Functions of Several Variables 27
2.2.2.1 Functions of Two Variables 27
2.2.2.2 Functions with Arbitrary Finite Number of Variables 28
2.2.2.3 Examples 30
2.3 Constrained Optimization 32
2.3.1 Functions of Two Variables 32
2.3.2 Functions with Arbitrary Finite Number of Variables 34
Reference 36
3 Elements of Variational Calculus 37
3.1 Short History 37
3.2 Preliminary Issues 38
3.2.1 Necessary Conditions for Extremization of Functionals 38
3.2.2 Dual Methods in Variational Calculus 40
3.3 Euler Extremization Procedure 43
3.4 The Basic Lemma 45
3.4.1 The Statement and Proof of the Fundamental Lemma 48
3.5 The Euler-Lagrange Equation for Other Cases of Practical Interest 49
3.5.1 Integrands Depending on Several Functions 49
3.5.2 Integrands Containing Higher Order Derivatives 52
3.5.3 Integrands Depending on Several Independent Variables 54
3.6 Analytical Solutions of Euler-Lagrange Equations 55
3.6.1 The Case When F = F/left( {x,u^{{/prime }} } /right) 55
3.6.2 The Case When F = F/left( {u,u^{{/prime }} } /right) 57
3.6.3 The Case When F/left( {x,y,y^{{/prime }} } /right) Is Total Derivative 59
3.7 Boundary Conditions 60
3.7.1 Natural Boundary Conditions 60
3.7.2 Transversality Conditions 61
3.8 Extremals and Isoextreme Curves 63
3.8.1 Another Interpretation of the Transversality Condition 63
3.8.2 The Regularity Assumption 65
3.8.3 Obtaining Extremals from Isoextreme and Vice Versa 66
3.8.4 Example 66
3.8.4.1 Euler-Lagrange Approach 67
3.8.4.2 Hamilton-Jacobi Approach 69
3.8.5 Corner Conditions (Erdmann-Weierstrass) 70
3.9 Variational Notation 71
3.10 Constrained Extremization 74
3.11 Isoperimetric Problems 78
3.11.1 Extreme with More Constraints 85
3.11.2 The Case of Multiple Dependent Variables 86
References 87
Theory 88
4 Generalities Concerning the Optimal Control Problems 89
4.1 Variational Problems with Differential Equations as Constraints 89
4.1.1 Generalization of Some Notions of Variational Calculus 89
4.1.2 Differential Equations Acting as Constraints. Consequences 91
4.1.3 Problems of Type Lagrange, Mayer and Bolza 94
4.2 Solving Optimal Control Problems 95
4.2.1 Constraints on the Solutions 96
4.2.2 The Principle of Optimality for Parts of the Optimal Trajectory 97
4.2.3 Direct and Indirect Methods 98
References 100
5 The Maximum Principle (Pontryagin) 101
5.1 Preliminaries 101
5.2 The Fundamental Theorem 103
5.3 Comments on the Fundamental Theorem 107
5.3.1 Strategies of Using the Necessary Conditions 107
5.3.2 The Case of Non-autonomous Systems 108
5.3.3 Functionals Depending on Parameters 109
5.4 Other Useful Theorems 110
5.4.1 Non-autonomous Systems: Processes with Unspecified Duration 110
5.4.2 Non-autonomous Systems: Optimal Rapid Reaction 112
5.4.3 Processes with Specified Duration 113
5.5 Linear Rapid Reaction Systems 114
5.6 The Synthesis Problem 117
5.7 Example 118
References 121
6 The Gradient Method 122
6.1 Common Extreme Problems 122
6.1.1 Unconstrained Optimization 122
6.1.2 Constrained Optimization 127
6.2 Simple Variational Problems 128
6.3 Optimal Control Problems 130
6.3.1 The Fundamental Equation 131
6.3.2 Process with Specified Duration but Without Final Conditions 135
6.3.3 Process with Specified Duration and One Final Condition 137
6.3.4 Process with Unspecified Duration and Without Final Conditions 138
6.4 Constraints for the Control Functions and State Variables 139
6.4.1 Constraints for the Control Functions 139
6.4.2 Constraints for the State Variables 141
6.5 General Approach 142
References 147
7 Dynamic Programming (Bellman Method) 148
7.1 Common Optimization Problems 148
7.1.1 The Grid Method 148
7.1.2 The Bellman Method 148
7.1.3 Example 151
7.2 Problems of Variational Calculus 154
7.3 Optimal Control Problems 157
7.3.1 Extension of the Variational Calculus Method 157
7.3.2 Bellman Equation 159
7.3.3 Example 163
7.4 Linear Processes and Quadratic Objective Functions 164
7.5 Comments 168
References 168
Applications: Heat Transfer and Storage 169
8 Heat Transfer Processes 170
8.1 Optimal Strategies for Common Heat Transfer Processes 170
8.1.1 Determination of Optimal Strategies 170
8.1.2 The Case When the Value of n Is Arbitrary 172
8.1.3 The Case When n = 1 173
8.1.3.1 Source Temperature Constant in Time 173
8.1.3.2 Thermal Flux Constant in Time 174
8.1.3.3 Comparison 174
8.1.4 The Case When n = ?1 175
8.1.5 The Case When n = 4 176
8.1.6 The Case of Entropy Generation at Constant Speed 177
8.2 Optimal Paths for Minimizing Lost Available Work 177
8.2.1 Introduction 177
8.2.2 Theory 178
8.2.2.1 Model 178
8.2.2.2 Measures of Dissipation 179
8.2.2.3 Optimization Problem 181
8.2.2.4 Dimensionless Formulation 182
8.2.3 Results 184
8.2.3.1 Newtonian Heat Convection ( n = 1 ) 187
8.2.3.2 Special Conduction Case ( n = - 1 ) 189
8.2.3.3 Radiative Heat Transfer ( n = 4 ) 190
8.2.4 Conclusions 192
Appendix 8A 193
Appendix 8B 195
References 196
9 Heat Exchangers 197
9.1 Simple Approach 197
9.1.1 Usual and Optimized Operation Strategies 198
9.2 Optimal Strategies for Steady State Heat Exchanger Operation 200
9.2.1 Introduction 200
9.2.2 Optimal Heating/Cooling Strategies 201
9.2.3 Optimization of Heat Exchanger Operation Based on Minimum Entropy Generation 203
9.2.4 Optimization of Steady-State Heat Exchanger Operation for Arbitrary Criteria 206
9.3 Conclusions 210
References 211
10 Storage of Thermal Energy and Exergy 212
10.1 Unsteady Operation of Storage Elements 212
10.2 The Exergy Loss During the Storage Process 214
10.3 Thermal Energy Storage in Stratified and Fully Mixed Water Tanks 216
10.3.1 Introduction 216
10.3.2 Stratified Liquid Storage Tanks 217
10.3.2.1 Model 217
10.3.2.2 Performance Indicator 222
10.3.2.3 Results and Discussion 225
10.3.3 Fully Mixed Liquid Storage Tanks 228
10.3.3.1 Model 228
10.3.3.2 Indicator of Performance 229
10.3.3.3 Results 229
10.3.4 Conclusions 231
Appendix 10A 233
Appendix 10B 234
References 235
11 Heating and Cooling Processes 237
11.1 Optimization of Heating and Cooling Processes by Variational Calculus 237
11.1.1 Cooling Process Without Time Limitation 237
11.1.2 Cooling Process in Limited Time 239
11.2 Optimal Control of Forced Cool-Down Processes 241
11.2.1 Introduction 241
11.2.2 Forced Cooling Processes with Minimization of Cooling Fluid Mass 241
11.2.3 Forced Cooling Processes with Minimization of Dissipation Measures 245
11.2.3.1 Dissipation Measures 245
11.2.3.2 Minimization of Dissipation Measures 246
11.3 Conclusion 251
References 251
12 Optimization of Thermal Insulation of Seasonal Water Storage Tanks 253
12.1 Optimization of the Distribution of Thermal Insulation 253
12.2 Optimization of the Total Volume of Thermal Insulation 259
Reference 261
13 Optimization of Pin Fin Profiles 262
13.1 Optimal Control Methods 263
13.1.1 Methodology 263
13.1.1.1 Geometry 263
13.1.1.2 Heat Transfer Model 265
13.1.1.3 Optimal Control Problem 267
13.1.1.4 Optimal Control Method 269
13.1.1.5 Implementation 270
Geometry 270
Reference Parameters 270
Technological Constraints 270
13.1.1.6 Particular Cases 271
Temperature Imposed at z = 0 (or /xi = 0 ) 271
Temperature Imposed at z = L (or /xi = 1 ) 271
13.1.2 Results 272
13.1.2.1 Expected Accuracy 272
13.1.2.2 Particular Cases 272
Temperature Imposed at z = 0 (or /xi = 0 ) 272
Temperature Imposed at z = L (or /xi = 1 ). 278
13.1.3 Conclusions 283
Appendix 13A 283
References 285
Applications: Solar Energy Conversion into Thermal Energy Part 287
14 Optimization of Solar Energy Collection Systems 288
14.1 General Approach 288
14.1.1 Determination of the Optimal Solution 289
14.1.2 Collectors with Uniform Properties 293
14.1.3 Collectors with Non-uniform Properties 295
14.1.4 Example and Discussion 296
14.2 More Involved Treatment 299
14.2.1 Introduction 299
14.2.2 Theory 300
14.2.2.1 The Optimization Problem 300
14.2.2.2 Time Averaged Energy Balance Equation 300
14.2.3 Solar Energy Applications 302
14.2.4 Economical Indices 303
14.2.5 Meteorological and Actinometric Data 306
14.2.6 Model Implementation 306
14.2.6.1 Computing Procedure 306
14.2.6.2 Model Validation 307
14.2.6.3 Input Values 308
14.2.7 Solar Collectors with Optimal Uniformly Distributed Parameters 309
14.2.8 Solar Collectors with Optimal Non-uniformly Distributed Parameters 314
14.2.9 Conclusions 318
References 318
15 Flat-Plate Solar Collectors. Optimization of Absorber Geometry 320
15.1 Optimization of Absorber Geometry by Using Economic Considerations 320
15.1.1 Absorber Plate of Uniform Thickness 321
15.1.1.1 Example 324
15.1.2 Absorber Plate of Variable Thickness 324
15.1.3 The Optimal Fin Width 327
15.1.3.1 Example 328
15.1.4 Discussion and Conclusions 329
15.2 More Realistic Approach 329
15.2.1 Introduction 329
15.2.2 Meteorological Data 330
15.2.3 Model Implementation 331
15.2.4 Uniform Fin Thickness 331
15.2.5 Variable Fin Thickness 337
15.2.6 Conclusions 345
Appendix 15A 345
15.A.1 Optical Efficiency 345
15.A.2 Overall Heat Loss Coefficient 347
15.A.3 Collector Heat Removal Factor 348
15.A.4 Iterative Procedure 349
15.A.5 Shape of Collection Area 350
Appendix 15B 350
References 351
16 Optimal Time-Dependent Operation of Open Loop Solar Collector Systems 352
16.1 Simple Variational Approach for Maximum Exergy Extraction 353
16.1.1 Model of Flat Plate Solar Collector Operation 353
16.1.2 Optimal Strategy for Maximizing the Collected Exergy 354
16.2 Optimal Control of Flow for Maximum Exergy Extraction 357
16.2.1 Introduction 357
16.2.2 Meteorological Database 358
16.2.3 Transient Solar Energy Collection Model 358
16.2.4 Optimum Operation 360
16.2.4.1 Variational Approach for a Simple Case 362
16.2.4.2 Variational Approaches for the General Case 363
16.2.4.3 Direct Optimal Control Approach 365
16.2.5 Optimum Operation 366
16.2.6 Aspects of Controller Design 370
16.2.7 Conclusions 372
References 373
17 Optimal Time-Dependent Operation of Closed Loop Solar Collector Systems 375
17.1 Classification and Simple Approach 375
17.1.1 Performance Criteria 376
17.1.2 Systems with Storage at Uniform Temperature 377
17.1.3 Systems with Stratified Storage Tanks 380
17.1.4 Comparison and Discussions 384
17.2 More Realistic Approach for Systems with Fully Mixed Water Storage Tanks 385
17.2.1 Introduction 385
17.2.2 Closed Loop System 385
17.2.3 Flow Controllers 386
17.2.4 Operation Model 387
17.2.4.1 Configuration of Fig. 17.2a 388
17.2.4.2 Configuration of Fig. 17.2b 389
17.2.4.3 Model Validation 390
17.2.5 Optimal Control 390
17.2.5.1 Configuration of Fig. 17.2a 391
17.2.5.2 Configuration of Fig. 17.2b 392
17.2.6 Model Implementation 392
17.2.6.1 Primary Circuit 392
17.2.6.2 Water Storage Tank 393
17.2.6.3 Secondary Circuit 394
17.2.6.4 Meteorological and Actinometric Data 394
17.2.6.5 Computational Procedures 395
17.2.7 Results and Discussions 396
17.2.8 Conclusions 405
Appendix 17A 406
Appendix 17B 406
Appendix 17C 408
17C.1 Computation of Pump Power 408
17C.2 Computation of Pressure Loss Coefficients 410
References 411
18 Optimal Flow Controllers 413
18.1 Optimal Control 413
18.2 Implementation 418
18.3 Comparison and Discussions 419
References 421
Applications: Heat Engines 422
19 Endoreversible Heat Engines 423
19.1 Endoreversible Heat Engine Model 423
19.2 Implementation of the Optimal Control Theory 425
19.2.1 Definitions 425
19.2.2 Formulation of the Optimal Control Problem 426
19.2.3 Application of the Maximum Pontryagin Principle 427
19.2.4 Properties of the Solutions of Optimal Control Problems 428
19.3 Optimal Performances 428
19.3.1 Maximum Power 429
19.3.1.1 Application of the Maximum Principle 429
19.3.1.2 Optimal Solutions 431
19.3.1.3 Switchings 432
19.3.1.4 Optimal Controls and Trajectories 433
19.3.2 Maximum Efficiency 438
19.3.2.1 Application of the Maximum Principle 438
19.3.2.2 Optimal Solutions 439
19.3.2.3 Switchings 441
19.3.2.4 Optimal Controls and Trajectories 441
19.3.3 Conclusion 444
References 444
20 Diesel Engines 445
20.1 Engine Model 445
20.1.1 Fuel Combustion at Finite Speed 445
20.1.2 Modeling of Losses 446
20.1.2.1 Friction Losses 447
20.1.2.2 Pressure Drops 447
20.1.2.3 Thermal Losses 447
20.1.2.4 Losses at Fuel Injection 448
20.1.2.5 Incomplete Combustion 448
20.1.2.6 Exhaust Pressure Losses 448
20.1.3 Conventional Piston Path 448
20.2 Optimization Procedure 449
20.2.1 Steps (1)–(3). Processes When Power Is not Generated 450
20.2.1.1 Unbounded Acceleration 450
20.2.1.2 Bounded Acceleration 451
20.2.2 Stage (4). Allocation of Time Durations for Processes When Power Is not Generated 452
20.2.3 (5) Expansion 453
20.2.3.1 Unbound Acceleration 454
20.2.3.2 Bounded Acceleration 456
20.2.4 (6) Maximizing the Net Mechanical Work 457
20.3 Optimal Trajectories and Controls 457
20.3.1 Heat Engine Configuration 457
20.3.2 Optimized Engine Operation 459
References 465
21 Optimization of Daniel Cam Engines 466
21.1 Introduction 466
21.2 Model 467
21.2.1 Daniel Cam Engine Representation 467
21.2.2 Mechanical and Thermal Model 468
21.2.2.1 Movement and Energy Laws. Work Production 468
21.2.2.2 Heat Loss Model 469
21.2.3 Dimensionless Formulation 472
21.2.4 Optimization 473
21.2.5 Numerical Procedure 474
21.2.6 Model Implementation 475
21.3 Results and Discussions 477
21.3.1 Present Model Versus Simpler Approaches 477
21.3.1.1 Comparison with Classical Rod-Crank System 477
21.3.1.2 Comparison with Simplified Treatment of Convection Heat Loss Process 480
21.3.1.3 Comparison with Simplified Treatment of the Overall Heat Loss Process 484
21.3.1.4 Comparison with Unconstrained Piston Acceleration 484
21.3.2 Optimal Solution. Dependence on Design and Operation Parameters 486
21.3.2.1 Cylinder Wall and Thermal Insulation. Materials and Thickness 486
21.3.2.2 Auto-ignition Moment 491
21.3.2.3 Cooling Convection Coefficient 495
21.4 Conclusions 498
Appendix 21A 499
21.A.1 Combustion 499
21.A.2 Heat Losses 500
21.A.3 Frictional Losses 502
Appendix 21B 502
21.B.1 Classical Rod-Crank System 502
Appendix 21C 504
References 509
22 Photochemical Engines 512
22.1 Engine Model 512
22.2 Engine Operation Mode 518
22.3 Optimal Trajectories of the System 519
22.3.1 Maximizing the Work Produced 521
22.3.2 Minimizing the Entropy Production 521
22.4 Results and Discussions 522
References 524
Applications: Lubrication 525
23 Optimization of One Dimensional Slider Bearings 526
23.1 Introduction 526
23.2 Model 527
23.3 Optimal Control 530
23.4 Optimum Design and Operation 534
23.4.1 Direct Optimal Control Method 536
23.4.1.1 Numerical Procedures and Implementation 536
23.4.1.2 Testing the Direct Optimal Control Method 537
23.4.1.3 Analytic Approach 537
23.4.1.4 Optimal Control Approach 539
23.4.1.5 Sensibility Analysis 540
23.4.2 Constraints and Approximations 541
23.4.2.1 Maximum Pressure 541
23.4.2.2 Maximum Temperature 543
23.4.2.3 Maximum Bearing Load 544
23.4.2.4 Minimum Bearing Height 545
23.4.2.5 Levels of Approximation 547
23.4.3 Design Parameters 548
23.4.3.1 Lubricant Type 548
23.4.3.2 Bearing Length 550
23.4.3.3 Bearing Inlet Height 551
23.4.3.4 Sliding Velocity 552
23.4.3.5 Inlet Lubricant Pressure 553
23.4.3.6 Inlet Lubricant Temperature2 555
23.5 Conclusions 556
Appendix 23A 556
23.A1 Sensibility Analysis 556
23.A2 Constraints and Approximation 561
23.A2.1 Maximum Temperature Constraint 561
23.A2.2 Maximum Bearing Load 564
23.A2.3 Levels of Approximation 567
23.A3 Design Parameters 567
23.A3.1 Lubricant Type 567
23.A3.2 Bearing Length 570
23.A3.3 Bearing Inlet Height 572
23.A3.4 Sliding Velocity 573
23.A3.5 Inlet Lubricant Pressure 575
References 576
Index 579