Design with Constructal Theory

About the Authors xi

Preface xiii

List of Symbols xvii

1. Flow Systems 1

1.1 Constructal Law, Vascularization, and Svelteness 1

1.2 Fluid Flow 6

1.2.1 Internal Flow: Distributed Friction Losses 7

1.2.2 Internal Flow: Local Losses 11

1.2.3 External Flow 18

1.3 Heat Transfer 20

1.3.1 Conduction 20

1.3.2 Convection 24

References 31

Problems 31

2. Imperfection 43

2.1 Evolution toward the Least Imperfect Possible 43

2.2 Thermodynamics 44

2.3 Closed Systems 46

2.4 Open Systems 51

2.5 Analysis of Engineering Components 52

2.6 Heat Transfer Imperfection 56

2.7 Fluid Flow Imperfection 57

2.8 Other Imperfections 59

2.9 Optimal Size of Heat Transfer Surface 61

References 62

Problems 63

3. Simple Flow Configurations 73

3.1 Flow Between Two Points 73

3.1.1 Optimal Distribution of Imperfection 73

3.1.2 Duct Cross Sections 75

3.2 River Channel Cross-Sections 78

3.3 Internal Spacings for Natural Convection 81

3.3.1 Learn by Imagining the Competing Extremes 81

3.3.2 Small Spacings 84

3.3.3 Large Spacings 85

3.3.4 Optimal Spacings 86

3.3.5 Staggered Plates and Cylinders 87

3.4 Internal Spacings for Forced Convection 89

3.4.1 Small Spacings 90

3.4.2 Large Spacings 90

3.4.3 Optimal Spacings 91

3.4.4 Staggered Plates, Cylinders, and Pin Fins 92

3.5 Method of Intersecting the Asymptotes 94

3.6 Fitting the Solid to the “Body” of the Flow 96

3.7 Evolution of Technology: From Natural to Forced Convection 98

References 99

Problems 101

4. Tree Networks for Fluid Flow 111

4.1 Optimal Proportions: T- and Y -Shaped Constructs 112

4.2 Optimal Sizes, Not Proportions 119

4.3 Trees Between a Point and a Circle 123

4.3.1 One Pairing Level 124

4.3.2 Free Number of Pairing Levels 127

4.4 Performance versus Freedom to Morph 133

4.5 Minimal-Length Trees 136

4.5.1 Minimal Lengths in a Plane 137

4.5.2 Minimal Lengths in Three Dimensions 139

4.5.3 Minimal Lengths on a Disc 139

4.6 Strategies for Faster Design 144

4.6.1 Miniaturization Requires Construction 144

4.6.2 Optimal Trees versus Minimal-Length Trees 145

4.6.3 75 Degree Angles 149

4.7 Trees Between One Point and an Area 149

4.8 Asymmetry 156

4.9 Three-Dimensional Trees 158

4.10 Loops, Junction Losses and Fractal-Like Trees 161

References 162

Problems 164

5. Configurations for Heat Conduction 171

5.1 Trees for Cooling a Disc-Shaped Body 171

5.1.1 Elemental Volume 173

5.1.2 Optimally Shaped Inserts 177

5.1.3 One Branching Level 178

5.2 Conduction Trees with Loops 189

5.2.1 One Loop Size, One Branching Level 190

5.2.2 Radial, One-Bifurcation and One-Loop Designs 195

5.2.3 Two Loop Sizes, Two Branching Levels 197

5.3 Trees at Micro and Nanoscales 202

5.4 Evolution of Technology: From Forced Convection to Solid-Body

Conduction 206

References 209

Problems 210

6. Multiscale Configurations 215

6.1 Distribution of Heat Sources Cooled by Natural Convection 216

6.2 Distribution of Heat Sources Cooled by Forced Convection 224

6.3 Multiscale Plates for Forced Convection 229

6.3.1 Forcing the Entire Flow Volume to Work 229

6.3.2 Heat Transfer 232

6.3.3 Fluid Friction 233

6.3.4 Heat Transfer Rate Density: The Smallest Scale 234

6.4 Multiscale Plates and Spacings for Natural Convection 235

6.5 Multiscale Cylinders in Crossflow 238

6.6 Multiscale Droplets for Maximum Mass Transfer Density 241

References 245

Problems 247

7. Multiobjective Configurations 249

7.1 Thermal Resistance versus Pumping Power 249

7.2 Elemental Volume with Convection 250

7.3 Dendritic Heat Convection on a Disc 257

7.3.1 Radial Flow Pattern 258

7.3.2 One Level of Pairing 265

7.3.3 Two Levels of Pairing 267

7.4 Dendritic Heat Exchangers 274

7.4.1 Geometry 275

7.4.2 Fluid Flow 277

7.4.3 Heat Transfer 278

7.4.4 Radial Sheet Counterflow 284

7.4.5 Tree Counterflow on a Disk 286

7.4.6 Tree Counterflow on a Square 289

7.4.7 Two-Objective Performance 291

7.5 Constructal Heat Exchanger Technology 294

7.6 Tree-Shaped Insulated Designs for Distribution of Hot Water 295

7.6.1 Elemental String of Users 295

7.6.2 Distribution of Pipe Radius 297

7.6.3 Distribution of Insulation 298

7.6.4 Users Distributed Uniformly over an Area 301

7.6.5 Tree Network Generated by Repetitive Pairing 307

7.6.6 One-by-One Tree Growth 313

7.6.7 Complex Flow Structures Are Robust 318

References 325

Problems 328

8. Vascularized Materials 329

8.1 The Future Belongs to the Vascularized: Natural Design Rediscovered 329

8.2 Line-to-Line Trees 330

8.3 Counterflow of Line-to-Line Trees 334

8.4 Self-Healing Materials 343

8.4.1 Grids of Channels 344

8.4.2 Multiple Scales, Loop Shapes, and Body Shapes 352

8.4.3 Trees Matched Canopy to Canopy 355

8.4.4 Diagonal and Orthogonal Channels 362

8.5 Vascularization Fighting against Heating 364

8.6 Vascularization Will Continue to Spread 369

References 371

Problems 373

9. Configurations for Electrokinetic Mass Transfer 381

9.1 Scale Analysis of Transfer of Species through a Porous System 381

9.2 Model 385

9.3 Migration through a Finite Porous Medium 387

9.4 Ionic Extraction 393

9.5 Constructal View of Electrokinetic Transfer 396

9.5.1 Reactive Porous Media 400

9.5.2 Optimization in Time 401

9.5.3 Optimization in Space 403

References 405

10. Mechanical and Flow Structures Combined 409

10.1 Optimal Flow of Stresses 409

10.2 Cantilever Beams 411

10.3 Insulating Wall with Air Cavities and Prescribed Strength 416

10.4 Mechanical Structures Resistant to Thermal Attack 424

10.4.1 Beam in Bending 425

10.4.2 Maximization of Resistance to Sudden Heating 427

10.4.3 Steel-Reinforced Concrete 431

10.5 Vegetation 442

10.5.1 Root Shape 443

10.5.2 Trunk and Canopy Shapes 446

10.5.3 Conical Trunks, Branches and Canopies 449

10.5.4 Forest 453

References 458

Problems 459

11. Quo Vadis Constructal Theory? 467

11.1 The Thermodynamics of Systems with Configuration 467

11.2 Two Ways to Flow Are Better than One 470

11.3 Distributed Energy Systems 473

11.4 Scaling Up 482

11.5 Survival via Greater Performance, Svelteness and Territory 483

11.6 Science as a Consructal Flow Architecture 486

References 488

Problems 490

Appendix 491

A. The Method of Scale Analysis 491

B. Method of Undetermined Coefficients (Lagrange Multipliers) 493

C. Variational Calculus 494

D. Constants 495

E. Conversion Factors 496

F. Dimensionless Groups 499

G. Nonmetallic Solids 499

H. Metallic Solids 503

I. Porous Materials 507

J. Liquids 508

K. Gases 513

References 516

Author Index 519

Subject Index 523