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Design with Constructal Theory

Design with Constructal Theory

Adrian Bejan, Sylvie Lorente

ISBN: 978-0-471-99816-7 September 2008 552 Pages


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Design course on the universal principle of configurations in nature and engineering-the constructal law

Design with Constructal Theory offers a revolutionary new approach based on physics for understanding and predicting the designs that arise in nature and engineering, from the tree and the forest to the cooling of electronics, urban design, decontamination, and vascular smart materials. This book shows how you can use the method of constructal theory to design human-made systems in order to reduce trial and error and increase the system performance.

First developed in the late 1990s, constructal theory holds that flow architecture arises from the natural evolutionary tendency to generate greater flow access in time and in flow configurations that are free to morph. It unites flow systems with solid mechanical structures, which are viewed as systems for the flow of stresses. Constructal theory unites nature with engineering, and helps us generate novel designs across the board, from high-density packages to vascular materials with new functionalities (self-healing, self-cooling), and from tree-shaped heat exchangers to svelte fluid-flow and solid structures.

Design with Constructal Theory starts with basic principles and then shows how these principles are applied to understanding and designing increasingly complex systems. Problems and exercises at the end of each chapter give you an opportunity to use constructal theory to solve actual design problems.

This book is based on a design course developed by the two authors for upper-level undergraduates and graduate students at Duke University and other universities all over the world. With the authors' expert guidance, students and professionals in mechanical, civil, environmental, chemical, aerospace, and biomedical engineering will understand natural systems, and then practice design as science, by relying on constructal strategies to pursue and discover novel and effective designs.

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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

"This book represents the outcome of over 12 years of research and teaching by the authors on constructal theory and its application. It provides comprehensive and elegant discussion of a revolutionary new approach for understanding and predicting the designs that arise in both nature and engineering, from the  tree and the forest to the cooling of electronics, urban design, decontamination, and vascular smart materials. This book is highly recommended for everyone, especially students and professionals in mechanical, civil, environmental, energy and power, chemical, aerospace, and biomedical engineering, as well in geophysics and biology." (International Journal of Energy Research, 2010)

"The constructal law provides a broad coverage of "designedness" everywhere, from engineering to geophysics and biology….it provides the student with strategy for how to pursue and discover design-the configurations or patterns-in both space and time. Constructal theory pushes design thinking closer to science and away from art. It tears down the walls between engineering and natural sciences." (Mechanical Engineering, September 2009)

"A balance between individual and institutional approaches is the best idea, according to a new theory by a Duke University engineer Adrian Bejan, who thinks institutions benefit most from the co-existence of large groups that self-organize naturally and lone scientists coming up with brilliant new ideas…. big thinkers didn't disappear. Bejan argues they continued to thrive. He thinks his "constructal theory," which he began describing in 1996, might explain why. The theory states that so-called flow systems evolve to balance and minimize imperfections, reducing friction or other forms of resistance, so that the least amount of useful energy is lost. Examples in nature include rivers and streams that make up a delta or the intricate airways of the lungs. In research done by humans, Bejan sees two main flows: those of ideas in the form of scientific findings, and those of support, measured by tangible factors such as funding and lab space." (Robert Roy Brit,,, December 2008)

"Design with Constructal Theory offers a revolutionary new approach to design based on physics for understanding and predicting the designs that arise in nature and engineering…This book shows how you can use the method of constructal theory to design human-made systems in order to reduce trial and error and increase the system performance. It is beautifully illustrated, in color and black & white. This book is highly recommended to professors, students and professionals in mechanical, civil, environmental, chemical, aerospace and biomedical engineering. It is recommended to all the readers interested in design in nature, and in design as science, strategy, and novel and effective designs." (International Journal of Heat and Mass Transfer, 11/12/08)