Treelike structures abound in natural as well as man-made transport systems, which have fascinated multidisciplinary researchers to study the transport phenomena and properties and understand the transport mechanisms of treelike structures for decades. The fluid flow and heat transfer in treelike networks have received an increasing attention over the past decade as the highly efficient transport processes observed in natural treelike structures can provide useful hints for optimal solutions to many engineering and industrial problems. This review paper attempts to present the background and research progress made in recent years on the transport phenomenon in treelike networks as well as technological applications of treelike structures. The subtopics included are optimization of branching structures, scaling laws of treelike networks, and transport properties for laminar flow, turbulent flow, heat conduction, and heat convection in treelike networks. Analytical expressions for the effective transport properties have been derived based on deterministic treelike networks, and the effect of branching parameters on the transport properties of treelike networks has also been discussed. Furthermore, numerical simulation results for treelike microchannel networks are presented as well. The proposed transport properties may be beneficial to understand the transport mechanisms of branching structures and promote the applications of treelike networks in engineering and industry.
Issue Section:
Review Article
References
1.
Mandelbrot
, B. B.
, 1982
, The Fractal Geometry of Nature
, Freeman, New York
.2.
MacDonald
, N.
, 1983
, Trees and Networks in Biological Models
, Wiley
, Chichester, UK
.3.
Bejan
, A.
, 2000
, Shape and Structure, From Engineering to Nature
, Cambridge University Press
, Cambridge, UK
.4.
Bejan
, A.
, and Zane
, J. P.
, 2012
, Design in Nature
, Doubleday, New York
.5.
Takahashi
, T.
, 2014
, Microcirculation in Fractal Branching Networks
, Springer
, Tokyo, Japan
.6.
Sherman
, T. F.
, 1981
, “On Connecting Large Vessels to Small. The Meaning of Murray's Law
,” J. Gen. Physiol.
, 78
(4
), pp. 431
–453
.7.
Banavar
, J. R.
, Maritan
, A.
, and Rinaldo
, A.
, 1999
, “Size and Form in Efficient Transportation Networks
,” Nature
, 399
(6732
), pp. 130
–132
.8.
Gafiychuk
, V. V.
, and Lubashevsky
, I. A.
, 2001
, “On the Principles of the Vascular Network Branching
,” J. Theor. Biol.
, 212
(1
), pp. 1
–9
.9.
Bejan
, A.
, 2015
, “Constructal Law: Optimization as Design Evolution
,” ASME J. Heat Transfer
, 137
(6
), p. 061003
.10.
Albert
, R.
, and Barabási
, A.-L.
, 2002
, “Statistical Mechanics of Complex Networks
,” Rev. Mod. Phys.
, 74
(1
), pp. 47
–97
.11.
Chen
, M.
, Yu
, B. M.
, Xu
, P.
, and Chen
, J.
, 2007
, “A New Deterministic Complex Network Model With Hierarchical Structure
,” Physica A
, 385
(2
), pp. 707
–717
.12.
Wang
, X. Q.
, Mujumdar
, A. S.
, and Yap
, C.
, 2010
, New Approaches to Micro-Electronic Component Cooling
, LAP Lambert Academic Publishing
, Saarbrücken, Germany
.13.
Bejan
, A.
, and Lorente
, S.
, 2013
, “Constructal Law of Design and Evolution: Physics, Biology, Technology, and Society
,” J. Appl. Phys.
, 113
(15
), p.151301
.14.
Yu
, B. M.
, Xu
, P.
, Zou
, M.
, Cai
, J.
, and Zheng
, Q.
, 2014
, Fractal Physical Transport in Porous Media
, Science Press
, Beijing, China
.15.
Young
, T.
, 1809
, “On the Functions of the Heart and Arteries
,” Philos. Trans. R. Soc. London
, 99
(0
), pp. 1
–31
.16.
Hess
, W. R.
, 1917
, “Über die periphere Regulierung der Blutzirkulation
,” Pflugers Arch. Eur. J. Phys.
, 168
(9), pp. 439
–490
.17.
Thompson
, D. W.
, 1917
, On Growth and Form
, 2nd ed., Cambridge University Press
, Cambridge, UK
.18.
Murray
, C. D.
, 1926
, “The Physiological Principle of Minimum Work—I: The Vascular System and the Cost of Blood Volume
,” Proc. Natl. Acad. Sci. U.S.A.
, 12
(3
), pp. 207
–214
.19.
Murray
, C. D.
, 1926
, “The Physiological Principle of Minimum Work Applied to the Angle of Branching of Arteries
,” J. Gen. Physiol.
, 9
(6
), pp. 835
–841
.20.
McDonald
, D. A.
, 1974
, Blood Flow in Arteries
, 2nd ed., Edward Arnold
, London, UK
.21.
Zamir
, M.
, 1976
, “Optimality Principles in Arterial Branching
,” J. Theor. Biol.
, 62
(1
), pp. 227
–251
.22.
Uylings
, H. B. M.
, 1977
, “Optimization of Diameters and Bifurcation Angles in Lung and Vascular Tree Structures
,” Bull. Math. Biol.
, 39
(5
), pp. 509
–520
.23.
Zamir
, M.
, 1978
, “Nonsymmetrical Bifurcations in Arterial Branching
,” J. Gen. Physiol.
, 72
(6
), pp. 837
–845
.24.
Horsfield
, K.
, 1978
, “Morphometry of the Small Pulmonary Arteries in Man
,” Circ. Res.
, 42
(5
), pp. 593
–597
.25.
Roy
, A. G.
, and Woldenberg
, M. J.
, 1982
, “A Generalization of the Optimal Models of Arterial Branching
,” Bull. Math. Biol.
, 44
(3
), pp. 349
–360
.26.
Riva
, C. E.
, Grunwald
, J. E.
, Sinclair
, S. H.
, and Petrig
, B. L.
, 1985
, “Blood Velocity and Volumetric Flow Rate in Human Retinal Vessels
,” Invest. Ophthalmol. Visual Sci.
, 26
(8), pp. 1124
–1132
.27.
Zhong
, J.
, and Nilsson
, G.
, 1993
, “An M-ary Fractal Tree Based Modeling of Microcirculation
,” 15th Annual International Conference of the
IEEE
, Piscataway, NJ, pp. 553
–554
.28.
Taber
, L. A.
, Ng
, S.
, Quesnel
, A. M.
, Whatman
, J.
, and Carmen
, C. J.
, 2001
, “Investigating Murray's Law in the Chick Embryo
,” J. Biomech.
, 34
(1
), pp. 121
–124
.29.
Masters
, B. R.
, 2004
, “Fractal Analysis of the Vascular Tree in the Human Retina
,” Annu. Rev. Biomed. Eng.
, 6
(1
), pp. 427
–452
.30.
Buijs
, J. O. D.
, Bajzer
, Z.
, and Ritman
, E. L.
, 2006
, “Branching Morphology of the Rat Hepatic Portal Vein Tree: A Micro-CT Study
,” Ann. Biomed. Eng.
, 34
(9
), pp. 1420
–1428
.31.
Chandran
, K. B.
, Udaykumar
, H. S.
, and Reinhardt
, J. M.
, 2011
, Image-Based Computational Modeling of the Human Circulatory and Pulmonary Systems
, Springer
, New York
.32.
Skoura
, A.
, Megalooikonomou
, V.
, Diamantopoulos
, A.
, Kagadis
, G. C.
, and Karnabatidis
, D.
, 2012
, “Classification of Tree and Network Topology Structures in Medical Images
,” Data Mining for Biomarker Discovery
, P. M.
Pardalos
, P.
Xanthopoulos
, and M.
Zervakis
, eds., Springer-Verlag
, New York
, pp. 79
–90
.33.
Perdikaris
, P.
, Grinberg
, L.
, and Karniadakis
, G. E.
, 2015
, “An Effective Fractal-Tree Closure Model for Simulating Blood Flow in Large Arterial Networks
,” Ann. Biomed. Eng.
, 43
(6
), pp. 1432
–1442
.34.
Weibel
, E. R.
, 1963
, Morphometry of the Human Lung
, Academic Press
, New York
.35.
Horsfield
, K.
, and Cumming
, G.
, 1968
, “Morphology of the Bronchial Tree in Man
,” J. Appl. Physiol.
, 24
(3), pp. 373
–383
.36.
Hooper
, G.
, 1977
, “Diameters of Bronchi at Asymmetrical Divisions
,” Respir. Physiol.
, 31
(3
), pp. 291
–294
.37.
West
, J. B.
, 1979
, Respiratory Physiology—The Essentials
, 2nd ed., Williams and Wilkins
, Baltimore, MD
.38.
Horsfield
, K.
, 1990
, “Diameters, Generations, and Orders of Branches in the Bronchial Tree
,” J. Appl. Physiol.
, 68
(2), pp. 457
–461
.39.
Kitaoka
, H.
, and Suki
, B.
, 1997
, “Branching Design of the Bronchial Tree Based on a Diameter-Flow Relationship
,” J. Appl. Physiol.
, 82
(3), pp. 968
–976
.40.
Mauroy
, B.
, Filoche
, M.
, Weibel
, E. R.
, and Sapoval
, B.
, 2004
, “An Optimal Bronchial Tree May Be Dangerous
,” Nature
, 427
(6975
), pp. 633
–636
.41.
Takaki
, R.
, 2005
, “Can Morphogenesis Be Understood in Terms of Physical Rules?
,” J. Biosci.
, 30
(1
), pp. 87
–92
.42.
Wang
, Z.
, Zhao
, M.
, and Yu
, Q.-X.
, 2001
, “Modeling of Branching Structures of Plants
,” J. Theor. Biol.
, 209
(4), pp. 383
–394
.43.
McCulloh
, K. A.
, Sperry
, J. S.
, and Adler
, F. R.
, 2003
, “Water Transport in Plants Obeys Murray's Law
,” Nature
, 421
(6926
), pp. 939
–942
.44.
West
, G. B.
, Brown
, J. H.
, and Enquist
, B. J.
, 1997
, “A General Model for the Origin of Allometric Scaling Laws in Biology
,” Science
, 276
(5309
), pp. 122
–126
.45.
Enquist
, B. J.
, Brown
, J. H.
, and West
, G. B.
, 1998
, “Allometric Scaling of Plant Energetics and Population Density
,” Nature
, 395
(6698
), pp. 163
–165
.46.
West
, G. B.
, Brown
, J. H.
, and Enquist
, B. J.
, 1999
, “The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms
,” Science
, 284
(5420
), pp. 1677
–1679
.47.
Brown
, J. H.
, and West
, G. B.
, 2000
, Scaling in Biology
, Oxford University
, New York
.48.
Bejan
, A.
, 2001
, “The Tree of Convective Heat Streams: Its Thermal Insulation Function and the Predicted 3/4-Power Relation Between Body Heat Loss and Body Size
,” Int. J. Heat Mass Transfer
, 44
(4
), pp. 699
–704
.49.
Dreyer
, O.
, 2001
, “Allometric Scaling and Central Source Systems
,” Phys. Rev. Lett.
, 87
(3
), p. 038101
.50.
Rau
, A. R. P.
, 2002
, “Biological Scaling and Physics
,” J. Biosci.
, 27
(5
), pp. 475
–478
.51.
Santillán
, M.
, 2003
, “Allometric Scaling Law in a Simple Oxygen Exchanging Network: Possible Implications on the Biological Allometric Scaling Laws
,” J. Theor. Biol.
, 223
(2
), pp. 249
–257
.52.
Bejan
, A.
, 2005
, “The Constructal Law of Organization in Nature: Tree-Shaped Flows and Body Size
,” J. Exp. Biol.
, 208
(9
), pp. 1677
–1686
.53.
Chaui-Berlinck
, J. G.
, 2006
, “A Critical Understanding of the Fractal Model of Metabolic Scaling
,” J. Exp. Biol.
, 209
(16
), pp. 3045
–3054
.54.
Rinaldo
, A.
, Rodriguez-Iturbe
, I.
, Rigon
, R.
, Ijjasz-Vasquez
, E. J.
, and Bras
, R. L.
, 1993
, “Self-Organized Fractal River Networks
,” Phys. Rev. Lett.
, 70
(6
), pp. 822
–825
.55.
Sun
, T.
, Meakin
, P.
, and Jøssang
, T.
, 1994
, “Minimum Energy Dissipation Model for River Basin Geometry
,” Phys. Rev. E
, 49
(6
), pp. 4865
–4872
.56.
Buchanan
, M.
, 2002
, “Prediction: A Game of Chance
,” Nature
, 419
(6909
), p. 787
.57.
Paik
, K.
, and Kumar
, P.
, 2008
, “Emergence of Self-Similar Tree Network Organization
,” Complexity
, 13
(4
), pp. 30
–37
.58.
Bejan
, A.
, 1997
, “Constructal-Theory Network of Conducting Paths for Cooling a Heat Generating Volume
,” Int. J. Heat Mass Transfer
, 40
(4
), pp. 799
–816
.59.
Bejan
, A.
, 1998
, “Constructal Theory: From Thermodynamic and Geometric Optimization to Predicting Shape in Nature
,” Energy Convers. Manage.
, 39
(16–18), pp. 1705
–1718
.60.
Bejan
, A.
, 2000
, “From Heat Transfer Principles to Shape and Structure in Nature: Constructal Theory
,” ASME J. Heat Transfer
, 122
(3
), pp. 430
–449
.61.
Bejan
, A.
, and Lorente
, S.
, 2004
, “The Constructal Law and the Thermodynamics of Flow Systems With Configuration
,” Int. J. Heat Mass Transfer
, 47
(14–16), pp. 3203
–3214
.62.
Bejan
, A.
, and Lorente
, S.
, 2006
, “Constructal Theory of Generation of Configuration in Nature and Engineering
,” J. Appl. Phys.
, 100
(4
), p. 041301
.63.
Reis
, A. H.
, 2006
, “Constructal Theory: From Engineering to Physics, and How Flow Systems Develop Shape and Structure
,” ASME Appl. Mech. Rev.
, 59
(5
), pp. 269
–282
.64.
Bejan
, A.
, and Lorente
, S.
, 2008
, Design With Constructal Theory
, Wiley
, Hoboken, NJ
.65.
Manjunath
, K.
, and Kaushik
, S. C.
, 2014
, “Second Law Thermodynamic Study of Heat Exchangers: A Review
,” Renewable Sustainable Energy Rev.
, 40
, pp. 348
–374
.66.
Bejan
, A.
, and Errera
, M. R.
, 2015
, “Technology Evolution, From the Constructal Law: Heat Transfer Designs
,” Int. J. Energy Res.
, 39
(7
), pp. 919
–928
.67.
Kearney
, M. M.
, 2000
, “Engineered Fractals Enhance Process Applications
,” Chem. Eng. Prog.
, 96
(12), pp. 61
–68
.68.
Tondeur
, D.
, and Luo
, L.
, 2004
, “Design and Scaling Laws of Ramified Fluid Distributors by Constructal Approach
,” Chem. Eng. Sci.
, 59
(8–9), pp. 1799
–1813
.69.
Fan
, Z.
, Zhou
, X.
, Luo
, L.
, and Yuan
, W.
, 2008
, “Experimental Investigation of the Flow Distribution of a 2-Dimensional Constructal Distributor
,” Exp. Therm. Fluid Sci.
, 33
(1
), pp. 77
–83
.70.
Tondeur
, D.
, Fan
, Y.
, and Luo
, L.
, 2009
, “Constructal Optimization of Arborescent Structures With Flow Singularities
,” Chem. Eng. Sci.
, 64
(18
), pp. 3968
–3982
.71.
Wang
, L.
, Fan
, Y.
, and Luo
, L.
, 2014
, “Lattice Boltzmann Method for Shape Optimization of Fluid Distributor
,” Comput. Fluids
, 94
, pp. 49
–57
.72.
Luo
, L.
, Fan
, Y.
, Zhang
, W.
, Yuan
, X.
, and Midoux
, N.
, 2007
, “Integration of Constructal Distributors to a Mini Crossflow Heat Exchanger and Their Assembly Configuration Optimization
,” Chem. Eng. Sci.
, 62
(13
), pp. 3605
–3619
.73.
Luo
, L.
, Tondeur
, D.
, Gall
, H. L.
, and Corbel
, S.
, 2007
, “Constructal Approach and Multi-Scale Components
,” Appl. Therm. Eng.
, 27
(10
), pp. 1708
–1714
.74.
Luo
, L.
, Fan
, Z.
, Gall
, H. L.
, Zhou
, X.
, and Yuan
, W.
, 2008
, “Experimental Study of Constructal Distributor for Flow Equidistribution in a Mini Crossflow Heat Exchanger (MCHE)
,” Chem. Eng. Process.
, 47
(2
), pp. 229
–236
.75.
Fan
, Y.
, Boichot
, R.
, Goldin
, T.
, and Luo
, L.
, 2008
, “Flow Distribution Property of the Constructal Distributor and Heat Transfer Intensification in s Mini Heat Exchanger
,” AIChE J.
, 54
(11
), pp. 2796
–2808
.76.
Lim
, D.
, Kamotani
, Y.
, Cho
, B.
, Mazumder
, J.
, and Takayama
, S.
, 2003
, “Fabrication of Microfluidic Mixers and Artificial Vasculatures Using a High-Brightness Diode-Pumped Nd: YAG Laser Direct Write Method
,” Lab Chip
, 3
(4
), pp. 318
–323
.77.
Emerson
, D. R.
, Cieślicki
, K.
, Gu
, X.
, and Barber
, R. W.
, 2006
, “Biomimetic Design of Microfluidic Manifolds Based on a Generalized Murray's Law
,” Lab Chip
, 6
(3
), pp. 447
–454
.78.
Barber
, R. W.
, and Emerson
, D. R.
, 2008
, “Optimal Design of Microfluidic Networks Using Biologically Inspired Principles
,” Microfluid. Nanofluid.
, 4
(3
), pp. 179
–191
.79.
Yue
, J.
, Boichot
, R.
, Luo
, L.
, Gonthier
, Y.
, Chen
, G.
, and Yuan
, Q.
, 2010
, “Flow Distribution and Mass Transfer in a Parallel Microchannel Contactor Integrated With Constructal Distributors
,” AIChE J.
, 56
(2), pp. 298
–317
.80.
Tesař
, V.
, 2011
, “Bifurcating Channels Supplying ‘Numbered-Up’ Microreactors
,” Chem. Eng. Res. Des.
, 89
(12
), pp. 2507
–2520
.81.
Zhang
, L.
, Peng
, D.
, Lyu
, W.
, and Xin
, F.
, 2015
, “Uniformity of Gas and Liquid Two Phases Flowing Through Two Microchannels in Parallel
,” Chem. Eng. J.
, 263
, pp. 452
–460
.82.
Su
, Y.
, Chen
, G.
, and Kenig
, E. Y.
, 2015
, “An Experimental Study on the Numbering-Up of Microchannels for Liquid Mixing
,” Lab Chip
, 15
(1
), pp. 179
–187
.83.
Damiri
, H. S.
, and Bardaweel
, H. K.
, 2015
, “Numerical Design and Optimization of Hydraulic Resistance and Wall Shear Stress Inside Pressure-Driven Microfluidic Networks
,” Lab Chip
, 15
(21
), pp. 4187
–4196
.84.
Nasharudin
, M. N.
, Kamarudin
, S. K.
, Hasran
, U. A.
, and Masdar
, M. S.
, 2014
, “Mass Transfer and Performance of Membrane-Less Micro Fuel Cell: A Review
,” Int. J. Hydrogen Energy
, 39
(2), pp. 1039
–1055
.85.
Tüber
, K.
, Oedegaard
, A.
, Hermann
, M.
, and Hebling
, C.
, 2004
, “Investigation of Fractal Flow-Fields in Portable Proton Exchange Membrane and Direct Methanol Fuel Cells
,” J. Power Sources
, 131
(1–2), pp. 175
–181
.86.
Senn
, S. M.
, and Poulikakos
, D.
, 2004
, “Tree Network Channels as Fluid Distributors Constructing Double-Staircase Polymer Electrolyte Fuel Cells
,” J. Appl. Phys.
, 96
(1
), pp. 842
–852
.87.
Senn
, S. M.
, and Poulikakos
, D.
, 2004
, “Laminar Mixing, Heat Transfer and Pressure Drop in Tree-Like Microchannel Nets and Their Application for Thermal Management in Polymer Electrolyte Fuel Cells
,” J. Power Sources
, 130
(1–2), pp. 178
–191
.88.
Senn
, S. M.
, and Poulikakos
, D.
, 2006
, “Pyramidal Direct Methanol Fuel Cells
,” Int. J. Heat Mass Transfer
, 49
(7–8), pp. 1516
–1528
.89.
Li
, P.
, Coopamah
, D.
, and Ki
, J.-P.
, 2008
, “Uniform Distribution of Species in Fuel Cell Using a Multiple Flow Bifurcation Design
,” ASME
Paper No. FuelCell2008-65106.90.
Li
, P.
, Coopamah
, D.
, and Dhar
, N.
, 2008
, “Analysis and Optimization of Flow Distribution Channels for Uniform Flow in Fuel Cells
,” ASME
Paper No. FEDSM2008-55180.91.
Liu
, H.
, Li
, P.
, and Lew
, J. V.
, 2010
, “CFD Study on Flow Distribution Uniformity in Fuel Distributors Having Multiple Structural Bifurcations of Flow Channels
,” Int. J. Hydrogen Energy
, 35
(17
), pp. 9186
–9198
.92.
Kjelstrup
, S.
, Coppens
, M.-O.
, Pharoah
, J. G.
, and Pfeifer
, P.
, 2010
, “Nature-Inspired Energy- and Material-Efficient Design of a Polymer Electrolyte Membrane Fuel Cell
,” Energy Fuels
, 24
(9
), pp. 5097
–5108
.93.
Cheng
, S. J.
, Miao
, J. M.
, and Tai
, C. H.
, 2012
, “Numerical Simulation Applied to Study the Effects of Fractal Tree-Liked Network Channel Designs on PEMFC Performance
,” Adv. Mater. Res.
, 488–489
, pp. 1219
–1223
.94.
Chen
, Y. P.
, Zhang
, C.
, Wu
, R.
, and Shi
, M.
, 2011
, “Methanol Steam Reforming in Microreactor With Constructal Tree-Shaped Network
,” J. Power Sources
, 196
(15
), pp. 6366
–6373
.95.
Chen
, Y. P.
, Yao
, F.
, and Huang
, X.
, 2015
, “Mass Transfer and Reaction in Methanol Steam Reforming Reactor With Fractal Tree-Like Microchannel Network
,” Int. J. Heat Mass Transfer
, 87
, pp. 279
–283
.96.
Bejan
, A.
, and Almogbel
, M.
, 2000
, “Constructal T-Shaped Fins
,” Int. J. Heat Mass Transfer
, 43
(12
), pp. 2101
–2115
.97.
Rocha
, L. A. O.
, Lorente
, S.
, and Bejan
, A.
, 2002
, “Constructal Design for Cooling a Disc-Shaped Area by Conduction
,” Int. J. Heat Mass Transfer
, 45
(8
), pp. 1643
–1652
.98.
Bonjour
, J.
, Rocha
, L. A. O.
, Bejan
, A.
, and Meunier
, F.
, 2004
, “Dendritic Fins Optimization for a Coaxial Two-Stream Heat Exchanger
,” Int. J. Heat Mass Transfer
, 47
(1
), pp. 111
–124
.99.
Matos
, R. S.
, Laursen
, T. A.
, Vargas
, J. V. C.
, and Bejan
, A.
, 2004
, “Three-Dimensional Optimization of Staggered Finned Circular and Elliptic Tubes in Forced Convection
,” Int. J. Therm. Sci.
, 43
(5
), pp. 477
–487
.100.
Lorenzini
, G.
, and Rocha
, L. A. O.
, 2006
, “Constructal Design of Y-Shaped Assembly of Fins
,” Int. J. Heat Mass Transfer
, 49
(23–24), pp. 4552
–4557
.101.
Biserni
, C.
, Rocha
, L. A. O.
, and Bejan
, A.
, 2004
, “Inverted Fins: Geometric Optimization of the Intrusion Into a Conducting Wall
,” Int. J. Heat Mass Transfer
, 47
(12–13), pp. 2577
–2586
.102.
Rocha
, L. A. O.
, Lorenzini
, E.
, and Biserni
, C.
, 2005
, “Geometric Optimization of Shapes on the Basis of Bejan's Constructal Theory
,” Int. Commun. Heat Mass Transfer
, 32
(10
), pp. 1281
–1288
.103.
Biserni
, C.
, Rocha
, L. A. O.
, Stanescu
, G.
, and Lorenzini
, E.
, 2007
, “Constructal H-Shaped Cavities According to Bejan's Theory
,” Int. J. Heat Mass Transfer
, 50
(11–12), pp. 2132
–2138
.104.
Xu
, P.
, Yu
, B. M.
, Yuan
, M. J.
, and Zou
, M. Q.
, 2006
, “Heat Conduction in Fractal Tree-Like Branched Networks
,” Int. J. Heat Mass Transfer
, 49
(19–20), pp. 3746
–3751
.105.
Xu
, P.
, and Yu
, B. M.
, 2006
, “The Scaling Laws of Transport Properties for Fractal-Like Tree Networks
,” J. Appl. Phys.
, 100
(10
), p. 104906
.106.
Yu
, B. M.
, and Li
, B. W.
, 2006
, “Fractal-Like Tree Networks Reducing the Thermal Conductivity
,” Phys. Rev. E
, 73
(6
), p. 066302
.107.
Kou
, J.-L.
, Lu
, H.-J.
, Wu
, F.-M.
, and Xu
, Y.-S.
, 2009
, “Analysis of Thermal Conductivity in Tree-Like Branched Networks
,” Chin. Phys. B
, 18
(4), pp. 1553
–1559
.108.
Mazloomi
, A.
, Sharifi
, F.
, Salimpour
, M. R.
, and Moosavi
, A.
, 2012
, “Optimization of Highly Conductive Insert Architecture for Cooling a Rectangular Chip
,” Int. Commun. Heat Mass Transfer
, 39
(8
), pp. 1265
–1271
.109.
Calamas
, D.
, and Baker
, J.
, 2013
, “Tree-Like Branching Fins: Performance and Natural Convective Heat Transfer Behavior
,” Int. J. Heat Mass Transfer
, 62
, pp. 350
–361
.110.
Calamas
, D.
, and Baker
, J.
, 2013
, “Behavior of Thermally Radiating Tree-Like Fins
,” ASME J. Heat Transfer
, 135
(8
), p. 082701
.111.
Dannelley
, D.
, and Baker
, J.
, 2013
, “Natural Convection Heat Transfer From Fractal-Like Fins
,” J. Thermophys. Heat Transfer
, 27
(4
), pp. 692
–699
.112.
Park
, K. T.
, Kim
, H. J.
, and Kim
, D.-K.
, 2014
, “Experimental Study of Natural Convection From Vertical Cylinders With Branched Fins
,” Exp. Therm. Fluid Sci.
, 54
(4), pp. 29
–37
.113.
Daneshi
, M.
, Shirani
, E.
, and Salimpour
, M. R.
, 2012
, “Constructal Nano-Scale Heat Trees Used for Electronics Cooling
,” 4th International Symposium on Heat Transfer and Energy Conservation
, Guangzhou, China, pp. 142
–146
.114.
Daneshi
, M.
, Zare
, M.
, and Salimpour
, M. R.
, 2013
, “Micro- and Nanoscale Conductive Tree-Structures for Cooling a Disk-Shaped Electronic Piece
,” ASME J. Heat Transfer
, 135
(3
), p. 031401
.115.
Flik
, M. I.
, and Tien
, C. L.
, 1990
, “Size Effect on the Thermal Conductivity of High-Tc Thin-Film Superconductors
,” ASME J. Heat Transfer
, 112
(4
), pp. 872
–881
.116.
Chen
, L.
, Feng
, H.
, Xie
, Z.
, and Sun
, F.
, 2013
, “Constructal Optimization for
,”Disc-Point” Heat Conduction at Micro and Nanoscales,” Int. J. Heat Mass Transfer
, 67
, pp. 704
–711
.117.
Feng
, H.
, Chen
, L.
, Xie
, Z.
, and Sun
, F.
, 2015
, “Constructal Entransy Dissipation Rate Minimization for Triangular Heat Trees at Micro and Nanoscales
,” Int. J. Heat Mass Transfer
, 84
, pp. 848
–855
.118.
Bejan
, A.
, and Errera
, M. R.
, 1997
, “Deterministic Tree Networks for Fluid Flow: Geometry for Minimal Flow Resistance Between a Volume and One Point
,” Fractals
, 5
(04
), pp. 685
–695
.119.
Wechsatol
, W.
, Lorente
, S.
, and Bejan
, A.
, 2003
, “Dendritic Heat Convection on a Disc
,” Int. J. Heat Mass Transfer
, 46
(23
), pp. 4381
–4391
.120.
Zimparov
, V. D.
, da Silva
, A. K.
, and Bejan
, A.
, 2006
, “Constructal Tree-Shaped Parallel Flow Heat Exchangers
,” Int. J. Heat Mass Transfer
, 49
(23–24), pp. 4558
–4566
.121.
Bello-Ochende
, T.
, Liebenberg
, L.
, and Meyer
, J. P.
, 2007
, “Constructal Cooling Channels for Micro-Channel Heat Sinks
,” Int. J. Heat Mass Transfer
, 50
(21–22), pp. 4141
–4150
.122.
Salimpour
, M. R.
, Sharifhasan
, M.
, and Shirani
, E.
, 2013
, “Constructal Optimization of Microchannel Heat Sinks With Noncircular Cross Sections
,” Heat Transfer Eng.
, 34
(10
), pp. 863
–874
.123.
Feng
, H. J.
, Chen
, L. G.
, Xie
, Z.
, and Sun
, F. R.
, 2013
“Constructal Optimization of a Disc-Shaped Body With Cooling Channels for Specified Power Pumping
,” Int. J. Low-Carbon Technol.
, 10
(3), pp. 229
–237
.124.
Stocks
, M. D.
, Bello-Ochende
, T.
, and Meyer
, J. P.
, 2014
, “Maximum Thermal Conductance for a Micro-Channel, Utilizing Newtonian and Non-Newtonian Fluid
,” Heat Mass Transfer
, 50
(6
), pp. 865
–875
.125.
Salimpour
, M. R.
, and Menbari
, A.
, 2015
, “Analytical Optimization of Constructal Channels Used for Cooling a Ring Shaped Body Based on Minimum Flow and Thermal Resistances
,” Energy
, 81
(2), pp. 645
–651
.126.
Pence
, D. V.
, 2000
, “Improved Thermal Efficiency and Temperature Uniformity Using Fractal-Like Branching Channel Networks
,” International Conference on Heat Transfer and Transport Phenomena in Micro Scale
, Banff, AB, pp. 142
–148
.127.
Pence
, D. V.
, 2002
, “Reduced Pumping Power and Wall Temperature in Microchannel Heat Sinks With Fractal-Like Branching Channel Networks
,” Microscale Thermophys. Eng.
, 6
(4), pp. 319
–330
.128.
Alharbi
, A. Y.
, Pence
, D. V.
, and Cullion
, R. N.
, 2003
, “Fluid Flow Through Microscale Fractal-Like Branching Channel Networks
,” ASME J. Fluids Eng.
, 125
(6
), pp. 1051
–1057
.129.
Alharbi
, A. Y.
, Pence
, D. V.
, and Cullion
, R. N.
, 2004
, “Thermal Characteristics of Microscale Fractal-Like Branching Channels
,” ASME J. Heat Transfer
, 126
(5
), pp. 744
–752
.130.
Chen
, Y. P.
, and Cheng
, P.
, 2002
, “Heat Transfer and Pressure Drop in Fractal Tree-Like Microchannel Nets
,” Int. J. Heat Mass Transfer
, 45
(13
), pp. 2643
–2648
.131.
Chen
, Y. P.
, and Cheng
, P.
, 2005
, “An Experimental Investigation on the Thermal Efficiency of Fractal Tree-Like Microchannel Nets
,” Int. Commun. Heat Mass Transfer
, 32
(7
), pp. 931
–938
.132.
Chen
, Y. P.
, Zhang
, C. B.
, Shi
, M. H.
, and Yang
, Y.
, 2010
, “Thermal and Hydrodynamic Characteristics of Constructal Tree-Shaped Minichannel Heat Sink
,” AIChE J.
, 56
(8), pp. 2018
–2029
.133.
Chen
, Y. P.
, and Deng
, Z.
, 2015
, “Gas Flow in Micro Tree-Shaped Hierarchical Network
,” Int. J. Heat Mass Transfer
, 80
, pp. 163
–169
.134.
Hong
, F. J.
, Cheng
, P.
, Ge
, H.
, and Joo
, G. T.
, 2007
, “Conjugate Heat Transfer in Fractal-Shaped Microchannel Network Heat Sink for Integrated Microelectronic Cooling Application
,” Int. J. Heat Mass Transfer
, 50
(25–26), pp. 4986
–4998
.135.
Wang
, X. Q.
, Mujumdar
, A. S.
, and Yap
, C.
, 2006
, “Thermal Characteristics of Tree-Shaped Microchannel Nets for Cooling of a Rectangular Heat Sink
,” Int. J. Therm. Sci.
, 45
(11
), pp. 1103
–1112
.136.
Wang
, X. Q.
, Yap
, C.
, and Mujumdar
, A. S.
, 2006
, “Numerical Analysis of Blockage and Optimization of Heat Transfer Performance of Fractal-Like Microchannel Nets
,” ASME J. Electron. Packag.
, 128
(1
), pp. 38
–45
.137.
Wang
, X. Q.
, Mujumdar
, A. S.
, and Yap
, C.
, 2006
, “Laminar Heat Transfer in Constructal Microchannel Networks With Loops
,” ASME J. Electron. Packag.
, 128
(3
), pp. 273
–280
.138.
Wang
, X. Q.
, Mujumdar
, A. S.
, and Yap
, C.
, 2007
, “Effect of Bifurcation Angle in Tree-Shaped Microchannel Networks
,” J. Appl. Phys.
, 102
(7
), p. 073530
.139.
Xu
, P.
, Wang
, X. Q.
, Mujumdar
, A. S.
, and Yap
, C.
, 2009
, “Thermal Characteristics of Tree-Shaped Microchannel Nets With/Without Loops
,” Int. J. Therm. Sci.
, 48
(11
), pp. 2139
–2147
.140.
Wang
, X. Q.
, Xu
, P.
, Mujumdar
, A. S.
, and Yap
, C.
, 2010
, “Flow and Thermal Characteristics of Offset Branching Network
,” Int. J. Therm. Sci.
, 49
(2
), pp. 272
–280
.141.
Ghaedamini
, H.
, Salimpour
, M. R.
, and Mujumdar
, A. S.
, 2011
, “The Effect of Svelteness on the Bifurcation Angles Role in Pressure Drop and Flow Uniformity of Tree-Shaped Microchannels
,” Appl. Therm. Eng.
, 31
(5
), pp. 708
–716
.142.
Ghaedamini
, H.
, Salimpour
, M. R.
, and Campo
, A.
, 2011
, “Constructal Design of Reverting Microchannels for Convective Cooling of a Circular Disc
,” Int. J. Therm. Sci.
, 50
(6
), pp. 1051
–1061
.143.
Hart
, R. A.
, and Silva
, A. K.
, 2011
, “Experimental Thermal-Hydraulic Evaluation of Constructal Microfluidic Structures Under Fully Constrained Conditions
,” Int. J. Heat Mass Transfer
, 54
(15–16), pp. 3661
–3671
.144.
Hart
, R. A.
, Ponkala
, M. J. V.
, and Silva
, A. K.
, 2011
, “Development and Testing of a Constructal Microchannel Flow System With Dynamically Controlled Complexity
,” Int. J. Heat Mass Transfer
, 54
(25–26), pp. 5470
–5480
.145.
Xu
, S.
, Qin
, J.
, Guo
, W.
, and Fang
, K.
, 2013
, “The Design of an Asymmetric Bionic Branching Channel for Electronic Chips Cooling
,” Heat Mass Transfer
, 49
(6
), pp. 827
–834
.146.
Xie
, G.
, Zhang
, F.
, Sundén
, B.
, and Zhang
, W.
, 2014
, “Constructal Design and Thermal Analysis of Microchannel Heat Sinks With Multistage Bifurcations in Single-Phase Liquid Flow
,” Appl. Therm. Eng.
, 62
(2
), pp. 791
–802
.147.
Li
, Y.
, Zhang
, F.
, Sunden
, B.
, and Xie
, G.
, 2014
, “Laminar Thermal Performance of Microchannel Heat Sinks With Constructal Vertical Y-Shaped Bifurcation Plates
,” Appl. Therm. Eng.
, 73
(1
), pp. 185
–195
.148.
Xie
, G.
, Shen
, H.
, and Wang
, C.-C.
, 2015
, “Parametric Study on Thermal Performance of Microchannel Heat Sinks With Internal Vertical Y-Shaped Bifurcations
,” Int. J. Heat Mass Transfer
, 90
, pp. 948
–958
.149.
Calamas
, D.
, and Baker
, J.
, 2015
, “Experimental Performance of a Solid Heat Exchanger With Tree-Like Flow Passages
,” Exp. Heat Transfer
, 28
(3
), pp. 205
–221
.150.
Miguel
, A. F.
, 2016
, “A Study of Entropy Generation in Tree-Shaped Flow Structures
,” Int. J. Heat Mass Transfer
, 92
, pp. 349
–359
.151.
Ghodoossi
, L.
, 2005
, “Thermal and Hydrodynamic Analysis of a Fractal Microchannel Network
,” Energy Convers. Manage.
, 46
(5
), pp. 771
–788
.152.
Escher
, W.
, Michel
, B.
, and Poulikakos
, D.
, 2009
, “Efficiency of Optimized Bifurcating Tree-Like and Parallel Microchannel Networks in the Cooling of Electronics
,” Int. J. Heat Mass Transfer
, 52
(5–6), pp. 1421
–1430
.153.
Camburn
, B.
, Wood
, K.
, and Crawford
, R.
, 2012
, “Novel Topological Approach to Designing Flow Channels
,” 14th International Design Engineering Technical Conference
, Chicago, IL, Aug. 12–15, p. 71448
.154.
Reddy
, B. V. K.
, Ramana
, P. V.
, and Narasimhan
, A.
, 2008
, “Steady and Transient Thermo-Hydraulic Performance of Disc With Tree-Shaped Micro-Channel Networks With and Without Radial Inclination
,” Int. J. Therm. Sci.
, 47
(11
), pp. 1482
–1489
.155.
Kim
, S.
, Lorente
, S.
, and Bejan
, A.
, 2009
, “Transient Behavior of Vascularized Walls Exposed to Sudden Heating
,” Int. J. Therm. Sci.
, 48
(11
), pp. 2046
–2052
.156.
Cho
, K.-H.
, and Kim
, M.-H.
, 2012
, “Transient Thermal-Fluid Characteristics of Vascular Networks
,” Int. J. Heat Mass Transfer
, 55
(13–14), pp. 3533
–3540
.157.
Zamfirescu
, C.
, and Bejan
, A.
, 2003
, “Constructal Tree-Shaped Two-Phase Flow for Cooling a Surface
,” Int. J. Heat Mass Transfer
, 46
(15
), pp. 2785
–2797
.158.
Daniels
, B. J.
, Liburdy
, J. A.
, and Pence
, D. V.
, 2007
, “Adiabatic Flow Boiling in Fractal-Like Microchannels
,” Heat Transfer Eng.
, 28
(10
), pp. 817
–825
.159.
Kwak
, Y.
, Pence
, D.
, Liburdy
, J.
, and Narayanan
, V.
, 2009
, “Gas–Liquid Flows in a Microscale Fractal-Like Branching Flow Network
,” Int. J. Heat Fluid Flow
, 30
(5
), pp. 868
–876
.160.
Daguenet-Frick
, X.
, Bonjour
, J.
, and Revellin
, R.
, 2010
, “Constructal Microchannel Network for Flow Boiling in a Disc-Shaped Body
,” IEEE Trans. Compon. Packag. Technol.
, 33
(1
), pp. 115
–126
.161.
Zhang
, C. B.
, Chen
, Y. P.
, Wu
, R.
, and Shi
, M. H.
, 2011
, “Flow Boiling in Constructal Tree-Shaped Minichannel Network
,” Int. J. Heat Mass Transfer
, 54
(1–3), pp. 202
–209
.162.
Sarkar
, A.
, 2014
, “Analysis of Constructal Two Phase Micro-Channel Heat Exchanger With Water and R 134a
,” Int. J. Eng. Res. Technol.
, 3
(6), pp. 809
–819
.163.
Lorente
, S.
, and Bejan
, A.
, 2006
, “Heterogeneous Porous Media as Multiscale Structures for Maximum Flow Access
,” J. Appl. Phys.
, 100
(11
), p. 114909
.164.
Xu
, P.
, Yu
, B. M.
, Feng
, Y. J.
, and Liu
, Y. J.
, 2006
, “Analysis of Permeability for the Fractal-Like Tree Network by Parallel and Series Models
,” Physica A
, 369
(2
), pp. 884
–894
.165.
Xu
, P.
, Yu
, B. M.
, Feng
, Y. J.
, and Zou
, M. Q.
, 2006
, “Permeability of the Fractal Disk-Shaped Branched Network With Tortuosity Effect
,” Phys. Fluids
, 18
(7
), p. 078103
.166.
Ronayne
, M. J.
, and Gorelick
, S. M.
, 2006
, “Effective Permeability of Porous Media Containing Branching Channel Networks
,” Phys. Rev. E
, 73
(2
), p. 026305
.167.
Xu
, P.
, Yu
, B. M.
, Qiu
, S. X.
, and Cai
, J. C.
, 2008
, “An Analysis of the Radial Flow in the Heterogeneous Porous Media Based on Fractal and Constructal Tree Networks
,” Physica A
, 387
(26
), pp. 6471
–6483
.168.
Tan
, X.-H.
, and Li
, X.-P.
, 2014
, “Transient Flow Model and Pressure Dynamic Features of Tree-Shaped Fractal Reservoirs
,” J. Hydrodyn.
, 26
(4
), pp. 654
–663
.169.
Fan
, J. T.
, Sarkar
, M. K.
, Szeto
, Y. C.
, and Tao
, X. M.
, 2007
, “Plant Structured Textile Fabrics
,” Mater. Lett.
, 61
(2
), pp. 561
–565
.170.
Sarkar
, M. K.
, Fan
, J. T.
, Szeto
, Y. C.
, and Tao
, X. M.
, 2009
, “Development and Characterization of Light Weight Plant Structured Fabrics
,” Fibers. Polym.
, 10
(3
), pp. 343
–350
.171.
Chen
, Q.
, Fan
, J. T.
, and Sarkar
, M. K.
, 2012
, “Biomimetics of Branching Structure in Warp Knitted Fabrics to Improve Water Transport Properties for Comfort
,” Text. Res. J.
, 82
(11
), pp. 1131
–1142
.172.
Doumanidis
, C. C.
, 2009
, “Nanomanufacturing of Random Branching Material Architectures
,” Microelectron. Eng.
, 86
(4–6), pp. 467
–478
.173.
Xu
, Z.
, and Buehler
, M. J.
, 2009
, “Hierarchical Nanostructures Are Crucial to Mitigate Ultrasmall Thermal Point Loads
,” Nano Lett.
, 9
(5
), pp. 2065
–2072
.174.
Darbari
, S.
, Abdi
, Y.
, Mohajerzadeh
, S.
, and Soleimani
, E. A.
, 2010
, “High Electron Emission From Branched Tree-Like Carbon Nanotubes Suitable for Field Emission Applications
,” Carbon
, 48
(9
), pp. 2493
–2500
.175.
Kou
, J.
, Qian
, H.
, Lu
, H.
, Liu
, Y.
, Xu
, Y.
, Wu
, F.
, and Fan
, J.
, 2011
, “Optimizing the Design of Nanostructures for Improved Thermal Conduction Within Confined Spaces
,” Nanoscale Res. Lett.
, 6
(1
), p. 422
.176.
Huang
, H.-F.
, Chu
, Q.-X.
, and Xiao
, J.-K.
, 2010
, “Fractal Power Network Based on Plant Vein for Power Integrity
,” IEEE Trans. Electromagn. Compat.
, 52
(3
), pp. 759
–762
.177.
Huang
, H.-F.
, Liu
, S.-Y.
, and Guo
, W.
, 2012
, “A Hierarchical Tree Shaped Power Distribution Network Based on Constructal Theory for EBG Structure Power Plane
,” Prog. Electromagn. Res. B
, 36
, pp. 173
–191
.178.
Vandewalle
, N.
, and Ausloos
, M.
, 1997
, “Construction and Properties of Fractal Trees With Tunable Dimension: The Interplay of Geometry and Physics
,” Phys. Rev. E
, 55
(1
), pp. 94
–98
.179.
Bejan
, A.
, and Lorente
, S.
, 2011
, “The Constructal Law Origin of the Logistics S Curve
,” J. Appl. Phys.
, 110
(2
), p. 024901
.180.
Bejan
, A.
, and Lorente
, S.
, 2012
, “The Physics of Spreading Ideas
,” Int. J. Heat Mass Transfer
, 55
(4
), pp. 802
–807
.181.
Zamir
, M.
, 1976
, “The Role of Shear Forces in Arterial Branching
,” J. Gen. Physiol.
, 67
(2
), pp. 213
–222
.182.
Woldenberg
, M. J.
, and Horsfield
, K.
, 1986
, “Relation of Branching Angles to Optimality for Four Cost Principles
,” J. Theor. Biol.
, 122
(2
), pp. 187
–204
.183.
Woldenberg
, M. J.
, and Horsfield
, K.
, 1983
, “Finding the Optimal Lengths for Three Branches at a Junction
,” J. Theor. Biol.
, 104
(2
), pp. 301
–318
.184.
Metzger
, R. J.
, and Krasnow
, M. A.
, 1999
, “Genetic Control of Branching Morphogenesis
,” Science
, 284
(5420
), pp. 1635
–1639
.185.
Jones
, E. A. V.
, Noble
, F.
, and Eichmann
, A.
, 2006
, “What Determines Blood Vessel Structure? Genetic Prespecification vs. Hemodynamics
,” Physiology
, 21
(6
), pp. 388
–395
.186.
Bejan
, A.
, Rocha
, L. A. O.
, and Lorente
, S.
, 2000
, “Thermodynamic Optimization of Geometry: T- and Y-Shaped Constructs of Fluid Streams
,” Int. J. Therm. Sci.
, 39
(9–11), pp. 949
–960
.187.
Cieślicki
, K.
, 1999
, “Resistance to the Blood Flow of a Vascular Tree: A Model Study
,” Pol. J. Med. Phys. Eng.
, 5
, pp. 161
–172
.188.
Durand
, M.
, 2006
, “Architecture of Optimal Transport Networks
,” Phys. Rev. E
, 73
(1
), p. 016116
.189.
Liu
, Y.
, and Kassab
, G. S.
, 2007
, “Vascular Metabolic Dissipation in Murray's Law
,” Am. J. Physiol. Heart Circ. Physiol.
, 292
(3), pp. H1336
–H1339
.190.
Luo
, L.
, Yu
, B. M.
, Cai
, J. C.
, and Mei
, M. F.
, 2010
, “Symmetry Is Not Always Prefect
,” Int. J. Heat Mass Transfer
, 53
(21–22), pp. 5022
–5024
.191.
Kou
, J.
, Chen
, Y.
, Zhou
, X.
, Lu
, H.
, Wu
, F.
, and Fan
, J.
, 2014
, “Optimal Structure of Tree-Like Branching Networks for Fluid Flow
,” Physica A
, 393
(1), pp. 527
–534
.192.
Bejan
, A.
, and Lorente
, S.
, 2011
, “The Constructal Law and Evolution of Design in Nature
,” Phys. Life Rev.
, 8
(3
), pp. 209
–240
.193.
Womersley
, J. R.
, 1955
, “Method for the Calculation of Velocity, Rate of Flow and Viscous Drag in Arteries When the Pressure Gradient is Known
,” J. Physiol.
, 127
(3
), pp. 553
–563
.194.
Loudon
, C.
, and Tordesillas
, A.
, 1998
, “The Use of the Dimensionless Womersley Number to Characterize the Unsteady Nature of Internal Flow
,” J. Theor. Biol.
, 191
(1
), pp. 63
–78
.195.
West
, B. J.
, 1990
, Fractal Physiology and Chaos in Medicine
, World Scientific
, Singapore
.196.
Shlesinger
, M. F.
, and West
, B. J.
, 1991
, “Complex Fractal Dimension of the Bronchial Tree
,” Phys. Rev. Lett.
, 67
(15
), pp. 2106
–2108
.197.
Majumdar
, A.
, 1992
, “Role of Fractal Geometry in the Study of Thermal Phenomena
,” Annual Review of Heat Transfer
, C. L.
Tien
, ed., Hemisphere Publishing
, New York
, pp. 51
–110
.198.
Kalda
, J.
, 1999
, “On the Fractality of the Biological Tree-Like Structures
,” Discrete Dyn. Nat. Soc.
, 3
(4
), pp. 297
–306
.199.
Kamiya
, A.
, and Takahashi
, T.
, 2007
, “Quantitative Assessments of Morphological and Functional Properties of Biological Trees Based on Their Fractal Nature
,” J. Appl. Physiol.
, 102
(6
), pp. 2315
–2323
.200.
Aguirre
, J.
, Viana
, R. L.
, and Sanjuán
, M. A. F.
, 2009
, “Fractal Structures in Nonlinear Dynamics
,” Rev. Mod. Phys.
, 81
(1
), pp. 333
–386
.201.
Min
, K.
, Hosoi
, K.
, Kinoshita
, Y.
, Hara
, S.
, Degami
, H.
, Takada
, T.
, and Nakamura
, T.
, 2012
, “Use of Fractal Geometry to Propose a New Mechanism of Airway-Parenchymal Interdependence
,” Open J. Mol. Integr. Physiol.
, 2
(01
), pp. 14
–20
.202.
Rian
, I. M.
, and Sassone
, M.
, 2014
, “Tree-Inspired Dendriforms and Fractal-Like Branching Structures in Architecture: A Brief Historical Overview
,” Front. Archit. Civ. Eng. China
, 3
(3), pp. 298
–323
.203.
Longo
, G.
, and Montévil
, M.
, 2014
, Perspectives on Organisms: Biological Time, Symmetries and Singularities
, Springer
, New York
, pp. 38
–41
.204.
Tuckerman
, D. B.
, and Pease
, R. F. W.
, 1981
, “High-Performance Heat Sinking for VLSI
,” IEEE Electron. Devices Lett.
, 2
(5
), pp. 126
–129
.205.
Wechsatol
, W.
, Lorente
, S.
, and Bejan
, A.
, 2005
, “Tree-Shaped Networks With Loops
,” Int. J. Heat Mass Transfer
, 48
(3–4), pp. 573
–583
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