Abstract
The size of the offspring vessels and airways in circulatory and respiratory trees can be predicted by theory. We first review the relationship connecting a parent tube to daughter tubes based on the application of optimization principles, such as minimizing energy expenditure, minimizing the total flow resistance.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
“Maximum flow access” corresponds to minimum travel time or minimum transfer time (Bejan and Ledezma 1998). Thus, it is equivalent to state that “for a finite-size flow system to persist in time, it must evolve such that it provides a minimum travel time to currents that flow through it.”
- 2.
Wechsatol et al. (2006) studied the effect of junction losses on the optimal connection of large vessels to small. They concluded that the junction losses have a sizeable effect on optimized geometry when a dimensionless parameter called svelteness, defined by the ratio between the external and internal length scales, is lower than the square root of 10.
References
Aragón AM, Wayer JK, Geubelle PH, Goldberg DE, White SR (2008) Design of microvascular flow networks using multi-objective genetic algorithms. Comput Methods Appl Mech Eng 197:4399–4410
Ashrafizaadeh M, Bakhshaei H (2009) A comparison of non-Newtonian models for lattice Boltzmann blood flow simulations. Comput Math Appl 58:1045–1054
Bejan A (1997) Constructal-theory network of conducting paths for cooling a heat generating volume. Trans ASME: J Heat Transfer 40:799–816
Bejan A (2000) Shape and structure, from engineering to Nature. Cambridge University Press, Cambridge
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 44:699–704
Bejan A (2005) The constructal law of organization in nature: tree-shaped flows and body size. J Exp Biol 208:1677–1686
Bejan A (2006) Advanced engineering thermodynamics, 3rd edn. Wiley, Hoboken
Bejan A, Ledezma GA (1998) Streets tree networks and urban growth: optimal geometry for quickest access between a finite-size volume and one point. Phys A 255:211–217
Bejan A, Lorente S (2008) Design with constructal theory. Wiley, Hoboken
Bejan A, Rocha LAO, Lorente S (2000) Thermodynamic optimization of geometry: T and Y-shaped constructs of fluid streams. Int J Therm Sci 39:949–960
Bejan A, Dincer I, Lorente S, Miguel AF, Reis AH (2004) Porous and complex flow structures in modern technologies. Springer, New York
Bejan A, Lorente S, Miguel AF, Reis AH (2006a) Constructal theory of distribution of river sizes. In: Bejan A (ed) Advanced engineering thermodynamics, 3rd edn. Wiley, Hoboken, pp 779–782
Bejan A, Lorente S, Wang K (2006b) Networks of channels for self-healing composite materials. J Appl Phys 100:033528
Bejan A, Lorente S, Lee J (2008) Unifying constructal theory of tree roots, canopies and forests. J Theor Biol 254:529–540
Bejan A, Lorente S (2013) Constructal law of design and evolution: Physics, biology, technology, and society. J Appl Phys 113:151301
Blaiszik BJ, Kramer SL, Olugebefola SC, Moore JS, Sottos RN, White SR (2010) Self-healing polymers and composites. Annu Rev Mater Res 40:179–211
Broe R, Rasmussen ML, Frydkjaer-Olsen U, Olsen BS, Mortensen HB, Peto T, Grauslund J (2014) Retinal vascular fractals predict long-term microvascular complications in type 1 diabetes mellitus: the danish cohort of pediatric diabetes 1987 (DCPD1987). Diabetologia 57:2215–2221
Burrowes KS, Hoffman EA, Tawhai MH (2009) Species-specific pulmonary arterial asymmetry determines species differences in regional pulmonary perfusion. Ann Biomed Eng 37:2497–2509
Cetkin E, Oliani A (2015) The natural emergence of asymmetric tree-shaped pathways for cooling of a non-uniformly heated domain. J Appl Phys 118:024902
Cetkin E, Lorente S, Bejan A (2011a) Vascularization for cooling and mechanical strength. Int J Heat Mass Transf 54:2774–2781
Cetkin E, Lorente S, Bejan A (2011b) Hybrid grid and tree structures for cooling and mechanical strength. J Appl Phys 110:064910
Cetkin E, Lorente S, Bejan A (2011c) Vascularization for cooling and mechanical strength. Int J Heat Mass Transf 54:2774–2781
Chen YP, Cheng P (2005) An experimental investigation on the thermal efficiency of fractal tree-like microchannel nets. Int Commun Heat Mass Transfer 32:931–938
Cohn DL (1954) Optimal systems: I the vascular system. Bull Math Biophys 16:59–74
Cohn DL (1955) Optimal systems: II the vascular system. Bull Math Biophys 17:219–227
Hess WR (1903) Eine mechanisch bedingte Gesetzmäßigkeit im Bau des Blutgefäßsystems. Archiv für Entwicklungsmechanik der Organismen 16:632–641
Hess WR (1917) Über die periphere Regulierung der Blutzirkulation. Pflüger’s Archiv für die gesamte Physiologie des Menschen und der Tiere 168:439–490
Horsfield K (1986) Morphometry of airways. In: Macklem PT, Mead J (eds) Handbook of physiology: the respiratory system III. American Physiological Society, Bethesda, pp 75–87
Horsfield K, Cumming G (1967) Angles of branching and diameters of branches in the human bronchial tree. Bull Math Biophys 29:245–259
Horsfield K, Cumming G (1968) Morphology of the bronchial tree in man. J Appl Physiol 24:373–383
Horton RE (1932) Drainage basin characteristics. Trans Am Gcophys Union 13:350–361
Horton RE (1945) Erosional development of streams and their drainage basins: hydrophysical approach to quantitative morphology. Geol Soc Am Bull 56:275–370
Huang W, Yen RT, McLaurine M, Bledsoe G (1996) Morphometry of the human pulmonar vasculature. J Appl Physiol 81:2123–2133
Hutchins GM, Miner MM, Boitnott JK (1976) Vessel caliber and branch-angle of human coronary artery branch-points. Circ Res 38:572–576
Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106:620–630
Kamiya A, Togawa T (1972) Optimal branching structure of the vascular tree. Bull Math Biophys 34:431–438
Kassab GS, Rider CA, Tang NJ, Fung YC (1993) Morphometry of pig coronary arterial trees. Am J Physiol Heart Circ Physiol 265:H350–H365
Kim S, Lorente S, Bejan A (2006) Vascularized materials: tree-shaped flow architectures matched canopy to canopy. J Appl Phys 100:063525
Kitaoka H, Ryuji T, Suki B (1999) A three-dimensional model of the human airway tree. J Appl Physiol 87:2207–2217
Lee J, Lorente S, Bejan A (2009) Transient cooling response of smart vascular materials for self-cooling. J Appl Phys 105:064904
Mandelbrot BB (1983) The fractal geometry of Nature. W.H. Freeman, New York
Miao T, Chen A, Xu Y, Yang S, Yu B (2016) Optimal structure of damaged tree-like branching networks for the equivalent thermal conductivity. Int J Therm Sci 102:89–99
Miguel AF (2010) Natural flow systems: acquiring their constructal morphology. Int J Des Nat Ecodyn 5:230–241
Miguel AF (2012) Lungs as a natural porous media: architecture, airflow characteristics and transport of suspended particles. In: Delgado J (ed) Heat and mass transfer in porous media. Advanced Structured Materials Series, vol 13. Springer, Berlin, pp 115–137
Miguel AF (2015) Fluid flow in a porous tree-shaped network: optimal design and extension of Hess–Murray’s law. Phys A 423:61–71
Miguel AF (2016a) Toward an optimal design principle in symmetric and asymmetric tree flow networks. J Theor Biol 389:101–109
Miguel AF (2016b) Scaling laws and thermodynamic analysis for vascular branching of microvessels. Int J Fluid Mech Res 43:390–403
Miguel AF (2016c) A study of entropy generation in tree-shaped flow structures. Int J Heat Mass Trans 92:349–359
Miguel AF (2017) Penetration of inhaled aerosols in the bronchial tree. Med Eng Phys 44:25–31
Miguel AF (2018) Constructal branching design for fluid flow and heat transfer. Int J Heat Mass Transf 122:204–211
Moledina S, de Bruyn A, Schievano S, Owens CM, Young C, Haworth SG, Taylor AM, Schulze-Neick I, Muthurangu V (2011) Fractal branching quantifies vascular changes and predicts survival in pulmonary hypertension: a proof of principle study. Heart 97:1245–1249
Moreau B, Mauroy B (2015) Murray’s law revisited: Quémada’s fluid model and fractal tree. J Rheol 59:1419
Murray CD (1926a) The physiological principle of minimum work. I. The vascular system and the cost of blood volume. Proc Natl Acad Sci U S A 12:207–214
Murray CD (1926b) The physiological principle of minimum work applied to the angle of branching of arteries. J Gen Physiol 9:835–841
Muzychka YS (2007) Constructal multi-scale design of compact micro-tube heat sinks and heat exchangers. Int J Therm Sci 46:245–252
Pence DV (2003) Reduced pumping power and wall temperature in microchannel heat sinks with fractal-like branching channel networks. Microscale Thermophys Eng 6:319–330
Phillips CG, Kaye SR (1997) On the asymmetry of bifurcations in the bronchial tree. Respir Physiol 107:85–98
Popel AS, Johnson PC (2005) Microcirculation and hemorheology. Annu Rev Fluid Mech. 37:43–69
Pries AR, Neuhaus D, Gaehtgens P (1992) Blood viscosity in tube flow: dependence on diameter and hematocrit. Dtsch Arch Klin Med 169:212–222
Pries AR, Reglin B, Secomb TW (2003) Structural response of microcirculatory networks to changes in demand: information transfer by shear stress. Am J Physiol Heart Circ Physiol 284:H2204–H2212
Reis AH (2006) Constructal view of scaling laws of river basins. Geomorphology 78:201–206
Reis AH, Miguel AF, Aydin M (2004) Constructal theory of flow architecture of the lungs. Med Phys 31:1135–1140
Revellin R, Rousset F, Baud D, Bonjour J (2009) Extension of Murray’s law using a non-Newtonian model of blood flow. Theor Biol Med Model 6:7
Rocha LAO, Lorente S, Bejan A (2006) Conduction tree networks with loops for cooling a heat generating volume. Int J Heat Mass Transfer 49:2626–2635
Rocha LAO, Lorente S, Bejan A (2014) Vascular design for reducing hot spots and stresses. J Appl Phys 115:174904
Rodríguez-Iturbe I, Rinaldo A (2001) Fractal river basins: chance and self-organization. Cambridge University Press, Cambridge
Rojas AMT, Romero AM, Pagonabarraga I, Travasso RDM, Poire EC (2015) Obstructions in vascular networks. Critical vs non-critical topological sites for blood supply. PLoS ONE 10:e0128111
Schneider W (2003) Cardiovascular fluid mechanics, CISM Courses and Lectures Series, vol 446. Springer, Berlin
Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27(379–423):623–656
Sherman TF (1981) On connecting large vessels to small, the meaning of Murray’s law. J Gen Physiol 78:431–453
Taber LA, Ng S, Quesnel AM, Whatman J, Carmen CJ (2001) Investigating Murray’s law in the chick embryo. J Biomech 34:121–124
Takayasu H (1990) Fractals in physical sciences. Manchester University Press, Manchester
Thoma R (1901) Über den verzweigungsmodus der arterien. Archiv für Entwicklungsmechanik der Organismen 2:352–413
Toksvang LN, Berg RM (2013) Using a classic paper by Robin Fahraeus and Torsten Lindqvist to teach basic hemorheology. Adv Physiol Educ 37:129–133
Toohey KS, Sottos NR, Lewis JA, Moore JS, White SR (2007) Self-healing materials with microvascular networks. Nat Mater 6:581–585
Uylings HBM (1977) Optimization of diameters and bifurcation angles in lung and vascular tree structures. Bull Math Biol 39:509–519
Wagner A (2007) From bit to it: how a complex metabolic network transforms information into living matter. BMC Syst Biol 1:33
Wechsatol W, Lorente S, Bejan A (2002) Optimal tree-shaped networks for fluid flow in a disc-shaped body. Int J Heat Mass Transf 45:4911–4924
Wechsatol W, Lorente S, Bejan A (2005) Tree-shaped networks with loops. Int J Heat Mass Transfer 48:573–583
Wechsatol W, Lorente S, Bejan A (2006) Tree-shaped flow structures with local junction losses. Int J Heat Mass Trans 49:2957–2964
West GB, Brown JH, Enquist BJ (1997) A general model for the origin of allometric scaling laws in biology. Science 276:122–126
White SR, Sottos NR, Geubelle PH, Moore JS, Kessler MR, Sriram SR, Brown EN, Viswanathan S (2001) Autonomic healing of polymer composites. Nature 409:794–797
Williams HR, Trask RS, Weaver PM, Bond IP (2008) Minimum mass vascular networks in multifunctional materials. J R Soc Interface 5:55–65
Xu P, Yu B, Yun M, Zou M (2006) Heat conduction in fractal tree-like branched networks. Int J Heat Mass Trans 49:3746–3751
Xu P, Wang XQ, Mujumdar AS, Yap C, Yu BM (2009) Thermal characteristics of tree-shaped microchannel nets with/without loops. Int J Therm Sci 48:2139–2147
Xu P, Sasmito AP, Yu B, Mujumdar AS (2016) Transport phenomena and properties in treelike networks. Appl Mech Rev 68: 040802-1–040802-17
Young T (1809) On the functions of the heart and arteries. Philos Trans R Soc Lond 99:1–31
Zamir M (1975) The role of shear forces in arterial branching. J Gen Physiol 67:213–222
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 The Author(s)
About this chapter
Cite this chapter
Miguel, A.F., Rocha, L.A.O. (2018). Tree-Shaped Flow Networks Fundamentals. In: Tree-Shaped Fluid Flow and Heat Transfer. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-73260-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-73260-2_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-73259-6
Online ISBN: 978-3-319-73260-2
eBook Packages: EngineeringEngineering (R0)