Constructal law is a theory of the generation of design (configuration, pattern, geometry) in nature. According to this theory, natural design and the constructal law unite all animate and inanimate systems. The constructal law was stated by Adrian Bejan in 1996 as follows: "For a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it." The constructal law is receiving increased acceptance within the scientific community.
"Constructal" is a word coined by Adrian Bejan, in order to describe the natural tendency of flow systems (rivers, trees and branches, lungs, tectonic plates  as well as engineered forms)  to generate and evolve structures that increase flow access.
The constructal law was proposed in 1996 as a summary of all design generation and evolution phenomena in nature, bio and non-bio. The constructal law represents three steps toward making “design in nature” a concept and law-based domain in science:
- Life is flow: all flow systems are living systems, the animate and the inanimate.
- Design generation and evolution is a phenomenon of physics.
- Designs have the universal tendency to evolve in a certain direction in time.
The constructal law is proposed as a first principle of physics accounting for all design and evolution in nature. It holds that shape and structure arise to facilitate flow. The designs that happen spontaneously in nature reflect this tendency: they allow entities to flow more easily – to measurably move more current farther and faster per unit of useful energy consumed. Rain drops, for example, coalesce and move together, generating rivulets, streams and the mighty river basins of the world because this design allows them to move more easily.
The constructal law asks the question: Why does this design arise at all? Why can't the water just seep through the ground? The constructal law provides this answer: Because the water flows better with design. The constructal law covers the tendency of nature to generate designs to facilitate flow.
The constructal law covers natural phenomena of organization, such as tree-shaped flows, round tubes and bones, scaling laws, etc. The lightning bolts that flash across the sky generate a tree-like structure because this is a good design for moving a current (electricity) from an area (the cloud) to a point (a church steeple or another cloud). The circulatory and nervous systems of biological creatures generate a similar tree-like design because they too are moving currents from a point to an area and from an area to a point.
Although treelike structures are a very common design in nature, they are only one manifestation of the constructal law. In a simple example, logs floating on a lake or icebergs at sea orient themselves perpendicular to the wind which increases the transfer of motion from the moving air body to the water body. A more complex example is the design of animals that have evolved to move mass more efficiently (to cover more distance per unit of useful energy) across the landscape.
This includes the seemingly “characteristic” sizes of organs, the shape of bones, the rhythm of breathing lungs and beating hearts, of undulating tails, running legs, and flapping wings. The constructal law proclaims that all these designs have arisen—and work together—to allow animals, like raindrops in a river basin, to move more easily across a landscape. Because human beings are not separate from but a part of nature, their designs are also governed by the constructal law.
The constructal law defines the time direction of all evolutionary design phenomena. It states that designs should evolve over time to acquire better configurations to provide more access for the currents that flow through them. It defines in physics terms what it means to be “better”, more "fit", to “survive”, and to be efficient. Not all changes are improvements, but those that stick measurably enhance flow.
Constructal design occurs at every scale. Each component of an evolving flow system—each rivulet, each tree, and each road—acquires evolving designs to facilitate flow access. As these elements coalesce into larger and larger structures (into evolving river basins, forests and transport networks), a hierarchy emerges such that the multi sized components and channels work together so that everything flows more easily. This is seen in the shape and structure of the neural networks in the brain, of the alveoli in the lung, the size and distribution of vegetation in the forest and of human settlements on the map.
In the big picture, all the mating and morphing flows on the largest system that surrounds us, the Earth itself, evolve to enhance global flow. For example, trees and other forms of vegetation that move moisture from the ground to the air are components of the larger global system, including forests, river basins and weather patterns, that have the tendency to equilibrate all the moisture on Earth. The constructal law states that every flow system is destined to remain imperfect. The direction of design evolution is toward distributing the imperfections of the system, such that the “whole” flows easier (e.g., river basin, animal body, human vehicle).
Evolution never ends. Optimality statements (minimum, maximum, optimum, static, end design, destiny) have only local, limited applicability. The constructal law covers them because it is about the time direction of all the evolutionary design phenomenon.
The constructal law is a law of physics - the law of design generation and evolution in nature. The natural phenomenon is not the elimination but the distribution (better and better over time) of imperfection. The distribution of imperfection generates the geometry (shape, structure) of the system. Today, optimization of many systems arising in thermal engineering such as conductive pathways (fins, cavities  and highly conductive inserts), convective channels and radiative systems  are inspired by constructal law.
For example, in point-area and point-volume flows, the constructal law predicts tree architectures, such flows displaying at least two regimes: one highly resistive and one with lower resistivity. The constructal-law tendency manifests itself at every scale.
|Application||What flows||Tree channels: Low Resistivity||Interstitial spaces: High Resistivity|
|Packages of electronics||Heat||High-conductivity inserts (blades, needles)||Low conductivity substrate|
|Urban traffic||People||Low-resistance street car traffic||Street walking in urban structure|
|River basins||Water||Low-resistance rivulet and rivers||Darcy flow through porous media|
|Lungs||Air||Low-resistance airways, bronchial passages||diffusion in alveoli tissues|
|Circulatory system||Blood||Low-resistance blood vessels, capillaries, arteries, veins||diffusion in capillaries tissues|
The constructal law provides a unifying theory of evolution. It holds that inanimate and animate phenomena generate evolving configurations to move more easily. The constructal law also provides the physics definition of life, of what it means to be alive. It states that life means flow and the free generation of design. If the flows stop, the system is dead (in thermodynamic equilibrium). The constructal law is the physics law of life, design and evolution.
Thermodynamics rests on two laws. Both are first principles: The first law commands the conservation of energy, and the second law commands irreversibility: the tendency of all currents to flow from high (temperature, pressure) to low. These two laws are about systems in the most general sense, viewed as black boxes, without shape and structure.
The two laws of thermodynamics do not account for nature completely. Nature is not made of black boxes. Nature’s boxes are filled with evolving, freely morphing configurations—even the fact that they have names (rivers, blood vessels) is due to their appearance, organization, or design. Where the second law commands that things should flow from high to low, the constructal law commands that they evolve in configurations that flow more and more easily over time.
|State||Flow architecture (flow structure)|
|Process||Change of structure (design change)|
|Properties||Global objective and global constraints|
|Equilibrium state||Equilibrium flow architecture|
|Fundamental relation||Fundamental relation|
|Constrained equilibrium states||Nonequilibrium architectures|
|Removal of constraints||Increased freedom to morph|
|Energy minimum principle||Evolution toward greater flow access|
In contrast to fractal models of observed objects in nature, the constructal law is predictive and thus can be tested experimentally. Many natural designs, animate and inanimate, have been explained and unified by the constructal law. For example:
|Global convective cells on Earth's atmosphere||River basin design: Horton's rules of stream numbers (~4) and lengths (~2), and all the other scaling rules (e.g., Melton, Hack) of river basins all over the world|
|The distribution of city sizes and numbers, i.e. Zipf's law relating log (size) versus log (rank)||The distribution of tree sizes and numbers on the forest floor|
|Vision, cognition, and the golden ratio phenomenon||Pedestrian movement, speeds, and patterns|
|The flow of education as a morphing vasculature on the globe, and the rigidity of university rankings||The entire architecture of vegetation: roots, trunks, canopies, branches, leaves, and the forest, including the prediction of Leonardo da Vinci's rule, Huber's rule, and the Fibonacci sequence.|
|The emergence of urban traffic design.||The entire morphogenesis of dendritic crystals (e.g., snowflakes), as a flow structure that facilitates the flow of the heat of solidification.|
|The scaling law of all animal locomotion (running, flying, swimming): speeds, frequencies, forces and the work spent per unit of mass moved and distance traveled.||The evolution of speed in sports.|
|Kleiber's law, the relationship between metabolic rate and body size.||The relationship between breathing and heart beating times and body size.|
|The human bronchial tree with 23 levels of bifurcation.||The relationship between the mass transfer contact area and body mass.|
|The dimensions of the alveolar sac.||The total length of the airways.|
|The total alveolar surface area.||The total resistance to oxygen transport in the respiratory tree.|
|The life time and life travel of animals, vehicles (human & machine species), rivers and air currents.||Aspect ratios of airplanes|
The main criticism of the constructal law is that its formulation is vague. The constructal law states that “For a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it”, but there is neither a mention of what these “currents” are nor an explicit definition of what “providing easier access” means, nor precisely formulating the relationship with math. Without defining the physical quantities or their exact relationships, it is not physical or mathematical.
The related criticism of the constructal law is that there has been no attempt to prove it from first principles. Contrarily to alternative theories of non-equilibrium thermodynamics, there is no proof of constructal theory based on simplified systems of statistical physics. The claim that constructal theory is a fundamental principle of thermodynamics itself has also been disputed.
Responses to criticisms
Bejan has responded to this criticism  by noting that one cannot prove a first principle based on other first principles. The constructal law is not about what flows, but about the physics phenomenon of how any flow system acquires its evolving configuration (design) over time. The constructal law is not about optimality (max, min, opt)—it is the definition of “life” in physics terms, and of the time direction of the changes in flow configuration.
Bejan  also noted that the phenomenon governed by the constructal law (design in nature) is macroscopic (finite size, not infinitesimal). It is the birth of design and evolution of design in all the parts together. It is dynamic, not static. The evolution never ends. There is no end design, no destiny (max, min, opt).
Bejan and Lorente expanded on this  by explaining the difference between a law of physics (e.g., the constructal law) and the many invocations of the law, which underpin many “theories” based on the law. In the section titled “The constructal law versus the second law of thermodynamics” they noted that the constructal law and the second law are first principles. The constructal law is a useful reminder not only of what was missing in physics and thermodynamics (the law of design and evolution) but also of what is present. For example, contrary to the critic’s view, the first and second laws of thermodynamics did not require any “proof based on simplified systems of statistical physics.”
The constructal law is a statement of a natural tendency—the time direction of the phenomenon. It is as non-mathematical as the original statements of the second law:
Clausius: No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.
Kelvin: Spontaneously, heat cannot flow from cold regions to hot regions without external work being performed on the system.
A new law does not have to be stated in mathematical terms. The mathematization of the second law statement (and of thermodynamics) came later. The constructal law underwent the same evolution. The 1996 statement was followed in 2004 by a complete mathematical formulation of constructal-law thermodynamics.
Flow leads to better flow. Like any other law of physics, the constructal law is a concise summary of observed facts: the natural tendency of flow systems to evolve toward configurations that provide easier access over time. The word “access” means the ability to move through a confined space such as a crowded room. This is why “finite size” appears in the constructal law statement. This mental viewing covers all the flow design and evolution phenomena, animate and inanimate, because they all morph to enter and to flow better, more easily.
- T. Basak, The law of life: the bridge between Physics and Biology. Phys Life Rev 8, 249-252 (2011)
- Bejan, Adrian (1997). “Advanced Engineering Thermodynamics,” (2nd ed.). New York: Wiley.
- A. Kremer-Marietti and J. Dhombres, L’Épistemologie (Ellipses, Paris, 2006)
- Errera, Marcelo (January 2, 2015). "Constructal Law of Design in Nature". scoop.it. Retrieved January 2, 2015.
- D. Tondeur, Y. Fan and L. Luo, Constructal optimization of arborescent structures with flow singularities. Chem Eng Sci 64, 3968−3982 (2009).
- S. Quéré, Constructal theory of plate tectonics. Int J Design & Nature Ecodyn 5, 242-253 (2010).
- D. Queiros-Conde and M. Feidt, Eds., Constructal Theory and Multi-scale Geometries: Theory and Applications in Energetics, Chemical Engineering and Materials (Les Presses de L’ENSTA, Paris, 2009).
- A. H. Reis, Constructal theory: from engineering to physics, and how flow systems develop shape and structure. Appl Mech Rev 59, 269-282 (2006).
- L. Wang, Universality of design and its evolution. Phys Life Rev 8, 257-258 (2011)
- N. Acuña, Mindshare. Igniting Creativity and Innovation Through Design Intelligence (Motion, Henderson, Nevada 2012).
- A. F. Miguel, The physics principle of the generation of flow configuration, Phys Life Rev 8, 243-244 (2011)
- MR Hajmohammadi, H Eskandari, M Saffar-Avval and A Campo, A new configuration of bend tubes for compound optimization of heat and fluid flow, Energy 62 (2013) 418-424.
- M. R. Hajmohammadi, S. Poozesh, A. Campo and Seyed Salman Nourazar, Valuable reconsideration in the constructal design of cavities, Energy Conversion and Management, 66 (2013) 33–40.
- A. F. Miguel. Constructal pattern formation in stony corals, bacterial colonies and plant roots under different hydrodynamics conditions. Journal of Theoretical Biology 242, 954-961 (2006)
- A. F. Miguel, A. Bejan. The principle that generates dissimilar patterns inside aggregates of organisms. Journal Physica A 388, 727-731 (2009)
- A. H. Reis and C. Gama, Sand size versus beachface slope – an explanation based on the Constructal Law, Geomorphology 114, 276-283 (2010).
- A. H. Reis, Constructal view of scaling laws of river basins. Geomorphology 78, 201-206 (2006).
- A. F. Miguel, Natural flow systems: acquiring their constructal morphology. Int J Design & Nature Ecodyn 5, 230-241 (2010).
- A. H. Reis, Design in nature, and the laws of physics. Phys Life Rev 8, 255-256 (2011)
- Bejan, A; Marden, James H. (2006). Constructing Animal Locomotion from New Thermodynamics Theory. American Scientist, July–August, Volume 94, Number 4
- G. Resconi, Morphotronics and Constructal theory, LINDI 2011, 3rd IEEE Int. Symp. Logistics and Industrial Informatics, Budapest, Hungary, August 25–27 (2011).
- L. C. Kelley and K. Behan, Empathy & evolution: how dogs convert stress into flow. Psychology Today 6 August 2012.
- L. C. Kelley and K. Behan, The canine mind bows to the Constructal Law. Psychology Today 16 October (2012).
- Bejan, A (2010). The Constructal Law Origin of the Wheel, Size, and Skeleton in Animal Design. American Journal of Physics, Vol. 78, No. 7: 692-699
- Miguel, Antonio F. (27 April 2009). "Constructal theory of pedestrian dynamics". Physics Letters A 373 (20): 1734–1738. doi:10.1016/j.physleta.2009.03.020.
- Bejan, A; Merkx, Gilbert A, eds. (2007). "Constructal Theory of Social Dynamics." New York: Springer.
- P. Kalason, Épistémologie Constructale du Lien Cultuel (L’Harmattan, Paris, 2007).
- Bejan, Adrian (2005). "The Constructal Law of Organization in Nature: Tree-shaped flows and body size". Journal of Experimental Biology 208 (9): 1677–1686. doi:10.1242/jeb.01487.
- Lorente, S; Bejan, A (2010). Few Large and Many Small: Hierarchy in Movement on Earth. International Journal of Design & Nature and Ecodynamics. Vol. 5, No. 3: 1–14.
- Bejan, Adrian; Lorente, Sylvie; Lee, J. (7 October 2008). "Unifying constructal theory of tree roots, canopies and forests". Journal of Theoretical Biology 254 (3): 529–540. doi:10.1016/j.jtbi.2008.06.026.
- Y. Ventikos, The importance of the constructal framework in understanding and eventually replicating structure in tissue. Phys Life Rev 8, 241-242 (2011).
- Bejan, A; Lorente, S (2008). “Design with Constructal Theory,” Hoboken: Wiley.
- M. R. Hajmohammadi, E. Shirani, M. R. Salimpour and A. Campo, Constructal placement of unequal heat sources on a plate cooled by laminar forced convection, International Journal of Thermal sciences, 60 (2012) 13–22.
- M. R. Hajmohammadi, S. Poozesh, S. S. Nourazar, Constructal design of multiple heat sources in a square-shaped fin, Journal of Process Mechanichal Engineering, 226 (2012) 324-336.
- M. R. Hajmohammadi, V. Alizadeh Abianeh, M. Moezzinajafabadi and M. Daneshi, Fork-shaped highly conductive pathways for maximum cooling in a heat generating piece, Applied Thermal Engineering, 61 (2013) 228–235
- M. Eslami and K. Jafarpur, Optimal distribution of imperfection in conductive constructal designs of arbitrary configurations. J Appl Phys 112, 104905 (2012).
- A. Pouzesh, M. R. Hajmohammadi and S. Poozesh, Investigations on the internal shape of constructal cavities intruding a heat generating body, Thermal Science, DOI: 10.2298/TSCI120427164P 2012.
- M.R. Hajmohammadi, O. Joneydi Shariatzadeh, M. Moulod and S.S. Nourazar, Phi and Psi shaped conductive routes for improved cooling in a heat generating piece, International Journal of Thermal Sciences 77 (2014) 66–74
- M. R. Hajmohammadi, M. Rahmani, A. Campo and O. Joneydi Shariatzadeh, Optimal design of unequal heat flux elements for optimized heat transfer inside a rectangular duct, Energy, 68 (2014) 609-616.
- M. R. Hajmohammadi, S. S. Nourazar, A. Campo and S. Poozesh, Optimal discrete distribution of heat flux elements for in-tube laminar forced convection, International Journal of Heat and Fluid Flow, 40 (2013) 89–96.
- M. R. Hajmohammadi, M. R. Salimpour, M. Saber and A. Campo, Detailed analysis for the cooling performance enhancement of a heat source under a thick plate, Energy Conversion and Management, 76 (2013) 691–700
- M. R. Hajmohammadi, S. Poozesh and R. Hosseini, Radiation effect on constructal design analysis of a T-Y-shaped assembly of fins, Journal of Thermal Science and Technology, 7 (2012) 677-692.
- M. R. Errera and C. A. Marin, A Comparison Between Random and Deterministic Dynamics of River Drainage Basins Formation, Engenharia Térmica (Thermal Engineering), Vol. 8, n. 01, 65-71 (2009).
- R. A. Hart and A. K. da Silva, Experimental thermal-hydraulic evaluation of constructal microfluidic structures under fully constrained conditions. Int J Heat Mass Transfer 54, 3661-3671 (2011).
- R. A. Hart, M. J. V. Ponkala and A. K. da Silva, Development and testing of a constructal microchannel flow system with dynamically controlled complexity. Int J Heat Mass Transfer 54, 5470-5480 (2011).
- Z. Fan, X. Zhou, L. Luo and W. Yuan, Experimental investigation of the flow distribution of a 2-dimensional constructal distributor. Exp Therm Fluid Sci 33, 77−83 (2008).
- K.-H. Cho, W.-P. Chang and M.-H. Kim, A numerical and experimental study to evaluate performance of vascularized cooling plates. Int J Heat Mass Transfer 32, 1186-1198 (2011).
- D. Haller, P. Woias and N. Kockmann, Simulation and experimental investigation of pressure loss and heat transfer in microchannel networks containing bends and T-junctions. Int J Heat Mass Transfer 52, 2678−2689 (2009).
- L. Chen, Progress in study on constructal theory and its applications. Science China, Technological Sciences 55 (3), 802-820 (2012).
- Bejan, Adrian; Reis, Antonio H. (25 March 2005). "Thermodynamic optimization of global circulation and climate". International Journal of Energy Research 29 (4): 303-316. doi:10.1002/er.1058.
- Bejan, Adrian (2009). "The Golden Ratio Predicted: Vision, Cognition And Locomotion As A Single Design In Nature". International Journal of Design & Nature and Ecodynamics 4 (2): 97–104. doi:10.2495/D&NE-V4-N2-97-104.
- Bejan, Adrian (2007). Int. Journal of Design & Nature and Ecodynamics 2 (4): 319-327. doi:10.2495/D&N-V2-N4-319-327.
- Bejan, Adrian (Summer 1996). "Street network theory of organization in nature". Journal of Advanced Transportation 30 (2): 85–107. doi:10.1002/atr.5670300207.
- Bejan, Adrian; Ledezma, Gustavo A. (15 June 1998). "Streets tree networks and urban growth: Optimal geometry for quickest access between a finite-size volume and one point". Physica A: Statistical Mechanics and its Applications 255 (1-2): 211-217. doi:10.1016/S0378-4371(98)00085-5.
- Bejan, A.; Lorente, S.; Yilbas, B.S.; Sahin, A.Z. (26 April 2013). "Why solidification has an S-shaped history". Nature Scientific Reports 3: 1711. doi:10.1038/srep01711.
- Bejan, Adrian; Marden, James H. (January 15, 2006). "Unifying constructal theory for scale effects in running, swimming and flying". Journal of Experimental Biology 209: 238-248.
- Charles, Jordan D.; Bejan, A. (August 1, 2009). "The evolution of speed, size and shape in modern athletics". Journal of Experimental Biology 212: 2419-2425.
- Bejan, Adrian (February 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". International Journal of Heat and Mass Transfer 44 (4): 699-704. doi:10.1016/S0017-9310(00)00138-1.
- Bejan, Adrian (June 1997). "Theory of Organization in Nature: Pulsating Physiological Processes". International Journal of Heat and Mass Transfer 40 (9): 2097-2104. doi:10.1016/S0017-9310(96)00291-8.
- Reis, A. H.; Miguel, A. F.; Aydin, M. (16 April 2004). "Constructal theory of flow architecture of the lungs". Medical Physics 31 (5): 1135-1140. doi:10.1118/1.1705443.
- Bejan, A. (24 Aug 2012). "Why the bigger live longer and travel farther: animals, vehicles, rivers and the winds". Nature Scientific Reports 2: 594. doi:10.1038/srep00594.
- Bejan, A.; Charles, J. D.; Lorente, S. (22 July 2014). "The evolution of airplanes". Journal of Applied Physics 116: 044901. doi:10.1063/1.4886855.
- Kleidon, A (2009). “Nonequilibrium thermodynamics and maximum entropy production in the Earth system: Applications and implications.” Naturwissenschaften, 96, 653-677.
- Kleidon, A; Malhi, Y; Cox PM (2010) “Maximum entropy production in environmental and ecological systems.” Philos Trans R Soc Lond B Biol Sci., 365, 1297–1302.
- Dewar, R (2003). “Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states.” J. Phys. A: Math. Gen., 36, 631-641.
- Dewar, RC (2005). “Maximum entropy production and the fluctuation theorem.” J. Phys. A: Math. Gen., 38, L371-L381.
- Bejan, A (2006). “Constructal theory of pattern formation.” Hydrol. Earth Syst. Sci. Discuss., 3, 1773-1807.
- Bejan, A. (2010). Design in nature, thermodynamics, and the constructal law. Comment on “Life, hierarchy, and the thermodynamic machinery of planet Earth” by A. Kelidon, Physics of Life Reviews Vol. 7, No. 4: 467-470.
- L. A. O. Rocha, Constructal law: from the law of physics to applications and conferences. Phys Life Rev 8, 245-246 (2011).
- Bejan, A & Lorente, S. (2011). “The constructal law and the evolution of design in nature,” Physics of Life Reviews, Vol. 8, No. 3: 209-240
- A. Bejan and S. Lorente, Constructal law of design and evolution: Physics, biology, technology, and society, Journal of Applied Physics 113, 151301 (2013); doi: 10.1063/1.4798429.
- Bejan, A; Lorente, S (2004). The Constructal Law and the Thermodynamics of Flow Systems with Conﬁguration. International Journal of Heat and Mass Transfer 47: 3203–3214
- Bejan, Adrian; Zane, Peder (January 24, 2012). Design in Nature: How the Constructal Law Governs Evolution in Biology, Physics, Technology, and Social Organization. New York: Doubleday. p. 11. ISBN 978-0-307-744340.
- How a single principle of physics governs nature and society: A. Bejan, TEDx Mid-Atlantic, 26 October 1992
- Why Every Snowflake is NOT Unique, 20 December 2013
- The Case for a 'Constructal' Law of Design in Nature, 23 March 2012
- A New 'Constructal' Law of Nature?: A. Bejan, 05 May 2009
- Constructal Theory and the Evolution of Living Systems, 05 Dec 2014