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Constructal law

constructal law, constructal law of design in nature
Constructal law is a theory of the generation of design configurations, patterns, 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" "Constructal" is a word coined by Bejan to describe the natural tendency of flow systems eg rivers, trees and branches, lungs, tectonic plates and engineered forms to generate and evolve structures that increase flow access


  • 1 Introduction
  • 2 Manifestations
  • 3 Evolutionary design
  • 4 Constructal thermodynamics
  • 5 Criticisms
  • 6 Responses to criticisms
  • 7 References
  • 8 External links


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:

  1. Life is flow:all flow systems are living systems, the animate and the inanimate
  2. Design generation and evolution is a phenomenon of physics
  3. 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

Evolutionary design

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 eg, 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, cavitiesand highly conductive inserts, convective channels and radiative systemsare 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

The tree is the natural flow design for achieving flow access between one point and a volume Alternating trees achieve flow access between two planes Natural porous media exhibit multi-scale flow structures consistent with the multiple scales and performance of alternating treesSome domains of application
ApplicationWhat flowsTree channels:Low ResistivityInterstitial spaces:High Resistivity
Packages of electronicsHeatHigh-conductivity inserts blades, needlesLow conductivity substrate
Urban trafficPeopleLow-resistance street car trafficStreet walking in urban structure
River basinsWaterLow-resistance rivulet and riversDarcy flow through porous media
LungsAirLow-resistance airways, bronchial passagesdiffusion in alveoli tissues
Circulatory systemBloodLow-resistance blood vessels, capillaries, arteries, veinsdiffusion 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

Constructal thermodynamics

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

Classical thermodynamics versus constructal thermodynamics
ThermodynamicsConstructal theory
StateFlow architecture flow structure
ProcessChange of structure design change
PropertiesGlobal objective and global constraints
Equilibrium stateEquilibrium flow architecture
Fundamental relationFundamental relation
Constrained equilibrium statesNonequilibrium architectures
Removal of constraintsIncreased freedom to morph
Energy minimum principleEvolution 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 atmosphereRiver basin design:Horton's rules of stream numbers ~4 and lengths ~2, and all the other scaling rules eg, Melton, Hack of river basins all over the world
The distribution of city sizes and numbers, ie Zipf's law relating log size versus log rankThe distribution of tree sizes and numbers on the forest floor
Vision, cognition, and the golden ratio phenomenonPedestrian movement, speeds, and patterns
The flow of education as a morphing vasculature on the globe, and the rigidity of university rankingsThe 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 designThe entire morphogenesis of dendritic crystals eg, 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 traveledThe evolution of speed in sports
Kleiber's law, the relationship between metabolic rate and body sizeThe relationship between breathing and heart beating times and body size
The human bronchial tree with 23 levels of bifurcationThe relationship between the mass transfer contact area and body mass
The dimensions of the alveolar sacThe total length of the airways
The total alveolar surface areaThe total resistance to oxygen transport in the respiratory tree
The life time and life travel of animals, vehicles human & machine species, rivers and air currentsAspect ratios of airplanes


One reservation toward 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 There has been no attempt to prove it from first principles Contrarily to alternative theories of non-equilibrium thermodynamics

There is also the claim that “the constructal law can be seen as a complement to MEP maximum entropy production”

Responses to criticisms

No criticism has been published in the peer reviewed literature against the constructal law as a law of physics Strictly speaking refs published after 1996 do not refer to the constructal law at all There is one scholar who questioned the clarity of the language used in the constructal law In ref, Kleidon wrote “It is as yet unclear how the term ‘access’ in the definition of the constructal theory sic can be quantified, but its original roots are undoubtedly in thermodynamics” Bejan has responded 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

Bejanalso 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 thisby explaining the difference between a law of physics eg, 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

Finally, Reis has showed that MEP maximum entropy production, like its complete opposite minimum entropy production is a special ad-hoc optimality principle that is covered by the constructal law


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External links

  • 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

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