Draft:Constructal Law

The Constructal Law, proposed in 1996 by Adrian Bejan, a thermal engineer at Duke University in the United States, is a physical law that pertains to the design and evolution of natural phenomena in thermodynamics.

The term 'constructal' is coined by Bejan from the Latin verb construere, meaning 'to construct,' and is described as a counterpart to the word 'fractal,' which derives from the Latin-based word for 'beraking things.' The constructal law asserts that the configuration (design) of matter filling non-equilibrium systems tends to evolve in a way that facilitates the flow of energy and material passing through it. This law declares a universal tendency for design changes (evolution) in non-equilibrium systems as a physical law. The constructal law stands as a new, independent physical law in thermodynamics, and it can not be derived from other physical laws. It allows for theoretical predictions and explanations of natural designs and evolutional phenomena, such as Horton's laws of river structures, scaling laws of animal lifespan and movement speed, the structure of the atmosphere, the formation of turbulent vortices and spray flows, power laws of urban scale and frequency, and Moore's Law.

History

The constructal law was proposed by Bejan in a paper submitted in 1996 (published in 1997). In this paper, dendritic structures were derived as the optimal conduction pathways for dissipating heat on electronic substrates. While pointing out that these dendritic structures, which are engineeringly optimized for heat flow, are commonly observed in nature, Bejan proposed the constructal law, suggesting that there is an unknown funcamental natural principle.

Bejan, originally a thermal engineer specializing in the optimization of systems through exergy analysis, recalls that the discovery of the constructal law led him to a career path he had never imagined. He says that listening to a lecture by physicist Ilya Prigogine at an international thermodynamics conference was the epiphany that led to the discovery. Prigogine, who received the Nobel Prize for his work on the theory of non-equilibrium thermodynamics, adhered to traditional physics view, stating that the similarity in 'forms' observed in nature, such as the structures of rivers and the lungs of living organisms, was merely 'Aléatoire' (the result of throwing dice) in his speech. It was at that moment that Bejan had the flash of insight that led to the constructal law.

Numerous academic papers focusing on the constructal law have been published, primarily by Bejan and his collaborators. Bejan was awarded the Benjamin Franklin Medal in the United States in 2018 and the Humboldt Prize in Germany in 2019, indicating a certain level of acceptance in the scientific community. Bejan's work has been published not only in academic papers but also through a series of books. Since 2006, an international conference on the constructal law has been held annually.

Fomulation of the Constructal Law

Initial Definition

The constructal law was first proposed in a 1996 paper, defined 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 (global) currents that flow through it."

This initial definition has largely remained unchanged, although other sources have expressed it slightly differently:

"For a finite-size flow system to persist in time (to live), its configuration must evolve in such a way that provides greater and greater access to the currents that flow through it."

The constructal law declares a universal tendency for non-equilibrium systems to evolve configurations over time that facilitate "greater access" (easier flow).

Terms

In classical thermodynamics, the contents of the system are typically treated as a black box. However, in real flow systems, there is material filling the system, which inherently has a configuration. The term 'configuration' used in the definition of the constructal law refers to the arrangement, pattern, or rhythm of the material filling the flow system. Depending on the context, Bejan's writings use various terms such as form, shape, geometry, structure, organization, and architecture interchangeably, all conveying similar meanings. Every real system has a configuration (form), and the constructal law uniquely focuses on this 'configuration'—a new system characteristic not addressed by classical thermodynamics.

The term 'access' is straightforwardly described as 'ease of flow.' Bejan likens it to the situation of a crowded room, where the opportunity to enter and move represents the concept of access. The features that determine access are precisely those of the system's configuration. The definition includes term 'finite-size' because if there were no constraints on the size of the room space, issues of crowding and mobility would not arise. The meaning of 'finite-size' is further clarified in later formulations based on 'sveltness'.

In physics, including the constructal law, 'evolution' is defined as the physical change in configuration over time. Bejan explains "Evolution is the sequence of flow configurations that the live system exhibits over time. This is where geometry meets thermodynamics. ...Think of evolution as 'geometric irreversibility'." The constructal law states that changes in configuration over time have a specific direction toward easier flow, and past forms do not match future forms. The law can be interpreted as defining the irreversibility of 'configuration (form)' which is the new characteristic of non-equilibrium systems.

The term 'flow system,' although not very familiar in thermodynamics, is used almost synonymously with non-equilibrium systems (systems with flows of energy or matter). However, the term emphasizes the flow of energy or matter and the configuration that guides it, which are main concerns of the constructal law.

In the perspective of the constructal law, 'design' refers to a configuration with a purpose (arrangement of parts, forms, colors, etc.). Under this view of nature, natural configurations are recognized as designs that facilite 'easier flow.'

Constructal Theory

The application of the constructal law to specific natural phenomena is known as constructal theory. Table 1 lists the constructal theories that have been presented. These theories design phenomena in every domain, from inanimate objects to biological organisms and artificial creations, all governed by the constructal law. Various constructal theories have been organized in review papers and books. In a similar way that various theories like the theory of falling bodies and fluid boundary layers are derived from Newton's laws of motion, a variety of theories also stem from the constructal law. In inanimate objects, there are constructal theories related to the minimum turbulence eddies as indicated by the Reynolds number, while in biology, there are theories on the lifespan and movement scaling laws of animals (Allometry), and in artifacts, there are theories like Zipf's law of city size and frequency occurrence. The correlation between predictions made by constructal theory and observed facts in these various design phenomena serves as empirical evidence supporting the validity (predictive power) of the constructal law. Additionally, constructal theory also covers theories about common design characteristics in flow structures, such as circular cross-sections, dendritic structures, economies of scale, hierarchy, and power laws.

Table 1: List of Constructal Theories

Domain	Theories
Inanimate	Minimum size of turbulence eddies, lifespan of spray flows, shape of lightning, river structures, composition of the atmosphere and oceans
Biological	Lifespan of animals, animals' travel distance, structure of lungs, structure of cerebral vessels
Artificial	City size and frequency of occurrence, urban transport networks, evolution of engines, evolution of airplanes
Design Characteristics	Circular cross-sections, dendritic structures, S-curves, economies of scale, hierarchy

Mathematical Formulation and Sveltness

Several Expressions of the Constructal Law

The constructal law, initially proposed as a lingusitic declaration in 1996, was subsequently formulated mathematically. The new formulation declares the constructal law using three system characteristics: 'performance (ease of flow, which is the reciprocal of resistance),' 'teritory (external size),' and 'compactness (the reciprocal of internal size).' The 1996 definition is an expression of the constructal law under conditions constrained by 'external size (L)' and 'internal size (V),' and is represented mathematically as follows. This represents the universal tendency of flow systems to 'increase in performance.'

dR ≤ 0 (constant L, V) (increase in performance) Next, under conditions where 'external size (L)' and 'flow resistance (R)' are constrained, the expression of the constructal law becomes the following, which signifies an 'increase in simplicity.'

dV ≤ 0 (constant L, R) (increase in compactness) "For a system with fixed global size and global performance to persist in time (to live), it must evolve in such a way that its flow structure occupies a smaller fraction of the available space."

Finally, under conditions constrained by 'flow resistance (R)' and 'internal size (V),' the expression of the constructal law becomes the following, which means 'increase in domain.'

dL ≥ 0 (constant R, V) (increase in teritory) "In order for a flow system with fixed global resistance and internal size to persist in time, the architecture must evolve in such a way that it covers a progressively larger territory." These expressions covers all evolutionary phenomena observed in nature. For example, the technological evolution in semiconductors known as Moore's Law corresponds to an 'increase in compactness,' while the spreading of species corresponds to an 'increase in teritory,' interpreted as a type of evolutionary phenomenon.

Sveltness

Additionally, as a physical characteristic dealing with system composition, a dimensionless number known as sveltness (Sv) was introduced. Sveltness is defined using the external size (L) and internal size (V) of the flow system as follows.

Sv = L / V^1/3

The 'increase in compactness' and 'increase in domain' in the constructal law both imply an 'increase in sveltness.' Thus, including these three expressions, the content of the constructal law can also be described as "the architecture of the flow system inevitably evolves in a direction that increases performance or sveltness."

Formulation Using Phenomenological Coefficients

One of Adrian Bejan's collaborators, Héitor Reis from the University of Évora, has formulated the constructal law using the phenomenological coefficients found in the linear phenomenological laws of non-equilibrium thermodynamics. Although this formulation is not by Bejan himself, Reis's paper is also cited by Bejan. Reis has defined 'large access (ease of flow)' in the constructal law as an increase in the phenomenological coefficient (L), and has formulated it as follows:

J = L・X (Linear phenomenological law: J is the thermodynamic flow, X is the thermodynamic force, L is the phenomenological coefficient)

dL ≥ 0 (Formulation of the constructal law using phenomenological coefficients)

Moreover, this formulation leads to interesting conclusions about the relationship between the constructal law and the Extremal principles of non-equilibrium systems. While some views suggest that there are no extremum principles in non-equilibrium systems, akin to the principle of increasing entropy in isolated systems (the second law of thermodynamics) (for instance, Prigogine), several empirical extremum principles have been proposed, and the debate continues. One well-known extremum principle is the Maximum Entropy Production Principle (MEP), which is actively studied in non-equilibrium systems like Earth's atmosphere and oceans. MEP hypothesizes that the atmosphere and oceans are organized such that entropy production is maximized, and predictions based on MEP regarding Earth's temperature distribution and oceanic currents are well-matched with observational facts. Although MEP is widely discussed in the field of Earth sciences, its origin lies in the hypothesis that the overall heat transport rate is maximized in Bénard convection, and it is considered a general extremum principle for non-equilibrium systems beyond just climatic systems. In contrast, there is also a well-known extremum principle called the Minimum Entropy Production Principle (mEP, Prigogine's theorem), proposed by Ilya Prigogine. Prigogine denied the existence of extremum principles in far-from-equilibrium systems, but supported the existence of the mEP, which states that entropy production is minimized in regions close to equilibrium.

Contradictory extremum principles have been proposed for non-equilibrium systems, both of which lack clear physical foundations. Reis argued that both of these extremum principles are manifestations under different constraint conditions of the constructal law<ref name="ref13">. Based on the fundamental equation of non-equilibrium thermodynamics by Lars Onsager and the aforementioned formulation of the constructal law, Reis demonstrated that under conditions with constraints on thermodynamic force (dX = 0), entropy production is maximized, and under conditions with constraints on thermodynamic flow (dJ = 0), entropy production is minimized. The constructal law could be a first principle integrating extremum principles in non-equilibrium systems.

σ = F・J (Fundamental equation of non-equilibrium thermodynamics: σ is entropy production)

dX = 0 → dσ > 0 (Maximum entropy production: MEP)

dJ = 0 → dσ < 0 (Minimum entropy production: mEP)