# Constructal theory

The constructal theory of global optimization under local constraints explains in a simple manner the shapes that arise in nature. It is the thought that flow architecture comes from a principle of maximization of flow access, in time, and in flow configuration that are free to morph.

The Constructal law proclaims a tendency in time about the generation of animate and inanimate flow systems: "the maximization of access for the currents that flows through a morphing flow system ". This theory replaces the belief that nature is fractal, and allow one to design and analyse systems under constraints in a quest for optimality.

This theory allows the design and understanding of natural systems, thermal dissipators, communication networks, etc.

The constructal theory was invented by Adrian Bejan.

## History

The constructal theory was developed by Adrian Bejan, Ph. D. MIT (1975) in the late 90's.

Professor Bejan taught at MIT until 1976 and is now J.A. Jones Distinguished Professor of Mechanical Engineering at Duke University, Durham.

Bejan's research areas cover: entropy generation minimization, exergy analysis, condensation, convection in porous media, transition to turbulence, etc.

"Constructal" is a word created by Adrian Bejan, coming from the latin verb construere, to construct, in order to designate, in the constructal theory's point of view, the naturally optimized forms such as rivers, trees and branches, lungs and also the engineered forms coming from a constructal optimization process.

## Principles

For example, in point-area and point-volume flows, constructal theory predicts tree architectures, such flows have displaying at least two regimes: one highly resistive and a less resisitive one, and it can be applied at any scale: from macroscopic to microscopic systems.

Some domains of application
Application What flows Tree channels Interstitial spaces
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 main principle of the constructal theory is that every system is destined to remain imperfect.

According to this, the best that can be done is to optimally distribute the imperfections of the system, and this optimal distribution of imperfection will generate the geometry or shape of the studied system.

The constructal way of distributing the system's imperfection is to put the more resistive regime at the smallest scale of the system. The constructal law is the principle that generates the perfect form, which is the least imperfect form possible.

### Constructal law

The constructal principle was enonced in 1996 by Adrian Bejan 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."

### Thermodynamical analogy

Analogies between thermodynamics and constructal theory
Thermodynamics Constructal theory
State Flow architecture (geometry, structure)
Process Change of structure
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 Maximization of flow access Bejan

## Achievements

The constructal theory is predictive and so has been be verified.

The constructal principle of optimized tree-flow architecture allowed to predict many allometric laws, e.g.:

• Kleiber's law, the proportionality between metabolic rate $\displaystyle q_{0}$ and body size $\displaystyle M$ raised to the power $\displaystyle 3/4$ :
$\displaystyle q_{0} \sim M^{\frac 3 4}$
• the proportionality between breathing and heart beating times $\displaystyle t$ and body size $\displaystyle M$ raised to the power $\displaystyle 1/4$ :
$\displaystyle t \sim M^{\frac 1 4}$
• mass transfer contact area $\displaystyle A$ and body mass $\displaystyle M$ :
$\displaystyle A \sim M^{\frac 7 8}$
• the proportionality between the optimal cruising speed $\displaystyle V_{opt}$ of flying bodies (insects, birds, airplanes) and body mass $\displaystyle M$ in kg raised to the power $\displaystyle 1/6$ :
$\displaystyle V_{opt} \sim 30.M^{\frac 1 6} m.s^{-1}$

Bejan's Constructal Law also explains why we have a bronchial tree with 23 levels of bifurcation. The constructal model of flow architecture of the lungs delivers in a pure deterministic way:

• 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.