## SUMMARY

The constructal law is the statement that for a flow system to persist in time it must evolve in such a way that it provides easier access to its currents. This is the law of configuration generation, or the law of design. The theoretical developments reviewed in this article show that this law accounts for (i) architectures that maximize flow access (e.g. trees), (ii) features that impede flow (e.g. impermeable walls, insulation) and (iii) static organs that support flow structures. The proportionality between body heat loss and body size raised to the power 3/4 is deduced from the discovery that the counterflow of two trees is the optimal configuration for achieving (i) and (ii) simultaneously: maximum fluid-flow access and minimum heat leak. Other allometric examples deduced from the constructal law are the flying speeds of insects, birds and aeroplanes, the porosity and hair strand diameter of the fur coats of animals, and the existence of optimal organ sizes. Body size and configuration are intrinsic parts of the deduced configuration. They are results, not assumptions. The constructal law extends physics (thermodynamics) to cover the configuration, performance, global size and global internal flow volume of flow systems. The time evolution of such configurations can be described as survival by increasing performance, compactness and territory.

## The broad view: biology, physics and engineering

The occurrence of flow configuration (shape, structure) is a phenomenon so universal that it unites the natural with the engineered, and the animate with the inanimate. From 1996, constructal theory has shown that flow architectures such as trees and round tubes can be deduced from a single law of maximization of access for currents (Bejan, 1996, 1997a, 2000). If this law is correct, then how do we account for the occurrence of configurations that obstruct the flow of currents? Hair and fur prevent the flow of heat from the body of the animal to the ambient surroundings. Pairs of blood vessels in counterflow serve a similar insulation function in the tissue under the skin. Insulation technologies are everywhere in engineering. Much more obvious examples come from the flow of fluids: impermeable walls are everywhere, in fact, without them there would be no `streams'. How can we reconcile such obvious contradictions with the maximization of flow access?

Even if the configurations that maximize flow access could be put with those that obstruct flow under the same theoretical tent, it would cover only flow systems. So how can we account for the occurrence of refined mechanical (non-flow) structures such as the skeletons of animals and the frames of aeroplanes? Principles of maximum stress uniformity and minimum weight have been invoked to account for the generation of architecture in mechanical support structures. How can such `static' principles be reconciled with the principle that unifies flow systems?

In this paper I answer these questions by reviewing a series of recent
developments based on constructal theory, where geometry is the big unknown—
the mechanism by which the global performance of the flow system is
maximized. The argument goes as follows: `what' flows (fluid, heat,
electricity, goods, people) is not as important as `how' the flow derives its
architecture from the competition between objectives and constraints. The
maximization of `access' means many things, depending on what flows; for
example, the minimization of flow resistance and pumping power in fluid flow
(from blood vessels to atmospheric circulation), the minimization of
electrical resistance and Joule heating in all electrical networks (computers,
power grids), and the minimization of travel time and cost in transportation
and business (the Fermat principle of urban design and economics). In isolated
thermodynamic systems, it means the acceleration of mixing *en route*
to equilibrium and no flow.

In engineering, where the heat engine was the stimulus for the discovery of thermodynamics, the constructal law delivers precisely what Sadi Carnot called for: the minimization of friction and shocks in fluid flow, and the avoidance of large temperature differences in heat flow. Such thermodynamic `imperfection' cannot be avoided, because of size and time constraints. Resistances will always be present. As Poirier (2003) put it recently, the only way up on the `staircase to heaven' envisioned by Sadi Carnot is by arranging and balancing the resistances against each other. To arrange and to distribute is to make the drawing, to deduce what was missing — the architecture. Optimal distribution of imperfection is the constructal law of geometry generation.

Natural flow systems exhibit the same tendency. The largest engine on earth— the wheels of atmospheric and oceanic circulation — achieves the same objective as birds, aeroplanes, and many other blobs of organized material movement (`streams') such as eddies of turbulence: to facilitate the movement of matter all over the globe, i.e. to maximize the mixing of the matter that is the globe. River basins are trees, in accordance with the constructal law of maximization of flow access. River cross-sections have a universal proportionality between width and depth, which can be deduced from the constructal law.

If animal design proceeds in accordance with the constructal law, then the
animal destroys less exergy (i.e. useful energy, fuel) and requires less food.
`Animal design' means currents that flow along maximum-access paths between
organs, currents that are guided by walls, not currents that leak directly to
the ambient. Fuel or food management means that the engine and the animal must
carry the optimal weight that makes the *whole* animal efficient, not
the individual organ. From this comes the need to spread maximum stresses
uniformly, and to support the flow structure with a mechanical structure
having minimal weight. There is no contradiction between the constructal law
of flow architecture and weight (size) as a constraint for the mechanical
structure that supports the flow structure.

Along this theoretical route the constructal law provides the physics that is missing from the Darwinian principle of the fittest animal being the one that survives. It provides the physics definition of what is meant by `the fittest,' or by its equivalent `the survivor,' not only in biology but also in engineering, geophysics and economics, where selection and evolution are also evident. According to constructal theory therefore, animals, river basins and all of us — the `man + machine species' — are the same.

## Constructal law

The constructal law was first published in 1996 in the context of optimizing the access to flow between one point and an infinity of points (area or volume, in two- or three-dimensional systems, respectively), with application to traffic (Bejan, 1996), the cooling of small-scale electronics (Bejan, 1997b), and living fluid trees (Bejan, 1997c,d):

*`For a finite-size open system to persist in time (to survive) it must
evolve in such a way that it provides easier and easier access to the currents
that flow through it.'*

The constructal law is about the time arrow of the phenomenon of flow architecture generation. It is a self-standing law, distinct from the second law of thermodynamics (see the concluding section of this article). Constructal theory has been reviewed in books (Bejan, 1997a, 2000, 2004; Rosa et al., 2004; Bejan et al., 2004) and in articles (Poirier, 2003; Lewins, 2003; Torre, 2004; Guerreri, 2004).

According to the constructal law, in the case of a flow between one point
and an infinity of points, the flow path was constructed as a sequence of
steps starting with the smallest building block, the size of which is fixed,
and continued in time with larger building blocks (assemblies, constructs).
The mode of transport with the highest resistivity (slow flow, diffusions,
walking and high cost) was placed at the smallest scale, filling completely
the smallest elements. Modes of transport with successively lower
resistivities (fast flow, streams, vehicles, and low cost) were placed in the
larger constructs, where they were used to connect the area-point or
volume-point flows integrated over the constituents. The geometry of each
building block was optimized for area-point access. The architecture that
emerged was a tree in which every geometric detail was a result — the
tree, as a geometric form *deduced* from a single principle.

A simple illustration of the discovery of the tree geometry is in the
minimization of travel time between an infinity of points (area *A*)
and one point (M, Fig. 1A). The
deduction of the flow architecture from the constructal law is atomistic: from
small to large, in time (see *A*_{1}, *A*_{2},
*A*_{3}...; Fig.
1A). There are at least two modes of locomotion: slow (walking,
speed *V*_{0}), and fast (vehicles,
*V*_{1}<*V*_{2}<*V*_{3},...).
The construction starts with the smallest elemental area,
*A*_{1}=*H*_{1}*L*_{1}, where the
*A*_{1} size is fixed but the shape
*H*_{1}/*L*_{1} may vary. The elemental size
*A*_{1} is dictated by the land property and culture of those
who live on *A*_{1}.

The minimization of the travel time between all the points of
*A*_{1} and the boundary point M_{1} generates four
geometric features: (i) the *V*_{1} street is placed on the
longer axis of the *H*_{1}*L*_{1} rectangle;
(ii) the slow movement covers *A*_{1} entirely, i.e. every
inhabitant of *A*_{1} must first walk before reaching the
street; (iii) the rectangle shape
*H*_{1}/*L*_{1}; and (iv) the approach angleα
_{1}. For example, when
*V*_{1}>>*V*_{0}, the optimal design of
the elemental area is
*H*_{1}/*L*_{1}=2*V*_{0}/*V*_{1}
and α_{1}=0.

Interesting is that the optimal *A*_{1} configuration can be
deduced in two ways (Bejan,
1996,
1997a). One is the altruistic
approach, in which the inhabitant with the worst geographical position is
considered (point P, Fig.
1B-D), and the travel time from P to M_{1} is minimized.
The other is the egotistical approach, where the travel time of every
inhabitant of the white area is calculated, averaged over
*A*_{1}, and then minimized. Both methods yield the same
geometry. The architecture that is good for the most peripheral member of the
assembly is good for the assembly as a whole: the urge to organize is an
expression of selfish behavior. We return to this important aspect of the
theory in the discussion of Fig.
6.

The opportunity to optimize geometry continues at progressively larger
scales. As shown in Fig. 1, a
first assembly (*A*_{2}) consists of a number
(*n*_{2}) of optimized elements (*A*_{1}). The
new street of speed *V*_{2} (>*V*_{1})
collects or distributes the traffic that covers the elements. The minimization
of travel time between the area *A*_{2} and the point
M_{2} calls for placing the *V*_{2} street on the axis
of *A*_{2}, touching every point of *A*_{2} with
slow movement, and optimizing two geometric features, the shape
*H*_{2}/*L*_{2} (or *n*_{2}) and
the approach angle α_{2}.

The generation of geometry continues with assemblies of progressively higher order and larger size. Every feature of the emerging drawing is the result of invoking one principle: the maximization of access for traffic that connects one point with the infinity of points that make up the area. Nothing is assumed, postulated or copied (modeled) from nature.

The optimized features of constructs of higher order
(*A*_{i}, *i*>2 in
Fig. 1D) fall into a pattern
that can be summarized as a simple algorithm. If these theoretical recurrence
formulas were to be repeated *ad infinitum*, only then would the
resulting image be a fractal (Bejan,
1997a; Avnir et al.,
1998). The image of Fig.
1 is not a fractal. In other words, constructal optimization of
volume-to-point accounts for why natural structures look like images generated
by fractal algorithms truncated at a small but finite length scale, why such a
cut-off scale exists, and why in a natural structure the algorithm breaks down
in the steps situated close to the smallest scale (e.g. step *i*=1 in
Fig. 1B).

Three-dimensional volume-to-point access has been optimized in the same manner, and the result is a tree architecture (for reviews, see Bejan, 1997a, 2000). In fluid trees, the structure is visible as channels and ducts if the flow possesses at least two regimes with dissimilar resistivities, high and low. The high-resistivity regime (e.g. viscous diffusion, Darcy flow) covers most of the space, as it fills the interstices formed between the smallest channels. The channels and streams are characterized by much lower resistivity (e.g. flow in ducts and streams). The structure is not visible — it is not even an issue — when only one flow regime (viscous diffusion) is present.

In the constructal fluid tree the dimension changes settle into a pattern
(e.g. dichotomy) after the order of the volume construct becomes high enough.
Dichotomy is not an assumption — it is an optimization result deduced
from the constructal law (Bejan et al.,
2000). Furthermore, the optimized diameter factor obeys Murray's
law (*D*_{i+1}/*D*_{i}=2^{1/3}) after
the second construct (Murray's law was originally derived for fully developed
laminar flow; the corresponding result for turbulent flow is discussed in
Bejan et al., 2000). The step
factor for tube lengths *L*_{i+1}/*L*_{i}
exhibits a cyclical pattern for each sequence of three construct sizes,
provided that *i*>2. The theoretical tree has a definite (finite,
known) beginning: the smallest scale and the optimized first construct. The
geometry and finite size of this beginning distinguish this theoretical
construction from the algorithms assumed and used in fractal geometry. The
inner cut-off, and the breakdown of the algorithm at small-enough scales, are
as important as any other geometric feature. The flow through the arms of the
tree is as important as the invisible flow across the armpits.

The formulae that accompanied the fluid tree construction were the minimum
necessary for making the drawing, and are tabulated in Bejan
(1997a,c).
As shown in Bejan (2000), more
information can be obtained from the tables, for example, the total volume,
the total internal surface of all the tubes (*A*), the volume-averaged
porosity, the total mass (*M*_{b}) and the cross-sectional area
of the tree root. Interesting relationships emerge when these quantities are
plotted against each other, for example, the near-proportionality between
*A* and *M*_{b} (see Eqn 9 below).

## Heat loss *vs* body size

Constructal theory predicted the proportionality between metabolic rate and
body mass raised to the power 3/4, by invoking the constructal law twice (cf.
the introductory section of this article): in the minimization of body heat
loss, and the minimization of blood pumping power (Bejan,
2000,
2001). The minimization of
pumping power yields the constructal fluid tree (Bejan,
1997a,c):
this can be derived more succinctly by optimizing a plane construct consisting
of a plane T-shaped junction (Bejan et
al., 2000; Fig.
2A). For simplicity, assume right angles and Hagen-Poiseuille flow
with constant properties in every tube. The stream *μ*_{i}
encounters the flow resistance of two *L*_{i+1} tubes in
parallel, which are connected in series with one *L*_{i} tube.
When the resistance is minimized by fixing the total tube volume, we find
*D*_{i+1}/*D*_{i}=2^{-1/3}, which is
independent of the tube lengths (*L*_{i},
*L*_{i+1}) and the relative position of the three tubes. Next,
we optimize the lengths when the space allocated to the construct is fixed,
2*L*_{i+1}*L*_{i}=constant. This yields the
optimal ratio
*L*_{i+1}/*L*_{i}=f=2^{-1/3}, where the
smallest length scale is labeled *i*=*n*, and largest
*i*=0.

The trees of blood vessels are an architectural feature under the skin, but
not the only one. The other is the *superposition* of the arterial and
venous trees, so closely and regularly that tube *i* of one tree is in
counterflow with tube *i* of the other
(Fig. 2C,D). This is a thermal
insulation feature.

The arterial stream is warmer than the venous stream: heat flows
transversally, from stream to stream. Because the enthalpy of the warmer
stream is greater than that of the colder stream, the counterflow convects
longitudinally the energy current
*q*_{i}=*μ*_{i}*c*_{p}Δ*T*_{t,i},
where *c*_{p} is the specific heat of blood, andΔ
*T*_{t,i} is the stream-to-stream temperature difference
at level *i*. It was first shown in heat transfer
(Bejan, 1979) and later in
bioengineering (Weinbaum and Jiji,
1985), that such a counterflow sustains a longitudinal temperature
gradient, Δ*T*_{i}/*L*_{i}, and that the
convective energy current is proportional to this gradient:
(1)

where *h*_{i} is are the overall stream-to-stream heat
transfer coefficient and *p*_{i} is the perimeter of contact
between the two streams. The stream-to-stream thermal resistance
*h*_{i}^{-1} is the sum of two resistances: the
resistance through the fluid in the duct
(∼*D*_{i}/*k*_{f}, where
*k*_{f} is the fluid thermal conductivity), plus the resistance
through the solid tissue that separates two tubes
(∼*t*_{i}/*k*, where *D*_{i} is
diameter and *k* is the tissue thermal conductivity;
*t*_{i} is defined in Fig.
2D: *t*_{i} is the average thickness of the tissue
that separates two adjacent *D*_{i} tubes). Even when the tubes
touch, *t*_{i} is of the same order as *D*_{i}.
In addition, because *k*_{f}∼*k*, we conclude that
*h*_{i}∼*k*/*D*_{i}, and Eqn 1
becomes:
(2)

The double-tree fluid structure is a single tree of convective heat leakage
with zero net mass flow. The convective tree stretches from the core
temperature of the animal (at *i*=0) to the skin temperature. The
latter is registered in many of the elemental volumes (*i*=*n*)
that are near the skin. The many counterflow pairs of the two fluid trees
sustain the overall temperature difference Δ*T*:
(3)

In going from Eqn 2 to Eqn 3, we used the continuity relations for fluid
flow
(*N*_{i}*μ*_{i}=*μ*_{0},
constant) and heat flow
(*N*_{i}*q*_{i}=*q*_{0},
constant). Recalling the *L*_{i+1}/*L*_{i}
constant f, we substitute
*L*_{i}=*L*_{0}f^{i},
*L*_{n}=*L*_{0}f^{n} and
*N*_{i}=2^{i} into Eqn 3:
(4)

The right side has quantities that are constant, and quantities that depend
on *n* (the number of construction steps). The ratio
*q*_{0}/*μ*_{0} is independent of body size
(*n*) because both *q*_{0} and *μ*_{0}
are proportional to the metabolic rate.

The volume inhabited by the tree is estimated by considering the stretched
tree as a cone in Fig. 2B. The
base of the cone (at *i*=*n*) has an area of size
*N*_{n}*L*_{n}^{2}∼2^{n}*L*_{n}^{2}.
The height of the cone is of the same order as the sum of all the tube
lengths,
*L*_{0}+*L*_{1}+...+*L*_{n}=*L*_{0}(1-f^{n+1})/(1-f),
and the volume scale is:
(5)

The relationship between metabolic rate and total volume is obtained by
eliminating *n* between Eqns 4 and 5. The result is visible in closed
form if *n* is sufficiently large so that (2f)^{n+1}>>1
in Eqn 4, and f^{n+1}<<1 in Eqn 5. In this limit
*q*_{0} is proportional to 2^{n}, and *V* to
(2/f)^{n}. From this follows:
(6)

It can be verified numerically that Eqn 6 also holds for small *n*.
In conclusion, the proportionality between metabolic rate and body size raised
to the power 3/4 is predictable from pure theory.

Constructal theory also anticipates the proportionality between breathing
(or heartbeating) time and body size raised to the power 1/4
(Bejan, 2000). In one of the
first constructal papers (Bejan,
1997d), it was shown that the pumping power required by the heart
for blood circulation and the thorax for breathing is minimal if (a) the flow
is intermittent (in and out, on and off), and (b) the `in' time interval
(*t*_{1}) is of the same order of magnitude as the `out' time
interval (*t*_{2}). Features (a) and (b) come from pure theory
(the constructal law), not from observations. The optimal time scale
(*t*_{1,2}∼*t*) is:
(7)

where *A* is the total internal contact area of all the tubes of the
tree, *D* is the mass diffusivity, Δ*C* is the
concentration difference that drives the mass transfer process, and
*μ* is the total mass flow rate of the tree (blood, air). The flow
rate *μ* is proportional to the metabolic rate of the animal. Eqn 7
shows that in order to predict *t* as a function of body mass
(*M*_{b}) we need expressions for
*μ*(*M*_{b}) and *A*(*M*_{b}).
From the optimized tree of convective currents we obtained Eqn (6), or:
(8)

To predict the relationship *A*(*M*_{b}), we argue
that the thickness of the tissue penetrated by mass diffusion during the
breathing or heartbeating time *t* is proportional to
*t*^{1/2}. The body volume (or mass) of the tissue penetrated
by mass diffusion during this time obeys the proportionality relationship
*M*_{b}∼*At*^{1/2}. Eliminating *t*
between *M*_{b}∼*At*^{1/2} and
*t*∼(*A*/*μ*)^{2}, and using Eqn 8, we
conclude that the contact area should be almost proportional to the body mass:
(9)

Finally, the proportionalities in Eqns 8 and 9 and Eqn 7 mean that: (10)

This allometric law is supported convincingly by the large volume of observations accumulated in the physiology literature (Schmidt-Nielsen, 1984).

The constructal law for body heat loss *vs* size, Eqn 6, stands out
on the background provided by the history of the theoretical attempts to
predict this relationship. The earliest was Rubner's heat transfer model:
because the convective heat loss is proportional to the body surface, the
metabolic rate must be proportional to the length scale
(*V*^{1/3}) squared, i.e. the body mass or volume raised to
power 2/3. The heat transfer model was discredited by more recent observations
of birds and mammals, which suggest an exponent closer to 3/4 than 2/3.
Because of such observations, heat transfer was not included as a feature in
recent fluid mechanics tree network models (e.g.
West et al., 1997), which, by
the way, is a good illustration of why modeling is empiricism. We return to
these models in the concluding section.

Why should anyone question the currently accepted models by resurrecting heat transfer? First, and this is key, modeling is not theory. Models are simplified descriptions (facsimiles) of objects observed in nature. Second, the minimization of pumping power, which is invoked by modelers, is the constructal law, because less pumping power means less exergy [useful (liberated) energy; see below] destruction, and less food for the animal to survive. But, a lower heat leak also means less food and less exergy destruction. This is why the minimization of pumping power goes hand-in-hand with the old heat-loss doctrine, not against it. Minimum pumping power consumption and minimum loss of body heat are parts of the same constructal law — how to be constructed to be the fittest (the survivor), how to perform best, how to flow best.

Additional support for the constructal theory of body heat loss comes from the allometric laws of the design of the hair coats of animals, such as the proportionality between the hair strand diameter and the animal body length scale raised to the power 1/2 (Fig. 3). This allometric law was predicted (Bejan, 1990a,b) by minimizing the body heat loss through the hair coat, and is reviewed in the book by Nield and Bejan (1999). The 1/2 exponent was predicted for both natural convection and forced convection.

Another common feature of animal hair coats is the porosity, which is high and nearly constant (between 0.95 and 0.99) for all animal sizes (Bejan, 1993). This feature was predicted by minimizing the combined heat loss by conduction and radiation through the hair air coat. This analysis and the most recent design applications of the constructal law are reviewed in Bejan et al. (2004).

Constructal theory predicts not only the 3/4 exponent for Eqn 6, but also
the gradual decrease of this exponent as the body size decreases. The 3/4
exponent is valid at the limit where the body heat loss is impeded primarily
by the convective resistance posed by the blood counterflow of perfectly
matched tube pairs (Fig. 2B).
As shown in Bejan (2000,
2001), heat-loss paths in
general are more complicated. The convective thermal resistance posed by the
trees in counterflow (*R*_{1}) resides inside the animal. This
resistance runs in parallel with a second internal resistance
(*R*_{2}) associated with the conductive heat leak through the
tissue. On the outside of the animal the heat current flows through the
convective resistance (*R*_{3}) associated with the body
surface exposed to the ambient (air, water). The conductive resistance
*R*_{2} is proportional to the body length scale
*V*^{1/3} divided by the body surface scale
*V*^{2/3}; hence
*R*_{2}∼*V*^{-1/3}. The convective tree
resistance *R*_{1} is proportional to
*V*^{-3/4}. The ratio
*R*_{2}/*R*_{1}∼*V*^{5/12}
shows that *R*_{2} becomes progressively weaker (i.e. the
preferred path) as the body size decreases. At that limit the exponent in the
power law between heat loss and body size becomes 1/3. In other words, from
constructal theory we should expect a gradual decrease in the power-law
exponent as the body size decreases.

The generality of the constructal deduction of the allometric law of metabolism (Eqn 6) is due to the view that the flow structure results from the clash between two objectives: the need to carry certain substances from the core to the periphery of the organism (e.g. nutrients, water, ions, waste products), and the need to avoid the direct leakage of these substances and energy (heat) into the ambient surroundings. All biological flow architectures are results of this clash, from microbes to plants and animals, including warm-blooded and cold-blooded vertebrates. The regulated temperature difference between the body core and the ambient surroundings (large, or small) is not the issue. According to constructal theory, the flow system (the animal and its movement) must evolve like any other engine-propelled body on the surface of the earth (e.g. Fig. 5), and this means that the exergy derived from the food must be channeled optimally through the motor (muscles), not dumped straight into the ambient surroundings.

## Flight and organ size

The design principle and results reviewed so far are relevant across the board, from biology to engineering. This point is pressed with vigor by the aircraft sketched in Fig. 4: if the word `fuel' is replaced by `food', then the same drawing is valid for a bird, and reveals how the energy liberated by food is destroyed by all the currents that flow around and through the animal. The useful (liberated) energy is known as exergy in thermodynamics (Bejan, 1997a), and as energy consumption in biology. The food or fuel exergy is destroyed completely by currents that overcome resistances.

The flying system becomes `more fit' when the total destruction of exergy
is minimized: more body mass flown, to longer distances. This is just like the
Gulf Stream: more ocean mass carried with less resistance, i.e. faster and for
longer distances. The mechanisms that destroy food exergy (e.g. air friction)
cannot be minimized individually and eliminated, because each such mechanism
serves the flying body as a whole. This is illustrated in most general terms
(Bejan, 2000) by minimizing
the sum (*Ẇ*) of the food exergies
required by air friction and lifting (supporting) the aircraft:
(11)

where *D* is the body linear dimension, *V* is cruising
speed, ** g** is gravitational acceleration, ρ

_{a}is air density and ρ

_{b}is body density (based on the total body mass scale

*M*

_{b}∼ρ

_{b}

*D*

^{3}). The terms on the right side of Eqn 11 cannot be eliminated. They can be minimized together, thus: (12)

when the cruising speed *V*_{opt} has the scale
(13)

Flying animals and machines are represented by the scalesρ
_{b}∼10^{3} kg m^{-3} andρ
_{a}∼1 kg m^{-3}, such that Eqn 13 becomes:
(14)

where *V*_{opt} is in m s^{-1} and
*M*_{b} is in kg.

Fig. 5 shows this
theoretical line next to flying speed data taken from extensive compilations
(Tennekes, 1996, and
references therein). The agreement between the line and the data is
remarkable, in view of the simplicity of the body model with one length scale
(*D*). Insects, birds and aeroplanes have multiple length scales, and
this may explain why some of the data fall above or below the line.

Agreement over such a wide diversity of sizes and types of flying flow systems shows that the constructal law — the optimal distribution of imperfection — unites the designs of all the flying systems, the animate with the engineered. This is stressed by the additional features of Fig. 5. Small animals (insects, hummingbirds) flap their wings all the time, and their engine propellers (the wings) also provide the lift. In this limit of small mass, the motor and the lift functions are performed by a single structure: the wings. At the other end of the body mass scale, large masses (aircraft) fly with separate motor and lift structures. The lift is provided by the wings proper, and the motor (thrust) by a different set of wings — the blades of the turbofan engine.

Between the `fully integrated' and `separate' motor and lift we find the `almost separate' distribution of motor and lift functions. We see this in the V-shaped flocks of migratory birds. The goose is the motor when it flies at or near the tip of the V; when it is not, the goose surfs on the waves generated by the geese working in front. Pterosaurs are also in-between. Their motor and lift functions were almost separate: they flapped their wings rarely, and glided most of the time under the hot sun (Frey et al., 2003).

More recently, we showed that the minimized flood consumption dictates the
physical sizes of the various flow components of a complex flying system
(Bejan and Lorente, 2002).
Consider the total food required for flying over a distance *L*. This
quantity is proportional to the total food exergy that is destroyed,
which, according to Eqns 12 and 13, is proportional to body weight and
distance:
(15)

In the aircraft industry this proportionality is known as the `take-off gross weight' criterion: the fuel penalty associated with placing a new component on board is proportional to the mass of that component. Smaller flow components are attractive. But, flow components function less efficiently when their sizes decrease: they pose greater resistance to flows, they destroy more exergy, and so the flying animal or machine requires more food or fuel to carry such components.

Constructal theory accounts for the existence of characteristic
(proportionate) organ sizes, in animals and engineering installations. The
fundamental trade-off in body size is illustrated in
Fig. 6. The total food exergy
required by a flow component is the sum of the food exergy destroyed by the
component and the food exergy required by the flying system to carry the
component on board. There is an optimal component size such that its impact
(penalty) on the total food required by the bird is minimum. This trade-off is
fundamental: it rules the optimization of organ sizes in every flow system,
animals and vehicles alike. A simple illustration of this basic phenomenon is
the optimization of the diameter *D* of a round duct (pipe or blood
vessel), when its mass flow rate *μ* is prescribed
(Bejan and Lorente, 2002).

## Survival by increasing performance, compactness and territory

In constructal theory, body size, architecture and complexity are results,
not assumptions. They are intrinsic parts of the drawing: the optimal
configuration to which the flow system tends in time, in accordance with the
constructal law. This tendency was recently put on an analytical basis, such
that the constructal law becomes a new extension of thermodynamics — the
thermodynamics of non-equilibrium (flow) systems with configuration
(Bejan and Lorente, 2004).
This formulation is condensed in Fig.
7. A flow system (e.g. a tree) has `properties' that distinguish
it from a static (non-flow) system. The properties of a flow system are: (1)
global external size, e.g. the length scale of the body bathed by the tree
flow, *L*; (2) global internal size, e.g. the total volume of the
ducts, *V*; (3) at least one global objective, or performance, e.g. the
global flow resistance of the tree, *R*; (4) configuration, drawing,
architecture; and (5) freedom to morph, i.e. freedom to change the
configuration.

The flow systems covered by the constructal law stated at the start of this
article populate and move in the *V*=constant plane shown in
Fig. 7. This plane houses a
Performance *vs* Freedom diagram: in time, and if the architecture is
free to change, *R* decreases (i.e. performance increases) at constant
*L* and *V*. The configuration with the smallest *R*
value represents the equilibrium flow structure. The configurations that
preceded it are non-equilibrium flow structures.

At equilibrium the flow configuration achieves the most that its freedom to morph has to offer. Equilibrium does not mean that the flow architecture stops changing. On the contrary, it is here at equilibrium that the flow geometry enjoys most freedom to change. Equilibrium means that the global performance does not change when changes occur in the flow architecture.

The evolution of configurations in the constant-*V* cut (also at
constant *L*, Fig. 7)
represents survival through increasing performance — survival of the
fittest. This is the physics principle that now underpins Darwin's argument,
the law that rules not only the animate flow systems but also the natural
inanimate flow systems and all the man and machine species. The constructal
law defines the meaning of `the survivor', or of the equivalent concept of
`the more fit'.

In the bottom plane of Fig.
7, the locus of equilibrium structures is a curve with negative
slope, (∂*R*/∂*V*)<0, because of flow physics: the
resistance decreases when the size of the internal space inhabited by the flow
increases. This slope means that the non-equilibrium flow structures occupy
the hypersurface suggested by the three-dimensional surface sketched in
Fig. 7. The time evolution of
non-equilibrium flow structures toward the bottom edge of the surface (the
equilibrium structures) is the action of the constructal law.

The same time arrow can be described alternatively with reference to the
constant-*R* cut through the three-dimensional space of
Fig. 7. Flow architectures with
the same global performance (*R*) and global size (*L*) evolve
toward compactness — smaller volumes dedicated to internal ducts, i.e.
larger volumes reserved for the working `tissue' (the interstices). This is
survival based on the maximization of the use of the available space. Survival
*via* increasing compactness is equivalent to survival *via*
increasing performance: both statements are the constructal law.

A third equivalent statement of the constructal law becomes evident if we
recast the constant-*L* design world of
Fig. 7 in the
constant-*V* design space of Fig.
8. In this new figure, the constant-*L* cut is the same
Performance *vs* Freedom diagram as in
Fig. 7, and the constructal law
means survival by increasing performance. The new aspect of
Fig. 8 is the shape and
orientation of the hypersurface of non-equilibrium flow structures: the slope
of the curve in the bottom plane
(∂*R*/∂*L*)_{V} is positive because of flow
physics, i.e. because the flow resistance increases when the distance traveled
by the stream increases.

The world of possible designs (the hypersurface) can be viewed in the
constant-*R* cut made in Fig.
8, to see that flow structures of a certain performance level
(*R*) and internal flow volume (*V*) morph into new flow
structures that cover progressively larger territories. There is a limit to
the spreading of a flow structure, and it is set by global properties such as
*R* and *V*. River deltas in the desert, animal species on the
plain, and the Roman empire spread to their limits. Such is the constructal
law of survival by spreading, by increasing territory for flow and
movement.

## Overview: theory *vs* modeling

This article began with a broad view of how the constructal law accounts for the generation of configuration everywhere, from flow architectures to mechanical structures, and from animate systems (biology), to inanimate systems (physics) and man + machine species (engineering, economics, business). The constructal literature listed in the References covers all these domains: this literature is recommended to the readers because, although known in engineering, it is largely unknown in biology.

The applications of constructal theory are not limited to the examples given in this article. In fact, the connections between constructal theory and the large volume of observations available in established fields such as biology are yet to be made. They deserve to be explored to depths much greater than in engineering books (Bejan, 1997a, 2000). In particular, because this special issue focuses on allometric laws and body size, it is important to see clearly the position of constructal theory relative to the allometric model of West et al. (1997).

Constructal theory (Bejan, 1996, 1997a-c) predates the model of West et al. (1997), and covers a physics that is much greater than the allometric behavior correlated by the model. As stated in the title of their paper, the work of West et al. is the modeling of certain biological systems, which they observe and then describe analytically. Modeling is empiricism, not theory.

The model of West et al. is based on at least three *ad hoc*
assumptions: (i) the existence of a `space-filling fractal-like branching
pattern' (read: tree), (ii) the final branch of the network is a
size-invariant unit, and (iii) the energy required to distribute resources is
minimized.

These three features were already present in 1996 constructal theory, not as convenient assumptions to polish a model and make it work, but as invocations of a single principle: the constructal law. Specifically, (iii) is covered by the constructal law, (i) is the tree-flow architecture that in constructal theory is deduced from the constructal law, and (ii) is the smallest-element scale that is fixed in all the constructal tree architectures. To repeat, in constructal theory the tree-shaped flow is a discovery, not an observation, and not an assumption.

Because features (i-iii) are shared by constructal theory and by the model of West et al., every allometric law that West et al. connect to their model is an affirmation of the validity of constructal theory. Every success of constructal theory in domains well beyond the reach of their model (e.g. river basins, flight, dendritic solidification, global circulation, mud cracks; see Bejan, 2000) is an indication that animal design is an integral part of a general theoretical framework— a new thermodynamics — that unites biology with physics and engineering.

Biology and `natural' selection have just been made a part of physics. There are two time arrows in this new thermodynamics. The old is the time arrow of the second law, i.e. the statement of irreversibility: everything flows from high to low. The new is the time arrow of the constructal law, or `how' everything flows: configurations morph toward easier flowing architectures, toward animal designs that are more fit, toward geophysical currents that flow along better paths, and toward man + machine species that are more efficient. All these macroscopic constructs mix the earth's crust better and better, that is much more effectively than in the absence of flow architecture.

Constructal theory strikes a balance between determinism and chance. In a constructal tree, for example, the position of the branches can be predicted (e.g. Fig. 1), but nobody knows exactly how the individual or the molecule will move across the interstices. Likewise, it is not known exactly how (i.e. in what vertical plane of Figs 7 and 8) a non-equilibrium flow structure migrates toward equilibrium. Chance and additional constraints will definitely play a role. There is no ambiguity, however, about the direction of the migration, and about the top performance level, which is achieved at equilibrium, where imperfections are distributed optimally.

- © The Company of Biologists Limited 2005