Constructal theory: from thermodynamic and geometric optimization to predicting shape in nature

CONSTRUCTAL THEORY: FROM THERMODYNAMIC

AND GEOMETRIC OPTIMIZATION TO PREDICTING

SHAPE IN NATURE

ADRIAN BEJAN$

$Department of Mechanical Engineering and Materials Science, Duke University, Durham, NorthCarolina, 27708-0300, USA

AbstractÐThis is a review of recent theoretical developments toward predicting macroscopic organiz-ation (the occurrence of shape and structure) in natural ¯ow systems, animate and inanimate. The start-ing point is the question of how to optimize the access between one point and a ®nite volume (i.e., anin®nite number of points). If the volume is an electronic device that generates heat uniformly, thenaccess optimization means minimum thermal resistance between the volume and a point-size heat sink.Similarly, if the volume must be bathed at every point by a ¯ow (e.g., air ¯ow in the lung, or blood¯ow in a capillary bed), optimal access means minimum ¯ow resistance between the volume and asource or sink. The main discovery is purely geometric: any ®nite-size portion of this composite canhave its shape optimized such that its overall resistance to ¯ow is minimal. Consequently, the optimal-access solution for the total volume is obtained by optimizing volume shape at every length scale, in asequence that begins with the smallest building block (elemental system), and proceeds toward largerbuilding blocks (assemblies, constructs). The solution is constructed, hence the ``constructal'' name ofthe associated theory. The paths form a tree network in which every single geometric detail is deter-mined theoretically. The tree network cannot be determined theoretically when the time direction isreversed, from large elements toward smaller elements. The constructal principle is further illustratedfor ¯uid ¯ow between a volume and one point, for minimum-time travel between an area and onepoint, and for minimum-cost economics structures. # 1998 Elsevier Science Ltd. All rights reserved

Constructal theory Self-organization Natural form Economic networks

NOMENCLATURE

A=Cross-sectional area (m2)Ap=Cross-sectional area occupied by kp material (m2)D=Thickness of kp blade (m)H=Height (m)k0=Low thermal conductivity (W/m K)kp=High thermal conductivity (W/m K)K=Permeability (m2)L=Length (m)

mÇ '0=Volumetric mass ¯ow rate (kg/s m3)nÇ=Traveling rate (people/s)nÇ0=Traveling rate per unit area (people/s m2)N=Number of tributariesP= Pressure (bar)q=Heat current (W)

q'0=Volumetric heat generation rate (W/m3)t=Thickness (m)u=Mean longitudinal velocity (m/s)v=Local vertical velocity (m/s)V=Velocity (m/s)W=Thickness (m)x,y=Cartesian coordinates (m)

Greek symbolsb=Angle (Fig. 7)

DT=Excess temperature (K)l=Lagrange multiplierm=Viscosity (kg/sm)n=Kinematic viscosity (m2/s)f=Volume fraction of kp materialF=Function

Subscripts( )0=Elemental volume( )1=First construct

Energy Convers. Mgmt Vol. 39, No. 16±18, pp. 1705±1718, 1998# 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0196-8904/98 $19.00+0.00

PII: S0196-8904(98)00054-5

1705

SPATIAL ORGANIZATION OF NATURAL SYSTEMS

In this paper is outlined a theory to predict some of the macroscopic shapes that we see inNature [1±3], in both animate and inanimate systems. This theory deserves the ``constructal''name for the reasons given in Section 6. I review only the issue of organization in space, forwhich I propose the problem of optimal access, or ¯ow with minimal resistance between onepoint and an in®nite number of points (a ®nite volume, or a ®nite area). Natural phenomenathat are organized in time are treated in Ref. [3].

We will see that the theory that ¯ourishes in answer to the optimal-access question is strik-ingly simple. It brings under the same deterministic umbrella a wide variety of natural phenom-ena the structure of which has so far been assumed to lie beyond the powers of determinism.The examples are literally everywhere: tress, roots, leaves, river deltas, river basins, lightning,streets, dendrites, and the pulmonary, nervous and vascular systems. As these spatially and tem-porally organized phenomena cover the living and the nonliving, the present theory provides alink between the physical and the biological worlds.

HEAT FLOW PATTERNS

Consider ®rst the engineering problem of cooling electronic components and packages. Theobjective is to install a maximum amount of electronics (heat generation) in a ®xed volume insuch a way that the peak temperature does not exceed a certain level. The work that has beendone to devise cooling techniques to meet this objective is enormous [4]. The frontier is beingpushed in the direction of smaller package dimensions. There comes a point where miniaturiza-tion makes convection cooling impractical, because the ducts through which the coolant must¯ow take too much space. The only way to channel the generated heat out of the electronicpackage is by conduction. The conduction path will have to be very e�ective (of high thermalconductivity, kp), so that the temperature di�erence between the hot spot (the heart of the pack-age) and the heat sink (on the side of the package) will not exceed a limit.

Conduction paths too take space: designs with fewer and smaller paths are better suited forthe miniaturization evolution. The fundamental problem addressed in this section can be statedas follows [1]: Consider a ®nite-size volume in which heat is being generated at every point, andwhich is cooled through a small patch (heat sink) located on its boundary. A ®nite amount ofhigh conductivity (kp) material is available. Determine the optimal distribution of kp materialthrough the given volume such that the highest temperature is minimized.

The most basic function of any portion of the conducting path kp is to be `ìn touch'' with thematerial that generates heat volumetrically. This material ®lls the volume (V), and its thermalconductivity is low (k0). The optimal-access problem reduces to the geometry problem of allo-cating conducting path length to volume of k0 material, or vice versa. The allocation cannot bemade at in®nitesimally small scales throughout V, because the kp paths must be of ®nite lengthso that they can be interconnected to channel the total heat current (q) to the heat-sink point.There is only one option, namely: (i) to optimize the allocation of conductive path to one sub-system (volume element size) at a time, and (ii) to optimize the manner in which the volume el-ements are assembled and their kp paths are connected.

The discovery that I made is purely geometric: any ®nite-size portion of the heat generatingvolume can have its shape optimized such that its overall thermal resistance is minimal. Thesimplest example is the two-dimensional volume element represented by the rectangle H0�L0 inFig. 1. The area H0L0 is ®xed but its shape may vary. The amount of kp material allocated toH0L0 is also ®xed, and is represented by the constant f0=D0/H0.

The current (q0=q'0H0 L0W) generated by this volume element is collected by a blade (D0,L0)of high-conductivity material, and taken out of the volume through the point M0. In Fig. 1, therest of the H0 �L0 boundary is adiabatic, and W is the dimension perpendicular to the planeH0�L0. The volumetric heat generation rate q'0 is uniform. The hot spot occurs at the point P,which is the farthest from the heat sink M0. Since the heat current q0 is ®xed, the minimizationof the thermal resistance of the volume element is equivalent to the minimization of the tem-perature drop DT0, which is measured between P and M0. If, for the sake of analytical ease weassume that k0/kp<<1 and f0<<1, then DT0 can be expressed as:

A. BEJAN: CONSTRUCTAL THEORY OF SHAPE IN NATURE1706

DT0

q 000H0L0=k0� 1

8�H0

L0� k0=kp

2f0

� L0

H0: �1�

The terms on the right side represent, in order, the temperature drop through the k0 material(from P to the right end of the kp insert) and the temperature drop along the kp insert.Equation (1) shows that DT0 has a minimum with respect to the shape H0/L0:

H0

L0

� �opt

� 2k0=kpf0

� �1=2

: �2�

At this optimum, the two terms of the DT0 expression (1) are equal: the minimized extreme tem-perature di�erence is divided exactly in half by the elbow in the path from P to M0. The tem-perature drop through the k0 material equals the temperature drop through the kp material.What we have here is another manifestation of the principle of equipartition [5], which rules thegeometry of many systems that have been optimized thermodynamically [6, 7].

If the given volume is considerably greater than the H0L0 element of Fig. 1, then we may beable to cover it with an optimal construct, or an assembly of H0L0W elements, as shown inFig. 2. The volume of this ®rst assembly (H1L1W) is ®xed but its shape (H1/L1) may vary. Theminimization of this temperature drop from the corner P to the heat sink M1 pinpoints the opti-mal shape of the assembly: this optimization step is completely analogous to what we saw at thesmallest scale (Fig. 1). When the volume fraction of kp material built into the assembly is ®xed(f1), the design has a second degree of freedom: the ratio D1/D0, i.e., the relative thickness of

Fig. 1. Elemental volume with volumetric heat generation and one high-conductivity path along its axisof symmetry [1].

Fig. 2. The ®rst construct: a large number of elemental volumes (Fig. 1) connected to a central high-conductivity path [1].

A. BEJAN: CONSTRUCTAL THEORY OF SHAPE IN NATURE 1707

the new kp path required by the assembly. In the end, the double minimization of thermal resist-ance determines every geometric feature of the assembly: its shape, or the number of constitu-ents, and the thicknesses of the ``nerves'' of kp material. The temperature-drop equipartitionprinciple applies here as well: the temperature drop across the optimized construct (from P toM1) is divided equally between conduction across the farthest corner element and conductionalong the centre nerve (kp, D1).

This geometric optimization can be continued with constructs of stepwise larger sizes until thekp material covers the entire volume V. For example, in Fig. 3 we see the optimized fourthassembly and its distribution of kp material. The square shape of this particular assembly is anoptimization result, not an arbitrary choice made by the graphic artist. Every single geometricfeature of the cooling scheme is the result of theory. The integer 2, which describes the ``pair-ing'' of two smaller assemblies into a larger one, is also a result of geometric optimization. Thispairing pattern continues toward higher-level assemblies [1].

The fact that the optimized high-conductivity paths resemble the bifurcated (dichotomous)structure of a natural tree network is an accident, because at no time did the designer borrowfrom Nature. This accident, however, has two revolutionary implications:

1. Technologically, it is possible to construct in a few simple geometric steps the optimal net-work for channeling a current that is generated volumetrically. This ®nding is extremely im-portant in practice: if the designer were to start with the given volume V, he would have toguess (postulate) an existing network, and then optimize (numerically and randomly) a prohi-bitively large number of parameters.

2. Theoretically, it means that at the basis of the tree architecture of many living and nonlivingsystems rests a previously unknown design principle: the minimization of the resistance to¯ow between one point and a ®nite volume (an in®nite number of points) when the ¯ow rateis imposed. Reliance on such a universal design principle makes the tree networks determinis-tic, contrary to the established view.

CONSTRUCTION, OR THE TIME ARROW FROM SMALL TO LARGE

The deterministic power of the constructal method stems from its directionÐfrom small partsto a larger assemblyÐwhich happens to be the correct direction of time. This direction was ®rstrecognized in 1981 in the buckling theory of turbulent ¯ow [8, 9], where the ¯ow structure was

Fig. 3. The complete structure of the fourth construct, containing the optimized third, second and ®rstconstructs: the striations represent the kp inserts of the elemental volumes [1].


predicted by starting with viscous di�usion, predicting the ®rst (smallest) eddy, and proceedingin steps with geometrically similar assemblies of eddies.

To see how important the direction of time is, let us re-examine the volume cooling problemby starting with the assumption that the volume is already ®lled by a network of high-conduc-tivity paths (Fig. 4). The network is a sequence of branching stages (j= 1,2, . . . ,n), which pro-ceeds toward smaller scales. At each branching stage, the unknown dimensions of each path areDj and Lj. The total number of paths of size (Dj, Lj) is also unknown, and is labeled Nj. Thedirections in which the smaller branches invade the volume V are unknown.

The conservation of the total heat current (q0) at each branching stage requires that q0=Nj qj,where qj is the heat current through one branch of the jth order,

qj � kpDjWDTj

Lj�3�

DTj is the temperature di�erence across the jth branching stage, and W is the dimension perpen-dicular to Fig. 4. The total temperature di�erence sustained by the network is

DT �Xnj�0

DTj � q0kpW

Xnj�0

Lj

NjDj: �4�

We are interested in minimizing DT, or the ratio DT/q0, subject to the material constraint

Ap �Xnj�0

Apj �Xnj�0

NjLjDj �5�

The optimization problem (4) and (5) is equivalent to ®nding the extremum of the function

F �Xnj�0

Lj

NjDj� lNjLjDj

!�6�

where l is a Lagrange multiplier. The function F has extrema with respect to each product oftype NjDj, but not with respect to the lengths Lj. The optimized products are equal to the sameconstant, (NjDj)opt=lÿ1/2, which is obtained by using the Ap constraint (5):

Fig. 4. An assumed network of branching high-conductivity paths [1].


�NjDj�opt � Ap

Xnj�0

Li

0@ 1Aÿ1: �7�

Equation (7) represents the most that the reverse-time approach can produce. Now, if we assumethat each path undergoes bifurcation, Nj + 1/Nj=2, then equation (7) means that each pathwidth must shrink by a factor of 1/2 from one stage to the next, smaller stage,

Dj�1Dj

!opt

� 1

2: �8�

Equation (8) agrees only with the path-width magni®cation sequence determined theoretically [1]for constructs of the third order and higher. It is important to keep in mind that, unlike thetheory [1], equation (8) is not a prediction, because bifurcation was assumed, not predicted. Inconclusion, what makes the constructal method deterministic is its time arrow, from small tolarge. It invites us to rethink the language for naturally organized systems. Con¯uence yes,branching no. Coalescence yes, bifurcation no. Construction yes, fracturing no.

FLUID FLOW PATTERNS

Constructal theory can easily be extended to ¯uid ¯ow patterns that occur naturally. This isthe class of organized phenomena that has been studied the most. The features of tree-like pat-terns in ¯uid ¯ow have been described quantitatively in surprisingly sharp (reproducible) terms,notably in the study of lungs [10], vascularized tissues [11], botanical trees [12] and rivers [13].In recent years, these natural phenomena have been visualized on the computer by means of re-petitive fracturing algorithms, which had to be postulated.

In this section I illustrate only the start of the geometric optimization sequence for ¯uid ¯owresistance. Figure 5 shows that the function of the path is to distribute the stream mÇ from thepoint M to every elemental volume DV. In some cases two superimposed paths are needed. Forexample, in the bronchial tree the ¯ow is periodic (in and out), and only one path is needed. Inthe circulatory system two identical paths (in counter¯ow) are needed, such that each elementalvolume DV receives arterial blood at the rate dictated by local metabolism, DmÇ . The elementalvolume returns the blood at the same rate to the veins (e.g., the dark network in Fig. 5). Similarpairs of networks in counter¯ow are encountered in trees, roots and leaves. A river basin needsonly one network because the volume V is ¯at (the basin surface) and DmÇ is proportional to therainfall per unit surface. In sum, the function of the path is well represented by the volumetricmass ¯ow rate density mÇ '0 = DmÇ /DV, which must be collected by the network over the volumeV, and channeled as a single stream (mÇ) to the point M.

We focus on only one of the paths of Fig. 5, namely the dark one, and recognize that DmÇ isdriven from the elemental volume to the origin by the pressure di�erence (Pÿ PM). This di�er-ence varies with the position of the DV element relative to M: of special interest is the largest

Fig. 5. Network of ducts connecting one point (M) with every point residing inside a ®nite volume (V).


pressure di�erence, DP = (PmaxÿPM), which is needed by the elemental volumes that are situ-ated the farthest from the end of the network. In the lungs and the circulatory system, forexample, DP is the pressure level that must be maintained by the thorax and heart muscles. Thetime-averaged mechanical power consumed by these pumps, or the entropy generation rate ofthe ¯ow system is proportional to the product mÇDP. The total ¯owrate mÇ is ®xed, because Vand mÇ '0 are ®xed. The thermodynamic optimization of the path is equivalent to minimizing themaximum pressure di�erence, in the same way that in Sections 2 and 3 it is equivalent to mini-mizing the maximum temperature di�erence.

The smallest volume element is shown in Fig. 6. The volume V1=H1L1t is ®xed because thethickness t and the area A1=H1L1 are ®xed. Variable is the shape of the element, H1/L1. Thevolume V1 is visited uniformly by the mass ¯ow rate mÇ1=mÇ '0V1. At this elemental level onlyone tube (diameter D1) is used to collect the mÇ1 stream and lead it to the (0,0) point on theboundary. The mass ¯ow rate through this tube is mÇ(x), with mÇ(0)=mÇ1 at the origin (0,0) andmÇ (L1) = 0. Except for the point of origin, the surfaces of the elemental volume V1 are imperme-able. The thickness t is assumed to be su�ciently small, t < (H1, L1), such that the pressure®eld is essentially two-dimensional, P(x,y). The rest of the ¯ow path between (0,0) and anyother point (x,y) is located in the material situated above and below the x axis. To account forthis ¯ow we model the material as an anisotropic porous medium in which Darcy ¯ow isoriented purely in the y direction

v � K

mÿ @P@y

� �: �9�

In this equation v and K are the volume-averaged velocity and, respectively, the permeability inthe y direction. The ¯uid may ¯ow in the x direction only along the x axis. The pressure ®eldP(x,y) can be determined by eliminating v between equation (9) and the local mass continuityequation @v/@y=mÇ0'/r, and imposing @P/@y = 0 at y = H1/2 and P = P(x,0) at y = 0:

P �x; y� � _m 000�2K�H1yÿ y2� � P �x; 0�: �10�

The pressure distribution along the x axis can be determined after making similar assumptionsabout the stream that eventually exits as mÇ1 through the origin. Let us assume as in the earlieststudies [11, 14] Hagen±Poiseuille ¯ow through a round tube of length L1 and diameter D1.First, we use the mean velocity in the x direction u = (D1/2)

2(ÿ@P/@x)y = 0/(8m), to estimate thelocal mass ¯ow rate mÇ(x), which points toward the origin, mÇ (x) = (pD4

1/128�)(@P/@x)y=0.Mass conservation requires that the mass generated in the in®nitesimal volume slice (H1tdx)contribute to the mÇ(x) stream:mÇ0' H1tdx =ÿ dmÇ . Integrating this equation away from theimpermeable plane x = L1 (where mÇ=0), and recalling that mÇ1=mÇ0'H1L1t, we obtain

Fig. 6. The ¯ow through the smallest subsystem of the volume V of Fig. 5.


mÇ (x)=mÇ '0H1t(L1ÿx)=mÇ1(1ÿ x/L1). Finally, these last two equations yield the pressure distri-bution along the x axis

P �x; 0� � P0 � 128_m000�H1t

pD41

L1xÿ x2

2

!: �11�

This result can be combined with equation (10) to determine the pressure distribution over therectangular domain H1�L1, where P(0,0) = P0. The resulting expression shows that the peakpressure occurs in the two farthest corners, Pmax=P(L,2H/2), namely

DP1

_m000�� H2

1

8K� 64tA2

1

pD41H1

�12�

where DP1=PmaxÿP0. The optimal shape for minimal pressure drop is represented byH1,opt=(256tK/p)1/3A2/3

1 /D4/31 .

The objective of this brief analysis was to demonstrate the existence of an optimal systemshape at the elemental level. This optimization principle is analogous to equation (2) of theheat-¯ow problem. Beyond this point, the method of covering the given volume (Fig. 5) withincreasingly larger constructs follows the steps outlined in Section 2, and is described in detail inRef. [15]. The result is a ``composite material'' for ¯uid ¯ow: Hagen±Poiseuille ¯ow in the ducts,coexisting with Darcy ¯ow in the spaces between ducts. The pressure di�erence between thefarthest point of this volume and the sink point is minimal. One geometric featureÐa resultÐofthe optimal-access solution is the tree network formed by the ducts. Dichotomy, or the coalesc-ence of two tubes into a larger one is another result of theory, not an act of empiricism.

STREET PATTERNS

The best known natural occurrence of the tree pattern is in the structure of urban areas [2].Why do streets emerge in clusters (grids) that look almost the same from block to block, andfrom city to city? Why are streets and street patterns a mark of civilization? Indeed, why dostreets exist? The constructal-theory answer to these questions is astonishingly simple and direct.It is the geometric solution to the following access-optimization problem: Consider a ®nite-sizegeographical area A, and a point M situated inside A or on its boundary. Each member of thepopulation living on A must travel between his arbitrary point of residence P(x,y) and a pointM. The latter serves as common destination for all the individuals who live on A. The rate atwhich individuals must travel to M is ®xed and described by nÇ0 [people/m2s]. This also meansthat the rate at which people are streaming into M is constrained, nÇ=nÇ0A. Determine theoptimal ``bouquet'' of paths that link the points P of area A with the common destination Msuch that the time of travel required by the entire population is the shortest.

This problem is the minimal-time analog of the minimal-resistance problems solved in heattransfer (Section 2) and ¯uid ¯ow (Section 4). The area A could be a ¯at piece of farm landpopulated uniformly, with M as its central market. It also helps to think in time, by beginningwith the most ancient type of community that faced this access optimization problem. The old-est solution to this problem was also the simplest: unite with a straight line each point P andthe common destination M, and you will minimize the total time spent by the population enroute to M. These paths form a radial pattern centered at M.

The radial pattern disappeared naturally in areas where settlements were becoming too denseto permit straight-line access to everyone. Why the radial pattern disappeared ``naturally'' is theheart of the problem. Another important development was the horse driven carriage: with itman had two modes of locomotion, walking (V0), and riding in a carriage with an average vel-ocity V1 that was greater than V0. It is as if the area A became a composite material with twoconductivities, V0 and V1. Clearly, it would be faster for every inhabitant to travel in straightlines to M with the speed V1. This would be impossible, however, because the area A would endup being covered by beaten tracks, leaving no space for the inhabitants and their land proper-ties.


The real, more modern problem then is one of bringing the street near a small but ®nite-sizegroup of inhabitants: this group would have to walk ®rst (V0) in order to reach the street. Theproblem is one of allocating a ®nite length of street (V1) to each ®nite patch of area (A1=H1L1

in Fig. 7), where A1<<A. The area patch cannot be smaller than the size ®xed by the livingconditions (e.g., property) of the individuals who will be using the street. For simplicity weassume that the smallest area (A1) is rectangular. Although A1 is ®xed, its shape or aspect ratioH1/L1 is not. Indeed, the main objective is to anticipate optimal form: the area shape that maxi-mizes the access of the A1 population to the street segment allocated to A1.

Symmetry suggests that the best position for the street segment is along the longer of the axesof symmetry of A1. See Fig. 7, where L1>H1. Each individual must travel from a point of resi-dence P(x,y) to the (0,0) end of the streetÐthis, in order to ``get out'' of A1. The individual cantravel at two speeds, a low (walking) speed V0 when o� the street, and a higher (vehicle) speedV1 when on the street. The angle between the V0 and V1 paths may vary. The maximum traveltime occurs between the farthest corner (P) and the common destination (M)

t1 � L1

V1� H1

2V0

1

cosbÿ V0

V1

sinbcosb

� ��13�

where b is the angle between the V0 path and the H1 side. The minimization of t1 has twodegrees of freedom, the geometric aspect ratio H1/L1 and the angle b. The optimal angle forminimum t1, bopt=sinÿ1(V0/V1), shows that V0 should be perpendicular to V1 (i.e., that bopt=0)when V0<<V1. The minimization with respect to H1/L1 subject to A1=H1L1 yields

H1

L1

� �opt

� 2V0

f1V1; f1 � 1

cosbÿ V0

V1tanb: �14�

The twice-minimized travel time is t1,min=[(2A1/V0V1) cos bopt]1/2. At this optimum, the two

terms on the right-hand side of equation (13) are equal. This equipartition of time principlemeans that the total travel time is minimal when it is divided equally between traveling alongthe street (V1) and walking to reach the street (V0).

The second degree of freedom (the optimized angle b) plays only a minor role as soon as V1

is greater than V0. In other words, the shift from V0 to V1 does not have to be dramatic inorder for the b = 0 design to perform nearly as well as the b = bopt design. We reach the im-portant conclusion that small, internal variations in the organization pattern have almost noe�ect on the performance (t1,min, in this case) of the organized system. The practical aspect ofthis observation is that a certain degree of variability (imperfection) is to be expected in the

Fig. 7. The smallest elemental area A1=H1L1, and the street segment allocated to it [15].


patterns that emerge naturally. These patterns are not identical, or perfectly similar: thisaccounts for the historic di�culty of attaching a theory to naturally organized systems. Naturalpatterns are quasi-similar, but only in the same sense in which no two human faces are identical.Their performance, however, is practically the same as that of the mathematically optimized pat-tern. The contribution of constructal theory is that the performance and the main geometric fea-tures (mechanism, structure) of the organized system can be predicted in purely deterministicfashion.

This observation is one reason why in Sections 2 and 4 minor re®nements such as varying theangle of each intersection (coalescence) were not included in the analysis. Another reason is thatthe angle optimization idea has already been recognized* in biology and river morphology,beginning with Murray [16] and Thompson [14]. Highlighting it here would have detracted fromthe ®rst degree of freedomÐthe form of each building blockÐwhich is new and solely respon-sible for the theoretical construction of the pattern.

The constructal sequence can be carried out analytically [2] by following the steps illustratedin Section 2 and Fig. 8. An interesting feature of this construction is that the same pattern

Fig. 8. Constructal minimization of travel time between a ®nite area and one point [15].

*This idea is actually a lot older. Beginning with Heron of Alexandria two thousand years ago, the minimization of tra-vel time has been `ìnvoked'' to predict that (i) light travels in straight lines, (ii) the incident and the re¯ected raysmake equal angles with the mirror surface, and (iii) the refracted ray is bent in an optimal way when passing fromone medium into another (Fermat's principle). Constructal theory unites these optics phenomena with many othernatural ¯ow patterns, and raises to the rank of law the principle of travel-time (or resistance) minimization (Section7).


emerges if, instead of minimizing the largest travel time across each area assembly [e.g.,equation (13)], we minimize the travel time averaged over all the points of the area assembly.This means that the optimization of the shape of each area element is of interest to every inhabi-tant: what is good for the most disadvantaged individual is good for every individual. Thisconclusion has profound implications in the spatial organization of all living groups, from bac-terial colonies to societies. The urge to organize is an expression of sel®sh behavior.

ECONOMICS STRUCTURE IN SPACE

The economic activity that covers a given area is the access-optimization principle, and theeconomic and business structure is the result. To see how constructal theory explains the originof structure in economics and business [15], consider a stream of goods that proceeds from onepoint (producer) to every point of a ®nite-size territory (consumers). The ¯ow may also proceedin the opposite direction (e.g., grain harvested, or carpets woven by individuals). The objectiveis to minimize the total cost associated with the given stream.

The economics principle of economies of scale tells us that the unit cost is lower when thegoods move in the aggregate; that is, when they are organized into thicker streams. The unitcost is also proportional to the distance traveled. Clearly, the unit cost plays the same role asthe local thermal resistance in heat trees, the inverse of the travel speed in street trees, or thelocal ¯uid-¯ow resistance in ¯uid trees. The given business territory is covered naturally bytrees; that is, links of decreasing unit cost, starting from the highest unit cost that is allocated tothe smallest area scale (the individual), and continuing with a sequence of intermediaries (distri-butors) who handle increasingly larger fractions of the total stream of goods.

CONSTRUCTAL THEORY VERSUS FRACTAL ALGORITHMS

So far we have seen how the theory works in two entirely di�erent settings: the natural pat-terns revealed by ¯uid ¯ow (Section 4) and streets (Sections 5 and 6), and the arti®cial patternsemerging in engineering (Section 2). To appreciate how much is new in this theory, it is import-ant to note that one portion of the network pattern (e.g., the highest-order kp paths in Fig. 3) isnot new. It was ®rst proposed in physiology as a three-dimensional heuristic model for the vas-cular system [11], where it was known empirically that each tube is followed by two smallertubes, i.e., each tube undergoes bifurcation [e.g., equation (8)]. It was also known that the tubediameter must decrease by a constant factor (2ÿ1/3) during each bifurcation: this result was de-rived based on ¯ow resistance minimization [14], and is the only theory-based notion present inthe algorithms used to simulate the pulmonary tree or other tree-shaped patterns that appear innature (trees, roots, leaves, river basins, deltas, lightning, lungs, nervous system, vascular sys-tem). The description of these geometric constructions was made popular through the advent offractal geometry: in fact, a two-dimensional version of Cohn's branching ¯uid network (similaronly to the third and fourth constructs of Fig. 3) appeared in the books of Mandelbrot [17] andPrigogine [18], where it is presented heuristically as a ``model of the lung''. These authors didnot say anything predictive or deterministic en route the algorithm assumed in making the draw-ing.

The fractal description of tree networks is only partially correct, because it misses the genesisof all such phenomena, which lies in the smallest (®nite, predictable) length scale. What iswrong with the fractal descriptionÐand by wrong I mean totally upside downÐis the timearrow of the description (Section 3). Some would argue that fractal geometry has nothing to dowith time, and they would be right, as far as descriptive geometry goes. The assumed algorithmcan be executed in both directions, from the largest scale to the smallest, and from the smallestto the largest. As a descriptive aid for natural phenomena, however, the fractal description rep-resents a clear choice, namely, from the largest scale all the way to size zero in an in®nite num-ber of steps. The word fractal has the concept of time built into it: the act of breakingsomething evolves in time from large pieces to smaller pieces. As a way to think about predict-ing the morphology of natural systems, the fractal paradigm is oriented backwards. The theore-tician must still predict the algorithm postulated by the mathematician, or created by the artist.Fractals are description, not explanation.


If ``fractal''* is an appropriate Latin word for breaking things [17], i.e., for the opposite ofthe direction in which natural systems evolve, then the appropriate word for the geometry andevolution of optimized and organized natural phenomena is constructal{ [1].

CONSTRUCTION, THERMODYNAMICS, TIME AND LIFE

There is a fundamental link between the optimized conglomerates constructed in Section 2,and the natural occurrence of similar structures, both living and not living. The principle thatrules the design of each building block, and the manner in which an optimal number of suchblocks are assembled into larger entities is the optimization of access, or the minimization of re-sistance.

One feature of this paper and Refs. [1±3] is the large number and the extreme diversity of theorganized phenomena that can be predicted and explained based on constructal theory. Thereader should ¯ip again through this paper, and review the topics and the ®gures. There is nodi�erence between the living and the nonliving when faced with the opportunity to ®nd a moredirect route subject to global constraints. If living systems can be viewed as engines in compe-tition for better thermodynamic performance, then physical systems too can be viewed as livingentities in competition for survival. This analogy is purely empirical: we have a very large bodyof case-by-case observations indicating that certain designs (living and nonliving) evolve andpersist in time, while others do not. Now we know the particular feature (optimal access) thatsets each surviving design apart, but we have no theoretical basis on which to expect that thedesign that persists in time is the one that has this particular feature. This body of empirical evi-dence forms the basis for a new law of nature that can be summarized as follows [1]: For a®nite-size system to persist in time (to live), it must evolve in such a way that it provides easieraccess to the imposed (global) currents that ¯ow through it.

This law brings life and time explicitly into thermodynamics, and creates a new bridgebetween physics and biology. It has two parts. The ®rst recognizes the natural tendency ofimposed currents to construct paths of optimal access through constrained open systems. Thesecond part refers to the evolution (i.e., improvements) of these paths, which occur in an ident-i®able direction that can be associated with time itself.

The fundamental optimal-access problem of constructal theory is related to what in math-ematics is known as Steiner's problem: how to connect with the shortest line several points of aspeci®ed ®nite area. This problem is very important to the design of telecommunication net-works and computer architecture. According to Bern and Graham's recent review [19], ``the sol-ution to this problem has eluded the fastest computers and the sharpest mathematical minds,''and its solutions ``defy analysis.'' Solutions have been found for connecting a moderate numberof points in a plane. As computers become more powerful this number will increase, but it willnever be able to cover the given area.

My alternative to Steiner's problem was the proposal [2] to minimize the time of travel, andto recognize that the traveler has access to at least two modes of locomotion, one slow and theothers considerably faster. The slow mode was placed below the smallest scale of assembly, sothat every single point of the area was touched by a shapeless (volumetric) ¯ow akin to di�u-sion. The faster channels were arranged optimally to collect the volumetric ¯ow of each elemen-tal building block of the network.

The ``composite material'' formed by slow and fast ¯ow regimes is a feature that unites all thevolume-to-point access phenomena covered in this paper. Optimal-access constructs are also uni-ted by the observation that the ®ner details of the architecture play a negligible role. The overallshape of the tree and the shape of each of its building blocks are the geometric optima thatdetermine the optimal number and arrangement of branches at each level of assembly. This isan important observation because it sheds light on the origin of the coexistence of order anddisorder in natural structures. If optimal access at the system level is what matters, then theinnermost details of the access structure may vary from case to case, perhaps due to incidental

*From the Latin verb frangere (to break), which survives unchanged in both Italian and Romanian.{From the Latin verb construÆeÆre (to build), which survives as construire in French, Italian and Romanian.


(local) factors. The main point of constructal theory is that the larger pictureÐthe optimal sys-tem performance, geometric structure, and working mechanismÐcan be predicted and describedin a purely deterministic manner.

The optimal-access law was described in this paper with reference to a system with imposedsteady ¯ow. If the system discharges itself to one point in unsteady fashion, then the constructalminimization of volume-to-point resistance is equivalent to the minimization of the time of dis-charge, or the maximization of the speed of approach to equilibrium (uniformity, zero ¯ow,death). If the volume is unbounded, the constructs compound themselves and continue to spreadinde®nitely. Complexity continues to increase in time. Examples are the jet injected into a ¯uidreservoir, and the dendritic crystal that grows into a subcooled liquid. All the structuredphenomena mentioned at various stages in this paper, including the round cross-section of theblood vessel and the proportionality between river width and depth, can be predicted based onthe constructal theory of access optimization [15].

How important is the present solution to the optimal access problem, i.e., this constructalprinciple that allows us to anticipate the tree networks of so many natural systems? In contem-porary physics, a signi®cant research volume is being devoted to the search for universal designprinciples that may explain organization in animate and inanimate systems. In this search, thetree network is a symbol for the challenge that physicists and biologists face (Kau�man [20], pp.13±14): `Ìmagine a set of identical round-topped hills, each subjected to rain. Each hill willdevelop a particular pattern of rivulets which branch and converge to drain the hill. Thus theparticular branching pattern will be unique to each hill, a consequence of particular contingen-cies in rock placement, wind direction, and other factors. The particular history of the evolvingpatterns of rivulets will be unique to each hill. But viewed from above, the statistical features ofthe branching patterns may be very similar. Therefore, we might hope to develop a theory ofthe statistical features of such branching patterns, if not of the particular pattern of one hill.''

Constructal theory provides an answer to the challenge articulated so well by Kau�man. Itintroduces an engineering ¯avor into the current debate on natural organization, which untilnow has been carried out mainly in physics and biology. As a result of their training, engineersbegin the design of a device by ®rst understanding its purpose. The size of the device is always®nite, never in®nitesimal. The device must function (i.e., ful®l its purpose) subject to certain con-straints. To analyze the device is not su�cient: to optimize it, to construct it, and to make itwork is the real objective. Finally, many designs that di�er in some of the ®ner details havenearly the same overall performance as the optimal design. All these featuresÐpurpose, ®nitesize, constraints, optimization, constructionÐcan be seen in the living and nonliving systemsoptimized in this paper. The resulting constructions are purely deterministic, and consequentlythey represent an alternative worthy of consideration within ®elds other than engineering. Theprogress made in this direction is reviewed in Ref. 15.

AcknowledgementsÐThis work was supported by the National Science Foundation.

REFERENCES

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FL. ch. 1. 1996.5. Tondeur, D. and Kvaalen, E., Ind. Eng. Chem. Res., 1987, 26, 50.6. Bejan, A., J. Appl. Phys., 1996, 79, 1191.7. Bejan, A., Entropy Generation Minimization, CRC Press, Boca Raton FL, 1996.8. Bejan, A., Lett. Heat. Mass Transfer, 1981, 8, 187.9. Bejan, A., Entropy Generation through Heat and Fluid Flow, Wiley, New York, 1982.

10. Weibel, E. R., Morphometry of the Human Lung, Academic Press, New York, 1963.11. Cohn, D. L., Bulletin of Mathematical Biophysics, 1954, 16, 59.12. MacDonald, N., Trees and Networks in Biological Models, Wiley, Chichester, UK, 1983.13. Scheidegger, A. E., Theoretical Geomorphology, Springer-Verlag, Berlin, 1970.14. Thompson, D'. A. W, On Growth and Form, Cambridge University Press, Cambridge, UK, 1942.15. Bejan, A., Advanced Engineering Thermodynamics. 2nd edn. Wiley, New York, 1997.16. Murray, C. D., in Proc. Acad. Nat. Sci., Vol. 12, 1926, p. 207.17. Mandelbrot, B. B., The Fractal Geometry of Nature, Freeman, New York, 1982.


18. Prigogine, I., From Being to Becoming, Freeman, San Francisco, 1980.19. Bern, M. W. and Graham, R. L., Scienti®c American, 1989, January, 84.20. Kau�man, S. A., The Origins of Order: Self-Organization and Selection in Evolution, Oxford University Press, New

York, 1993.


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Constructal theory: from thermodynamic and geometric optimization to predicting shape in nature