Equipartition, optimal allocation, and the constructal approach to predicting organization in nature

Rev Gin Therm (I 998) 37, 165-l 80 0 Eisevier, Paris

Equipartition, optimal allocation, and the constructal approach to predicting organization in nature

Adrian Bejan a*, Daniel Tondeur b a Department of Mechanical Engineering and Materials Science, 80x 90300, Duke University, Durham, NC 27708-0300, USA

b Laboratoire des sciences du genie chimique, CNRS, Ensic-INPL, E.P. 451, 1, rue Grandville, 54001, Nancy cedex, France

(Received 24 July 1997; accepted 18 November 1997)

Abstract-This is a review of recent engineering developments in thermodynamic optimization, which shed light on a universal design principle that accounts for macroscopic organization in nature. It is shown that the optimal performance of a finite-size system with purpose is always characterized by the equipartition of driving forces or the optimal allocation of material subject to overall constraints. Examples are drawn from natural inanimate systems (river basins, turbulent flow) and animate systems (living trees). It is shown that this principle also governs the architecture of tree networks. Tree networks can be obtained in purely deterministic fashion by minimizing the flow resistance (or the time of travel) between one point and a finite area or a finite volume (an infinite number of points). The shape of each volume element can be optimized for minimal flow resistance. The network is ‘constructed’ by assembling the shape-optimized building blocks, and proceeding in time from the smallest volume element toward larger constructs. In constructal theory small size and shapeless flow (diffusion) come first, and larger sizes and geometrical form (channels, streams) come later. 0 Elsevier, Paris

thermodynamics / optimization / tree network / constructal / fractal / equipartition of driving forces

R&urn6 - Equipartition, allocation optimale et approche constructale pour predire I’organisation dans la nature. Cet article constitue une mise au point sur des d&eloppements r&ems en optimisation thermodynamique, qui contribuent ?I proposer un principe general regissant I’organisation macroscopique de systemes naturels. On montre que la performance optimale d’un systeme fini est caract&iGe par une Cquipartition des forces matrices ou par I’allocation optimale des ressources soumises A des contraintes globales. On presente des exemples de systemes inanimes (bassins hydrographiques, ecoulements turbulents) et de systemes anim6s (arbres). On montre que ce principe gouverne egalement I’architecture des structures arborescentes. Cette architecture peut Ctre g&Me d’une maniere purement dbterministe, en minimisant la r&istance a I’6coulement (ou le temps de transfert) entre un point et une surface ou un volume fini. La forme de chaque Gment de volume qui minimise la resistance globale a I’&oulement peut Ctre dCtermin6e. Le reseau est alors “construit” en assemblant ces Gments, optimises aux diffkrentes &helles, en partant de I’&helle la plus petite. C’est I’approche <<constructale>,, que nous opposerons A I’approche fractale. 0 Elsevier, Paris

thermodynamique / optimisation / reseau arborescent / constructal / fractal / equipartition des forces

Nomenclature

A area ............................ Cl.2 coefficients ......................

c thermal conductance. ...... D thickness :. .............. Dh hydraulic diameter. ........

f conjugate force ............ H height ....................

j flux.. .................

* Correspondence and reprints.

. . . . . . . . . .

. . .

m2 equation (6)

J.s-l.K-'

m m

K-l m

J.s-1.m-2

k

ko

k, L n,

N3 11,

i Sgen 4exl t

constant........................ equation (11)

low thermal conductivity.. . J.s-l.rn-‘K-’

high thermal conductivity . J.s-‘.rn-’ K-l length.......................... m number of constituents in a construct number of k, paths

volumetric heat generation rate . . . J.s-1.m-3

heat transfer rate.. JY1

local entropy generation density. . J.s-’ .K-’ ‘me2

entropy generation rate J.s-l.K-'

time............................ S

165

A. Bejan, D. Tondeur

T temperature..................... AT temperature difference ucc free stream velocity. V volume vo small velocity VI, 2.3 larger velocities. W’ power X conductance allocation ratio

Greek letters

8, travel time. . x Lagrange multiplier XB buckling wavelength v kinematic viscosity u2 variance or volume fraction occupied by the k,

material

Subscripts

B buckling C cold c Carnot, reversible h hot

H high L low

K K

111.5 -1

m3 rn.s -I

111.5 -1

J.s-i

I1

m2.s -1

1. OBJECTIVE

Natural self-optimization and self-organization is one of the most researched topics in physics and biology today. In this paper we draw attention to a series of recent results obtained in engineering thermodynamics. The pattern revealed by these new developments casts a new light on natural organization in both animate and inanimate systems. It injects a new flavor-a dose of determinism-in a debate in which natural order is widely considered influenced by nondeterministic processes, for example, the result of chance and necessity [l] or of fluctuations and stability [2].

Much of the current interest in the origin of order in nature is fueled by signs and ideas that point toward the existence of universal design principles. The search for such principles acquired the status of research starting with Thompson’s treatise [3]. It continued with the thermodynamic theory of dissipative structures [ill and the fractal-geometry description of virtually every known form of natural organization [5].

Two recent reviews [6, 71 of the progress made during the past two decades provide a captivating account of why principles of ‘universality’ deserve to be taken seriously. The issue is no longer ‘how’ similar are the shapes of different entities such as the lung and the river basin fractal geometry has gone a long way to provide a common description for natural structures.

The fascinating issue is ‘why’ certain structures are common in nature. In other words, are there general principles laws that govern the formation of these structures. and, if so, can we invoke these laws to predict natural organization?

The challenge to discover design rules and universality on the line where order and disorder coexist is aptly described by Kauffman [6) in his 1993 treatise.

“The question we must address is whether there might be statistical order within such historical processes. A loose analogy makes this point. Imagine a set of identical round topped hills. each subjected to rain. Each hill will develop a particular pattern of rivulets which branch and converge t,o drain the hill. Thus the particular branching pattern will be unique to each hill, a consequence of particular contingencies in rock placement, wind direction, and other fact,ors. The particular history of the evolving patterns of rivulets will be unique t,o each hill. But, viewed from above, the statistical features of the branching pat,terns may be very similar. Therefore. we might hope to develop a theory of the statistical features of such branching patterns. if not of the particular pattern on one hill.”

Indeed, we will show that by adopting an engineering (design optimization) perspective it is possible to predict in a, purely deterministic manner the tree-shaped characteristic of the river basin and other examples of natural organization. The key is the engineering point of view. Engineers begin the design of a device by first understanding it,s ‘purpose’. The size of the device is always finite, never infinitesimal. The device must function (i.e., fulfill its purpose) subject to overall constraints. To analyze the device is not sufficient: to optimize it. to construct it, and to make it work is the real objective. Finally, many designs that differ from one another in some of the finer details have nearly the same overall performance as the opt,imal design. All these features- purpose, finite size, constraints, optimization. construction -arc present in the living and non-living examples discussed in this paper. The theory that emerges is purely deterministic, and consequently it represents an alternative worthy of consideration within fields o&r than engineering.

Our paper is organized in three parts. In tlrc first we review the current status of thermodynamic optimization in engineering. this in order to illustrate the emergence of a pattern among the optimal solutions to highly diverse engineering problems. In brief, ea.& problem consists of specifying the open (flow) system and its outer constraints (e.g. finite size. finite time, materials. currerns, purpose). and asking whether ai1 optimal configuration exists (e.g. geometry. allocation of materials) such that a certain figure of merit is maximized or minimized (e.g. power output, entropy generation. cost). The pattern that, is revealed by solutions is one of ‘equipartition’ [8, 91 or optimal allocation [lO] of finite material resources.

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In the second part of the paper we move away from engineering (i.e. man-made, artificial systems with purpose) and show that the same pattern of equipartition characterizes natural phenomena and systems. The examples cover the wide range from inanimate systems (e.g. turbulent flow structure, river basin morphology) to the even more diverse world of animate systems (various living trees, allometric laws in physiology).

In the third section we put the two classes of results together: the artificial and the natural. We show that by optimizing analytically the access between one point and a finite-size volume (an infinite number of points) it is possible to predict (construct) the detailed pattern of the living tree. This ‘constructal theory’ and principle of optimal access [ll] are a way to understand nature as a conglomerate that is engineered, not random. It also represents a dramatic shift from describing order to predicting order.

2. EQUIPARTITION IN ENGINEERING

2.1. Power plants

It is fitting that we start this brief engineering overview with the application that gave birth to the science of thermodynamics: the heat engine. Figure 1 shows the simplest model [lo, 12-151 of an actual (irreversible) heat engine that operates steadily or in an integral number of cycles. The optimization of such models is a significant activity in the engineering field of thermodynamic optimization, which in some applications is also known as entropy generation minimization [16]. In the model of figure 1 the heat input Qn is fixed. It is assumed that the irreversibility is due solely to the temperature differences TH - THC and TLC - TL, which drive the heat input and the rejected heat Qn. The rest of the heat engine-the compartment shown between THC and TLC-is irreversibility free. The simplest way to account for the relationship between heat transfer rate and temperature difference is to assume the proportionalities

QH = CH(TH -THc) QL = CL(TLC -TL) Cl>21

In these relations Cn and CL are the thermal conductances of the two heat exchangers. Since each thermal conductance is proportional to the size of the heat exchanger surface, i.e. proportional to the size of the piece of hardware, it makes sense to regard Cn and CL as partners in an overall size constraint,

CH + CL = c (constant) (3)

The power output W = QH (1 - TLC/THC) can be combined with equations (1) and (2) to see how it is

TLC

TL

QH G TH

THC

reversible compartment -W

Figure I. Model of an irreversible heat engine with hot-end and cold-end heat exchangers.

affected by the way in which the C inventory is allocated between the two heat exchangers

W -=I- TL/TH

QH QH (4) l---

( 1 1+-

THC x l-x >

In this expression x = CHIC is the conduction allocation fraction. It is easy to see that when Cn and CL vary subject to the size constraint equation (3) the power output is maximized when x = l/2, i.e.:

CH = CL (5)

This equipartition principle can also be derived by minimizing the entropy generation rate of the power plant. This result is astonishing because of its simplicity and generality. The engineering literature continues to show that equation (5) holds for increasingly complex models of irreversible power plants, starting for example with the model of figure 1 in which the heat input Qn may vary, in addition to the variable size ratio Cn/Cn [17]. The equipartition principle also holds for the corresponding class of models of irreversible refrigeration plants [18].

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2.2. Heat exchangers

Further examples of equipartition arc found in the thermodynamic optimization of heat exchangers which represent an extremely important class of engineering systems and a sizeable literature. In most configurations the heat exchanger surface separates two streams. In a Rankine-cycle power plant, for example, the two streams are the working fluid (boiling water), and the hot stream (combustion gases) that serves as a source of energy. Each side of the heat exchanger surface is characterized geometrically by a flow length L and a hydraulic diameter Dh, or by the slenderness ratio L/Dh. The entropy generation rate associated with each side of the heat transfer surface is due to two effects heat transfer across a finite temperature difference and flow with friction. For example. in the case of the counterflow heat exchanger with ideal gases on both sides the contributions to the entropy generated on one side of the surface appear explicitly in a relation of the type

(‘3)

where cl and c2 account for the properties and flow regime of the ideal gas! as shown [lo]. Of interest in the present discussion is the geometric implication of equation (6): changes in t,he slenderness ratio LIDh induce changes of opposite signs in the two terms of the &, expression. There is an optimal flow channel slenderness such that ,$‘,,, is minimal: (LIDI,),,, = (c1/c2) ‘12. At this optimum the entropy generation rate is due equally to heat transfer and fluid friction. In other words: the equipartition of entropy generation is the answer to the thermodynamic and geometric optimization of each flow passage.

The preceding example was about the distribution of entropy generation between two different dissipative mechanisms that are present in a heat exchanger. Let us take now a different point of view, namely, that of distributing the entropy generation between different parts of a process when there is only one dissipative mechanism (heat transfer).

In the language of irreversible thermodynamics [19], the local density of entropy generation ( >gPn) caused by heat transfer is the product of the local flux j and the conjugate force f.

Sgen = j f (7)

where the units are Sgen[J.mP2.KP1.s-i], j [J.m-‘.s-i] and f [K-l]. The conjugate force is the difference between the inverse temperatures of the hot side (Th) and cold side (Z’,) of the heat transfer medium,

The total entropy generation per unit time exchanger: S,,,[J.K-‘.s-‘1, is obtained by sgrn over the entire heat transfer area A,

in the heat integrating

(‘3)

Assume now that we impose a constant area A and a specified heat transfer ‘duty’ for the heat exchanger. Jspec [J.s’]. The duty is a common constraint in thermal engineering; for example. the function of the heat exchanger may be to isobarically heat a given mass flow rate (ti) of a single-phase fluid (C,) by a specified number of degrees (AT). In this case. the heat transfer duty is:

Jspec = s

jdA=tic,AT (10) A

We also assume a linear heat transfer constitutive relation

.i=kf (11)

where k is a constant. Note that k is not the ordinary heat transfer coefficient used in thermal engineering. because f is based on inverse temperatures.

We arc interested in the distribution of the force f over the fixed area A such that the total entropy generation rate S,,, is minimized subject to the Jspec constraint. This variational problem is equivalent to minimizing the quantity j f + X j with respect to f. where X is a Lagrange multiplier.

~(jf+xjj=~(kf’+Xkf)=o (12)

The solution is simply:

.f=-; (contant) (13)

In conclusion, the distribution of driving forces that minimizes the generation of entropy when the surface area and heat transfer duty are specified, is the ‘uniform (equipartitioned) distribution’.

Let us focus on the engineering implications of this conclusion. It is difficult: if not impossible, to design and operate a heat exchanger in which the temperature difference between fluids (or rather AT-‘) is the same on all the points of the heat transfer surface. Real heat exchangers are not equipartitioned. It is important then to compare the performance of real configurations with the optimal (equipartitioned) design. For this we calculate the difference in the total rate of entropy production and find:

jlrpal - i’,,,jn = A k o”(f) (14)

where a”(f) is the variance of the distribution of driving force, which is a positive quantity when f is not uniform.

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This means that a heat exchanger with specified duty is less dissipative when the driving force is more uniform. The best known practical example of this design principle is the two-stream counterflow heat exchanger, which is less dissipative than the equivalent co-current (parallel flow) heat exchanger. Not surprisingly, the performance of the crossflow configuration falls between the counterflow and the parallel flow.

What do these conclusions mean to the engineer? To begin with, when the area is fixed the heat exchanger can meet the specified duty by employing a smaller flow rate of heating fluid. On the other hand, if the area constraint is relaxed and, instead, the fluid flow rate is constrained, then equipartition means a smaller heat transfer area, i.e. a smaller capital investment.

To appreciate the generality of the force equipartition principle, consider the following observations.

a) The preceding demonstration was based on the linear relation between flux and force, equation (11). It can be extended to phenomena covered by non- linear monotonic concave relations, such as the power law j = b f a with a < 1, or the harmonic dependence j = of/(1 + bf). The preceding conclusions hold even if the j(f) relation has slight convexity [9].

b) The linear relation (11) implies that if the force is constant so is the flux. Then, according to equation (7), the local entropy generation rate too is constant. In conclusion, the equipartition principle can be restated in terms of flux or local entropy generation.

c) The equipartition principle was illustrated here by considering the process of heat transfer. It applies unchanged to mass transfer or any other type of process that can be described by the formalism of equations (7)~(14).

d) The time variable was not used in the preceding demonstration. This is equivalent to assuming that the steady-state regime prevails: or that the results apply at only one instant in time. It can be shown that the same analysis may be performed with time as the integrating variable, instead of, or in addition to surface area. In this case it is found that the force should be equipartitioned in time, which means that the steady state is optimal. In this way we recover the classical result of Prigogine [20, 211 according to which the steady states of homogenous systems are states of minimal entropy generation.

e) The principle of the equipartition of driving forces (but not of fluxes and entropy generation) has been recently shown to apply to heterogeneous systems, where the transfer occurs along several paths with different but constant conductances [22].

f) The equipartition principle has also been developed for multicomponent systems [8, 91. These analyses are considerably less straightforward than the present demonstration, because of the nature of the constraints. Usually, the constraint is placed not on the overall flux but on one or the other comoonents of the svstem. Other applications in which forces has been identified as

I

the uniformity of driving an optimization principle

include: diabatic distillation (where heat transfer is distributed along the column) [23, 241, chromatography (where temporal sequences of cuts and recycle may be optimized) [25], evaporators and condensers.

Thermodynamic optimization is a considerably wider and more diverse field, with fundamental developments in cryogenics, heat transfer, storage systems, power plants and refrigeration plants, heat pumps, distillation and other separation methods. The field was reviewed first in the book [26] and most recently in references [lo] and [16]. One can take another look at these developments to discover that each optimum is another manifestation of the equipartition and allocation trends illustrated in this section.

3. EQUIPARTITION IN NATURE

Nature offers an even more impressive list of examples of equipartition, or optimal allocation. 111 this section we focus on two areas with multiple forms of geometric similarity, which for many years have intrigued researchers: the morphology of river basins and organs of living systems (e.g. lungs, vascularized tissues). A property of these forms is that their features can be correlated (summarized) as formulas known as power laws in geomorphology [27-301 and allometric laws in physiology [31-331. The power law is also a characteristic of fractal geometry-indeed, fractal geometry is now a popular description of such observations.

It is important to recognize from the outset that improvements in how we record and store empirical observations do not represent new theory, i.e. an enhancement of our predictive powers. This point was stressed two decades ago by Bloom [29] with reference to the tree network of the river basin:

“The techniques of quantitative fluvial geomorphology give an excellent ‘description’ of drainage networks but no ‘explanation’.”

Bloom’s remark applies with peculiar force to the allometric laws of physiology and the images generated by repeated algorithms in fractal geometry. Fractal algorithms are descriptive, not predictive. The challenge is to predict the algorithm postulated by the mathematician.

3.1. River networks

River basins all over the world reveal the dendritic structure (‘dendron’ = tree, in Greek) in which channels merge into larger channels downstream. The tree network connects a finite rainfall area with one point- the lowest point, which serves as sink. The network is without loops, i.e. its links do not cross. In line with the problem stated by Kauffman at the end of Section 2,

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the tree structure is always the same. whether found in nature, after a long history, or created in the laboratory (.&we 21.

Rodriguez-Iturbe et al. [35] showed recently that the most important structural characteristics of the drainage network can be derived if the network is optimized according to three ‘postulates’:

1) the energy dissipated in any link of the network is minimal:

2) the dissipation of energy per unit area is distributed uniformly over the network:

3) the energy dissipated by the network as a whole is minimal.

Figure 2. The development of an artificial river drainage basin over a 15.2 x 9.1 m2 rainfall erosion area [341.

In the graphic examples given in reference [35] the network was optimized numerically in a rectangular region. The procedure was initiated by assuming an initial network configuration, which was then perturbed randomly. Subsequent studies reinforced the connection between postulates (l)-(3) and the tree structure [36]: suggesting that these optimization principles are universal [37]. Noteworthy in this ensemble is postulate (a), which is an example of the equipartition principle illustrated in section 2.

Even more important from a theoretical point of view is the ‘zeroth-order postulate’, which is never mentioned in river morphology and physiology:

0) a network of channels ‘exists’ at the start of the random optimization process.

Why should ‘channel flow’ occur naturally over certain lines of a finite area on which rain falls at every point,? In other words, why should the flow become ‘organized’ (bundled) into channels (or tubes in the lung) when, originally, this flow is clearly ‘diffused’. The flow originates from and bathes every point of the given area. The diffused flow that fills the interstitial spaces between the smallest links of the network is overlooked entirely by simulations based on postulates (O)-(3). In section 4 we will show that it is possible to predict features (0). (I), (2) and (3) by relying entirely on the constructal approach to minimizing the resistance to

volume-to-point flow. Of course, this does not mean that a single formula predicts every geometrical detail in the entire world, especially in the case of rivers. Lack of certainty continues to rule the local and temporal conditions of the river basin, as stated by Kauffman in the paragraph quoted in section 1.

3.2. Turbulent flow

Another example of natural equipartition is found at the transition between laminar flow and turbulent flow. To see why equipartition can be expected in this case, recall that tree flows such as figure 2 have the property to cover a given space in two regimes, not one. Each flow path consists of a portion with high resistance (viscous diffusion) placed at the smallest scale. and several portions with low resistance (streams) at larger scales. Turbulent flows are known to combine the same two regimes--viscous diffusion and streams (eddies)--- therefore they should exhibit ‘structure’ in the same way as any other multi-regime flow that covers a given volume.

This view of the origin of turbulent flow was advocated in engineering by Bejan [26] and Mikic [38]. As illustration, consider the growth of a shear layer in a viscid fluid. In figure 3 the observer rides on the shear layer as the fluid reservoirs slide to the right and to the left with the velocity U,/2. Immediately following the t = 0 start of the shearing motion, the horizontal midplane is occupied by a one-dimensional laminar shear layer. The thickness of this shear layer D(t) increases as t1j2, as shown in detail (a) of figure 3. The size of D/2 is known from the similarity solution to the problem of viscous diffusion in the vicinity of an impulsively started wall [39]

The question is one of optimal access (resistance minimization), or fastest mixing, or the fastest route to uniformity: how can the shear flow spread over the given space in the shortest time possible? The laminar regime is the most effective regime only in the beginning, when the derivative dD/dt is large. As time increases, the viscous diffusion swelling of the shear region slows down monotonically, because dD/dt decreases as te1j2. It is essential that the shear flow has access to growth via a second regime known as eddy formation, or buckling and roll-up. The growth (transversal) velocity in this second regime is constant, dD/dt N Use/2: because the peripheral speed of the eddy is set by the two fluid reservoirs. This second regime is indicated by the straight line in figure 3a.

How far can the shear layer grow by rolling up once’? The spacing between two adjacent eddies, i.e. the diameter of the first roll formed by the layer of thickness D is dictated by the buckling wavelength.

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I /--- strprns f----y I I t

D2 Dl I i I - ’ \ / \ /

1 I I I

eddy formation

_ _ Eq. (15), viscous diffusion

0 e 0 t

Figure 3. The viscous-diffusion growth of a two-dimensional shear layer, and the stepwise expansion that follows.

which is practically equal to the wavelength for neutral inviscid (Kelvin-Helmholtz) instability [40, 411:

Xgw2D (W

This means that the shear-layer thickness increases from D to 20 during a roll-up time of order:

XB XB - - 40 u,/a - u, (17)

The rate of lateral expansion through eddy formation is (20 - D)/tB - U,/4. whereas the viscous swelling rate indicated by equation (15) is dD/dt N 2(v/t)‘/2 N G/D. The most rapid expansion occurs when the viscous regime is followed by eddy formation at the time when the viscous growth rate is just outpaced by the eddy growth rate, i.e. when DU,lv N 32, or, in an order of magnitude sense:

Equation (18) is the ‘local Reynolds number criterion’, which was found to predict and correlate the transition to turbulence in all known configurations (for a review, see reference [41]). The same criterion is obtained by equating the time of viscous diffusion over the distance D [equation (15)] with the time of eddy travel over the same distance [equation (17)]. This equipartition of time principle is also behind Mikic et al.;s [42] work on fine-tuning sustained flow oscillations in the forced convection cooling of electronics.

In figure 3 the size of the first eddy is labeled DI. In the analysis of equations (15))(18) this length scale was called 20 (see also figure 3a). The order of magnitude of D1 is given by the local Reynolds number criterion (18): the smallest eddy in a turbulent flow field-the size of the smallest structural system- is such that its Reynolds number is of order 102. Beyond the first eddy formation event (Dl), the flow continues to expand in steps, by rolling and forming eddies. Each step leads to the doubling of the mixing region, cf. equation (16). Viscous diffusion does not have

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time to act-to competePover larger distances such as Dz, D3, etc. The outer boundaries of the mixing region are extended now by the second flow regime: streams (eddies). In a frame attached to one of the reservoirs this sequence of events is represented by the structure sketched in figure 4. The shear layer begins with a laminar tip (figure 4> top), and is followed by a sequence of geometrically similar building blocks, the size of which doubles in the downstream direction (figure 4, bottom). In the actual flow this growth is the result of eddy-to-eddy interaction over multiple scales. and the result of scale covariance. as we discuss in section 5.

Figure 4. The shear flow of figure 4 in a frame attached to the lower fluid reservoir. Top: stable laminar length in the beginning of the shear layer [26]. Bottom: the constant-angle growth of the shear layer, as the repeated coalescence of geometrically similar building blocks [41].

The structures of figures 3 and 4 are supported by numerous independent studies, for example, the experiments of Brown and Roshko [43] and the theory of Gibson [44]. These images----the repeated aggregation of geometrically similar building blocks-appeared in engineering at about the same time as fractal geometry [5], and when they were certainly unrelated to the work that was being done in mathematics. One

important difference is that in figures 3 and 4 the structures are theoretical and develop in time, from small to large; as in the constructal theory described in the next section. The constructal approach merits further consideration in turbulence research, where the constrained optimization may predict other features of turbulent flow. not just the transition and the growth rate of the mixing region. This will move the turbulence question from the list of examples of natural organization to the field of engineering applications.

4. CONSTRUCTAL THEORY

In this section we review a new theory that predicts the spatial organization of a wide variety of natural phenomena, both living and not living [ll. 45. 461. The theory deserves the ‘constructal’ name for the reasons given at the end of section 4.2. It is based on seeking the optimal access, or flow with minimum resistance between one point and a.11 ‘infinite’ member of points (finite area: or finite volume). We will show that the theory that flourishes in answer to the optimal- access question is strikingly simple. It brings under the same deterministic umbrella a wide variety of natural phenomena, the ‘tree’ structure which so far has been assumed to lie beyond the powers of determinism. The examples are literally everywhere: trees? roots. leaves: river deltas. river basins. lightning. streets; and the pulmonary. nervous and vascular systems.

4.1. How to cool a finite-size space that generates heat volumetrically

The theory was first formulated in heat t,ransfer engineering. with application to the optimal cooling of electronic components and packages. In such devices the objective is to install a maximum amount of electronics (i.e. heat generation rate) in a given volume such that the maximum temperature does not exceed a certain level. The technological frontier is being pushed toward smaller package dimensions. There comes a point where miniaturization makes convection cooling impractical. because the cooling channels would take too much space. The only option left is to channel the generated heat by conduction. along paths of very high thermal conductivity (&).

Conduction paths too take space: designs with fewer and smaller paths are better suited for the miniaturization evolution. The fundamental optimal- access problem in heat conduction is [ll].

Consider a finite-size volume in which heat is being generated at every point, and which is cooled through a small patch (heat sink) located on its boundary. A finite amount of high-conductivity (Ic,) material is available. Determine the optimal distribution of L+ material through the given volume such that the highest temperature is minimized.

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The most basic function-the purpose-of any portion of the conducting path k, is to be ‘in touch’ with the material that generates heat volumetrically. This material fills the volume (V), and its thermal conductivity is low (Ica). The optimal-access problem reduces to the geometric problem of allocating conducting path length to volume of Ica material, or vice versa. A key observation is this: the allocation cannot be made at infinitesimally small scales throughout V, because the /c, paths must be of finite length so 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 subsystem (volume element size) at a time; and

ii) to optimize the manner in which the volume elements are assembled and their k+ paths are connected.

The discovery made in reference [ll] is purely geometric: any finite-size portion of the heat generating volume can have its ‘shape’ optimized such that its overall thermal resistance is minimal. Optimized volume elements are then assembled into a larger volume the shape of which is also optimized. This assembly and geometric optimization sequence is repeated in steps, from the smallest volume element to the largest assembly, until the given volume is covered. One of the features of the structure that emerges-a byproduct of the construction-is a network of 5 paths that is shaped as a tree. All the features of the structure of the (/co, IcP) composite material that covers the volume, and all the features of the associate tree of k, paths are the result of a purely theoretical, deterministic process guided by a single principle.

The simplest example of shape optimization is pro- vided by the two-dimensional volume element represented by the rectangle Ha x LO in figure 5. The area HO x LO is fixed but its shape may vary. The amount of k, material allocated to HoLo is fixed, and is represented by the manufacturing parameter 40 = Do/Ho. The heat current (qo = q”‘HoLoW) generated by this volume element is collected by a blade (DO, LO) of high-conductivity material, and taken out of the volume through the point n/ro. In figure 5, the rest of the HO x LO boundary is adiabatic, and W is the dimension perpendicular to the plane HO x LO. The volumetric heat generation rate q “’ is uniform. The ‘hot spot’ occurs at the point P, which is the farthest from the heat sink Ms. Since the heat current qo is fixed, the minimization of the thermal resistance of the volume element is equivalent to the minimization of the temperature drop from P to MO. The minimal-resistance design is represented by a special shape (geometric aspect ratio) of the HO x LO rectangle:

(55),,, = 2 (%!!$)li? (19)

Y

I

Hd21p k,. 9”’ I

c7 0 9”’ H&W

kg, q”’

I -Hd2

Figure 5. Elemental conduction volume with heat generation and a high-conductivity path along its axis of symmetry [l 11.

In figure 5 the maximum temperature difference (A T,s) occurs between one of the right-side corners and the origin (0,O). When the shape of the elemental volume is optimal, the minimized AL!‘,0 is divided equally between vertical conduction through the ko material and horizontal conduction along the k, insert. The ‘equipartition’ of temperature difference (or thermal resistance) is equivalent to the geometric optimization of the elemental system.

If the given volume is greater than the HoLoW element of figure 5, then we may be able to cover it with an optimal assembly of HoLoW elements, as shown in figure 6. The shape (HI/LI) or the number of constituents (no = HILI/HoLo) may vary. The minimization of the temperature drop from the corner P to the heat sink 441 pinpoints the optimal shape (or size ni) of the assembly: this optimization step is completely analogous to what we saw at the smallest scale (figure 5). Minimal resistance is achieved through optimal shape, or optimal growth.

When the volume fraction of k, material built into the first assembly is fixed (c&), the design has a second degree of freedom: the ratio Dl/Do, i.e. the relative thickness of the new k, path required by the first assembly. In the end, the double minimization of thermal resistance determines every geometric feature of the first assembly: its shape, or the number of

Y

H,/2 I +I Ho t-

0 0 9”’ H,L,W

-H,/2

Figure 6. The first construct: a number of optimized elemental volumes (figure 5) are connected to a central high-conductivity path [l 11.

173


constituents, and the thicknesses of the ‘nerves‘ of k, material.

This optimization can be continued with assemblies of stepwise larger sizes until the k, material is spread over the entire volume V. For example, in figure 7we see the optimized fourth assembly and its distribution of k, material. The square shape of this particular assembly is an ‘optimization result’, not an arbitrary choice made by the graphic artist. Again, every single geometric feature of the cooling scheme shown in Jigure 7 is the result of analysis. The fact that at higher orders of assembly the optimized high-conductivity paths exhibit the ‘bifurcated’ (dichotomous) structure of natural tree networks is also a result, i.e. not an assumption. At no time did the designer borrow from nature.

P

M4

Figure 7. The optimized fourth construct [l 11.

The evolution from assemblies with many elements (e.g. figure S) to higher-order assemblies with just two constituents (e.g. figure 7 and, later, figure 8) is due to having assumed that the same high-conductivity material (kp) is installed in the main fiber of each new assembly starting with i = 1. In other words, pairing (dichotomy) is a result of geometric optimization in a volume that is covered by inserts of a single (high) conductivity. In general, a different high-conductivity material (k,,,,i > 1) can be used to make the main nerve of each new assembly. An example of this kind will be constructed in section 4.3, where each new main street is characterized by its own, higher speed.

The geometric optimization principle illustrated in figures 5-7 has far reaching implications.

1) Technologically, it is possible to construct in a few simple geometric steps the optimal network for channeling a current that is generated volumetrically.

174

This finding is extremely important in practice: if the designer were to start with the given volume V, he or she would have to guess (postulate) an existing network; and then optimize (numerically and randomly) a proh_lbltively large number of parameters, as done in the numerical simulations of river drainage basins [35, 361.

2) Theoretically, it means that at the basis of the tree architecture of many living and nonliving systems rests only one design principle: the minimization of the resistance to flow between one point and a finite volume (an infinite number of points) when the flow rate is imposed. Reliance on such a universal design principle makes the tree network structure ‘deterministic’, contrary to the prevailing doctrine.

The geometrically optimal construction started in figures 5-7 can be continued to a higher order of assembly, until the structured composite (ko, kp) covers the given space. One interesting feature in this limit is that the construction settles into a pattern of pairing (or bifurcation, from the reverse point of view), in which the integer 2 is a result of geometric optimization. For example, figure 8 shows this pairing and size doubling pattern. which is also visible starting with the fourth assembly (i 2 4) in table I. If the shaded corner of figure 8 is one of the optimized fourth assemblies of figure 7, then the large square domain of figure 8 represents the optimized eighth assembly. All the fibers that are visible in figure 8 have the same conductivity (Ic,), hence the dichotomous structure [ll]. In table I we used ATai,min for the minimized value of the maximum temperature difference that occurs across the optimized assembly

P

Figure 8. The optimized tree network formed by the higher- order constructs of the conduction composite (lo, k,) [l 1 I.


TABLEAU I / TABLE I The main features of the first six optimized assemblies of the conduction heat tree [l 11

Assembly level Assembly height factor Assembly Path width factor Path length factor ATaz,min i K/H,-1 shape” LA/D,-1 Li/Li-1 q”‘H,2/ko

1 0 r

4 1

1 Do k,

( >

l/2

Ho ko >> 1 R ( Do k, -

Ho ko >

L/2 >> 1 1

4 2 1 S ~ = 31/z 2.31 1 j/&j 2@2&,

3 2 R 2 1 ;(I+&) 2(1+&J

4 1 S 2 2 :(I+&) 2(1+&)

5 2 R 2 1 ;(l+&) 2(1+&)

6 1 s 2 2

a S = square; R = rectangle (aspect ratio = 2); r = rectangle (aspect ration < 1).

(e.g. between the origin and the upper-right corner in figure 6). The number 4% is the fraction of the volume occupied by k, material in each assembly.

To appreciate how much is new in this geometric construction, it is important to note that one portion of the network pattern (namely, the portion formed by the higher-order assemblies, figure 8) is not new. It was first proposed in physiology as a three-dimensional ‘heuristic model’ for the vascular system [47], where it was known ‘empirically’ that each tube is followed by two smaller tubes, i.e. each tube undergoes bifurcation. It was also known that the tube diameter must decrease by a constant factor (2-l’“) during each bifurcation: this result had been derived based on flow resistance minimization [3] coupled with the assumption that each tube is continued by ‘two’ tubes. The description of these geometric constructions was made popular through the advent of fractal geometry: in fact, a two- dimensional version of Cohn’s branching fluid network appeared in the books of Mandelbrot [5] and Prigogine [48], where it is presented heuristically as a ‘model of the lung’. These authors did not say anything predictive (i.e. deterministic) on the way to the algorithm assumed in making the drawing.

4.2. The time direction: from small to large

The time direction in which the space is filled, or the network constructed, is the key to being able to predict the optimal structure in a purely theoretical, deterministic manner. In the geometric approach illustrated in figures 5-8 and table I the time direction is from small to large, with geometric

optimization (and no empiricism) employed at every volume scale. The approach used in physiology proceeds in the opposite direction, from large to small, as each tube is broken down into smaller tubes. This sequence is said to be carried out ad infinitum in fractal geometry [5], even though every natural tree network exhibits very clearly a critical volume scale below which ‘branches’ do not exist.

It was shown in reference [ll] that if the same conduction problem is pursued in the opposite direction, from large to small, it is not possible to anticipate theoretically any of the geometric features of the optimal structure. In other words, to apply the physiology method to the heat generating volume is to begin with the assumption that the volume is covered already by an arbitrary network of high-conductivity paths. The thicknesses, lengths, relative positions, and numbers of branches at each branching stage are unknown. All these numerous parameters represent degrees of freedom in the design. The postulated network proceeds toward smaller scales, indefinitely. The only result that is produced by the classical method of minimizing the overall thermal resistance using Lagrange multipliers is [ll]

(Djiv,),,, = constant (20)

where Nj is the unspecified number of k, paths of thickness Dj. Now if we ‘assume that in going from large to small each path undergoes bifurcation (N,+i/Nj = 2), then equation (20) means that each path width must shrink by the same factor from one branching stage to the next:

(21)

175


Equation (21) is the ‘conduction’ analog of the tube diameter reduction factor (2-l’“) known in physiology [3]. Equation (21) also agrees with the large-size limit (i > 3) of the theoretical D, results summarized in table 1. It is important to keep in mind that, unlike in table 1, the approach that led to equation (21) is not predictive. because in this approach bifurcation was assumed, not predicted.

The fractal algorithms that are used to generate patterns that look like natural tree networks align them- selves with the time direction favored in physiology: from large to small. It is true, of course, that the postulated algorithm can be executed in both directions, from the largest scale to the smallest; and from the smallest to the largest. As a descriptive aid for ‘natural’ phenomena, however, the fractal description represents a clear choice, namely, from the largest scale all the way to size zero in an infinite number of steps. The word ‘fractal’ has the concept of time built in it: the act of breaking something evolves in time from large pieces to smaller pieces.

In conclusion what makes the access-optimization construction of figures 5-8 deterministic is the time arrow, from small to large. It invites us to rethink the way in which we describe systems that are organized naturally. Confluence, yes; branching, no. Coalescence, yes; bifurcation, no. Construction! yes; fracturing, no. To emphasize this direction and the difference between it and the prevailing point of view, the theory of reference [ 1 l] was named ‘constructal’.

4.3. Street pattern formation

The constructal approach becomes even more trans- parent if we think literally about optimizing the access between a finite-size entity and one point. Consider the problem of minimizing the travel time between a finite- size area (A) and a point (A&) situated on the boundary of A [45]. The density of travellers, i.e. the rate at which people travel from each point of A to A4 is uniform, ti” [people.m-2 .s-i].

If the people can travel at only one speed (e.g. by walking) then the optimal travel paths constitute a bundle of rays (straight lines) drawn between A4 and each point situated inside A. If at least two modes of locomotion are available. for example, walking with speed VO and riding in a faster vehicle (VI). then the area A becomes a ‘composite material’ with two distinct ‘conductivities’ in a way that is analogous to the (ka, k,) composite of figures 5-8. The problem is one of allocating optimally a VI length (street) to a small but finite area or group of inhabitants. This group would have to walk first in order to reach the street: because the slower mode (Vo in figure 9, or I;0 in figure 5: or viscous diffusion in figure 3) is assigned to cover every point of the volume element of the smallest size.

The size of the smallest area element (A, in figure 9) is fixed by the living conditions (e.g. property) of the

Y t I -4

it- --~~ L, --- tl Figure 9. The smallest area element, Al, and the street segment allocated to it [45].

individuals who have access to the VI path. Although the elemental size A1 is fixed, A1 = Hi&, its shape or aspect ratio (HiILl) is not. The time of travel between the farthest corner (Li,Hi/2) and the origin (0,O) is:

t,=+g 0

This time is minimal when the shape of A1 is

(22)

The same conclusion is reached by minimizing the average travel time, that is, the time from the arbitrary point P(x,y) to (0, 0) averaged over all the points of type P. In other words, what is good for the most disadvantaged inhabitant (the farthest corner of Al) is good for the community as a whole. This conclusion sheds light on the natural origin of spatial organization in all living groups, from bacterial colonies to our own social structures.

When the A1 shape is optimal, the two terms on the right side of equation (22) are equal to each other. Once again, we find that the ‘equipartition’ principle rules the optimized physical configuration: the time of travel is divided equally between travelling at VO and travelling at VI.

Larger areas can be covered with successive assemblies of smaller areas. The constructal sequence is analogous to the one illustrated for the heat tree in figures 5 8. For example. the optimal shape of the second area element (A2 = Hz&) is given by another simple result [45] :

Hz

( >

Vl - =- L2 opt v2 (24)

where V2 is the speed of travel along the new street. relative to which the VI streets are tributaries. The number of elemental areas AI assembled into the

176


optimally shaped AZ is assumed large (figure lo), and is given by

A2 vz n2=---=2- Al Kl

(25)

The construction continues with increasingly larger assemblies in which every new geometric feature is determined by minimizing the time of travel out of (or into) each assembly. One byproduct of this construction is the pattern of the emerging access routes: a tree network, which is completely deterministic. When i 2 3, the optimized geometry is represented by the formulas:

(26)

Figure 10. The second area element, AZ, as a construct of area elements of size A1 [45].

As in figure 10, these results are based on the assumption that each assembly contains a large number of constituents, which means that the speeds increase sensibly from one assembly to the next [cf. equation (28)] and that the optimal shape of each area assembly is slender [cf. equation (26)]. Figure 11 illustrates the emerging street pattern when VI/VO = 5 (hence n2 = 20) and Vi/V,-1 = 2 for i>2 (hence H%/Li = I/2 for i 2 2, and n2 = 16 for i 2 3). On the other hand, if the strength of the inequality sign (VO < VI < V2 < . . .) decreases as the area assemblies become larger, the number of constituents in each new assembly decreases, and the optimal shapes become closer to square. The construction is approximate because at each level of assembly the number ni of equation (28) is rounded off to the closest even integer.

A

I I I I I I I I 1

Figure 11. The optimized geometry of the first three area constructs when VI/V0 = 5 and V2/VI = V3/V2 = 2.

Figures 9-11 and equations (22)-(29) were based on the simplifying assumption that all the streets intersect at right angles [45]. It can be shown that in each assembly there exists an optimal angle between each new street and its tributaries, in the same way that an optimal angle exists between each tube and its branches in the pulmonary and/or circulatory systems [2, 491. This optimal angle was not included in this discussion because it would have detracted from the degree of freedom that was highlighted-the shape (form) of each building block-which is new and solely responsible for the theoretical construction of the tree pattern.

If the algorithm of equations (26)-(29) is repeated ad infinitum in both directions with constant ratio V,/Vi-l then the resulting pattern of streets would be a fractal. The present pattern is not fractal for two reasons. First, the details of the geometry are not the same at all scales. For example, the number of constituents (Ai-I) in a larger construct (Ai) varies with i, because it depends on the velocity ratios at the preceding scales [i,i-l,i-2,...; see equation (ZS)]. This property is called scale covariance (section 5), in contrast to the scale invariance that characterizes fractals. The second reason is that unlike in fractal geometry where the infinity of points of A is touched by repeating the algorithm infinitely down to the area size A1 = 0, in constructions such as figure 10 the algorithm starts from a finite-size area (AI). The infinity of points of area A is connected to the street network by placing and using the slowest speed (VO) all over the smallest areas (AI), and optimizing the shape of AI.

An even simpler way to illustrate the area shape optimization and street pattern formation is to assume that (unlike in figure 9) the street VI serves as one of the Lr-long boundaries of the area element AI. This is

177


what farmers do on their properties: they use streets as boundaries. The VO traffic generated by A1 is collected by the VI street, but only from the side of the street that faces Al. The other side of the street is fenced. This choice leads to HI. opt = (AI V~/VI)‘/~ and

Hl VI (-> =- Ll opt Vl

(36)

which differs slightly from equation (23). Starting with the second area element (i > 2) the geometric optimization continues in the way shown in figure 10. by placing the new street on the centerline of the new assembly. The results obey the recurrence relations listed in equations (26)-(29): the advantage is that now these relations cover the case i = 1 as well.

The formulas for travel time minimization and street pattern formation; equations (23)-(29), can be generalized to any number of levels of assembly starting with i = 1. We define the time of travel 0, along the main path of each area element.

Bt=$ (i=O.l....) (31) z

with O0 1

= Lo/h and Lo = G HI. In this new terminology, the results of minimizing the travel time at each area level can be expressed as follows: optimal width of the first area element

optimal aspect ratio

optimal area ratio

optimal (minimized) mean travel time

t, = 0, = 2”-‘Oo (i> 1)

optimal (minimized) maximum travel time

ti = 2 8, = 2” l9(, (i > 1)

optimal variance of the travel time distribution

(32)

(33)

(34)

(35)

(36)

(37)

The maximum travel time t, is defined as in equation (22) and figure 9: it is the time of travel between the origin (the end of the main path) and one of the most distant corners of the area element. The mean travel time Et is calculated by averaging the travel times

associated with all the points of an area element (e.g. the travel from point P to the origin in figure 9). Fi- nally, the variance of the travel time distribution is - uf = (tz - tiy.

The review published recently by Stanley et al. [7] shows that urban growth is an active field in which the universality of structures such as figure 11 is recognized. In this section we illustrated the principle behind the two-dimensional area-to-point access, because it is the simplest. The corresponding problem in three dimensions can be stated easily: minimize the time of travel from all the points P of a volume V to one common destination point M, subject to the constraint that the travelling population rate is fixed. One application is the sizing and shaping of the floor plan in a multistory building, and the selection and placement of the optimal number of elevator shafts and staircases. The floor plan optimization proceeds according to the method illustrated based on figure 9; where Vu corresponds to the travel through (movement, out of) the rooms, and VI is the travel along the corridor. The unknown in this first building block is the shape of the floor area AI. Next, we stack a certain number (n2) of A1 elements on the vertical, such that the next path (V2) is vertical and accounts for the elevator or the staircase. The optimization sequence may be taken to a third assembly (or even higher order assemblies) if the towers optimized in this fashion must be integrated into a larger building.

The access optimization theory can be extended to cases where the travelling population is distributed unevenly, for example, to highways, railroads, telecom- munications, and air routes (e.g. the organization of such comicctions into hubs, or centrals). Another clear application is in operations research and manufacturing. where the invention of the first auto assembly line is analogous to the appearance of the first street (figure 9).

5. CONCLUSION: A THEORY OF MACROSCOPIC ORGANIZATION IN NATURE?

The area-to-point and volume-to-point access routes reviewed in section 4.3 are analogous to the minimal- resistance constructs for heat transfer (section 4.1) and the minimal-dissipation networks for river basins (section 3.1). These tree networks share the same deterministic umbrella with other forms of natural organization that in the past have been rationalized by minimizing the time of travel: light travels in straight lines, the incident and reflected rays make equal angles with the mirror surface (Heron of Alexandria), and the refracted ray is bent in an optimal way when passing from one medium into another (Fermat). Additional applications of constructal theory are reviewed in

178


reference [46], for example, the dendritic crystal, the pattern of cracks in a shrinking solid, and the proportionality between depth and width in rivers of all sizes.

There is also an important connection between constructal theory and fractal geometry. The classical fractal approach postulates a priori self-similarity, that is, scale invariance of the geometry of the objects, or ‘invariance in the zoom direction’. Constructal theory establishes instead relations between successive scales (‘scale covariance’, not invariance) as a ‘deterministic result’ of constrained optimization. The quotations in this paragraph are notions introduced by Nottale [50] in his book ‘Fractal Space-Time and Microphysics’. Nottale writes:

“One of the possible ways to understand fractals would be to look at the fractal behavior as the result of an optimization process... Such a combina- tion...may come from a process of optimization under constraint, or more generally of optimization of several quantities sometimes apparently contradictory

%ume .” or example) maximizing surface while minimizing

. .

Constructal theory is an example of constrained optimization, as envisioned by Nottale. It takes one additional step to realize that this optimization may not yield self-similarity (scale invariance) but rather the more general property of scale covariance.

Acknowledgment

The authors’ work reviewed in this paper was supported by CNRS and the National Science Foundation. One of the reviewers of the original manuscript made extremely pertinent comments on the deterministic and teleologic aspects expressed or implied in our paper. We thank him for this and for drawing our attention to possible misunderstandings that may result from some of our statements. Hence the following clarification.

One of the aims of this work is indeed to inject a ‘dose of determinism’ in some problems of organization in nature, which are widely regarded as being governed by probabilism. This determinism is introduced by formulating the problems as optimization problems, i.e. by recognizing and then meeting an ‘objective’, in analogy with engineering problems in which man sets the objective. The act of formulating such an objective (minimizing resistance, or finding the shortest path, etc.) does not represent belief in metaphysical order. It is a recognized and useful property that light travels in a heterogeneous medium according to the shortest time; there is no need to resort to metaphysics to accept this thought as ‘law’. The determinism that sprigs out of the present work resembles the search for order in chaos.

We would repeat that the natural geometrical forms addressed in this paper (e.g. the river basin) are not determined entirely by our approach. Only the overall structure is deterministic, not every fine detail, which

remains influenced by local and temporal conditions that are unknown. Determinism prevails at certain scales, which were shown to be expressible by predictive laws, while lack of certainty remains a characteristic of other scales.

ill

PI

[31

[41

151

KY

[71

P31

M

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Documents

Equipartition, optimal allocation, and the constructal approach to predicting organization in nature