Constructal tree-shaped paths for conduction and convection

INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2003; 27:283–299 (DOI: 10.1002/er.875)

Constructal tree-shaped paths for conduction and convection

Adrian Bejann,y

Department of Mechanical Engineering and Materials Science, Duke University, Box 90300, Durham,

NC 27708-0300, U.S.A

SUMMARY

This lecture reviews a series of recent results based on the geometric minimization of the resistance to flowbetween one point (source, sink) and a volume or an area (an infinity of points). Optimization is achievedby varying the geometric features of the flow path subject to volume constraints. The method is outlined byusing the problem of steady volume-point conduction. Optimized first is the smallest elemental volume,which is characterized by volumetric heat generation in a low-conductivity medium, and one-dimensionalconduction through a high-conductivity ‘channel’. Progressively larger volumes are covered by assembliesof previously optimized constructs. Tree-shaped flow structures spring out of this objective and constraintsprinciple. Analogous problems of fluid flow, and combined heat and fluid flow (convection, trees of fins)are also discussed. The occurrence of similar tree structures in nature may be reasoned based on the sameprinciple (constructal theory) (Bejan, 2000). Copyright # 2003 John Wiley & Sons, Ltd.

KEY WORDS: constructal theory; dendrites; fractal; topology; architecture; optimization; design;complexity; fins; flow resistance minimization

1. CONSTRUCTAL TREES FOR VOLUME-POINT AND AREA-POINT FLOW

Traditionally, the discipline of heat transfer was concerned with the description (measurements,or predictions) of the relationship between the heat interaction between two or more entities,and the temperatures, materials and geometries of these entities. The most studied entities aretwo bodies, or two surfaces, or a surface and an external or internal stream of fluid (Bejan, 1993,p. 6). Most of the fundamentals that we possess today refer to configurations where heat flowsfrom a finite-size entity to another finite-size entity, where ‘finite size’ means a line, surface orvolume, i.e., an infinity of points. Classical exceptions are the solutions for conduction(diffusion) around concentrated heat sources (Bejan, 1993, pp. 177–184).

In this paper, I review a newly emerging body of work that describes the flow between a finite-size volume (or area) and a single point (source, or sink). This work goes well beyond thedescription (calculation) of the flow resistance associated with the flow path: the new andprimary focus is on optimizing geometrically the flow path, so that the global volume-point

Received 10 March 2001Accepted 8 June 2001Copyright # 2003 John Wiley & Sons, Ltd.

nCorrespondence to: Adrian Bejan, Department of Mehcanical Engineering and Materials Science, Duke University,Box 90300, Durham, NC 27708-0300, U.S.A.yE-mail: [email protected]

resistance is minimum. The flow path that results from geometric optimization is shaped as atree.

What flows along the tree links is not nearly as important as how the geometric form ‘‘tree’’ isgenerated by the geometric optimization principle. This is important: geometric form is aphenomenon that is generated (i.e. accounted for) by a principle. The examples given in thispaper refer to trees for heat flow, fluid flow, and combined heat and fluid flow. Trees forelectricity (e.g. lightning, circuitry), people and goods (e.g. streets, highways), and communica-tions are reviewed in a new book (Bejan, 2000). The thought that the geometric optimizationprinciple accounts not only for engineered flows but also for the billions of tree-shaped flows ofnature was named constructal theory (Bejan, 1997a,b).

Tree flows emerged as solutions to the fundamental heat transfer problem of how to connect(with minimal thermal resistance) a finite-size heat-generating volume to a concentrated heatsink. This problem has many and diverse applications that go beyond the field of heat transfer,for example in physiology, river morphology, and electrical engineering. In heat transfer, wherethis problem was first posed (Bejan, 1997a,b), the main application is in miniaturized deviceswhere cooling can be provided only by conduction. The trend toward smaller sizes andconduction-cooling at the smallest scales is illustrated in recent reviews of the field (Aung, 1988;Peterson and Ortega, 1990; Kakac et al., 1994).

The geometric optimization of assemblies of heat-generating components is also anestablished method in the cooling of electronics (Bejan, 1984; Bar-Cohen and Rohsenow,1984; Knight et al., 1991; Anand et al., 1992). The most fundamental results of geometricoptimization of heat transfer, in electronics packages, heat exchangers and thermal design ingeneral, are reviewed in Bejan (1995).

In the constructal optimization of the volume-to-point path for heat flow (Bejan, 1997a,b) itwas shown that a volume subsystem of any size can have its external shape and internal detailsoptimized such that its own volume-to-point resistance is minimal. This principle is repeated inthe optimization of volumes of increasingly larger scales, where each new volume is an assemblyof previously optimized smaller volumes. The construction spreads, as the assemblies coverlarger spaces.

To the engineer, the result of this process is the design with minimal resistance. To thephysicist, the end result is geometrical form (shape, structure) deduced from a singledeterministic principle. The visible part of the emerging structure is the tree network formedby the low-resistance portions of the medium. The invisible part}the shapeless, low-conductivity flow regime}covers the interstitial spaces and touches every point of the volume.The invisible part is as important as the visible structure. The construction has a definitedirection in time: from small to large. Shapelessness (diffusion) comes first, and structure(channels) comes later.

2. TREES FOR HEAT CONDUCTION

To see what is new in volume-to-point flow, we review some of the steps and conclusions of theconstructal approach. Unlike in Bejan (1997a,b), the present analysis emphasizes the growth

aspect of the construction (Bejan and Dan, 1999a). The heat-flow geometry is two dimensional.The low-conductivity material (k0) generates heat volumetrically at the rate q000, which is

Copyright # 2003 John Wiley & Sons, Ltd. Int. J. Energy Res. 2003; 27:283–299

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assumed uniform. A small (fixed) amount of high-conductivity material (kp) is to be distributedthrough the kp material.

The construction begins with the smallest volume scale (the elemental system), which isrepresented by the rectangular area A0=H0L0. The start of this sequence of volume sizes isshown at the top of Figure 1. The A0 size is known and fixed; for example, in the conductioncooling of an electronic material, A0 is the smallest size that is allowed by manufacturing andelectrical design constraints. The A0 system is ‘elemental’ because it has only one insert of high-conductivity material. This blade has the thickness D0, and is positioned on the long axis of theH0�L0 rectangle. The heat current q000A0 is guided out of A0 through the left end of the D0

channel, which is the heat sink. The hot spots occur in the right-hand corners.The global volume-to-point resistance is the ratio DT0=ðq000A0Þ; where DT0 is the temperature

difference between the hot spot and the heat sink. This measure is ‘global’ because the heatcurrent q000A0 is integrated over the system A0, and the maximum temperature difference DT0 isthe excess temperature of the hot spot, or hot spots of the entire system: the location of the hotspots is not an issue, as long as they reside inside the system. The global resistance refers to theentire system and its ability to accommodate the volume-to-point flow without violating therequirement that the hot spot not exceed a certain temperature ceiling.

Figure 1. Constructal growth: spatial growth through the minimization of thermal resistance (Bejan andDan, 1999a).


CONSTRUCTAL TREE-SHAPED PATHS 285

The composite material that fills A0 is characterized by two dimensionless numbers, theconductivity ratio *kk ¼ kp=k0; and the volume fraction of high-conductivity material, f0=D0L0/H0L0=D0/H0. It was shown that when *kk41 and f051 the volume-to-point resistance is givenby the two-term expression (Bejan, 1997a,b)

DT0k0q000A0

¼1

8

H0

L0þ

1

2 *kkf0

L0H0

ð1Þ

This resistance is minimal when the shape of A0 is

H0=L0� �

opt¼ 2 *kkf0

� ��1=2ð2Þ

The construction continues with the first assembly (A1 in Figure 1), which contains n1elemental systems, A1=n1A0. The resulting structure is shown in greater detail in Figure 2. Theheat currents produced by the elemental systems are collected by a new high-conductivityinsert of thickness D1, the left end of which is the heat sink. The hot spot is again in oneof the right-hand corners. The volume-to-point resistance is DT1=ðq000A1Þ; where DT1 is theexcess temperature at the hot spot. It was shown that the resistance is given by the two-termexpression

DT1 k0q000A1

¼H1

8 *kkf0L1þ

A1

2 *kkD1H1

ð3Þ

which was later minimized with respect to the shape parameterH1/L1. In the present analysis werewrite this expression in terms of n1, to illustrate the effect of size (n1) on the resistance of thefirst assembly:

DT1 k0q000A1

¼1

4n1 *kkf0

� �1=2 þ n1A1=20

23=2 *kk D1*kkf0

� �1=4 ð4Þ

The trade-off role played by n1 is clear. In the beginning, as the size of the assembly increasesfrom a small n1 value, the overall resistance decreases. In other words, in the beginningresistance minimization is achieved through growth. This trend ends at a certain (optimal) value

Figure 2. The first construct: a large number of elemental volumes connected to a central high-conductivity path (Bejan, 1997a,b).


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of n1, when the second term of Equation (4) takes over, and the resistance starts to increase. Theoptimal size is represented by

n1;opt ¼ *kk *DD1

� �1=2= 21=4 *kkf0

� �1=8h ið5Þ

where *DD1 ¼ D1=A1=20 : The corresponding minimal value of the first-assembly resistance is

DT1k0q000A1

� �m¼

1

23=4 *kk7=8f3=80

*DD1=21

ð6Þ

A second minimization of this resistance is possible when the volume fraction of kp materialallocated to A1 is fixed, f1=Ap1/A1, where Ap1=L1D1+n1f0A1,

f1 ¼ 2�1=2 *DD1*kkf0

� ��1=4þf0 ð7Þ

The minimization of resistance (6) subject to constraint (7) yields f0,opt=f1/2 and *DD1;opt ¼2�3=4 *kk

1=4f5=41 ; which can also be written as

D1=D0

� �opt

¼ *kkf1=2� �1=2

41 ð8Þ

Working these results back into Equations (5) and (6) we obtain

n1;opt ¼ *kkf1=2� �1=2

41 ð9Þ

DT1k0=q000A1

� �mm¼

*kkf1

� ��1ð10Þ

where the subscript ‘mm’ is a reminder that the resistance has been minimized twice. It can beverified that at this optimum the first-assembly shape and dimensions are exactly the same as in(Bejan, 1997a,b), namely (H1/L1)opt=2, H1,opt=(2A1)

1/2 and L1,opt=(A1/2)1/2.

Examined from the point of view of natural sciences, Equation (9) represents an importantstep: growth, assembly, or aggregation emerges for the first time as a mechanism of globalresistance minimization. The number optimized in Equation (9) represents growth when the opti-mized geometry has been preserved (memorized) at the elemental level. Growth, assembly, oraggregation is no longer a natural behaviour that we take for granted. It is not ‘self’-organization. It is not haphazard. It is a result}a consequence of a deterministic principle}theoptimization of access to internal currents, subject to global constraints.

The same geometric optimization principle applies at larger scales. The next scale is thesecond construct A2 (Figure 1), which contains a number (n2) of first constructs (Figure 2),A2=n2A1. The optimal external aspect ratio of the second construct is H2/L2=2, as in the twoexamples given in Figure 3. Note the optimized internal ratios of high-conductivity bladethicknesses, D1/D0 and D2/D1, where D2 is the thickness of the central (thickest, newest) blade.

The geometrically optimal construction started in Figures 1–3 can be continued to higherorders of assembly, until the structured composite (k0, kp) covers the given space. Oneinteresting feature in this limit is that the construction settles into a recurring pattern of pairing(or bifurcation, from the reverse point of view), in which the integer 2 is a result of geometricoptimization. For example, the left side of Figure 3 shows this pairing and size doubling pattern.The geometric parameters, volume fractions occupied by high-conductivity material ðfiÞ; andthe global resistances of constructs up to the sixth level, are reported analytically in tabular formin Bejan (1997a,b).



In summary, the geometric optimization of flow access between a volume and one point takesus to the geometric structure called ‘tree’. The method has technological and theoreticalimplications.

Technologically, it is possible to construct in a few simple geometric steps a near-optimalnetwork for channelling a current that is generated volumetrically. This finding is important inpractice: 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 prohibitivelylarge number of parameters, as is done currently in the numerical simulations of river drainagebasins and vascularized tissues (Rodriguez-Iturbe and Rinaldo, 1997; Meakin, 1998).

Theoretically, it means that at the basis of the tree architecture of many animate andinanimate systems rests on one design principle: the volume-constrained minimization of theglobal resistance to flow between one point and a finite volume (an infinite number of points).Reliance on such a universal design principle makes the tree network structure}its mainfeatures}predictable, contrary to the prevailing doctrine.

3. IMPROVED HEAT TREES

The heat trees of Figures 1–3 do not look entirely ‘natural’. This is due to the simplifyingassumptions on which their derivation was based. For example, the high-conductivity insertswere always drawn with constant thickness and perpendicular to their tributaries. These features

Figure 3. Second construct geometry optimized numerically (f2=0.1, *kk ¼ 300; n1=8; left side, n2=2; rightside, n2=4) (Almogbel and Bejan, 1999).


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served their purpose. They kept the number of geometric degrees of freedom to a minimum,and in this way they made possible the closed-form presentation of the geometricoptimization.

Constructal trees look more and more natural if their freedom to provide easier access to theirinternal currents is expanded. In the elemental system of Figure 1 (top) it was assumed that thekp channel stretches all the way across the volume. When this assumption is not made, we findnumerically that there is an optimal spacing between the tip of the kp channel and the adiabaticboundary of the elemental volume (Almogbel and Bejan, 1999). Figure 4 shows five cases ofoptimized elemental volumes with spacings at the tips, i.e. with k0 material all around the tips ofthe kp blades. This figure also shows how the optimized elemental volume}the smallest buildingblock of the constructal design}changes as the proportions of the composite material change:the elemental shape becomes more slender as either *kk or f0 increases.

When the angle formed between each tributary channel and its central stem is allowed to vary,numerical calculations of the two-dimensional heterogeneous conduction field show that thereexists an optimal angle for minimal volume-to-point resistance at the construct level (Ledezmaet al., 1997). This effect is illustrated for a first construct in Figure 5, where, for simplicity it wasassumed that all the tributaries are tilted at the same (variable) angle. The optimal inclination issimilar to that of tributaries in nature: pine needles, rivulets and bronchial ramifications pointaway from the root of the tree.

Figure 5 also shows that the volume-to-point resistance of the construct, #TT ¼ Tmax � Tminð Þk0=q000A1 decreases only marginally (by 5.8%) as the angle a changes from the perpendicularposition (a=08) to the optimal position (affi48). The optimized tip spacings of Figure 4 producereductions of 20% in the global elemental resistance.

These relatively unimportant improvements tell a very important story: the tree design isrobust with respect to various modifications in its internal structure. This means that the globalperformance of the system is relatively insensitive to changes in some of the internal geometric

Figure 4. Optimal shapes of elemental volumes with spacings at the tips of the high-conductivity channels:the effect of varying *kk ¼ kp=k0 and f0=D0/H0 (Almogbel and Bejan, 1999).



details. Trees that are not identical have nearly identical performance, and nearly identicalmacroscopic features such as the internal shape (Bejan, 2000).

The robustness of the tree design sheds light on why natural tree flows are never identicalgeometrically. They do not have to be, if the maximization of global performance is theirguiding principle. The ways in which the details of natural trees may differ from case to case arewithout number because, unlike in the constructions presented in this paper, the number ofdegrees of freedom of the emerging form is not constrained. Local details differ from case tocase because of unknown and unpredictable local features such as the heterogeneity of thenatural flow medium, and the history and lack of uniformity of the volumetric flow rate that isdistributed over the system. Marvelous illustrations of this element of ‘chance’ are provided byseemingly irregular river drainage basins all over the world (Errera and Bejan, 1998). The pointis that the global performance and structure (tree) are predictable, and the principle that takesthe system to this level of performance is deterministic (Bejan, 2000).

Robustness continues to impress as we increase the number of degrees of freedom of thegeometric design. In Figure 6 we see the results of a fully numerical optimization of the secondconstruct with perpendicular and constant-thickness inserts (D0, D1, D2), where all the othergeometric parameters were allowed to vary}the aspect ratios of all the rectangles, large andsmall, the number of elemental volumes in each first construct (n1), and the number of firstconstructs in each second construct (n2). The three designs shown in Figure 6 have beenoptimized with respect to all the free parameters except n2, and they have been drawn to scale[n1,opt=8, (D1/D0)opt=5, D2/D0=10]. The figure shows the effect of fine-tuning the number offirst constructs incorporated in the second construct (n2).

Figure 5. The optimization of the angle of confluence between tributaries and their common stem in a firstconstruct (f1=0.1, *kk ¼ 50; n1=4) (Ledezma et al., 1997).


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The same effect is documented numerically in Table I. The size of the second construct(A2=H2L2) is fixed. The cold spot (Tmin) is at the root of the tree, and the two hot spots are inthe farthest (left-side) corners. The aspect ratio H2/L2 refers to the vertical/horizontal externaldimensions of the largest rectangle. The best second construct is the one with n2=4: however,the neighbouring designs (n2=2, 6) perform nearly as well. The global resistances of all thesedesigns agree within 7% even though their internal structures are markedly different. Theiroptimized external shapes are also nearly the same. The case of Figure 6 was optimized furtherin Figure 3, by allowing spacings (k0 material) all around the tips of the elemental (D0) inserts(Almogbel and Bejan, 1999).

Once again, the global optimization principle leads us to very robust (invariant) features suchas the global performance level and the external shape of the construct. Surprisingly, we willarrive at the same conclusions via a completely different optimization in the next section.Changes in the internal tree structure, such as overgrowth and surgery (adding or cuttingbranches) has almost no effect on the globally optimized features.

4. TIME-DEPENDENT DISCHARGE FROM VOLUME TO POINT

In Figures 1–6 we saw how the tree structure is derived when the flow is steady from volume topoint, or point to volume. Practically the same structure is deduced when the objective is to

Figure 6. The second construct optimized numerically for minimum resistance in volume-point steadyflow, and the effect of changing the number of first constructs, n2; see also Table I (Ledezma et al., 1997).

Table I. The effect of increasing the number of first constructs (n2) in theoptimized second construct when f2=0.1 and *kk ¼ 300 (Ledezma et al., 1997).

n2 (Tmax–Tmin)mink0/(q0 0 0A2) ðH2=L2Þ

1=2opt

2 0.0379 1.4124 0.0354 1.3756 0.0374 1.360



minimize the time of discharge of a volume to a sink, as in lightning, exhaling, and river basinconstruction after a sudden downpour. The maximized performance in steady flow is the sameas the minimization of the time of approach to equilibrium: the mechanism that allows thesystem to reach this objective is the construction of optimal flow shape and structure.

We demonstrated the construction of the tree for volume-point discharge by minimizinggeometrically the cooldown time of an initially isothermal conducting solid (k0, without heatgeneration), which is placed suddenly in contact with a boundary sink point (Tmin) (Dan andBejan, 1998). With reference to the elemental volume of Figure 1 (top) this new problem meansthat q000=0, and at the time t=0 the entire A0=H0L0 rectangle is at the temperature Ti. Theenergy stored in this volume flows out through the origin. All the temperatures decrease,however the slowest to do so are the instantaneous hot spots [Tmax(t)] located in the farthestcorners (x=L0, y=�H0/2). The effective cooldown time tc is defined as the time when thehighest temperature has reached within 10% its final (equilibrium) level,

½TmaxðtcÞ � Tmin�=ðTi � TminÞ ¼ e ð11Þ

where e=0.1. The cooldown time has a sharp minimum with respect to the rectangle shape H0/L0. Numerical minimization of tc in the range 0.034f040.3 and 34 *kk4300 showed that theoptimal elemental shape is given by the same formula as Equation (2) in which the factor 2 isreplaced by 2.11}not a significant change.

That the optimized structure in volume-point discharge is the same as in volume-point steadyflow is stressed further by Figure 7. Compared are three optimized constructs of the secondkind, for which we minimized the cooldown time by varying all the geometric features except n2.The optimized structures are drawn to scale. Listed under each drawing are the assumed numberof first constructs in the structure n2, the optimized external shape #HH2;opt ¼ H2=L2

� �1=2opt

; and theminimized dimensionless cooldown time, #ttc;min ¼ tc;min a0=A2; where a0 is the thermal diffusivityof the k0 material. The three designs have practically the same global performance #ttc;min

� �and

external shape. Their robustness and internal details support in every respect the conclusions

Figure 7. The second construct optimized for minimum time of discharge from a volume to one point, andthe effect of changing the number of first constructs, n2(f2=0.1, *kk ¼ 300; D1/D0=5, D2/D0=10) (Dan and

Bejan, 1998).


A. BEJAN292

reached on the basis of maximizing global performance in steady flow between a volume andone point (Figure 6).

The volume-point discharge optimization shows clearly why the principle recognized in thiswork is new on the currently accepted background. The approach to equilibrium, the directionof time, or the difference between before and after, is the domain covered by the second law ofthermodynamics (Bejan, 1997b). The objective and constraints (constructal) principle invokedin this work accounts for an equally important part of nature, that is, of everything: themechanism for reaching equilibrium faster is the construction of macroscopic flow architecture,i.e., the generation of geometric form (Bejan, 2000).

5. CONSTRUCTAL DESIGN: INCREASING COMPLEXITY IN A VOLUME OFFIXED SIZE

An alternative to the ‘growth’ route to optimizing volume-point flow (Figure 1) is presented inFigure 8. This time the size of the volume (A) is fixed, although its external shape may vary inorder to accommodate optimally the internal structure that we will determine: the design. The

Figure 8. Constructal design: the minimization of volume-point resistance through the increase in theinternal complexity of a volume of fixed size (Bejan and Dan, 1999a). Note A0=A1=A2, etc.



volume fraction of high-conductivity material f=Ap/A is also fixed. The design objective is todistribute this material over A such that the resistance from A to the heat sink ðMÞ is minimal.

We contemplate using several designs for the internal structure of the same volume, A. InFigure 8 each is indicated by the subscript i=0,1,2,. . ., which represents, in order, the elementaldesign, the first-construct design, the second-construct design, etc. These designs have the samesize (A=A0=A1=A2), and are indicated by the labels put in the lower-left corner of each frame.More specifically, by first-construct design we mean that the A1 frame of Figure 8 has the sameinternal structure as in Figure 2, even though that structure is not shown in detail in Figure 8.Similarly, the A2 frame of Figure 8 has the same internal tree structure as Figure 6.

Next, we optimize the internal dimensions of each design (Ai, fi), and then see how well theyperform against each other. In other words, given the system size (A) and amount of kp-material(f), which design}what degree of internal complexity}is the best for minimizing the volume-to-point flow resistance.

This question was investigated in detail in Bejan and Dan, (1999a), from which Figure 9 isreproduced as an example. The first-construct structure is preferable to the elemental structurewhen the group *kkf is greater than 9. All the designs of Figure 9 are being compared on an equalbasis, e.g. volume fraction occupied by high-conductivity material ðf0 ¼ f1 ¼ f2 ¼ � � � ¼ fÞand overall size (A0 ¼ A1 ¼ A2 ¼ � � � ¼ A). The group *kkf; which accounts for the properties ofthe composite material, plays a critical role in the optimization of the design. As this groupincreases, i.e. as the kp inserts become more conductive and/or more voluminous, there comes apoint where the internal architecture makes a transition from one structure (A0) to a morecomplex one (A1). The transition is toward lower resistance via higher complexity, and it isanalogous to eddies in turbulence (Bejan, 2000) and rolls in B!eenard convection (Nelson andBejan, 1998).

Figure 9 shows further that the second-construct design performs better than the first-construct design when n241: the transition occurs at n256. The design route was pursued to the

Figure 9. Constructal design: the minimized global resistances of the elemental, first-construct and secondconstruct designs (Bejan and Dan, 1999a).


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third-construct level in Bejan and Dan (1999a), where the designs were also compared with fullnumerical simulations of conduction in the heterogeneous domain. Most important is theconclusion that the growth (Figure 1) and design (Figure 8) approaches lead to designs that arecomparable not only in overall performance but also in geometric appearance. Being thesimplest and most direct, the original growth method provides a useful shortcut to and anapproximation of the architecture that would result at the end of the application of the fixed-volume design method (Bejan and Dan, 1999a).

6. TREES FOR CONVECTION

In this section, we turn our attention to the more complex configurations where the flow of heatbetween a finite volume and one point is aided by the flow of a fluid. The resulting structures aretrees in which convection plays an important role, but not the complete role. In every elementalvolume convection is intimately coupled with pure conduction, in a phenomenon of conjugateheat transfer. The key to developing the optimal flow architecture, from the smallest elementalvolumes to larger and larger constructs, is to find this optimal coupling, or optimal balancebetween convection and conduction.

Convective flow architectures are of two types, depending on which portions of the structureare reserved for convection. In the first type the interstices are occupied by solid that generatesheat at every point, and conducts the heat by diffusion in the manner of the k0 material analysedin Figure 1. Convection is located in the branches of a tree formed by ducts filled with flowingfluid. The ramifications of two trees of this type visit each elemental volume. One tree deliverscold fluid to each element. The other tree collects the fluid heated by the element, andreconstitutes it into a single stream that eventually leaves the volume (Bejan and Errera, 2000).

The circulatory system performs its mass transfer function and secondary heat transferfunction by using a convective double-tree structure of the first type (Bejan, 2000). In therespiratory system the two flow trees are superimposed, as they rely on the same network ofbronchial tubes, one tree during inhaling, and the other during exhaling. Engineeringapplications of trees of convective channels abound in the cooling of virtually every enclosedelectrical heat-generating system, e.g. windings of electrical machines, computers, and electricaland electronic packages of many types and sizes. All must be cooled at every point, using forcedor natural convection.

In the convective tree of the second type the spaces occupied by conduction and convectionare reversed. Convection is in the interstices, and is coupled with pure conduction in solid parts,which form trees. Every interstitial space serves as sink or source for the current that passesthrough the root of the tree. Numerous applications for this flow structure are found in thedesign of heat transfer-enhanced surfaces for heat exchangers, and for cooling small scaleelectronics. In the latter, the tree structures are better known as fin trees and fin bushes(Hamburgen, 1986; Kraus, 1999).

We review the constructal fin-tree problem statement with reference to the general geometrysketched in Figure 10. Consider the two-dimensional volume of frontal area A and fixed lengthW, where W is aligned with the free stream ðU1; T1Þ: The problem consists of distributingoptimally through this volume a fixed amount of high-conductivity (kp) material, which takesheat from one spot on the boundary and discharges it throughout the volume. We may think ofthe boundary spot (root) as the external surface of an electronic module that must be cooled. In



this case the volume AW is the space that is allocated for the purpose of cooling the module byforced convection. This volume constitutes a global constraint.

As in the pure-conduction applications of the constructal method (Figures 1–9), we start thespace filling optimization sequence from the smallest, finite-size scale. The elemental systemconsists of a two-dimensional volume H0L0W, in which there is only one blade of kp material(Figure (10b)). The thickness of this blade is D0. Heat is transferred from one boundary spot(T0, at the root of the fin) to the entire elemental volume. If we neglect the heat transfer throughthe fin tip, and use the unidirectional fin conduction model (Gardner, 1945) the elemental heatcurrent is

q0T0 � T1ð ÞW

¼ 2kpD0h0� �1=2

tanh2 h0kpD0

� �1=2

L0

" #ð12Þ

The thickness of the elemental volume (H0) is fixed because it is the same as the optimal spacingbetween two successive D0-thick plate fins. The spacing is optimal when the laminar boundarylayers that develop over the swept lengthW became thick enough to touch at the trailing edge ofeach plate fin (Figure 10(c)). The optimal spacing (H0–D0�H0) is determined uniquely by thelength W and the pressure difference DP maintained across the swept volume (Bejan, 2000, 1995;Bejan and Sciubba, 1992),

H0=W ffi 2:7P� 1=4W ¼ 2:7 m a=W 2 DP

� �1=4ð13Þ

In forced convection the Bejan number ðDPW 2=maÞ plays the same role that the Rayleighnumber plays in natural convection (Petrescu, 1994). It can be shown that the minimized

Figure 10. (a) Volume (AW) serving as convective heat sink for a concentrated heat current (q). (b) Thesmallest volume element defined by a single plate fin. (c) Merging boundary layers in a channel with

optimal spacing (Bejan and Dan, 1999b).


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thermal resistance that corresponds to this spacing is also characterized by an average heattransfer coefficient that is given approximately by (Bejan and Sciubba, 1992)

h0 ffi 0:55kW

W 2 DPm a

� �1=4

ð14Þ

Here k is the thermal conductivity of the fluid. The factor DP that appears in Equation (14)refers to the pressure difference that is maintained in the W direction (e.g. by a fan), whereH0�L0 is the cross-section of one duct. If the H0-wide channel is open to one side (the side thatwould connect the tips of two successive fins), and if the entire assembly is immersed in a streamof velocity U1, then Equations. (13) and (14) are adequate if DP is replaced by the dynamicpressure associated with the free stream, DP ffi rU2

1=2:We now proceed toward larger scales by recognizing that at the elemental level the h0 and H0

values are two known constants. The next volume is the first assembly of constant frontal areaH1L1=A1, which is shown in Figure 11. The external shape H1/L1 is free to vary. The assemblyis defined by a central blade of thickness D1, which is connected to all the elemental volumes thatare needed to fill the A1W volume. The D1 blade connects the roots of all the D0 fins. When thenumber of elemental volumes in this assembly is large, the cooling effect provided by the D0 finsis distributed almost uniformly along the D1 stem. In this limit the D1 stem performs as a finimmersed in a convective medium with constant heat transfer coefficient. The effective heattransfer coefficient of this medium (h1) can be deduced from Equation (12) by noting that eachq0 current flows out of the D1 blade through an area of size H0W. In other words, we combine

Figure 11. First construct of plate fins (Bejan and Dan, 1999b).



h1=q0/[H0W(T0�T)] with Equation (12) and L0=H1/2, and obtain

h1 ¼1

H0ð2kpD0h0Þ

1=2 tanh2h0kpD0

� �1=2H1

2

" #ð15Þ

The D1 blade functions as a fin with insulated tip, therefore we write, cf. Equation (12)

q1ðT1 � T1ÞW

¼ ð2kpD1h1Þ1=2 tanh

2h1kpD1

� �1=2

L1

" #ð16Þ

Two constraints must be satisfied, the total volume, or frontal area (A1=H1L1), and the volumeoccupied by solid, i.e. the frontal area of all the solid (Ap,1=D1L1+n1D0L0), or the volumefraction f1=Ap,1/A1. The geometry of the construct has two degrees of freedom. In thenumerical work detailed in Bejan and Dan (1999b) the two optimized features were the externalshape H1/L1 and the internal ratio D1/D0. These were developed by maximizing the globalconductance (16). Associated results are the number of elemental fins (n1) and the twice-maximized conductance: they are not reproduced here because of space limitations.

Two invariant features of the fin trees (Bejan and Dan, 1999b) are worth noting. The overallshape of the volume occupied by the assembly does not change much as the volume sizeincreases. Another feature is the thickness of the elemental fin, D0,opt, which is insensitive tochanges in the total volume occupied by the construct.

Future work may address various improvements and refinements of the classical finconduction model that was used in setting up the example of Figures 10 and 11. For example,the effects of temperature-dependent thermal conductivity, radiative heat transfer, and spatiallyvarying heat transfer coefficient can be incorporated at the elemental level, while paying apenalty through the increased complexity of the analysis, and the need for numerical work evenat the elemental level.

Trees of fins may also be pursued in simpler configurations. When there are only twoelemental plate fins in the first construct, the assembly has the shape of a T or, when theelemental tips are bent downward, the shape of ‘tau’ (Kraus, 1999; Bejan and Almogbel, 2000).

Constructal trees of fins can also be developed in cylindrical geometries. The first constructconsists of a number of elemental circular plate fins mounted on a cylindrical stem (Alebrahimand Bejan, 1999). Fluid is forced to flow through the spaces between adjacent fins. Again, thereare two geometric features that can be optimized, the external aspect ratio of the cylindricalconstruct (diameter/length), and the ratio between the elemental plate-fin thickness and the stemdiameter. Cylindrical trees consisting of cylindrical (pin) fins are optimized geometrically inAlmogbel and Bejan, (2000). Trees of fins will continue to attract interest as heat transferaugmentation techniques in heat exchangers, e.g. the cooling of electronics.

The derivation of optimal geometric form in designs and in nature can be pursued based onmore detailed, computer-aided formulations. In pure heat conduction, the method has beenextended to three-dimensional tree flows and leaf-like shapes for the elemental volumes(Ledezma and Bejan, 1998; Neagu and Bejan, 1999a,b). In pure fluid flow, constructal treearchitectures have been developed for networks of round tubes (Bejan, 1997b,c), networks ofparallel-plate fissures (Bejan and Errera, 1997; Errera and Bejan, 1999), and river basins inerodable porous media (Errera and Bejan, 1998). Further extensions and opportunities to newwork, and a new mission for engineering science are proposed in a new book (Bejan, 2000).


A. BEJAN298

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Constructal tree-shaped paths for conduction and convection