Constructal multi-scale and multi-objective structures

INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2005; 29:689–710Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/er.1100

Constructal multi-scale and multi-objective structures

Adrian Bejan1,n,y and Sylvie Lorente2

1Department of Mechanical Engineering and Materials Science, Duke University, Box 90300, Durham,

NC 27708-0300, U.S.A.2Laboratory of Materials and Durability of Constructions, National Institute of Applied Sciences (INSA),

135 Avenue de Rangueil, Toulouse 31077, France

SUMMARY

This is a review of recent progress on constructal design made in two directions: multi-scale flow structures,and multi-objective design. The first direction is associated with the maximization of heat transfer ratedensity in a fixed volume in the limit of decreasing length scales, where boundary layers touch, andoptimized channels are no longer slender. In the first example of this type, spacings are optimized based onthe intersection of asymptotes method. In the second, the heat transfer density is further increased byplacing progressively smaller plates in the entrances of the channels formed by the first generation ofplates. In the third example, the placement of discrete heat sources on a vertical wall with naturaland forced convection is optimized. The second direction is the discovery of architectures that resultfrom two competing objectives, for example, mechanical strength and thermal insulation. It is shown thatthe internal configuration of a cavernous brick wall can be deduced from the clash between the twoobjectives. The same concept can be used to optimize the shape and structure of support beams that mustbe strong and, at the same time, must resist sudden attack by intense heating. Copyright # 2005 JohnWiley & Sons, Ltd.

KEY WORDS: constructal theory; spacings; multi-scale; hierarchical; packing; heat sources; multi-objective; mechanical strength; resistance to terrorist attack

1. CONSTRUCTAL THEORY AND DESIGN

A newly emerging body of work (Bejan, 2000; Bejan et al., 2004) is focusing attention on theprinciple-based generation of optimal geometry (configuration, architecture) in flow systemsendowed with global objectives and global constraints. In the beginning of this process, thesystem geometry is missing. The acquisition of geometry is the mechanism by which the systemmeets its global objectives under constraints. This mechanism is at work not only in engineeredsystems but also in naturally occurring systems, animate and inanimate. The view that geometry

Copyright # 2005 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: A. Bejan, Department of Mechanical Engineering and Materials Science, Box 90300, DukeUniversity, Durham, NC 27708-0300, U.S.A.

is generated by the pursuit of global performance under global constraints has been namedconstructal theory.

The earliest work was devoted to the simplest type of geometry generation: systems, thedevelopment of which is driven by a single objective. For example, in the tree-shaped constructsgenerated for cooling a volume, the single objective is the minimization of the global thermalresistance (Bejan, 2000). In tree-shaped constructs for distributing a fluid stream over an area(or collecting a stream from an area), the single objective is the minimization of the globalresistance to fluid flow, or the global pumping power (Bejan et al., 2004). In the design ofmechanical structures with prescribed loading and stiffness, the single global objective is theminimization of the material used to build the structure.

Constructal theory and design (Bejan, 2000; Bejan et al., 2004) serve as a reminder that flowsystems that must be designed (configured) must be treated as malleable, i.e. as morphingstructures that are as free to change as possible. Configurations that are assumed based on pastpractice, handbooks and design rules, are not necessarily the best. The only rule worthremembering is that geometry must not be taken for granted. Geometry matters, in fact,geometry is the result, not an assumption. It is geometry that endows the flow system with theability to serve its purpose, in spite of the constraints.

In this paper, we review some of our group’s most recent developments in two directions,which build on the constructal-theory basis described above. One direction is the march towardssmaller and smaller scales, compactness and maximum use of space. This is also a marchtowards optimized complexity, multiple scales, hierarchy and ‘construction’. The reason is thatwhen components become smaller, they also become more numerous and more interconnectedas they coalesce into devices that are useful at the human scale. Constructal design is theunderside of the miniaturization coin.

The other direction was recommended by the observation that the great diversity andapparent lack of ‘correlation’ of the structures that emerge in nature and engineering can beattributed to the fact that even the simplest element of a complex system has more than oneobjective. This is why a systematic extension of the constructal approach to multi-objectivesystems is necessary and timely. A first step in this direction was described in Lorente and Bejan(2002), where the internal structure of a cavernous wall of a building was deduced by pursuingtwo objectives: thermal insulation and mechanical strength. In this review, we explore thecombined ‘flow and strength’ constructal method in the direction of systems that must bemechanically strong and, at the same time, must retain their strength and integrity duringthermal attack. Mechanical structures become weaker and may collapse if they are exposed tointense heating. The collapse of the World Trade Centre is a reminder of how dangerous theeffect of sudden intense heating can be. Large buildings, highway overpasses and industrialinstallations are vulnerable.

2. OPTIMAL SPACINGS

The field of heat transfer has demonstrated for many years how the principle of generating flowgeometry works. First and foremost, this principle is about objective, or purpose: the need tomake things compact, to use space, to install the most into the available space. Good examplesare the spacings that are distributed uniformly through flow volumes, for example, fins in heatexchangers, and stacks of boards in electronic packages. Organization makes the volume

Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Energy Res. 2005; 29:689–710

A. BEJAN AND S. LORENTE690

perform best in a global sense. It spreads the hot spots through the volume. It distributesimperfection optimally.

The key to packing maximum convective heat transfer rate per unit volume is the observationthat every fluid packet and every volume element must be used for the purpose of transferringheat. Fluid flow regions that do not ‘work’ in a heat transfer sense must be avoided. Flowregions that work too much, and become ineffective (filled with ‘used’ fluid), must be eliminated.This activity of arranging and rearranging the volumetric distribution of flow and heat transferleads to the construction of internal structure}optimal flow architecture for maximal globalperformance subject to constraints.

The principle of forcing the fluid to ‘work’ everywhere can be illustrated by considering theperformance of a large volume that is filled with parallel-plate channels with forced convection.One channel is shown in Figure 1, where the designer may contemplate two extremes. First, ifthe channel length L is made shorter than the thermal entrance length XT, then the fluid thatoccupies the core of the duct does not participate in the heat transfer enterprise. Such fluid mustnot be allowed to leave the channel without having interacted thermally with the walls.

In the other extreme, when L is made longer than XT, the fluid is so saturated with heating orcooling from the wall that it can accommodate further heating or cooling only by overheating,i.e. by changing its bulk temperature in the downstream direction. This extreme (the fullydeveloped regime) must be avoided. It is important to note that the decision to avoid thethermally fully developed flow regime is new relative to current trends in micro-scale heatexchanger design, where laminar fully developed flow is a routine design feature.

Figure 1. Maximal heat transfer rate per unit volume is achieved when the channel length matches thethermal entrance length of the flow.


MULTI-SCALE AND MULTI-OBJECTIVE STRUCTURES 691

The best choice is in-between, L � XT; because in this configuration all the fluid of thechannel cross-section is active in a heat transfer sense. The fluid leaves the channel as soon as itcompletes its mission, i.e. as soon as the boundary layers have merged. In this configuration, thechannel volume is used to the maximum for the purpose of transferring heat between the streamand the walls. This principle has been used to optimize spacings in several channelconfigurations with natural and forced convection: volumes filled with a stack of continuousor staggered parallel plates, volumes filled with parallel cylinders in crossflow, and three-dimensional pin fin arrays with impinging flow. These results have been summarized in Bejan(2000).

The simplest example of geometric optimization for forced convection was the optimizationof the spacing between parallel-plates that fill a larger volume (Bejan and Sciubba, 1992). Withreference to the lower drawing of Figure 1, when the pressure difference between the two ends ofthe channel is specified (DP), maximal heat transfer rate per unit volume is achieved when

Dopt

Lffi 2:73Be�1=4 ð1Þ

where Be is the pressure difference number (Bhattacharjee and Grosshandler, 1988; Petrescu,1994; Furukawa and Yang, 2003),

Be ¼DPL2

mað2Þ

The corresponding maximal average heat transfer rate per unit of channel volume is

q-max 5�

0:62k

L2ðTw � T0ÞBe1=2 ð3Þ

where (Tw�T0) is the temperature difference between the plate surfaces and the entering single-phase coolant.

Equations (1)–(3) result from the intersection of asymptotes method (Lewins, 2003), in whichq- is estimated analytically in two extremes: (a) narrow spacings, in which the flow through thechannel is in the Hagen–Poiseuille regime, and (b) wide spacings, where the channel is thickerthan the boundary layers. The intersection of asymptotes (a) and (b) yields Equations (1)and (3). The inequality sign in Equation (3) is a reminder that the intersection of (a) and (b)provides a q-max estimate that falls above the actual peak of the q-ðDÞ curve.

According to Equation (3), the density of heat transfer rate can be increased by decreasing L,because q-max is proportional to L–1. Smaller and smaller dimensions are attractive. Accordingto Equation (1), however, D is proportional to L1/2. This means that the aspect ratio D/L isproportional to L–1/2, and the channel becomes less slender when the size L becomes smaller. Inthis limit of increasing D/L ratios, the boundary layer slenderness assumption on whichasymptote (b) and Equations (1)–(3) is based breaks down. The optimal spacings for channelswith maximal heat transfer density when dimensions are so small that Equations (1)–(3) do notapply are reported in Bejan (2004a), and are discussed next.

3. THE SMALL-SCALE LIMIT

Consider the limit of dimensions (L, D) as so small that D/L is not much smaller than 1. Theboundary layer theory does not hold. If the channels are sufficiently wide to be surrounded



completely by cold fluid (T0), then the heat transfer from each plate to the fluid is by quasi-radialconduction. This regime is shown in Figure 2(b). In the other extreme, Figure 2(a), dimensionsare again small, but, in addition, the channels that fill the fixed volume of the package (HL2) aremany, and the flow through each channel is in the Hagen–Poiseuille regime. In regime (a), themean velocity is

U ¼D2

12mDPL

ð4Þ

The total flow rate through the HL2 volume is ’mm ¼ rHLU: The total enthalpy increaseexperienced by the ’mm stream is q ¼ ’mmcPðTw � T0Þ; because in this limit the spacing D is so tight,and the wall–fluid thermal contact so good, that the outlet temperature of the ’mm streamis essentially equal to the plate temperature Tw. The volumetric heat transfer density isq- ¼ q=ðHL2Þ; or

q- ¼D2DP12nL2

cPðTw � T0Þ ð5Þ

In the opposite extreme, Figure 2(b), each plate is surrounded by fluid of temperature T0

because convection is negligible. In the limit of small linear dimensions, the rate of heat transferfrom one plate to the fluid (q1) is proportional to Lk(Tw�T0), where L is the linear dimension ofthe plate. We can use a more accurate q1 estimate by noting that if q1 emanates from a Tw disc of

Figure 2. Stack of parallel plates: (a) The small-D limit; (b) the large-D limit.



diameter d immersed in a medium at T0, then q1 is given by

q1 ¼ 4dkðTw � T0Þ ð6Þ

In the configuration of Figure 2(b) the plate area is L2 instead of pd 2/4, which means that inEquation (5) we may replace d approximately with (4/p)1/2L. Next, the total heat transfer rateemanating from the stack is q ¼ Nq1; where N ¼ H=D is the number of plates in the stack.Finally, the volumetric rate of heat transfer is q- ¼ q=ðHL2Þ; which becomes

q- ¼8kðTw � T0Þ

p1=2DLð7Þ

In summary, asymptotes (5) and (7) show that q- can be maximized with respect to D.Equation (5) holds when D is small, and shows that q- decreases as D decreases. Equation (7)holds when D is large, and shows that q- decreases as D increases. This asymptotic behaviourguarantees that q- reaches a maximum in the vicinity of the intersection of Equations (5) and(7). The optimal spacing and maximal heat transfer density are

Dopt

Lffi 3:78Be�1=3 ð8Þ

q-max 5�

1:2k

L2ðTw � T0ÞBe1=3 ð9Þ

Equation (9) shows that q- increases as L�4/3 as L decreases. This increase is faster incomparison with the behaviour of q- at larger scales, Equation (3), where q-max increases as L

–1

as L decreases. This change in the behaviour of q-max stresses not only the importance of seekingsmaller dimensions but also the importance of knowing the correct scaling laws whendimensions have become small enough.

The transition from the large-scales optimum to the small-scales optimum is obtained byintersecting Equations (1) and (8), or Equations (3) and (9). In either case, the transition isfound to occur at

Be1=2 � 8 ð10Þ

where Be1=298 is the domain of validity of the small-scales solution, Equations (8, 9). Thenumber Be1=2 represents a dimensionless flow length L when DP is specified. This observationrecommends the use of the dimensionless variables

ð *DD; *LLÞ ¼ ðD;LÞDPma

� �1=2

*qq- ¼q-ma

DPkðTw � T0Þð11Þ

where *LL is the same as Be1=2: These dimensionless variables transform Equations (1) and (8) into

*DDopt ffi 2:73 *LL1=2 ð12Þ

*DDopt ffi 3:78 *LL1=3 ð13Þ

and Equations (3) and (9) into

*qq-max 5�

0:62 *LL�1 ð14Þ

*qq-max 5�

1:2 *LL�4=3 ð15Þ



The two sets of asymptotes are displayed in Figure 3. The heat transfer density increases inaccelerated fashion as the length scale *LL decreases. In the same direction, the decrease of *DDopt

slows down. The transition from large scales to small scales occurs in the vicinity of *LL � 8;where the channel slenderness ratio is *DDopt= *LL � 1; in accordance with the threshold belowwhich the boundary layers in the entrance region (Figure 1) breakdown.

4. MULTI-SCALE STRUCTURES

Consider again the optimal spacing when boundary layers are distinct (Figure 1). Assume thatthe fluid Prandtl number is of order 1, so that the velocity and thermal boundary-layerthicknesses are both represented by (Bejan, 2004b)

d ffi 5xðUx=nÞ�1=2 ð16Þ

In this expression, U and n are the free stream velocity and the kinematic viscosity, and x ismeasured downstream from the entrance (Figure 2).

Is the stack of Figure 4 the best way to pack heat transfer into a fixed volume? It is, but onlywhen a single length scale is used, that is, if the structure is uniform. The structure of Figure 4 isuniform, because it does not change from x ¼ 0 to L0. At the most, the geometries of single-spacing structures vary periodically, as in the case of arrays of cylinders and staggered plates.

The structure of Figure 4 can be improved if more length scales (D0, D1, D2,. . .) are available(Bejan and Fautrelle, 2003). The technique consists of placing more heat transfer in regions ofthe volume HL0 where the boundary layers are thinner. Those regions are situated immediatelydownstream of the entrance plane, x ¼ 0: This observation is the same as taking the argument ofFigure 1 to a finer level: regions that do not work in a heat transfer sense must either be made to

Figure 3. The effect of the decreasing length scale on the optimal spacing andmaximal heat transfer rate density.



serve or be left out. In Figure 4, the wedges of fluid contained between the tips of opposingboundary layers are not involved in transferring heat. They can be involved, if heat-generatingblades of shorter length (L1) are installed on their planes of symmetry. This new design is shownin Figure 5.

Each new L1 blade is coated by boundary layers described by Equation (16). Since d increasesas x1/2, the boundary layers of the L1 blade merge with the boundary layers of the L0 blades at adownstream position that is approximately equal to L0/4. The approximation is due to theassumption that the presence of the L1 boundary layers does not affect significantly thedownstream development of the L0 boundary layers. By choosing L1 such that the boundarylayers that coat the L1 blade merge with surrounding boundary layers at the downstream end ofthe L1 blade, we invoke one more time the optimal packing principle of Figure 1. We are beingconsistent, and, because of this, every structure with merging boundary layers will be optimal,no matter how complicated.

The wedges of isothermal fluid (T0) remaining between adjacent L0 and L1 blades can bepopulated with a new generation of even shorter blades, L2 ¼ L1=4: Two such blades are shownin the upper-left corner of Figure 5. The length scales become smaller (L0, L1, L2), but the shapeof the boundary layer region is the same for all the blades, because the blades are all swept bythe same flow (U). The merging and expiring boundary layers are arranged according to the

Figure 4. Package of parallel plates with one spacing (Bejan and Fautrelle, 2003).



algorithm Li ¼ ð1=4ÞLi�1; Di ¼ ð1=2ÞDi�1 (i ¼ 1; 2; . . . ;m) where m is finite, not infinite. In otherwords, as in all the constructal tree structures (Bejan, 2000), the image generated by thealgorithm is not a fractal. The sequence of length scales is finite, and the smallest size (Dm, Lm) isknown, cf. Equation (18). In the first tree structures produced by constructal theory, the smallestsize is the starting step in time, i.e. the size of the elemental building block.

The number of blades of a certain size increases as the blade size decreases. Let n0 be thenumber of L0 blades in the uniform structure of Figure 4, n0 ¼ H=D0; where D0 ffi 2dðL0Þ ffi10ðnL0=UÞ1=2: The number of L1 blades is n1 ¼ n0; because there are as many L1 blades as thereare D0 spacings. At scales smaller than L1, the number of blades of one size doubles with everystep, ni ¼ 2ni21:

Two conflicting effects emerge as the structure grows in the sequence started in Figure 5. Oneis attractive: the total surface of temperature Tw that is installed in the HL0 volume increases.

Figure 5. Multi-scale package of parallel plates (Bejan and Fautrelle, 2003).



The other is detrimental: the flow resistance increases, the flow rate driven by the fixed DPdecreases, and so does the heat transfer rate associated with a single boundary layer. The keyquestion is how the volume is being used: what happens to the heat transfer rate density ascomplexity increases? The analysis (Bejan and Fautrelle, 2003; Bejan et al., 2004) shows that theheat transfer density, or dimensionless global thermal conductance, is

q0

kDTffi 0:36

H

L0Be1=2S1=2 ð17Þ

where DT ¼ Tw � T0; and Be is defined by Equation (2), in this case Be ¼ DPL20=ðmaÞ: The

geometric dimensionless group S refers strictly to structure (silouette):

S ¼ 1þn1

n0

L1

L0

� �1=2

þn2

n0

L2

L0

� �1=2

þ � � � þnm

n0

Lm

L0

� �1=2

¼ 1þm

2ð18Þ

The alternative to using the global conductance is the heat transfer rate density, q- ¼ q0=HL0:Both quantities increase with the applied pressure difference (Be) and the complexity of the flowstructure (S).

In conclusion, the effect of increasing S is beneficial from the point of view of packing moreheat transfer in a given volume. Optimized complexity is the route to maximal globalperformance in a morphing flow system (Bejan, 2000; Bejan et al., 2004). Optimized complexityshould not be confused with maximized complexity.

How large can m and the number S be? The answer follows from the observation that the flowgeometry of Figure 3 is valid when boundary layers exist, i.e. when they are slender. Figure 5makes it clear that boundary layers are less slender when their lengths (Li) are shorter. Theshortest blade length Lm below which the boundary layer heat transfer mechanism breaks downis Lm � Dm: This leads to 2mð1þm=2Þ1=4�0:17Be1=4; which establishes m as a slowly varyingmonotonic function of Be1/4. This function can be substituted in Equation (15) to see thecomplete effect of Be on the global heat transfer performance.

The required complexity (m) increases monotonically with the imposed pressure difference(Be). The monotonic effect of m is such that each new length scale (m) contributes to globalperformance less than the preceding length scale (m� 1). More flow means more length scales,and smaller scales. The structure becomes not only more complex but also finer. This is similarto how mud cracks change when the wind becomes stronger (Bejan, 2000). If the constructionstarted in Figure 5 is arbitrarily continued ad infinitum, then and only then would the resultingimage would be a fractal. Such images are not part of nature.

The multi-scale structure of Figure 5 was optimized numerically based on completesimulations of the flow and temperature fields (Bello-Ochende and Bejan, 2004), and the resultsconfirm the architecture and performance described analytically in this section.

5. MULTI-SCALE HEAT SOURCES: NATURAL CONVECTION

Consider a vertical wall of height H, which is in contact with a fluid reservoir of temperature T1

(Figure 6). The wall is heated by horizontal line heat sources (da Silva et al., 2004a). Each sourcehas the strength q0 ½W=m�: Each line heat source extends in the direction perpendicular to thefigure. The flow is two-dimensional and by natural convection in the boundary layer regime.The number of sources per unit of wall height is N 0ðyÞ:



According to constructal design, the global system (the wall) will perform best when all itselements work as hard as the hardest working element (Bejan, 2000). This means that if Tmax isthe maximal temperature that must not be exceeded at the hot spots that occur on the wall, thenthe entire wall should operate at Tmax. The problem is to determine the distribution ofheat sources on the wall, N0(y), such that the wall temperature is near the allowed limit,TwðyÞ ffi Tmax; constant. In da Silva et al. (2004a), we have assumed that the density of linesources is sufficiently high so that we may regard the distribution of discrete q0 sources as anearly continuous distribution of non-uniform heat flux,

q00ðyÞ ¼ q0N 0 ð19Þ

Figure 6. The multiple length scales of the non-uniform distribution of heat sources on a vertical wall withlaminar natural convection (da Silva et al., 2004a).



The heat flux distribution that corresponds to Tw ¼ Tmax and Pr01 is Nu ffi 0:5Ra1=4y ; or

q00

Tmax � T1

y

kffi 0:5

gbðTmax � T1Þy3

an

� �1=4ð20Þ

By eliminating q00(y) between Equations (19) and (20), we obtain the required distribution ofheat sources:

N 0ðyÞ ffi 0:5k

q0ðTmax � T1Þ5=4

gban

� �1=4

y�1=4 ð21Þ

This function shows that the heat sources must be positioned closer when they are near the startof the boundary layer. The total number of q0 sources that must be installed on the wall of heightH is

N ¼Z H

0

N 0dy ffi2

3

k

q0ðTmax � T1ÞRa1=4 ð22Þ

where Ra ¼ gbH3ðTmax � T1Þ=ðanÞ: The Rayleigh number accounts for two global constraints,the wall height H and the maximal allowable excess temperature at the hot spots. The total heattransfer rate from the q0 sources to the T1 fluid is

Q0 ¼ q0N ffi 23 kðTmax � T1ÞRa1=4 ð23Þ

This represents the global performance level to which any of the optimized non-uniformdistributions of concentrated heat sources will aspire.

The physical implementation of the preceding results begins with the observation that thesmallest scale that can be manufactured in the heating scheme of Figure 6 is the D0 height of theline heat source. The fixed, smallest scale is an intrinsic feature (a first step) in the constructionguided by constructal theory. The local spacing between two adjacent lines is S(y). This varieswith altitude in accordance with the N0 distribution function (21). The wall height interval thatcorresponds to a single line heat source is D0 þ SðyÞ: This means that the local number of heatsources per unit of wall height is

N 0ðyÞ ¼1

D0 þ SðyÞð24Þ

The strength of one heat source (q0) is spread uniformly over the finite height of the source,q000 ¼ q0=D0: The heat flux q000 is a known constant, unlike the function q00(y) of Equation (20),which will be the result of design. By eliminating N0(y) between Equations (21) and (24), weobtain the rule for how the wall heating scheme should be constructed

SðyÞH

ffi2q0Ra�1=4

kðTmax � T1Þy

H

� �1=4�D0

Hð25Þ

This S function has negative values in the vicinity of the start of the boundary layer. Thesmallest physical value that S can have is 0. This means that there is a starting wall section(05y5y0) over which the line sources should be mounted flush against each other. The heightof this section (y0) is obtained by setting S ¼ 0 and y ¼ y0 in Equation (25),

y0

Hffi Ra

D0

H

� �4kðTmax � T1Þ

2q0

� �4ð26Þ



From y ¼ 0 to y0, the wall is heated with uniform flux of strength q000 ¼ q0=D0: The number ofinfinite-height sources that cover the height y0 is N0 ¼ y0=D0: Above y ¼ y0; the wall is heatedon discrete patches of height D0, and the spacing between patches increases with height.

These basic features of the optimal design are illustrated in Figure 6. The design has multiplelength scales: H, D0, y0 and S(y). The first two are constraints. The last two are interrelated, andare results of global maximization of performance, subject to the constraints. Taken together,the lengths represent the constructal design}the flow architecture that out of an infinity ofpossible architectures brings the entire wall to the highest performance level possible.

The global heat transfer performance of the optimal design can be estimated in the limitwhere the number of heat-source strips D0 is sufficiently large. In this limit, integral (22) appliesonly in the upper region of the wall (y05y5H), where the concentrated sources are spacedoptimally according to Equation (21). In the lower region of height y0, the D0 strips are mountedwithout spacings between them, and their number is N0 ¼ y0D0 ¼ RaðD0=HÞ3½kðTmax2T1Þ=2q0�4: The total number of D0 strips on the H wall is N ¼ N0 þ

RH

y0N 0 dy: It can be shown

(da Silva et al., 2004a) that the total rate of heat transfer from the wall to the fluid (Q0 ¼ q0N) is

Q0 ffi 23kðTmax � T1ÞRa1=4 1�

3

16

kðTmax � T1Þ2q0

� �3D0

H

� �3

Ra3=4

" #ð27Þ

By comparing Equation (27) with Equation (23), we see that when D0 is finite the total heattransfer rate is less than in the limit of line heat sources (D0 ¼ 0). The analytical multi-scaledesign results were confirmed by numerical simulations and optimization of natural convectionin enclosures (da Silva et al., 2004a) and vertical channels (da Silva et al., 2004b).

6. MULTI-SCALE HEAT SOURCES: FORCED CONVECTION

The corresponding analysis for optimizing the placement of multi-scale heat sources on a wallwith laminar forced convection was presented in da Silva et al. (2004c). Here, we review themain results for Pr00:5; with reference to Figure 7 and the analytical steps detailed in thepreceding section. If the heat sources are sufficiently numerous, the optimal spacing mustincrease in the longitudinal direction according to the formula

SðxÞL

ffi3q0Pr�1=3Re�1=2

kðTmax � T1Þx

L

� �1=2�D0

Lð28Þ

Figure 7. The multiple length scales of the non-uniform distribution of heat sources on a wall with laminarforced convection (da Silva et al., 2004c).



where Re ¼ U1L=n: The starting section length x0 must be covered with heat sources with zerospacings,

x0

L

� �1=2ffi 0:664

D0k

Lq0ðTmax � T0ÞPr1=3Re1=2 ð29Þ

The total heat transfer rate from the wall with optimally distributed heat sources is

Q0 ffi 0:664kðTmax � T0ÞPr1=3Re1=2 1�x0

L

� �1=2� �

ð30Þ

These trends were confirmed by numerical optimization results (da Silva et al., 2004c) based onfull simulations of the flow and heat transfer in a large number of configurations of the classdefined in Figure 7.

7. FLOW AND STRENGTH OPTIMIZATION OF GEOMETRY

The two-objective (thermal and mechanical) optimization of geometry proposed in Lorente andBejan (2002) occurred on a rich background of research where the thermal and mechanicalobjectives have been pursued separately. The field of strength of materials is rich in examples ofoptimal shapes and structures for prescribed stiffness with minimum mass, or prescribed masswith maximum stiffness (Beer et al., 2002), e.g. the cantilever beam (Galilei, 1960) and thecolumn in end compression (Wilson et al., 1971). In heat transfer, there are many examples ofshape optimization with a single objective. For example, the minimization of global thermalresistance was the driving force in the shaping of fins (Kraus et al., 2001) and two-dimensionalenclosures with natural convection (Bejan, 1980). The maximization of thermal resistance indeformable enclosures with natural convection was described in Lartigue et al. (2000) andLorente (2002).

In Lorente and Bejan (2002), we proposed to combine thermal and mechanical objectives inthe discovery of optimal internal structure for walls that must have stiffness and thermalinsulation capabilities simultaneously. The fundamental problem is sketched in Figure 8. A wallis made out of terra-cotta bricks with air-filled caverns.

Here is how the internal structure of the cavernous wall results from the competitionbetween the two objectives. For a large stiffness in bending and buckling (end-to-endcompression), the wall design requires a large cross-sectional area moment of inertia. To meetthis objective, the wall must have all its solid material placed in two outermost layers. Thisarrangement, however, leaves one large air cavity in the middle, and the heat leak by naturalconvection through such a cavity is large. The competing extreme is one with numerous andvery narrow air slots and solid sheets. Such a configuration offers large thermal resistance andinadequate stiffness.

In summary, we face two extremes (one airspace vs many narrow airspaces), and bothextremes are bad. Both extremes highlight the imperfections that plague the system. Theconstructal way out of this difficulty is to ‘optimally distribute the imperfection’ (Bejan, 2000),i.e. to balance one imperfection against the other, or to intersect the asymptotes (Lewins, 2003).There is an in-between architecture that meets both objectives, mechanical stiffness and thermalresistance.



Figure 9 shows how to maximize the global thermal resistance ( *RR) when the stiffness of theentire wall is specified. The Rayleigh number on each curve (Ra ¼ gbðTleft2TrightÞH3=ðan))indicates the thermal and global constraints: the temperature difference and wall height arespecified. The specified area moment of inertia *II indicates that the figure is drawn for a class ofarchitectures with the same thickness. The dimensionless stiffness *II is defined as *II ¼ I=ðL3W=12Þ; where I is the area moment of inertia of the wall section A-A. The optimal configuration isindicated by the peak of the thermal resistance curve, which occurs at a certain number of airgaps, n. The dimensionless number b is shorthand for (L/H)Ra1/4. This optimal air cavitynumber (or size) leads to the remaining dimensions of the wall structure, when the wall thicknessand relative solid volume (solidity) are specified. As Ra increases, the optimal wall structurebecomes finer, with more and narrower air cavities. Details for how to determine and use theoptimal dimensions of the cavernous structure are provided in Bejan et al. (2004) and Lorenteand Bejan (2002).

Figure 8. Two objectives: stiffness and thermal insulation in a brick wall with verticalair cavities (Lorente and Bejan, 2002).



8. MECHANICAL STRUCTURES UNDER THERMAL ATTACK

The classical approach to providing a structure with thermal resistance against intense heating isby coating the structure with a protective layer after the structure has been designed (Lawson,2001). Gosselin et al. (2004) changed the conceptual approach to optimal structures, away fromthe single-objective lessons of the past, and in line with the two-objective morphing of structuresshown in Lorente and Bejan (2002). In this section, we illustrate this approach by using twoclasses of structures exposed to sudden heating: beams in pure bending and beams of concretereinforced with steel. In both classes, the solid structure is penetrated by time-dependentconduction heating. The mechanical and thermal objectives compete, and this competitiongenerates the optimal geometry of the system.

Consider a beam simply supported at each end (Figure 10). The beam geometry is two-dimensional, with the length L and symmetric profile H(x). The total load F [Nm–1] isdistributed uniformly over the beam length L. The force F is expressed per unit length in thedirection perpendicular to the plane of Figure 10. The weight of the beam is assumed to benegligible in comparison with the load. The beam profile is sufficiently slender so that itsdeformation in the y-direction is mainly due to pure bending.

The beam is initially isothermal at the ambient temperature T1, where it behaves elasticallythroughout its volume. The modulus of elasticity is E, which for simplicity is assumed constant.

Figure 9. The maximization of global thermal resistance when the mechanical stiffness is specified in theoptimization of the cavernous structure of Figure 8 (Lorente and Bejan, 2002).



Thermal attack means that at the time t ¼ 0 the beam is exposed on both surfaces to theuniform heat flux q00. Temperatures rise throughout, but they rise first in the subskin regions.These are also the first regions where the material behaviour changes from elastic to plastic. Thelast to undergo this change is the core region, in which the material behaves elastically.

We considered a family of beam shapes that are smooth and thicker in the middle, Figure 10:

*HH ¼ C½ *xxð1� *xxÞ�m ð31Þ

where ð *HH; *xxÞ ¼ ðH;xÞ=L: The shape parameters C and m are related through the size constraint

A=L2

C¼

Z 1

0

½ *xxð1� *xxÞ�m d *xx ð32Þ

where A is the area of the profile shown in Figure 10. The geometry is characterized by oneshape parameter (m), which plays the role of degree of freedom, and by three constructionparameters: *AA; *bb and

*ssy;ref ¼syðT1ÞF=L

ð33Þ

where sy is the yield stress of the beam material. The yield stress decreases as the temperatureincreases, sy=sy;ref ¼ 12bðT2Tref Þ; where b is a property of the material. The dimensionless *bb isequal to bq00L/(2k).

The calculation of the vertical beam deflection at its midpoint ð*ddmÞ was performed from t ¼ 0until the elastic core disappeared at a location x. The numerical example given in Figure 11shows that the deflection increases in accelerated fashion in time, and that *ddm can be minimizedby selecting the shape parameter m. This is the key result: the beam geometry can be selected insuch a way that the beam as a whole is most resistant to thermal attack. This is a result of howthe whole beam performs}a global result}because *ddm is a global feature. All the strained fibrescontribute to *ddm:

Figure 10. Beam in bending with uniform loading and sudden heating fromabove and below (Gosselin et al., 2004).



A second example is the opportunity for optimizing the internal structure of a beam ofconcrete reinforced with steel bars (Gosselin et al., 2004). Once again, the objective is maximalsurvivability to thermal attack. The beam is in pure bending, and its cross-section is shown inFigure 12. The steel bars run in the direction perpendicular to the figure, and are modelled as aslab with cross-section hs � b: The beam is loaded such that the steel slab is in tension, while theconcrete situated above the neutral line is in compression.

Thermal attack is modelled as a uniform heat flux (q00), which is imposed suddenly on theperiphery of the beam cross-section. The most critical part that is vulnerable under thermalattack is the steel; therefore, in the simplest model (Gosselin et al., 2004) the focus was on theheating that is applied on the bottom of the cross-section, which is the closest to the steel. Alayer of concrete of thickness l protects the steel against the thermal wave driven by theimposed heat flux q00. The thickness l plays an important role. In order for the beamto support a large load, l must be small: the steel must be positioned as far as possiblefrom the top of the beam cross-section. On the other hand, a high resistance to thermalattack requires a large l. The competition between these two requirements represents anoptimization opportunity, which is amply demonstrated by the results reported in Gosselin et al.(2004).

Figure 11. The effect of the beam lifetime (*tt) on the minimization of the mid-lengthdeflection (Gosselin et al., 2004).



9. CONCLUDING REMARKS

This paper reviewed the constructal-theory progress made by our group in two fundamentaldirections. One direction was the basic question of how to maximize the density of heat transferrate in a finite-size volume. We gave three examples. In the first, we saw that it is possible tooptimize the internal spacings of volumes with forced convection when boundary layersdisappear. This was demonstrated for one configuration: a volume filled with parallel plates. Avolume packed with spheres of one size can be optimized by using the same method (Bejan,2004a).

The second example reviewed the constructal principle of generating a multi-scale flowstructure that maximizes the heat transfer density installed in a fixed volume. The number ofscales of the multi-scale flow structure (m) increases slowly as the flow becomes stronger. Theflow strength is accounted for by the pressure difference maintained across the structure (Be).Two trends compete as the number of length scales increases. The structure becomes lesspermeable, and the flow rate decreases. At the same time, the total heat transfer surfaceincreases. The most important result is that the heat transfer density increases as the number of

Figure 12. The cross-section of a beam of concrete reinforced with steel and heatedsuddenly from below (Gosselin et al., 2004).



length scales increases. This increase occurs at a decreasing rate, meaning that each new(smaller) length scale contributes less to the global enterprise than the preceding length scale.There exists a characteristic length scale below which heat transfer surfaces are no longer linedby boundary layers. This smallest scale serves as cutoff for the theory-based algorithm thatgenerates the multi-scale structure.

In the third example of heat transfer density maximization, we considered the problem of howto arrange a number of discrete heat sources on a vertical wall with natural convection. Weshowed analytically that an optimal arrangement exists. The spacings between heat sources arenot uniform, and depend on the Rayleigh number. The optimal distribution of heat sourcesleads to maximal global performance}the minimal global thermal resistance between the walland the fluid. Heat sources must be positioned closer together when they reside near the start ofthe boundary layer. There is a region near the tip of the boundary layer where the heat sourcesmust be positioned flush against each other. Downstream from this region, the spacing betweenheat sources increases.

These conclusions have wide applicability because many engineering applications call for thejudicious distribution of ‘currents’ through volumes. This requirement stems from the need tomaximize the global performance of the macroscopic system, and the thought that every volumeelement should function at the same (the highest) level of performance as any other volumeelement. Because of this, uniform distribution of flow rate is often a requirement in thedesign of banks of parallel tubes in heat exchangers. Uniformity means single scale. Thework reviewed in this paper is a jolt in the opposite direction, where non-uniformly distributedmulti-scale structures (optimally maldistributed structures) promise greater heat transferdensity and compactness. This theme is explored in greater depth in a new book (Bejan et al.,2004).

The second fundamental direction is that the optimization of shape and structure in multi-objective systems is numerous and manifold. To address simultaneously their objectives calls fortruly interdisciplinary research. The optimal architecture of the multi-objective system is aconsequence of the competition between objectives. For example, in Section 8, the competitionis between the requirement of high strength in the absence of thermal attack, and the call forthermally insulated structures that resist thermal attack. The beam geometry is generated byconflict. The steel bars in a beam of reinforced concrete must be placed as far as possible fromthe top of the beam cross-section, in order to support the largest bending moment. On the otherhand, the steel bars must be positioned far from all the exposed surfaces (including the bottomof the beam cross-section) in order to maximize the resistance to the heat wave that penetratesthe beam.

The reviewed work showed that the ‘combined heat flow and strength’ method proposed inLorente and Bejan (2002) can be used in a wide domain of great contemporary importance:structures that combine mechanical strength with thermal resistance. At a fundamental level,this method is a proposal to change the conceptual design of structures, such that all theobjectives are pursued from the start. For beams with strength and thermal resistance, thispaper shows how to combine two disciplines before the start of conceptual design: strength ofmaterials and heat transfer.

More realistic models can be used in conjunction with the methods reviewed in this paper, inthe pursuit of optimal multi-scale and/or multi-objective architectures. Structures of greatercomplexity promise to benefit from the multi-scale and multi-disciplinary approach advocatedin this review.



NOMENCLATURE

A = beam cross-sectional area (m2)Be = pressure drop number, Equation (2)cP = specific heat at constant pressure (J kg–1K–1)C = factor, Equation (31)d = disc diameter (m)D = spacing (m)F = force per unit length (Nm–1)g = gravitational acceleration (m s–2)H = height (m)I = area moment of inertia (m4)k = fluid thermal conductivity (Wm–1K–1)L = length (m)m = smallest scalem = exponent’mm = mass flow rate (kg s–1)ni = number of platesN = number of heat sourcesN0 = number of sources per unit length (m–1)Pr = Prandtl number (n/a)q = heat transfer rate (W)q0, Q0 = heat transfer rate per unit area (Wm–1)q00 = heat flux (Wm–2)q- = heat transfer rate density (Wm–3)Ra = Rayleigh number, Equation (22)Re = Reynolds number, Equation (28)S = structure function, Equation (18)S = spacing (m)Tw = wall temperature (K)T0 = entrance temperature (K)T1 = free stream temperature (K)U = mean velocity (m s–1)x = longitudinal coordinate (m)XT = entrance length (m)y = vertical coordinate (m)

Subscripts

max = maximumopt = optimum

Superscripts

(�) = dimensionless variables, Equations (11), (31) and (33)

Greek letters

a = thermal diffusivity (m2 s–1)



b = coefficient of thermal expansion (K–1)b = factor, Equation (33)d = boundary layer thickness (m)DP = pressure difference (Nm–2)m = viscosity (kg s–1m–1)n = kinematic viscosity (m2 s–1)r = density (kgm–3)sy = yield stress (Nm–2)

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Constructal multi-scale and multi-objective structures