constructal multi-scale structure for maximal heat transfer density - bejan

8/13/2019 Constructal Multi-scale Structure for Maximal Heat Transfer Density - Bejan

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Constructal multi-scale structure for maximal

heat transfer density

A. Bejan, Durham, North Carolina, andY. Fautrelle, Grenoble, France

Received January 3, 2003Published online: June 12, 2003 Springer-Verlag 2003

Summary. This paper presents a new concept for generating the multi-scale structure of a finite-size

flow system that has maximum heat transfer densitymaximum heat transfer rate installed in a fixed

volume. Laminar forced convection and parallel isothermal blades fill the volume. The spacingsbetween adjacent blades of progressively smaller scales are optimized based on constructal theory: the

goal is maximum heat transfer density. The smaller blades are installed in the fresh-fluid regions that

sandwich the tips of the boundary layers of longer blades. The overall pressure difference is con-

strained. As the number of length scales increases, the flow rate decreases and the volume averaged

heat transfer density increases. There exists a smallest (cutoff) length scale below which heat transfer

surfaces are no longer lined by distinct (slender) boundary layers. Multi-scale flow structures for

maximum heat transfer rate density can be developed in an analogous fashion for natural convection.

The constructal multi-scale algorithms are deduced from principles, unlike in fractal geometry where

algorithms are assumed.

1 Geometry

A key result of constructal theory is the prediction of optimal spacings for the internal flow

structure of volumes that must transfer heat and mass to the maximum. This body of work

comprises both forced and natural convection, and is reviewed in [1], [2]. Optimal spacings have

been determined for several configurations, depending on the shape of the heat transfer surface

that is distributed through the volume: stacks of parallel plates, bundles of cylinders in cross-

flow, and arrays of staggered plates. In each configuration, the reported optimal spacing is a

single value, i.e., a single length scale that is distributed uniformly through the available vol-

ume.

Figure 1 shows qualitatively why an optimal spacing exists. The heat-generating blade shown

at the top of the figure is one of many parallel blades that fill a much larger package. Isolated in

Fig. 1 is the volume allocated to a single blade. The size of this volume is fixed, and is repre-

sented by the rectangular space V shown in Fig. 1ac. The shape ofV is not fixed. The fluid

sweeps the blade and convects heat. The boundary layers are regions that work (are active) in a

heat transfer sense. WhenVis considerably wider than the boundary layers (Fig. 1a), most ofV

is occupied by coolant that does not help the heat transfer enterprise. WhenVis much narrower

than the boundary layers (Fig. 1c), most ofVis occupied by thermally fully developed flow, i.e.,

by coolant that warms up in the downstream direction. Such a fluid is overworked, and

becomes poorer as a coolant.

Acta Mechanica 163, 3949 (2003)

DOI 10.1007/s00707-003-1008-3

Acta MechanicaPrinted in Austria


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The best configuration is in-between, Fig. 1b. Here the volume is just wide enough to house

the flow regions that work. The boundary layers merge in the plane of the outlet. This trans-

versal dimension ofV represents the optimal spacing between the parallel blades in the stack.

Three such blades are shown in Fig. 2, whereL0D0has replaced the two-dimensional volumeV

of Fig. 1. The spacingD0is the single length scale that is distributed uniformly through the two-

dimensional flow structure of lengthL0 and widthH. For concreteness, assume laminar forced

convection driven by the imposed pressure difference DP. Assume also that the blades are

isothermal, Tw, and have negligible thickness. Furthermore, the fluid Prandtl number is of

order 1, so that the velocity and thermal boundary-layer thicknesses are both represented by the

Blasius thickness (e.g., [3]),

dx ffi 5x Ux

m

1=2: 1

In this expressionUand m are the free stream velocity and the kinematic viscosity.

Is the stack of Fig. 2 the best way to pack heat transfer into a fixed volume? It is, but only

when a single length scale is to be used, that is, if the structure is to be uniform. The structure of

Fig. 2 is uniform, because it does not change fromx 0 to x L0. At the most, the geometries

of single-spacing structures vary periodically, as in the case of arrays of cylinders and staggered

plates.

The key observation is that the structure of Fig. 2 can be improved if more length scales

(D0;D1;D2;. . .) are available. The technique consists of placing more heat transfer in regions of

the volumeHL0where the boundary layers are thinner. Those regions are situated immediately

downstream of the entrance plane, x 0. This observation is the same as taking the argument

of Fig. 1 to a finer level: regions that do not work in a heat transfer sense must either be put to

work, or eliminated. In Fig. 2, the wedges of fluid contained between the tips of opposing

cold fluid

convective body

V

V

V

heat-generating blade

unused volume

overworked fluid

a

b

cFig. 1. The optimal volume shape for

the flow associated with one blade

40 A. Bejan and Y. Fautrelle


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boundary layers are not involved in transferring heat. They can be involved if heat-generating

blades of shorter length (L1) are installed on their planes of symmetry. This new design is shown

in Fig. 3.

Each new L1 blade is coated by boundary layers described by Eq. (1). Becaused increases

as x1/2, the boundary layers of the L1 blade merge with the boundary layers of the L0

blades at a downstream position that is approximately equal to L0/4. The approximation is

due to the assumption that the presence of the L1 boundary layers does not affect signifi-

cantly the downstream development (x > L0/4) of the L0 boundary layers. This assumption

is made for the sake of simplicity. The order-of-magnitude correctness of this assumption is

clear, and it comes from geometry: the edges of the L1 and L0 boundary layers intersect

when

L1 1

4L0: 2

Note that by choosing L1 such that the boundary layers that coat the L1 blade merge with

surrounding boundary layers at the downstream end of theL1 blade, we invoke one more time

the optimal packing principle of Fig. 1. We are consistent, 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 be

populated with a new generation of even shorter blades, L2 L1/4. Two such blades are shown

in the upper-left corner of Fig. 3. The length scales become smaller (L0,L1,L2), but the shape of

the boundary layer region is the same for all the blades, because the blades are all swept by the

same flow (U). The merging and expiring boundary layers are arranged according to thealgorithm

Li 1

4Li1; Di

1

2Di1 i 1; 2;. . .;m; 3

where we shall see that m is finite, not infinite. In other words, as in all the constructal tree

structures [1], the image generated by the algorithm (3) is not a fractal. It is a Euclidean image

[4], [5]. The sequence of decreasing length scales is finite, and the smallest size (Dm, Lm) is

known, as we will see in Eq. (21).

U, T0, P

0 x L0

TW

D0

Hd(x)

Fig. 2. Optimal package of parallel

plates with one spacing

Multi-scale structure for maximal heat transfer density 41


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To complete the description of the sequential construction of the multi-scale flow structure,

we note that the number of blades of a certain size increases as the blade size decreases. Let n0

be the number ofL0 blades in the uniform structure of Fig. 2,

n0 H

D0; 4

where

D0 ffi 2dL0 ffi 10

mL0

U 1=2

: 5

The number of L1 blades is n1 n0, because there are as many L1 blades as there are D0spacings. At scales smaller thanL1, the number of blades of one size doubles with every step,

ni 2ni1; i 2; 3;. . .;m: 6

Two conflicting effects emerge as the structure grows in the sequence started in Fig. 3. One is

attractive: the total surface of temperatureTw installed in theHL0 volume increases. 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 important question is

how the volume is being used: what happens to the heat transfer rate density as complexity

increases?

2 Heat transfer

The total heat transfer rate from the Tw surfaces to the T0 fluid can be estimated by

summing up the contributions made by the individual blades. The heat transfer rate through

one side of the L0 blade is equal (in an order of magnitude sense) to the heat transfer rate

associated with a laminar boundary layer, cf. the Pohlhausen solution for Prandtl numbers

of order 1 [3],

D0

L2

L1

L0

D2

D1

0 x Fig. 3. Optimal multi-scale package of

parallel plates



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qq000

DT

L0

k ffi 0:664

UL0

m

1=2: 7

Here qq000

[W/m2] is the L0-averaged heat flux, DT Tw T0, and k is the fluid thermal con-

ductivity. There are 2n0 such boundary layers, and their combined contribution to the totalheat transfer rate of the package of Fig. 3 is

q00

2n0 qq000L0 ffi 1:328kDT n0

UL0

m

1=2: 8

The same calculation can be performed for any group of blades of one size, Li. Their total heat

transfer rateq0i [W/m] is given by a formula similar to Eq. (8), in whichn0 andL0 are replaced

by ni and Li,

q0iffi 1:328kDT niULi

m

1=2: 9

The heat transfer rate of all the blades is the sum

q0 Xmi0

q0iffi 1:328kDT n0UL0

m

1=2

S; 10

whereS is the dimensionless geometric parameter

S 1 n1

n0

L1

L0

1=2

n2

n0

L2

L0

1=2

nm

n0

Lm

L0

1=2

11

or, in view of Eqs. (3) and (6),

S 1 m

2 : 12

This analysis confirms the trend noted at the end of the preceding section: the total heat transfer

rate increases monotonically as the complexity of the structure (m) increases.

3 Fluid friction

It is necessary to evaluate the flow resistance of the multi-scale structure, because the velocityU

that appears in Eq. (10) is not specified. The pressure difference DPis specified, and it is related

to all the friction forces felt by the blades. We rely on the same approximation as in the case of

heat transfer, and estimate the friction force along one face of one blade by using the solution

for the laminar boundary layer (e.g., [3]),

si ffi Cfi1

2qU2; Cfi

1:328

UL0=m 1=2

: 13

Heresiand Cfiare the averaged shear stress and skin friction coefficient, respectively. The total

force felt by the blades of size Li is

Fi 2nisiLiffi 1:328q mLi 1=2

niU3=2: 14

The total force for the multi-scale package is

FXmi0

Fi ffi 1:328q mL0 1=2

n0U3=2S: 15



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This force is balanced by the longitudinal force imposed on the control volume,

DPH F: 16

Finally, by combining Eqs. (15), (16) withn0 H=D0 and the D0 formula (5), we obtain the

order of magnitude of the average velocity of the fluid that bathes the structure:

Uffi 2:7 DP

qS

1=2: 17

This result confirms the second trend anticipated at the end of Sect. 1: the flow slows down as

the complexity of the structure (S, or m) increases.

4 Heat transfer rate density: the smallest scale

Putting together the results of the heat transfer and fluid flow analyses, we find how the

structure performs globally, when its constraints are specifiedDP;DT;H;L0. Eliminating U

between Eqs. (10) and (17) yields the dimensionless global thermal conductance,

q0

kDTffi 0:36

H

L0Be

1=2S1=2; 18

where Be is the dimensionless pressure drop that Bhattacharjee and Grosshandler [6] and

Petrescu [7] termed the Bejan number,

Be DPL2

0

la : 19

In this expressionland aare the fluid viscosity and thermal diffusivity. The alternative to using

the global conductance is the heat transfer rate density, q000 q0=HL0. Both quantities increase

with the applied pressure difference (Be) and the complexity of the flow structure (S). In

conclusion, in spite of the conflicting effects ofS in Eqs. (10) and (17), the effect of increasing S

is beneficial from the point of view of packing more heat transfer in a given volume. Optimized

complexity is the route to maximal global performance in a morphing flow system [1].

How large can the factorSbe? The answer follows from the observation that the geometry of

Fig. 3 and the analyses of Sects. 2 and 3 are valid when boundary layers exist, i.e., when they

are distinct. To be distinct, boundary layers must be slender. Figure 3 makes it clear that

boundary layers are less slender when their longitudinal scales (Li) are shorter. The shortest

blade lengthLm below which the boundary layer heat transfer mechanism breaks down is

Lm Dm: 20

In view of Eqs. (3), this means that

L0 2mD0: 21

Finally, by using Eqs. (5) and (17) we find the smallest scale, which occurs at the levelm given by

2m

1 m

2

1=4 0:17Be1=4: 22

In view of the order-of-magnitude character of the analysis based on Eq. (20), the right side of

Eq. (22) is essentially (Be/103)1/4. Equation (22) establishes m as a slowly varying monotonic

function ofBe1/4. This function can be substituted in Eq. (18) to see the complete effect ofBe on

the global heat transfer performance,



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q0

kDTffi 0:36

H

L0Be

1=21

1

2m

1=2: 23

In conclusion, the required complexity (m) increases monotonically with the imposed pressure

difference (Be). More flow means more length scales, and smaller smallest scales. The structurebecomes not only more complex but also finer. The monotonic effect ofmis such that each new

length scale (m) contributes to global performance less than the preceding length scale (m 1).

If the construction started in Fig. 3 is arbitrarily continued ad infinitum, then the resulting

image is a fractal andm and the heat transfer density (23) tend to infinity.

5 Symmetric growth

Imagine that the blades of the multi-scale structure of Fig. 3 are connected solidly at their

downstream end connected to the neighboring blade of immediately larger scale (see Fig. 4a).

This is certainly not a flow design recommendation, because the fluid must be able to flow

through the structure. It is a useful way to think, because it shows that the population ofLi

blades can be viewed as the branches of a tree structure, and that in this case the tree is

asymmetric. In Fig. 4a, the new (smaller) branches grow on only one side of the tips of the

existing branches.

Nature strikes us with images of tree-shaped structures endowed with symmetry, more like

the drawing shown in Fig. 4b. Here, new branches form on both sides of the tip of an existing

blade. Is this type of growth more advantageous from the point of view of packing maximal

heat transfer into a given volume?

The symmetric-growth alternative to Fig. 3 is shown in Fig. 5. In oneD0 spacing we placed

two L1 blades separated by the distance D1 D0=3. The new boundary layers merge with

neighboring boundary layers whenL1 L0=9. This rule follows from the nonlinear shape of

each boundary layer region, Eq. (1). If the number ofL0 blades in the package is n0, then thenumber ofL1 blades is n1 2n0.

a

b

Fig. 4. Asymmetric (a) versus sym-

metric (b) trees built with blades of

decreasing lengths and increasing

numbers



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The construction can be continued toward smaller scales, by placing two new blades in the

near-entrance region of every spacing. The dimensions of the growing structure follow the

algorithm:

Li1

9Li1; Di

1

3Di1; 24

ni 3i1

2n0 i 1; 2;. . .;ms: 25

The heat transfer and fluid mechanics analyses lead again to Eqs. (10), (11), (15), (17) and (18),

with the difference that Eq. (11) now yields

S 1 2

3ms; 26

wherems is the number of gap filling steps used in the symmetric construction. The subscripts

refers to the symmetric construction of Fig. 5, and is a reminder that the number of con-

struction steps (ms) is not the same as in the asymmetric construction (m), Fig. 3. The number

mscan be determined based on the same argument as in Eq. (20). The smallest blade lengthLmscannot be smaller than

Lms Dms ; 27

whereLms 9msL0and Dms 3

msD0. After using Eq. (5) forD0and Eq. (17) forU, we arrive

at

3ms 1

2

3ms

1=4

Be=1031=4: 28

Equation (28) is plotted in Fig. 6. We see that the complexity (ms) increases as the flow

becomes stronger (i.e., asBe increases). This trend is the same as in the asymmetric structure,

Eq. (22). Furthermore, by comparing Eqs. (28) and (22) when Be is specified, we conclude

that ms < m. The optimal symmetric structure is simpler because it is based on fewer con-

struction steps than the asymmetric structure. The relation between ms and m is roughly

ms=m ln 2= ln 3 0:63, and is only weakly dependent on Be.

The performance of the symmetric structure is given by the same relation as Eq. (23), withSs

in place ofS, namely

q0skDT

ffi 0:36H

L0Be

1=21

2

3ms

1=2; 29

for which the function ms(Be1/4) is furnished by Eq. (28). The performance relative to the

asymmetric case of Eq. (23) is measured by the ratio

L0

D0

D1

L1Fig. 5. Optimal multi-scale package of

parallel plates with two smaller blades

around each larger blade



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q0sq0

ffi 1 2ms=3

1 m=2

1=2

30

which is close to 1, but smaller than 1. The symmetric structure of Fig. 5 is only marginally less

effective than the asymmetric packing of Fig. 3. Figure 6 shows that the ratio q0s=q0 is a function

ofBe, because m and ms are functions ofBe.

6 Conclusions

In this paper, we described a new concept for generating a multi-scale flow structure that

maximizes the heat transfer density installed in a fixed volume. The method consists of ex-

ploiting every available flow volume element for the purpose of transferring heat. In laminar

forced convection, the working volume has a thickness that scales with the square root of the

length of the streamwise heat transfer surface. Larger surfaces are surrounded by thicker

working volumes. The starting regions of the boundary layers are thinner. They are surrounded

by fresh flow that can be put to good use: heat transfer blades with smaller and smaller lengths

can be inserted in the fresh fluid that enters the smaller and smaller channels formed betweenexisting blades.

The number of scales of the multi-scale flow structure (m) increases slowly as the flow

becomes stronger. The flow strength is accounted for by the pressure difference maintained

across the structure (Be). Boundary layers become thinner asBeincreases, and this means that

more small-scale heat transfer blades can be inserted in the interstitial spaces of the entrance

region of the complex flow structure.

Two trends compete as the number of length scales increases. The structure becomes less

permeable, and the flow rate decreases. At the same time, the total heat transfer surface

6

5

4

3ms ms

m m Fig.3

Fig.5

(Be / 103)1/4

(Be / 103)1/4

2

1

01 10 100

1

1

0.95

0.910 100

qs(Fig.5)

qs(Fig.3)

Fig. 6. The number of length scales

(m, ms) as functions of the imposedpressure difference (Be). The bottom

graph shows the relative heat transfer

density of the symmetric vs. asymmet-

ric multi-scale structures



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increases. The net and most important result is that the heat transferdensity increases as the

number of 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. We showed that there exists a characteristic length scale below which heat transfer

surfaces are no longer lined by boundary layers. This smallest scale serves as cutoff for thealgorithm that generates the multi-scale structure.

Forced convection was used in this paper only as a working example, i.e., as a flow mech-

anism on which to build the multi-scale structure. A completely analogous multi-scale structure

can be deduced for laminar natural convection. The complete analogy that exists between

optimal spacings in forced convection and natural convection was described by Petrescu [7]. In

brief, if the structure of Fig. 2 is rotated by 90 counterclockwise, and if the flow is driven

upward by the buoyancy effect, then the role of the overall pressure differenceDPis played by

the difference between two hydrostatic pressure heads, one for the fluid column of height L0

and temperatureT0, and the other for theL0fluid column of temperatureTw. If the Boussinesq

approximation applies, the effectiveDP due to buoyancy is

DP qgbDT L0; 31

where DT is Tw T0, b is the coefficient of volumetric thermal expansion, and g is the

gravitational acceleration aligned vertically downward (againstx in Fig. 2). By substituting the

DP expression (31) into Eq. (19) we find that the dimensionless group that replaces Be in

natural convection is the Rayleigh number

RagbDTL3

0

am : 32

Other than the Be ! Ratransformation, all the features due to the generation of multi-scale

blade structure for natural convection should mirror, at least qualitatively, the features de-

scribed for forced convection in this paper. The heat transfer augmentation effect due to theinsertion of a single plate in the entrance region of the parallel-plates channel was noted and

documented numerically by Aihara et al. [8].

Finally, the flow architecture constructed in this paper is a theoretical comment on fractal

geometry. Fractal structures are generated by assuming (postulating) certain algorithms. In

much of the current fractal literature, the algorithms are selected such that the resulting

structures resemble flow structures observed in nature. For this reason, fractal geometry is

descriptive, not predictive [1], [9]. It is not a theory.

Contrary to the fractal approach, the constructal theory used in this paper generated the

construction algorithms (Eqs. (3), (6), (24) and (25)), including the smallest scale cutoffs, Eqs.

(22) and (28). The algorithms were generated by the constructal principle of optimization of

global performance subject to global constraints [1]. This principle was invoked every time the

optimal spacing between two blades was used. Optimal spacings were assigned to all the lengthscales, and were distributed throughout the volume. With regard to fractal geometry and why

some fractal structures happen to resemble natural flow structures, the missing link has been

the origin of the algorithm [10]. Constructal theory delivers the algorithm as an optimization

result, from the constructal principle [1].



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References

[1] Bejan, A.: Shape and structure, from engineering to nature. Cambridge, UK: Cambridge

University Press 2000.

[2] Kim, S. J., Lee, S. W.: Air cooling technology for electronic equipment. Boca Raton: CRC Press

1996.

[3] Bejan, A.: Convection heat transfer, 2nd ed. New York: Wiley 1995.

[4] Bejan, A.: Advanced engineering thermodynamics, 2nd ed, p. 765. New York: Wiley 1997.

[5] Avnir, D., Biham, O., Lidar, D., Malcai, O.: Is the geometry of nature fractal? Science279, 3940

(1998).

[6] Bhattacharjee, S., Grosshandler, W. L.: The formation of a wall jet near a high temperature wall

under microgravity environment. ASME HTD 96 , 711716 (1988).

[7] Petrescu, S.: Comments on the optimal spacing of parallel plates cooled by forced convection. Int.

J. Heat Mass Transfer 37, 1283 (1994).

[8] Aihara, T., Ohara, T., Sasaco, A., Akaku, M., Gori, F.: Augmentation of free-convection heat

transfer between vertical parallel plates by inserting an auxiliary plate. 2nd European Thermal-

Sciences and 14th UIT National Heat Transfer Conference, Rome, Italy, 1996.

[9] Bradshaw, P.: Shape and structure, from engineering to nature. AIAA J. 39, 983 (2001).

[10] Nottale, L.: Fractal space-time and microphysics. Singapore: World Scientific 1993.

Authors addresses: A. Bejan, Department of Mechanical Engineering and Materials Science, Duke

University, Box 90300, Durham, North Carolina, NC 27708-0300, U.S.A.; Y. Fautrelle, Institut

National Polytechnique de Grenoble, EPM-MADYLAM Laboratory ENSHMG, B.P. 95, 38402 Saint

Martin dHeres Cedex, France


constructal multi-scale structure for maximal heat transfer density - bejan

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