mefte bragança (a. heitor reis)

Emergence of shape and flow structure in engineering and Nature in the light of Constructal Theory

A. Heitor Reis1,21 Department of Physics, University of Évora, Apartado 94, 7002-554 Évora, Portugal

2Geophysics Centre of Évora, Rua Romão Ramalho 59, 7000-671 Évora, Portugal

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Outline

The constructal Law The Constructal Law as a Minimum Time Principle The Constructal method Examples of application

•Heat and Fluid flow

•running, swimming and flying

•Flow architectures of the lungs

•Global Circulation and Climate

• Scaling Laws of River Basins

• Scaling Laws of Street Networks

• Beachface Morphing

Towards equilibrium flow structures Conclusions

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"For a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the currents that flow through it.“ (Adrian Bejan, 1996)

CONSTRUCTAL LAW



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Flow architectures are ubiquitous in Nature

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• A. Bejan, Shape and Structure, from Engineering to Nature Cambridge University Press, Cambridge, UK, 2000.• A. Heitor Reis, 2006, “Constructal Theory: From Engineering to Physics, and How Flow Systems Develop Shape and Structure”, Applied Mechanics Reviews, Vol.59, Issue 5, pp. 269-282.• A. Bejan and S. Lorente, Design with Constructal Theory, Wiley, 2008.

Reviews

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(U, S, V, N)

(U0, S0, V0, N0)

I

For some fixed XЄ(U,S,V,N) transfer between the two systems, “maximum flow access” (I max. ) corresponds to minimum transfer time” (Δt min.)

For some fixed XЄ(U,S,V,N) transfer between the two systems, “maximum flow access” (I max. ) corresponds to minimum transfer time” (Δt min.)

For some fixed XЄ(U,S,V,N) transfer between the two systems, “maximum flow access” (R min. ) corresponds to “minimum transfer time” (Δt min.)For some fixed XЄ(U,S,V,N) transfer between the two systems, “maximum flow access” (R min. ) corresponds to “minimum transfer time” (Δt min.)

Constructal law as a Minimum Time principle



• A. Heitor Reis, 2006, “Constructal Theory: From Engineering to Physics, and How Flow Systems Develop Shape and Structure”, Applied Mechanics Reviews, Vol.59, Issue 5, pp. 269-282.

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Constructal sequence of assembly and optimization, from the optimizedelemental area Ao, to progressively larger area-point flowsA. Bejan, Advanced EngineeringThermodynamics Sec. 13.5.

This global measure of flow imperfection can be minimizedwith respect to the shape of the area element. The optimalelemental shape is:

A. Bejan, Shape and Structure, from Engineering to Nature CambridgeUniversity Press, Cambridge, UK, 2000

The Constructal MethodArea-to-point flows

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Choice between too much fluid-flow resistance and too much heat-transfer resistance. The correct choice is to balance the two negative features so that their global effect is minimum. This balance yields the optimal spacing

Be is the dimensionless group

and are the fluid viscosity and thermal diffusivity, respectively.

Balancing global resistances Flow spacings

A. Bejan, Convection Heat Transfer, 3rd ed. Wiley, Hoboken, NJ, 2004

A. Bejan, Shape and Structure, from Engineering to Nature Cambridge University Press, Cambridge, UK, 2000

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To optimize the arrangement of heat-generating components, plate spacing and number of columns can vary. If spacing is too large or too small, the hot-spot temperature is high (in red). The optimal spacing is shown in the middle frame, where the hot spots are the coolest

A. Bejan, Shape and Structure, from Engineering to Nature Cambridge University Press, Cambridge, UK, 2000.

L fixed

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LINE-TO-LINE TREE FLOW



maximization of flow access between the points of one line and the points of a parallel line

Explanation for the occurrence of large pores and fissures through natural porous structures. These large features are strangely oriented, at an angle, not directly across the porous layer. Now we see why. The large channel P–P has two duties, to carry fluid across the layer directly and to feed the neighboring trees, which also carry fluid across the layer. The appearance of raggedness and disorganization is an illusion: such features come from the same principle as all the other features of the tree drawings.

S. Lorente and A. Bejan, JOURNAL OF APPLIED PHYSICS 100, 2006

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Making engineering design a Science

Flow architecture is the result and never is assumed in advance

Counterflow heat exchanger with two point-circle flow trees

A. K. da Silva, S. Lorente, and A. Bejan, J. Appl. Phys. 96, 1709 2004

A. Bejan and S. Lorente, Design with Constructal Theory, Wiley, 2008.

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A. Bejan and J. H. Marden, 2006, The Journal of Experimental Biology 209, 238-248

constructal theory for scale effects in running, swimming and flying

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A. Bejan, J.H. Marden. The constructal unification of biological and geophysical design. Physics of Life Reviews (2008)

Constructal theory of Constructal theory of flow architectures of flow architectures of the lungsthe lungs A. H. Reis, A. F. Miguel and M. Aydin, 2004 “Constructal theory of flow architectures of the lungs”, Medical Physics, V. 31 (5) pp.1135-1140.

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The respiratory treeThe respiratory tree• Starts at the trachea;

• Channels bifurcate 23 times before reaching the alveolar sac.

Has this special flow architecture been developed by chance or does it represent the optimum structure for the lung’s purpose, which is the oxygenation of the blood?



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Constructal law (principle) the flow architectures evolve in time in order to maximize the flow access under the constraints posed to the flow

Minimization of the global resistance to oxygen access with respect to level of bifurcation

Nopt = 23 (23.4)

Minimization of the global resistance to carbon dioxide removal with respect to level of bifurcation

Nopt = 23 (23.2)

1DL

TRD1035.2ln164.2N

ox

0ox

ox20

oxg40

4

opt



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Total resistance to oxygen and carbon dioxide transport between the entrance of the trachea and the alveolar surface is plotted as function of

the level of bifurcation (N).



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Constructal laws of the human respiratory tree

(I)

If the number Nopt=23 is common to mankind then a constructal rule

emerges:

m106.1.constL

D 3

0

20

“the ratio between the square of the trachea diameter and its length is constant and a length characteristic of mankind:”



20

2/1

ox0oxoxg

oxox

0

20

))(TR

D

V

AL63.8

L

D

V – Global volume of the respiratory tree; L – average lengthA – Area allocated to gas (O2 and CO2) exchange

(II)

The non-dimensional number AL/V, determines the characteristic length =D02/L0,

which determines the number of bifurcations of the respiratory tree by:

“The alveolar area required for gas exchange, A, the volume allocated to the respiratory system, V, and the length of the respiratory tree, L, which are constraints posed to the respiratory process determine univocally the structure of the lungs, namely the bifurcation level of the bronchial tree.”



CONSTRUCTAL THEORY OF GLOBAL CIRCULATION AND CLIMATE

A.Heitor Reis1 and Adrian Bejan2

1University of Évora, Department of Physics and Geophysics Center of Évora, Colégio Luis Verney, Rua Romão Ramalho, 59, 7000-671, Evora, Portugal

2Department of Mechanical Engineering and Materials Science, Duke University, Box 90300, Durham, NC 27708-0300 USA

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Paris, 11 Juin 2009Paris, 11 Juin 2009Théorie constrctale et géométries multi-échelleThéorie constrctale et géométries multi-échelleEmergence of shape and flow structure in engineering and

Nature in the light of Constructal TheoryEmergence of shape and flow structure in engineering and

Nature in the light of Constructal Theory

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Streamfunction lines of atmospherical long-term meridional circulation. The isoline unit is 10 Sverdrups and the ordinate represents altitude (adap. Phys. of, Climate, Peixoto and Oort, 1991)

Long-term meridional circulation

Haddley, Ferrel and Polar Cells

0 0

0 0

2

0

2

4

6

-6

-4

-2

-2

-1

20º 20º 40º 40º 60º 60º N S

10

8

6

4

2

PR

ES

SU

RE

(1

04P

a)

0º

LATITUDE 23



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Fig. 3 Zonal mean zonal velocity in ms–1 (shaded contours) and mass stream function in 1010 kg/s (lines) for a) DTR = 20K, b) 30 K, c) 60 K, d) 130 K and e)190 K ([9]).

Stenzel and Storch (2004)Model ECHAM (Modified) of ECMRWF

Earth’s rotation rate constant and equal to 24 h/dayEquator-to-pole temperature gradient (TR) allowed to vary

Question:

What induces the formation of the three-cell system of earth’s meridional circulation, really?

• For TR = 20 K only one cell develops;• For TR = 30 K two cell develop;• For TR = 60 K (actual earth’s conditions) and TR = 130 K

three cell develop; • For TR = 190 K only two cell develop; the Polar cell disappears!



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Constructal LawConstructal LawMaximization of heat flow leaving the equatorial belt – To each TH corresponds infinity surface partitions x=AH/A. One maximizes the heat flow q)

0x

T,0

x

T,0

x

q2L

2L

TH

With TH fixed, each curve q(x)TH, shows a maximum, which defines TL.. Then, with both TL and TH, x may be determined from the equation:

4.3

4.4

4.5

4.6

0.4 0.42 0.44 0.46 0.48x

qX

101

5W

TL=275.56K

TL=275.51K

TH=294.1K

TH=294.0K

TH=293.9K

TL=275.53K

B2

T)x1(Tx 4L

4H

The optimum point is determined from the minimum TL and corresponds to TL=275.51K and TH=294.0K, with xopt.= 0.434, which corresponds to latitude 25º40´



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0x

T,0

x

T,0

x

q2H

2H

TL

B2

T)x1(Tx 4L

4H

The optimum point is defined by the maximum of TH and corresponds to TH =288K e TL=265.5K, with xopt.= 0.8, which corresponds to latitude 53º10´

5.8

6.2

6.4

6.6

0.75 0.8 0.85

x

q (

X10

15W

) 6.0

TL=267.0K

TL=266.5K

TL=266.0K

TH=287.97K

TH=288.03K

TH=288.01K

Maximization of heat flow reaching the polar zone – To each TL corresponds infinity surface partitions x=AH/A.

Constructal LawConstructal Law

With TL fixed, each curve q(x)Tl, shows a maximum, which defines TL.. Then, with both TL and TH, x may be determined from the equation:



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LAT=25º40´N

LAT=25º40´S

x=0.43 q

q

LAT=53º10´N

LAT=53º10´S

x=0.80

q q

I II

T L=275.7K

T L=275.7K

T H=293.9K

TL=265.5K

TL=265.5K

TH=288.0K

240

250

260

270

280

290

300

0 0.2 0.4 0.6 0.8 1

x

T(K

)

LA

T=

25

º 4

0´

LA

T=

53

º 1

0´

TL

TL

TH

TH



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Average temperature on earth’s surface<T>I = 0.434 293.9 K + (1 – 0.434) 275.5 = 283.5 K<T>II = 0.800 288 K + (1 – 0.800) 265.5 = 283.5 K

Average temperature in the zone between the heat source and the heat sink (Ferrel Cell), THL=281.5K

0.434 293.5 K + (0.8 – 0.434)THL = 0.8 288 K

3.86

3.87

3.88

3.89

3.90

0 10 20 30 40 50 60 70 80 90LAT (º)

Sg

en (

101

4 W

/K)

x=0.

43

x=0.

8

Entropy generated as function of surface

partition, x

HLsH

20gen T

1

T

1q

T

1

T

1R)1(SS



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The constructal optimization provided the equator to pole temperature difference and the three-cell regime of meridional circulation. The poleward heat transfer seems to be determinant of the flow structure in line with the results of Stenzel

and von Storch (2004)

At slow rotation, the role of Earths’ rotation rate is determinant because it reduces the temperature gradient between dark and illuminate

hemispheres, but for values of day rotation period lower that 24 h it does not affect significantly the position of the three cells, as shown by

Jenkins (1996)

What is behind the three-cell system ?Earth’s rotation rate and/or equator to pole temperature difference?



River basins -River basins -Constructal Constructal theorytheory

A. Heitor Reis (2006) Geomorphology, 78, 201-206 Bejan, A., Lorente, S., Miguel, A. F. and Reis, A. H.: “Constructaltheory of distribution of river sizes”, §13. 4, pp. 774-779, in Bejan, A.: Advanced Engineering Thermodynamics, 3rd ed., Wiley, Hoboken, NJ, 2006

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River Douro

Length (km): 897 Drainage Area (km2): 97290Discharge (m3/s): 488



River networks are self-similar structures over a range of scales

L1ii RLL Horton’s law of stream lengths:

1.5 <RL<3.5

Bi1i Rnn Horton’s law of stream numbers:

3 <RB<5

L = (A)b b ~0.56 – 0.5

Hack’s law

2ωs D694.0F Melton’s law:

D = LT/A ; Fs= N/A



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CONSTRUCTAL THEORY (Bejan)

River basins as area-to-point flows

First construct made of elemental areas, A0 = H0L0.

A new channel of higher permeability collects flow from the elemental areas.



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D2Do

D1

AoAo

Do

D1

D2

Lo

52iiii DLmz

(a) (b)

Paris, 1 Juin 2009Paris, 1 Juin 2009Théorie constructale et géométries multi-échelleThéorie constructale et géométries multi-échelle

Minimization of z

Total area, total channel volume fixed

7410 2 /DD

7221 2 /DD

7310 2 /DD

7321 2 /DD

m m



Turbulent flow

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D2Do

D1

Ao

D3

AoDo

D1

D2

D3

D4

8150.zz dc

Quadrupling is better than assembling 8 elements

(c)

(d)

Pais, 11 Juin 2009Pais, 11 Juin 2009Théorie constructale et géométries multi-échelleThéorie constructale et géométries multi-échelle

m

m



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Conclusions

The scaling laws of geometric features of river basins can be anticipated based on Constructal Theory, which views the pathways by which drainage networks develop in a basin not as the result of chance but as flow architectures that originate naturally as the result of minimization of the overall resistance to flow (Constructal Law).

The ratios of constructal lengths of consecutive streams match Horton’s law for the same ratio.

Agreement is also found with the number of consecutive streams that match Horton’s law of ratios of consecutive stream numbers.

Hack’s law is also correctly anticipated by Constructal Theory, which provides Hack’s exponent accurately.

Melton’s law is verified approximately by Constructal Theory, which indicates 2.45 instead of 2 for the Melton’s exponent.

It was also shown that also with turbulent flow quadrupling is better than assembling eight elements, therefore anticipating Horton’s law of ratios of consecutive stream numbers

As has been demonstrated with many other examples either from engineering or from animate structures, the development of flow architectures is governed by the Constructal Law.



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Scaling laws of street networks

A. Heitor Reis (2007), Physica A, 387, 617–622

Paris, 11 Jui 2009Paris, 11 Jui 2009Théorie constructale et géométries multi-échelleThéorie constructale et géométries multi-échelle



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Lisboa, Portugal



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• Cities possess self-similar structures that repeat over a hierarchy of scales (Alexander, 1977, Krier, 1998).

IdealismIdealism: Because cities are complex man-made systems self-similarity springs out of the congenital ideas of beauty and harmony shared by mankind.

Constructal theoryConstructal theory: As with every living system, city networks have evolved in time such as to provide easier and easier access to flows of goods and people.

• Are city structures fractal objects? (Salingaros, 1995, Salingaros and West 1999). If yes, then why? Fractal geometry provides descriptions, rather than explanations!



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FRACTALFRACTAL DESCRIPTIONDESCRIPTION

• Hierarchically organized structures: a structure of dimension x is repeated at the scales rx, r2x, r3x, ...,

• Self-similarity at various scales

• The number of pieces N(X) of size X seems to follow an “inverse-power distribution law” of the type

orm – fractal dimension

n0n rXX

nm0n rCX)X(N

mCXXN

xrrx m



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FLOW STRUCTURES IN A LIVING CITYFLOW STRUCTURES IN A LIVING CITY



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Rivers of People and goods and “drainage” basins

Self-similarity at various scales

The same scaling laws hold

nL0n RLL

nNn RN



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Notice that the scale factor is given by 2~rL

L

1n

n

Therefore

LN2L2N m

, therefore m = 24~N

N

n

1n

m = fractal dimension xrrx m

mnn r)L(N nm

0n rCX)X(N



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• Chen and Zhou (2006) found that the fractal dimension of some German cities range from 1.5 (Frankfurt) to 1.8 (Stuttgart).

• Batty and Longley (1994) have determined the fractal dimension of cities by using maps from different years. They found values typically between 1.4 and 1.9. London’s fr actal dimension in 1962 was 1.77, Berlin’s in 1945 was 1.69, and Pittsburgh’s in 1990 was 1.78. Dimensions closer to 2 stand for denser cities.

• Shen (2002), carried out a study on the fractal dimension of the major 20 US cities and found that the fractal dimension, m, varied in between 1.3 for Omaha (population – 0.86 million) and 1.7 for New York City (population – 16.4 million).

•In the same study it is also shown that in the period 1792–1992 the fractal dimension of Baltimore has increased from 0.7 to 1.7, which indicates that the city network has been optimized in time.



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m0nn LL)L(N

1000

500

250

125

62,5

31,25

15,625

7,8125

1

10

100

1000

10000

100000

0 200 400 600 800 1000

Dimension (m)

nu

mb

er o

f p

iece

s

scale of the largest street1000m



Medieval BolognaPlan of a medieval city

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CONCLUSIONS

• Systems out of equilibrium with internal freedom to morph develop internal structured flow structures that are predicted by the Constructal Law.

• The internal flow structures evolve in time so as to provide easier and easier access to the currents that flow through them.

• “easier and easier access” means progress towards minimum global resistance (and flow time) that match the system constraints.

• The Constructal Law is a law of Nature.



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Sand size versus beachface slope – a constructal explanation

A. Heitor Reis and Cristina Gama, Geomorphology, (2009)



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Paris, 11 Juin 2009Paris, 11 Juin 2009Théorie construcale et géométries multi-échelleThéorie construcale et géométries multi-échelle

Darcy flow

Kozeny-Carman

Manning’s eq.

Mass conservation:

SWASH FLOWS uprush backwash

Iribarren Number

H0 - Open ocean wave height

Plunging waves Spilling waves

d - sand grain sizeβ = tan θ (beachface slope)



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Constructal law (equivalent formulation)Equivalent to flow maximization under the existing constraints is minimization of total time for a fixed amount of water to complete a swash cycle

Constructal law (equivalent formulation)Equivalent to flow maximization under the existing constraints is minimization of total time for a fixed amount of water to complete a swash cycle

H0 and d are fixed

Equilibrium beachface slope

ξ0 – Iribarren Number; H0 – Open ocean wave height; S – grain sphericity ; - Sand bed porosity; viscosity (water ); h – equivalent channel wetted perimeter: n – Manning’s coefficient (sand bed)

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A – B : “normal” beachface evolutionA – B : “normal” beachface evolution

A – B” : beachface evolution at constant sand grain size (same sand bed implies decreasing slope)

A – B” : beachface evolution at constant sand grain size (same sand bed implies decreasing slope)

A – B’ : beachface evolution with modification of sand bed (coarser sands allow keeping steeper beachface )

A – B’ : beachface evolution with modification of sand bed (coarser sands allow keeping steeper beachface )

Plunging waves

Spillingwaves

Sand grain size against equilibrium beachface slope

As the response to “wave climate” beachface morph in time by adjusting both its slope and sand bed composition



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Paris, 11 Juin 2009Paris, 11 Juin 2009Théorie constructale et géométries multi-échelleThéorie constructale et géométries multi-échelleSand grain size against equilibrium beachface slope

Spilling waves

Spilling waves are common in “pocket” beaches

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Beachface Engineering Beachface Engineering

E – F’ : “normal” beachface evolutionE – F’ : “normal” beachface evolution

E – F: beachface evolution at constant sand grain size (if the bed rock does not allow reaching the equilibrium slope all sand will be removed to the sea bottom)

E – F: beachface evolution at constant sand grain size (if the bed rock does not allow reaching the equilibrium slope all sand will be removed to the sea bottom)

F’ – G: beachface slope can be restored if the is fed with coarser sandF’ – G: beachface slope can be restored if the is fed with coarser sand

F’– H : sand bar may be formed that decreases wave height F’– H : sand bar may be formed that decreases wave height



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Towards equilibrium flow structures

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Performance vs freedom to change configuration, at fixed global external size

L - length scale of the body bathed by the tree flowV - global internal size, e.g., the total volume of the ductsL - length scale of the body bathed by the tree flowV - global internal size, e.g., the total volume of the ducts

L

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Performance vs freedom to change configuration, at fixed global internal size

L - length scale of the body bathed by the tree flowV - global internal size, e.g., the total volume of the ductsL - length scale of the body bathed by the tree flowV - global internal size, e.g., the total volume of the ducts

L

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Obrigado pela vossa atenção!

Thanks for your attention !

Paris, 11 Juin 2009Paris, 11 Juin 2009Théorie constructle et géométries multi-échelleThéorie constructle et géométries multi-échelleEmergence of shape and flow structure in engineering and

Nature in the light of Constructal TheoryEmergence of shape and flow structure in engineering and

Nature in the light of Constructal Theory

mefte bragança (a. heitor reis)

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