A Teoria Constructal no contexto da Física

8/2/2019 A Teoria Constructal no contexto da Fsica

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OutlineShape and structure in natural systems The constructal Law The Constructal Law as a Minimum Time Principle

The Constructal method

Examples of applicationHeat 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

Conclusions2

The Constructal Law in the framework of Physics


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Which feature is shared by all

these stuctures?

PEM Fuel Cell



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Constructal Law*(A. Bejan, 1996)

For a finite-size system to persist in time (to live), it must evolve in such a way that it provides

easier and easier access to the (global) currents that flow through it.

Structuralcomplexit

y

Time

System

System

*A. Bejan, Advanced Engineering Thermodynamics, 2nd Edition, Wiley, New York, 1997



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Constructal Theory

Structure optimization at every scale

From the simplest structure to higher complexity structures



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Area to point flows

The optimal structure is constructed by optimizing volume

shape at every length scale, in a hierarchical sequence thatbegins with the smallest building block and proceeds towardslarger building blocks (which are called constructs)*

Currents

Heat: q(x, y, z, t, , f,Pj)Mass: J(x, y, z, t, , f, Pk)

Constraintsgeometry: f(x, y, z)=0

Parameters: Pi=Const.

- optimization target

A. Bejan, Shape and Structure, from Engineering to Nature, Camb. Un. Press, 2000


h l h f k f h


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Rn/Ri InIi

n

1iiR/1R/1 iiIRV

It follows:

Total current fixed, I

The optimized structure minimizes total entropy generation:

TRITIRSn

iiigen

2

1

2

*A. Heitor Reis, 2004, Constructal Theory: From Engineering to Physics, or How Flow

Systems Develop Shape and Structure, a aparecer em Applied Mechanics Reviews,

i

Minimization of the global resistance at constant I*

0dII/IR2 in

1i

22

ii

0dI)nI/(R i

n

1i1i

2

gen ISTR

T/VISgen I/VR

)nI(IRR n1i ii

2n1i

2ii IIRR


Th C l L i h f k f Ph i


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"Among the principles that I could mention I noticed that most of them sprang in

one way or another from the least action principle of mechanics and

electrodynamics. But there is also a class that does not. As an example, if

currents are made to go through a piece of material obeying Ohm's law, the

currents distribute themselves inside the piece so that the rate at which heat is

generated is as little as possible. Now, this principle also holds, according to

classical theory, in determining even the distribution of the velocities of the

electrons inside a metal which is carrying a current.

R. P. Feynman (The Feyn. Lect. on Phys., vol II, 19-14, Add.-Wesley, Mass., 1963)

Feynman (1962)

Theorem of Minimum Entropy Production In the linear regime the total entropy

production in a system subject to flow of energy and matter reaches a minimumvalue at the nonequilibrium stationnary state.

I. Prigogine(Introd. to Thermod. of Irrev. Proc.,J. Wiley, N. York, 1962)

Prigogine (1962)


Th C t t l L i th f k f Ph i


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The MEP Hypothesis Climate like other open systems maximize the entropy

production rate, under the existing constraints .

G. W. Paltridge(Global Dynamics and Climate, 1975, Quat. J. R. Met. Soc. 101, 475-484)

Has been confirmed with relation to oceanic circulations (Shimokawa and Osawa 2001,2002) and atmospheric circulations (Kleidon et al. 2003).

Paltridge (1975)

MEP versus mEP (minimum entropy production)!!

The MEP Hypothesis applies to systems with many degrees of freedom while mEP

applies to systems with fixed boundary conditions and few degrees of freedom

R. Dewar(Information theory explanation of the fluctuation theorem, maximum entropyproduction and self-organized criticality in non-equilibrium stationary states, 2003,J. Phys. A,36, 631-641). See also J. Withfield, Complex systems, order out of chaos, 2005, Nature, 436,908-908.

Open point:

When and where to use MEP or mEP ?

Dewar (2003)


Th C t t l L i th f k f Ph i


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MEP versus mEP (minimum entropy production)!!

Constructal law

Global I fixed total entropy generation is :

TRITIRS

n

i iigen

2

1

2

Constructal law

V fixed total entropy generation is :



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Constructal Theory

Global currents fixed* mEP

Potential differencefixed* (driving force) MEP

Constraints

+Flow architectures (system design)

*Ver:A. Heitor Reis, 2006, Constructal Theory: From Engineering to Physics, or How

Flow Systems Develop Shape and Structure, Applied Mechanics Reviews .




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(U, S, V, N)

(U0, S0, V0, N0) For some fixed X(U,S,V,N) transfer between thetwo systems, maximum flow access (I max. )

corresponds to minimum transfer time (t min.)

For some fixed X(U,S,V,N) transfer between the two systems, maximum flow access (Rmin. ) corresponds to minimum transfer time (t min.)

Constructal law as a Minimum Time principleA. Heitor Reis, 2006, Constructal Theory: FromEngineering to Physics, and How Flow SystemsDevelop Shape and Structure, Applied MechanicsReviews, Vol.59, Issue 5, pp. 269-282.




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Flow architectures are ubiquitous in Nature



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A. Bejan, Shape and Structure, from Engineering to Nature CambridgeUniversity Press, Cambridge, UK, 2000. A. Heitor Reis, 2006, Constructal Theory: From Engineering to Physics, and How Flow Systems Develop Shape andStructure, 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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Constructal sequence of assembly andoptimization, from the optimizedelemental areaAo, to progressivelylarger 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-flowresistance and too much heat-transferresistance. The correct choice is tobalance the two negative features sothat their global effect is minimum. Thisbalance yields the optimal spacing

Be is the dimensionless group

and are the fluid viscosity andthermal diffusivity, respectively.

Balancing global resistancesFlow spacings

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

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




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

A. Bejan, Shape and Structure, from Engineering to Nature CambridgeUniversity 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 andfissures through natural porous structures. Theselarge features are strangely oriented, at an angle,not directly across the porous layer. Now we seewhy. The large channel PP has two duties, to carryfluid acrossthe layer directly and to feed theneighboring trees, which also carry fluid across thelayer. The appearance of raggedness anddisorganization is an illusion: such featurescome from the same principle as all the otherfeatures of the tree drawings.

S. Lorente and A. Bejan, JOURNAL OF APPLIED PHYSICS100, 2006




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e Co st ucta a t e f a e o of ys cs



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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.

f f y



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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

f f y



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A. Bejan, J.H. Marden. The constructalunification of biological and geophysical

design. Physics of Life Reviews(2008)

f f y



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Constructal theory of flow

architectures of the lungsA. 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 tree Starts at the trachea;

Channels bifurcate 23 timesbefore reaching the alveolar sac.

Has this special flow architecture been developed by chance ordoes it represent the optimum structure for the lungs purpose,

which is the oxygenation of the blood?


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Total resistance to oxygen and carbon dioxide transport between theentrance 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 ruleemerges:

m106.1.constL

D 3

0

2

0

the ratio between the square of the trachea diameter and its length isconstant and a length characteristic of mankind:



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2/1

ox0oxoxg

oxox

0

2

0

))(TR

D

V

AL63.8

L

D

ff

f

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 therespiratory system, V, and the length of the respiratory tree, L, which areconstraints posed to the respiratory process determine univocally the structure ofthe lungs, namely the bifurcation level of the bronchial tree.



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CONSTRUCTAL THEORY OF GLOBAL CIRCULATION AND CLIMATE

A.Heitor Reis1

and Adrian Bejan2

1University of vora, Department of Physics and Geophysics Center of vora, Colgio Luis Verney, RuaRomo 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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Streamfunction lines of atmospherical long-term meridionalcirculation. The isoline unit is 10 Sverdrups and the ordinaterepresents altitude (adap. Phys. of, Climate, Peixoto and Oort,1991)

Long-term meridional circulationHaddley, Ferrel and Polar Cells

00

00

2

0

2

4

6

-6

-4

-2

-2

-1

20 20 4040 6060 NS

10

8

6

4

2

PRESSURE(

104Pa)

0

LATITUDE

39



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Fig. 3 Zonal mean zonal velocity in ms1 (shaded contours) and mass stream function in1010 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) ofECMRWF

Earths 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 earths meridionalcirculation, really?

ForTR = 20 K only one cell develops;

ForTR = 30 K two cell develop;

ForTR= 60 K (actual earths conditions) and TR = 130 Kthree cell develop;

ForTR = 190 K only two cell develop; the Polar celldisappears!



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Constructal Law

Maximization of heat flow leaving the equatorial belt To each TH correspondsinfinity surface partitions x=AH/A. One maximizes the heat flow q)

0x

T,0

x

T,0

x

q

2L

2L

TH

With TH fixed, each curve q(x)TH, shows a maximum,

which defines TL.. Then, with both TL and TH, x maybe determined from the equation:

4.3

4.4

4.5

4.6

0.4 0.42 0.44 0.46 0.48x

qX1015W

TL=275.56K

TL=275.51K

TH=294.1K

TH=294.0K

TH=293.9K TL=275.53K

B2

T)x1(Tx 4L4H

The optimum point is determined from the

minimum TL and corresponds to TL=275.51KandTH=294.0K, with xopt.= 0.434, which corresponds

to latitude 2540



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0x

T,0

x

T,0

x

q

2H

2H

TL

B2

T)x1(Tx 4L4H

The optimum point is defined by the maximum ofTHand corresponds to TH =288K e TL=265.5K, with

xopt.= 0.8, which corresponds to latitude 5310

5.8

6.2

6.4

6.6

0.75 0.8 0.85x

q(X1015W)

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 correspondsinfinity surface partitions x=AH/A.

Constructal Law

With TL fixed, each curve q(x)Tl, shows a maximum,

which defines TL.. Then, with both TL and TH, x maybe determined from the equation:



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LAT=2540N

LAT=2540S

x=0.43

q q

LAT=5310N

LAT=5310S

x=0.80

q q

I II

TL=275.7K

TL=275.7K

TH=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)

LAT=2540

LAT=5310

TL

TL

TH

TH



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Average temperature on earths surfaceI = 0.434 293.9 K + (1 0.434) 275.5 = 283.5 K

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=281.5K0.434 293.5 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 ()

Sgen

(1014

W/K

)

x=0.4

3

x=0.8

Entropy generated as function ofsurface 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 differenceand the three-cell regime of meridional circulation. The poleward heat transferseems 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 itcontrols the temperature gradient between dark and illuminate

hemispheres, but for values of day rotation period lower that 24 h it doesnot affect significantly the position of the three cells, as shown byJenkins (1996)

What is behind the three-cell system ?

Earths rotation rate and/orequator to pole temperature difference?



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River basins -

Constructal theory

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

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River Douro

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



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

L1ii RLL Hortons law of streamlengths:1.5


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D2Do

D1

AoAo

Do

D1

D2

Lo

52iiii DLmz

(a) (b)

Paris, 1 Juin 2009

Minimization of z

Total area, total channelvolume fixed

74

10 2/

DD

7221 2

/DD

73

10 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)

m

m



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Conclusions

The scaling laws of geometric features of river basins can be anticipated based on ConstructalTheory, 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 Hortons law for the same ratio.

Agreement is also found with the number of consecutive streams that match Hortons law ofratios of consecutive stream numbers.

Hacks law is also correctly anticipated by Constructal Theory, which provides Hacks exponent

accurately.

Meltons law is verified approximately by Constructal Theory, which indicates 2.45 instead of 2

for the Meltons exponent.

It was also shown that also with turbulent flow quadrupling is better than assembling eight

elements, therefore anticipating Hortons law of ratios of consecutive stream numbers

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

Paris 11 Jui 2009

Thorie constructale et gomtries multi-chelle



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Scaling laws of street networks

A. Heitor Reis (2007), Physica A,387, 617622

Paris, 11 Jui 2009



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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).

Idealism: Because cities are complex man-made systemsself-similarity springs out of the congenital ideas of beauty andharmony shared by mankind.

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

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



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FRACTALDESCRIPTION

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

Self-similarity at various scales

Thenumber of pieces N(X) of size Xseems to follow an inverse-powerdistribution law of the type

orm fractal dimension

n0n rXX

nm

0n rCX)X(N

mCXXN

xrrx mff


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Rivers of People and goodsand drainage basins

Self-similarity at various scales

The same scalinglaws hold

nL0n RLL

n

Nn RN



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Notice that the scale factor is given by 2~rL

L

1n

n

Therefore

LN2L2Nm

, therefore m = 24~

N

N

n

1n

m = fractal dimension xrrx mff

mnn r)L(N

nm0n rCX)X(N


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m0nn LL)L(N

1000500

250

125

62,5

31,25

15,625

7,8125

1

10

100

1000

10000

100000

0 200 400 600 800 1000Dimension (m)

numberofpi

eces

scale of the largest street1000m



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Medieval BolognaPlan of a medieval city

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Paris, 11 Juin 2009

Thorie construcale et gomtries multi-chelle



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Darcy flow

Kozeny-Carman

Mannings eq.

Mass conservation:

SWASH FLOWSuprush backwash

Iribarren Number

H0 - Open ocean wave height

Plunging waves Spilling waves

d - sand grain size = tan (beachface slope)



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

H0 and d are fixed

Equilibrium beachface slope

0Iribarren Number; H0 Open ocean wave height; S grain sphericity ; f - Sand bed porosity; viscosity (water ); h equivalent channel wetted perimeter: nMannings coefficient (sand bed)



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A B : normal beachface evolution

AB : beachface evolution at constantsand grain size (same sand bed impliesdecreasing slope)

AB : beachface evolution withmodification of sand bed (coarser sandsallow 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 bedcomposition

Sand grain size against equilibrium beachface slope



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g g q p

Spilling waves

Spilling waves are commonin pocket beaches



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Beachface Engineering

EF : normal beachface evolution

E F: beachface evolution at constantsand grain size (if the bed rock does notallow reaching the equilibrium slope allsand will be removed to the sea bottom)

F G: beachface slope can be restored ifthe is fed with coarser sand

F H : sand bar may be formed that

decreases wave height


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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 flow

V - 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 ducts

L



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TERMODINMICA TEORIA CONSTRUCTAL

Estado Arquitectura de escoamentosProcesso Mudana de estruturaPropriedades (U, S, V, ...) Objectivo e constrangimentos globais (R, L, V,...)Estado de equilbrio Arquitectura de escoamentos de equilbrioRelao fundamental U(S, V, ...)

Relao fundamental R(L, V, ...)

Estados de no-equilbrio Arquitectura de escoamentos de no-equilbrioRemoo de constrangimentos Liberdade de evoluo de forma e estrutura2 Lei Lei Constructal

S mximo a U, V, ... constantes;U mnimo a S, V, ... constantes;V mximo a S, U, ... constantes

R mnimo a L, V, ... constantes;V mnimo a R, L, ... constantes;L mximo a R, V, ... constantes;



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Obrigado pela vossa ateno!

Thanks for your attention !

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A Teoria Constructal no contexto da Física