A talk “Constructal Theory and Its Applications” by renowned Professors A Faculty Based Seminar was presented by Professor Adrian Bejan and Professor Sylvie Lorente on 27 January 2010. Over 140 audiences attended.

Power point file of Prof. Bejan Power point file of Prof. Lorente

The last two decades have marked important changes in how thermodynamics is taught, researched and practiced. The generation of flow configuration was identified as a natural phenomenon. The new physics principle that covers this phenomenon is the constructal law, which was formulated in 1996: “For a finite-size flow system to persist in time (to survive) its configuration must evolve (morph) in time in such a way that it provides easier flow access to its currents.” The geometric structures derived from this principle for engineering applications have been named constructal designs. The thought that the same principle serves as basis for the occurrence of geometric form in natural flow systems is constructal theory. The origin of the generation of geometric form rests in the balancing (or distributing) of the various flow resistances through the system. A real system owes its irreversibility to several mechanisms, most notably the flow of fluid, heat and electricity. The effort to improve the performance of an entire system rests on the ability to balance all its internal flow resistances, together and simultaneously, in an integrative manner. This seminar will present Professor Bejan’s recent work and breakthrough on this subject area.

The Speakers :

Professor Adrian Bejan, Duke University Professor Bejan got his BS, MS and PhD from Massachusetts Institute of Technology (MIT). He is now the J.A. Jones Professor of Mechanical Engineering. His research covers a wide range of topics in thermodynamics, heat transfer, fluid mechanics, convection and porous media. More recently, he developed the constructal law of design in nature. He is ranked among the 100 most highly cited authors worldwide in engineering (all fields, all countries), the Institute of Scientific Information, 2001. Professor Bejan has received 15 honorary doctorates from universities in 10 countries. Lecture by Professor Bejan Professor Sylvie Lorente, INSA Toulouse, France Professor Lorente got her BS, MS and PhD from INSA Toulouse, France. She is at Laboratory of Materials and Durability of Constructions, INSA-UPS, Department of Civil Engineering, Nathional Institute of Applied Science, Touloue, France. Her research interests encompass vascularized materials, constructal theory, porous media, fluid mechanics, heat and mass transfer. Lecture by Professor Lorente

Presentation of Souvenir

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

www.constructal.org

Adrian BejanJ. A. Jones Distinguished Professor

Duke University

Constructal Theory and Applications

Sylvie LorenteUniversity of Toulouse

France

AnimateInanimate

“Design in Nature” is flow

2

“The design” vs. “to design”

1. The generation of “design” is a physics phenomenon.

2. The phenomenon is summarized by the constructal law (1996)

“For a finite-size flow system to persist in time (to live) it must evolve in such a way that it provides greater and greater accessto its currents”.

Time

The sense of the movie tape of design in nature

3

Nature flows with design (configuration)

Animate + inanimate Geo + Bio + Socio4

00

L

WW 21+

00

V

Larger animals

travel faster, V ~ M1/6

wave less frequently, t–1 ~ M–1/6

and are stronger, F = 2gM

move more mass to greater distances, W ~ MgL

Flying

5

Vertical loss: W1 ~ MgLb

Horizontal loss: W2 ~ FdragL

Constructal Flow of Animal Mass

A. Bejan, Shape and Structure, from Engineering to Nature,

Cambridge Univ. Press, 2000.

Constructal animals, “human & machine species”, technology evolution

6

J. Exp. Biol. 209 (2006) 238-248.

Pour la Science, August 2006.7

Fig. 2. Running world records for 100 m dash, men: (A) speed (V) vs time (t); (B) body mass (M) vs t; (C) V vs M. The world record data for all the figures cover the period 1929–2008.

The evolution of speed, size and shape in modern athleticsJ. D. Charles and A. Bejan, The Journal of Experimental Biology 212 (2009), 2419-2425

8

Fig. 1. Swimming world records for 100 m freestyle, men: (A) speed (V) vs time (t); (B) body mass (M) vst; (C) V vs M. The world record data for all the figures cover the period 1912–2008.

9

Constructal origin of “characteristic sizes”

10

Fixed mass: flying with less fuel

Fixed fuel: flying more mass, farther

More mass = faster

11

The size of every component that functions inside a complex flow system

2

P 32 = U

L D

∆ µ

Round duct with specified flow rate

is the mean fluid velocity,U ( )2m / D / 4ρπ&

1

p

W m P =

L p L

∆η

& &2W m

2 gVL L

≅&

21 2 124

W W c + c D

L D

+≅

& &

( )2 21 pc 128 m /= µ πη ρ& 2 wc 2 gV= περ

1/ 61/ 6 21

opt 2 22 p w

c 128 mD 2

c gV

⎛ ⎞⎛ ⎞ µ= = ⎜ ⎟⎜ ⎟ ⎜ ⎟π η ρ ρ⎝ ⎠ ⎝ ⎠

&

0 0

0 1

0 1

H V= 2

L V

t = t

Constructal Flow of People and Goods

~5 minutes Atlanta airport

V0: walking

V1: riding

12

NICE

PARIS

LONDON

13

2002

2050

14

1. The generation of “design” is a physics phenomenon.

2. The phenomenon is summarized by the constructal law (1996)

“For a finite-size flow system to persist in time (to live) it must evolve in such a way that it provides greater and greater accessto its currents”.

Time

The sense of the movie tape of design in nature

15

Constructal law: The time direction of the movie tape ofgeneration of design in nature,

not about optimality, min, max, “entropy”, destiny, or end-design.

Ad-hoc optimality principles, whose results were also deduced from

Minimization of entropy generation (EGM)Maximization of entropy generation (MEP)Minimization of flow resistanceMaximization of flow resistanceMinimum time, costMinimum weightOptimal organ sizeUniform stressesSurvival of the fittestSurvival of the most adaptableMaximum growth rate of disturbances : hydrodynamic instability

the constructal law:

16

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

www.constructal.org17

Thermodynamics: systems viewed as black boxes, without configuration:

“Energy” defined

“Entropy” defined

Carnot limit, exergy, availability

Gouy & Stodola theorem stability (U min, S max)

Note: Entropy generation minimization (EGM or maximization of efficiency) is not one of the derived statements. It is a self-standing ad-hoc statement.

System: closed, cycles (or steady).

18

L H

L H

Q Q0

T T− ≥

H LQ Q W 0− − =

Lrev H

H

TW Q 1

T

⎛ ⎞= −⎜ ⎟

⎝ ⎠

st1 law :

nd2 law :

Derived :

Open system,steady state :

Engine, onvehicle

Animal

19

Evolution, in time:

Configuration

Performance

20

Q: What happened to the produced work?

Mass was moved to a distance = Mixing

A:

Work = (“friction factor”) (weight) (distance)water

W land Mg Lair

21

The whole Earth is an “engine + brake” system

22

System: AnimalHuman & machineEarth

23

24

Water flow + flow of stresses

Leonardo da Vinci’s ruleh constant

x=

A. Bejan, S. Lorente and J. Lee, Journal of Theoretical Biology, Vol. 254 (3), 7 October 2008

Constructal Design of Roots, Trunk and Canopy:

Fibonacci sequence 25

Tree mass flow rate ~ tree length scale

Zipf’s distribution, predicted,

predicted earlier for city size vs. rank (2006)

Constructal Design of the Forest

26

(b)

(a)

The Pyramids, from Egypt to Central America

( ) ( )( )12 23

1/ 22 21 2

W W W

N R r N cos N sin H r

= +

= µ − + µ α + α +

( )⎡ ⎤⎢ ⎥⎣ ⎦

-1opt 2 1

1α = tan µ - µ2 27

Design with Constructal Theory, Vascularization

Sylvie Lorente, Adrian Bejan

Université de Toulouse, INSA Toulouse, France.

Duke University, Durham, USA

1

Adrian Bejan, Duke University, North Carolina, USA1996

“For a finite-size open system to persist in time (to survive) it must evolve in such a way that it provides easier and easier access to the currents that flow through it ”.

2

Introduction

Two visions

• Understand and predict natural phenomena (river basin, circulation in atmosphere, swimming, flying…)

• Design for the engineers

Constructal Theory :

Minimization of flow resistances Geometry

3

4

Tree-shaped networks in a disc-shaped body

Line-to-line tree flow

Vascularized materials for self healing

Vascularized materials for self cooling

Outline

Micro-channelsElectronics cooling

objectives

constraints Disc-shaped area Number of outletsTotal volume of the tubes

To deliver a fluid from a source to a given number of outlets (users)

Minimum ∆P

Tree-shaped networks in a disc-shaped body

5

We obtain the connecting angles by minimizing the function ‘overall pression’

(48 outlets)

The shape of the network is the result.It is « given » by the angles.

6

When optimized complexity is beneficial

N = constantpairing is a useful feature if N sufficiently large

N increases the level of pairing increases

complexity increasesN constant

7

Line-to-line tree flow

8

Laminar flowOptimal diameter ratio

3/11ii 2D/D =+

Hess-Murray law

9

°=α 45

For a given mass flow rate

d

We fix the volume of tubes

Or

10

Laminar flow, pressure drop

4D

LPo

8mP νπ

=∆ &

Poiseuille constant, 16 for

round tube

n, bifurcation levelα =45°

10 L2L =21 L2L =

etc, etc…

1ii m2m += &&

11

nref 2

14

P

P≅

∆∆

1P

P

ref

<∆∆

When are tree-shaped flows better than parallel flows?

Level of bifurcation: n > 4

refP / P 0.1∆ ∆ < n 7>12

Vascularized materials for self healing

• Objective: Recover the initial strength after healing

• First attempt: micro capsules or tubes embedded in the material. Today: vascular structures

• Cracks loss of mechanical performances

13

14

Pressurized network

( )dV m m m mi, j i, j N W S E

dtρ = + − −& & & &

Objective: to minimize the discharge time of the network

Pressurized network

0

0,4

0,8

1,2

8*8 20*20 40*40Grid

t_m

in /

t_1D

15

16

17

Result: D2/D1 optimal leading to a discharge time divided by 2

15

Direct injection of the fluid in the network

Trees matched canopy to canopy

trunk

canopy

trunk

canopy

16

17

Example of 6 elements

• Diameter ratio corresponding to a minimum flow resistance • Choice of the most performing network

Optimal diameter ratio and optimal aspect ratio

Systematic study: trend?

0

10

20

30

40

50

0 10 20 30 40 50 60

number of elements

redu

ctio

n of

the

flow

resi

stan

ce

Optimal aspect ratio: close to 1

D1/D2 optimizedvs single D

18

L = nd1P∆

H

m&

m&

d

p

L

py

3P∆

3D

Hm&

m&

1D

H

p

L r

py

1

2

r

3D

py3D

1D

1

2

r

3D4D

4D

m&

m&

First construct: one-dimensional stack

Second construct, 3D

Third construct, 3D

Trees matched canopy to canopy:the superiority of vascular design

19

First construct (1D stack) and second construct

Growth & transitions: larger and more complex vasculatures

610

1

210

1 10 310210

3P~∆

1p =

1P~∆

410

2

3

10

20

30

y

dN

1

210

1 10 210

P~∆

410

1P~∆

( )envP~

3∆

( )envP~

4∆

5p ≅

1p =

2p =

3p =

4p = 3r ≅

y

dN

First, second, and third constructs

Scaling up: not an enigma when the principle of configuration generation is known

20

Vascularized materials for self cooling

21

Effect of the pumping power on the temperature distribution

9103.1Be ×=

10107.1Be ×=

22

23

For a given pumping power, increasing n increases the thermal performance.

If n>4, trees are better.

Global thermal resistance vs pressure drop number, steady state24

25

Time delay before the start of cooling

Conclusion

• Optimal micro-vascularization through constructal theory designed porous media

• Application to different scales

26

2727

Toulouse