# Adrian Bejan - Constructal Law

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.

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. These recent developments are reviewed in the books Advanced Engineering Thermodynamics, 2nd ed. (Wiley, 1997) and Shape and Structure, from Engineering to Nature (Cambridge University Press, 2000).

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 minimize all its internal flow resistances, together and simultaneously, in an integrative manner.

Numerical temperature distribution of vertical channels in natural convection with the three level of optimized complixity: m=0, 1 and 2, at Ra=10^6. Reference: A. K. da Silva and A. Bejan, Constructal multi-scale structure for maximal heat transfer density in natural convection, Int. J. Heat Fluid Flow 2004. (in press)

Resistances cannot be minimized individually and indiscriminately, because of constraints: space is limited, streams must connect components, and components must fit inside the greater system. Resistances compete against each other. The route to improvements in global performance is by balancing the reductions in the competing resistances. This amounts to spreading the entropy generation rate through the system in an optimal way, so that the total irreversibility is reduced. Optimal spreading is achieved by properly sizing, shaping and positioning the components. Optimal spreading means geometry and geography. In the end, the geometric system-its architecture-emerges as a result of global, integrative optimization.

This method was developed first for heat flow, with application to the cooling of heat-generating volumes (e.g., packages of electronics) by using concentrated heat sinks and small amounts of high-conductivity insert material. The resulting structures were tree-shaped paths of high-conductivity material. Most fundamental was the geometric form 'tree' deduced from principle. The tree structure unifies the extremely wide class of engineered and natural flows that connect an infinity of points (volume, area) to one or more discrete points (sources, sinks). Natural examples are the river basins and deltas, lungs, vascularized tissues, lightning, botanical trees, and leaves. Manmade flows shaped as trees are found in the cooling systems of electronics packages and windings of electric machines, finned surfaces, regenerative heat exchangers, routes for

The transition from one tree structure to one with more level of pairing, as the flow resistance (f) is minimized while N increases. Dendritic convection on a disc, W. Wechsatol, S. Lorente & A. Bejan, Int. J. Heat Mass Transfer 46 (2003) 4381-4891 minimum-time and minimum-cost transportation, and networks for the collection and distribution of electricity, water and sewage. All these examples (fields, really) belong under one theoretical umbrella-constructal theory-as shown in a new book:

## Shape and Structure, from Engineering to Nature

**Adrian Bejan**

**Duke University **

Cambridge University Press, 2000

Breaking new ground in engineering and the natural sciences, this book explores the geometry of flow systems, both artificial and natural. It looks at the design and optimization of man-made systems, while theorizing on the generation of geometric form in natural systems. Similarities abound between the two, such as the presence of tree networks in both computers and the human lung.

Shape and structure spring from the struggle for better performance. Termed constructal theory, the thinking is that the "objective and constraints' principle used in engineering is the same mechanism from which the geometry in natural flow systems emerges. This book takes a close look at the derivation of geometric from from principle-the struggle for meeting the objective of better performance while being subject to global and local constraints. Such an examination of the purposeful and constrained optimization of engineering enables us to make better sense of nature's architecture.

From river channels to economic flows, this book draws many parallels between the man-made and natural worlds. Among the topics covered are mechanical structure, heat trees, ducts and rivers, structure in power systems, and transportation and economics structure. Well-illustrated and containing problems at the end of each chapter, this text is suitable for advanced undergraduate or graduate design courses. It will also appeal to a broad range of readers in engineering, the natural sciences, economics, and business.

**IMAGE GALLERY FROM "Shape and Structure, from Engineering to Nature" by Adrian Bejan:**

PLATE I. Tree and roots on the lower Danube (photograph courtesy of Teresa M. Bejan)

PLATE II. Three designs for the internal structure of a fixed total heat generating rate and a stream of coolant flowing vertically. The objective is to maximize the global thermal conductance between the volume and the stream, which is equivalent to minimizing the red areas (hot-spot temperatures) that occur inside the volume.

PLATE III. Deriving optimal shape from the minimization of global resistance between a volume and one point. Three competing designs are shown. The volume and the heat generation rate are fixed. The aspect ratio of the rectangular domain is variable. The resistance is proportional to the peak temperature difference, which is measured betweeen the hot spots (red) and the heat sink (blue). The middle shape minimizes the areas covered by red and has the smallest volume-point resistance.

PLATE IV. The temperature field in a first construct containing four elemental volumes. Note the formation of hot spots (red) in the corners situated the farthest from the heat sink (blue).

PLATE V. The optimal shapes of elemental volumes with spacings at the tips of the high-conductivity channels: the effect of varying . Note that the optimal shape becomes more slender and the global resistance decreases (the red disappears) as the volume and conductivity of the central channel increase.

PLATE VI. The optimization of the angle between the first-construct stems and the stem of a second construct. In this example the second construct has four first constructs, and each first construct has eight elemental volumes. When the stems are perpendicular the hot spots are concentrated in the farthest corners relative to the heat sink (blue). When the angle is too large (the first frame), the hot spots jump to the corners that are on the same side as the heat sink. In the optimal configuration (the second frame), the hot spots are spread most uniformly around the periphery of the system. More points work the hardest in this configuration. We pursue this idea further in the shapes of constant resistance constructed in Chap. 12.

PLATE VII. Channels with the same cross-sectional area, different shapes, and different wetted perimeters. The top row shows threeexamples with bottoms shaped as arcs os circles. The number under each drawing represents the flow resistance relative to the resistance of the channel with semicircular cross section. The four cross sections shown on the bottom row have had their depth/width ratios optimized for minimum resistance. note that the resistance does not vary appreciably from one shape to the next. Channel cross-sectional designs are robust.

PLATE VIII. Streaklines at time n=400 inside the system optimized in Fig. 6.16.

PLATE IX. Color displays of inflowing (blue) and outflowing (red) streams in the first construct (top), second construct (middle), and third construct (bottom).

PLATE X. the allocation of heat transfer surface and its effect on temperature differences and power output.

PLATE XI. Flock of pelicans flying overhead at Phinda, KwaZulu-Natal, South Africa (photograph courtesy of William A. Bejan).

PLATE XII. Three examples of the growth of street patterns as the minimization of travel time between a finite-size area and one point. Velocities increase as the constructs become larger. Each construct has been optimized twice, for shape and angle.

PLATE XIII. The color red shows the migration of the hot spots as the aspect ratio of the triangles changes. There is an optimal, intermediate slenderness that marks the moment when the hot spot jumps from the tip to the two base corners. In this design the entire long sides of the triangle are at the hot-spot temperature.

PLATE XIV. the third construct obtained when four of the second constructs of Fig. 12.5 are joined.

PLATE XV. The temperature field in a plate fin with adjustable crest inclination.

University Press, Cambridge, UK, 2000, 344 pages

Hardback (ISBN 0-521-79049-2) $110/£70

Paperback (ISBN 0-521-79388-2) $39.95/£24.95

### Constructal Theory Honors

**Adrian Bejan and Sylvie Lorente**, The 2004 Edward F. Obert Award of the American Society of Mechanical Engineers (ASME), for the paper "Thermodynamic formulation of the constructal law".

**Sylvie Lorente**, The 2005 Bergles-Rohsenow Young Investigator Award in Heat Transfer, American Society of Mechanical Engineering (ASME) for "Creativity and early impact on heat transfer research on buildings thermal design, dendritic conduction and convection, and ionic transport".

**Cesare Biserni**, The 2005 Prize of the Italian Union of Thermo-fluido-dynamics (UIT) for "Particolarmente interessante e completa, in riferimento all'Effetto Vapotron, la descrizione dell'apparato sperimentale messo a punto e realizzato, ed il ricco set di risultati sperimentali, ottenuti attraverso questa apparecchiatura. Interessante risulta essere l'applicazione della "Constructal Theory" all'ottimizzazione di alcune superfici alettate".

**Adrian Bejan and Sylvie Lorente**, The 2007 James P. Hartnett Award from the International Center for Heat and Mass Transfer, for the paper "Constructal tree-shaped flow structures", which was judged as the best paper presented in 2006 at a conference, symposium or seminar sanctioned world wide by the ICHMT.

**Attachments**

"Constructal theory explains why a river looks like a tree"

La Loi Constructale

Ingenieria Mecanica

Conference de Adrian BEJAN

J. A. Jones Distinguished Professor Duke University, USA

Ecole des Mines, Paris, 7 Feb. 2003

Université Henri Poincaré, Nancy, 12 Feb. 2003

**TITRE**

Théorie Constructale et Conception de Systémes et Procéds: Configurations Optimales des Ecoulements, de l'Ingénierie à la Nature.

**RESUME**

Des similtudes sont courantes entre les systémes en écoulement dans la nature et en Ingénierie par exemple, les arborescences existent en informatique, dans le corps humain, dans la croissance des cristaux, dans le développement des réseaux urbains ou de communication.

Dans cette conférence, l'auteur en partant de la conception et de l'optimisation des systémes manufacturés, propose un principe déterministe de structuration géométrique des systémes naturels; la théorie constructale suppose que les contraintes et objectifs de l'ingénierie sont aussi ceux qui gouvernent la géométrie des flux dans la nature.

Les systémes étant par essence imparfaits (second principe de la Thermodynamique), la fléche du temps impose aux formes et structures observées, d'optimiser les objectifs sous l'évolution des contraintes globales et locales.

Des exemples sont donnés, allant des échangeurs de chaleur (monde de l'ingénierie) en passant par l'économie jusqu'aux écoulements naturels des canaux, comme il est montré dans le dernier livre de l'auteur.

Cette conférence est donnée dans le cadre de la section française de l'ASME, en relation avec l'Association Française de Mécanique et la Société Française des Thermiciens.