(Évora, 20th November 2003)

(Proceedings of the Symposium “Bejan's Constructal Theory of Shape and Structure”, 2004, Rui N. Rosa, A. Heitor Reis et Antonio F. Miguel (Editeurs), Centro de Geofísica de Évora, Univ. d'Évora, Portugal.)

Prof. Adrian Bejan is the author of 14 books on engineering, energy and environmental topics, and the author of more than 350 scientific articles. He is part of the editorial board or is editor for 21 international journals. He is member of several scientific societies, receiving numerous honours from many of them. He holds 12 honorary doctorates from prestigious universities outside of the United States.

Adrian Bejan is distinguished as much by the great originality and versatility of his scientific work, widely renowned and praised, as by his exceptional didactic aptitude, repeatedly illustrated in his academic textbooks, several of them serving as primary references texts for engineering studies throughout the world. Much of the contents of his textbooks resulting from his abundant and innovative research work, Adrian Bejan conveys in them the imaginative approach of a exceptional researcher. Conversely, we are led to suppose that he finds, in his engineering lectures at university, a good deal of motivation and inspiration that he later conveys into in his research work.

He has attracted the attention of the scientific community in recent years as the author of the Constructal Theory, in which the relationship between form and structure in engineering and form and structure in nature is depicted. The optimization of the overall functioning of systems subjected to constraints has been identified as the essential principle in the generation of form and structure in both natural and artificial systems with internal fluid circulation. The geometric structures that result from the application of this principle in engineering are called ‘constructs’. The recognition that the same principle is the basis for the generation of the shape of systems with internal fluid circulation, whether living or inanimate, led to the thesis of the Constructal Theory, whose most recent developments were systematized in his book, Shape and Structure, from Engineering to Nature (Cambridge University Press, 2000).

In addition to engineering, this theory has since revealed its inquisitive and explanatory potential in other areas, such as biology and medicine (allometric laws, structure of the respiratory and circulatory systems, bodily rhythms, organ and tissue structures) and earth sciences (circulation of planetary fluids, structures of river basins, etc.). It also reveals some promising developments in social sciences.

Let’s examine the relationship between artistic activity and technical activity. We have a borderline concept – design - which lies between making a thing and giving it an appropriate shape. This tradition comes back from the Renaissance. Artist-engineers would draw a machine both to show how to construct it and to give it a harmonious shape. They searched inspiration in the shapes observed in Nature – as they considered natural laws as perfection rules. Today, with computer assisted and multimedia art concepts, people are combining their artistic vision with the latest advances of information and communication technology; and in doing so, artists sometimes make their own contributions to further technology advance. Nowadays, as in the XV century, knowledge is being combined between different domains.

In the beginning of Modern Science age, mathematically formulated theories were introduced as premises for the deduction of factual conclusions. These conclusions provided both applications of and tests to the theories. The key features were the introduction of descriptive, mathematical models and their application by means of deduction. The next step in the scientific method would be the transition from informal mathematical theories to formal analytic models, one of the benefits of formalisation being that it makes the models more accessible to machine processing.

Science can be thought of as a process of Observation-Hypothesis-Test. According to Karl Popper, the essence of scientific proof is testing under the Principle of Falsification.In other words a hypothesis cannot be proven to be true, but can be proven to be false or not, by confronting its predictions against the real world. Under these logical conditions, any test that confirms a hypothesis establishes "truth" on a conditional basis only, always subject to further testing and possible falsification. The result is a gradually expanding edifice of conditional truth.

Engineering is a similar self-correcting search process. However, in this case, it can be viewed as a trial-and-error process of Observation - Design - Test. The emphasis on design in the place of hypothesis confers to engineering a different motive force, but its method is essentially the same as that of science. In contrast to science, Engineering can be envisaged as a creative search process for what works by combining existing scientific principles and technologies into new products or processes to achieve defined purposes. Engineering is therefore the practical application of the scientific method, where a design replaces a hypothesis, and the principle of falsification takes the form of testing a prototype design.

From these convergent and overlapping views, one draws the conclusion that the foundations of technical creation and of scientific discovery are coincident and that, intertwined, the two creative processes reflect the dialectic relation of the very natural phenomena and of our understanding of them. Da Vinci, Bacon, Galileo, Pascal, Newton, Faraday, Darwin, are just a few of the exponents of that epistemological progress in which invention and discovery can hardly by differentiated, and only by methodological intention can be separated.

The revolutionary advances in biochemistry since the 1950’s, and of nanoscience in the 1990’s, have perpetuated that metaphor of Nature as machine and is feeding the dream of the Machine akin to Nature. Today, we confront new domains which challenge traditional distinctions between machines and non-machines. Nanostructures provide insight into both the emergence of life and the fabrication of innovative materials.

We face again the puzzling challenge of the identity of machine, the real or apparent distinction between Machine and Nature. Machine/Nature duality appears grounded in that scientific progress is assisted by technical advances and technological innovation is assisted by scientific discovery, whereas both share in having in Nature inspiration and the object of inquiry or action. In human evolution, our understanding of Nature and technical achievements appear closely associated. Throughout history no single demarcation line between machines and non-machines has withstood the emergence of major technological innovations or scientific findings.

Some
highlight the harmony observed in the natural world to argue that the whole
Universe and living beings in particular are necessarily products of external
actions. Biological structures, their optimal arrangements and organized
societies, would point to some sort of purpose or design. On the contrary,
others appeal to chance and a huge span of time and argue that the harmony and
optimal arrangements have arisen by necessity out of chance. These appeal to
natural forces alone, albeit unknown, acting over huge spans of time, to attain
those ordered
configurations.

In classical times, teleologists were those appealing to the role of external
causes in achieving a purpose, thereby making sense of otherwise unaccountable
phenomena. Among them were wise men such as Socrates, Plato and Aristotle. The
non-teleologists were men equally wise such as Democritus, Leucippus and
Epicurus. Their views would later influence the European philosophy from the
XVII century onwards. This debate, more that 25 centuries old, has captured the
minds of the greatest thinkers of all times. The debate assumed later the form
of opposition between idealists and materialists and continues up to the present
time.

Kant did not regard teleology and mechanism as opposites, but rather as
explanatory modes complementary to each other. He was committed to the idea that
all objects in Nature, either organic or inorganic, are completely controlled by
mechanical laws. However, a mechanical explanation could be given only when the
separation between cause and effect would be clear. In complex systems, causes
and effects are inextricably mixed, and explanations require external or unknown
causes to be invoked. External causes are required only to the extent that our
understanding of the whole context is incomplete, their epistemological role
being
transitory.

Even
the evolution of a deterministic world cannot be completely understood in terms
of its governing laws alone. The initial conditions and the boundary conditions
must be specified too. These constraints might be unknown in which case its
change and evolution cannot be predicted. Not predicted yet.

The law
of reflection of light was known since classical times. Heron of Alexandria
realized that as in free propagation, the reflection path obeys likewise a
minimal principle of length. The law governing the refraction of light proved
more difficult to discover. Only in 1621,
Willebrord
Snell stated it in
correct mathematical form.

Pierre de
Fermat (1657-62) assumed that light travels at a finite speed and at slower
pace in a denser medium. He realised that both the laws of reflection and refraction
could then be derived from the same principle that is, the light travelling
along a path would take the least possible transit time. Fermat placed the
propagation of light into the arena of four-dimensional space-time, like
geodesic paths, a remarkable premonition of important elements of both special
and general relativity.

Nearly a
century later (1747) Pierre Louis Moreau de Maupertuis invoked his “principle of least action” to derive Snell's law of light
refraction. In his view, the “least action” principle would be an
all-encompassing principle governing all areas of physics. Natural processes
should proceed in just such a way as to “minimize the quantity of
action”, an assertion closely akin to Fermat's principle of “least
time”.

The notion
that the phenomena of nature must follow some “best” course was not
new. Plato already quoted Socrates stating what we could now call a “best
possible” principle for the natural phenomena. The innovation of
Maupertuis was to offer a quantitative measure for the vague notion of what is
“best" for nature, and to demonstrate that this kind of reasoning can
produce predictive results. Lagrange later clarified the concept and named
it.

The
modern formulation of Fermat’s Principle has proven to be a remarkably
useful approach to the formulation of all kinds of physical problems involving
motion and change. Newton's Mechanics is contained in Hamilton's principle
of least action. Water running downhill seeks the steepest descent; and when
running into a basin distributes itself so as its surface rests at the lowest
level, thereby minimizing the potential energy. The path of a body in a
gravitational field is a geodesic in relativistic mechanics. Feynman's
formulation of quantum mechanics is based on a least-action principle applied
to path integrals.

All
processes of macroscopic change are irreversible. The entropy concept has been
coined in thermodynamics to capture just this fundamental feature of nature.
At the same time it allows to make quantitative statements about the efficiency
of energetic and material transformations. Its origins goes back to the XIX
century, when practitioners, engineers and scientists like Watt, Carnot and
Clausius meant to understand and increase the efficiency of steam engines.
By now, the original notion of entropy has been greatly generalized and applied
in many different contexts outside classical thermodynamics. The principles
of entropy maximization and of
free-energy
minimization, notwithstanding being formulations of the laws of thermodynamics
in two particular contexts of closed systems, are further examples of optimal
principle formulations, which had great historical reach.

Thermodynamics
is a powerful tool that scientists and engineers can use to attain a deeper
understanding of how naturally organized systems arise and evolve. More
important, engineers have applied the principles of thermodynamics, fluid
mechanics, and heat and mass transfer to construct models that account for the
inherent irreversibility of processes executed by systems, both natural and
man-made. In particular, entropy-generation minimization is an approach to
thermodynamic optimization that provides considerable insight into the
organization of the natural world. In the process of performing such analyses,
engineers determine the entropy that a system generates as a function of its
parameters, including size, shapes, and materials. Engineers can go on to
optimize the system's performance given their constraints.

When
engineers design a device or system, they must first understand its purpose. The
device must perform a function, subjected to given constraints. The engineer
conceives and designs it, optimizes its design, constructs it, makes it work and
optimizes its performance. The unique understanding that engineers can offer in
the search for the origins and evolution of naturally occurring structures is
that many designs for such structures have nearly the same overall performance
as the optimal design engineers could conceive, even though they may differ in
their details. This engineering insight helps to account for the evolution of
naturally occurring systems governed by energy and mass flow and subjected to
geometric and size constraints.

When a
heat source and current is imposed, minimization of the entropy generation
amounts to minimizing the resistance to the heat flow. In an elementary passage
of a heat exchanger, the generation of entropy is due to both heat differential
and pressure drop, which compete against one another. The hydraulic diameter
must be selected such that the sum of the two irreversibilities is minimized. To
cool an electronic package, both the volume and the heat-generation rate over
its volume are given. Heat is removed by a stream under natural or forced
convection. The geometric arrangement of heat-generating components can be
optimized such that the hot-spot temperature is minimal. Optimal package
architecture can be attained.

In the
case of minimizing the overall thermal resistance between a fixed
heat-generating volume and one sink point, every subsystem of the given volume
can have its shape optimized. This principle is applied first at the smallest
volume scale, where a single high-conductivity fibber removes the heat generated
by the low-conductivity material from each smallest element. The same
geometric-optimization principle applies at larger scales. The next volume is an
optimized assembly of optimized volume elements. The process of construction and
shape optimization continues stepwise to larger scales until the given volume is
fully covered. The high-conductivity paths form a tree, and the low-conductivity
paths fill the infinity of points of the given volume.

Tree
networks, as the one stated above, abound in nature in both animate and
inanimate flow systems. We find them everywhere: plants, leaves, roots, lungs,
vascular tissues, river drainage basins and dendritic crystals clusters. Every
detail of every natural tree can be anticipated through the construction and
optimization method illustrated by the heat flow tree. In fluid trees, the
smallest-scale volumetric flow is by slow viscous diffusion, while the larger
scale flow is organized into faster channelled streams which form a tree like
structure. Most important, the method of defining each feature of the evolving
tree is deterministic.

Volume-to-point constructs have a definite time direction: from small to large,
and from diffusion to structure (ramifications, channels and streams).
Determinism results only if this time arrow is respected. The optimized geometry
formed by combining low- and high-conductivity flow regimes unites all the
volume-to-point flows. Artificial constructs as well as natural structures, such
as the internal arrangement of components in a computer or the arterial-vascular
system in animals, require the same cooperation between slow and fast heat
transfer mechanisms, with the slow mode operating at the smallest scale.

Tree
networks in living communities are also observed and can be explained by the
principle of minimization of the travel time between one point and a finite-size
area of infinite destinations. Travellers have access to more than one mode of
locomotion, starting at the slowest speed (walking) from lanes to streets and
from streets to boulevards and proceeding toward faster modes of locomotion over
longer distances. The given area is covered in steps of increasingly larger
constructs. Each construct can be optimized for overall shape and angle between
assembly and constituents. This predicts urban growth, from alleys to streets,
avenues, and highways and drives multimodal transport systems that are set up to
minimize travel time over long distances. Minimizing time travel of people flow
determines geometric shape and flow regimes.

The
minimization of travel time has been invoked to account for the trajectory
of light and urban organization. Minimizing travel time such as minimizing
resistance to flows, suggests a generalization, that is: For a finite-size
system to persist it must evolve and organize in such a way that it provides
easier access to the imposed currents that flow through it. This statement
recognizes the natural tendency of imposed currents to construct paths of
optimal access through constrained open systems. It also accounts for the
evolution of these paths, which occurs in an identifiable direction that can
be aligned with time. This is the Constructal Principle that Adrian Bejan proposes
to the scientific community.

For
some years now, Prof. Adrian Bejan has maintained a close relationship with
the University of Évora, a relationship by which we feel greatly honoured. In
addition to his scientific collaboration with some of this University’s
faculty members, Prof. Adrian Bejan was the coordinator of the “School on
Porous and Complex Flow Structures in Modern Technologies” that took
place at the University in June 2002. And in May 2004, he will return as Chairman
of the 2nd International Conference on Applications of Porous
Media.

Our University highly appreciates the generous availability and the continued
collaboration of Prof. Adrian Bejan not only in terms of scientific research
but also in terms of the promotion of conditions affirming this University
as a nationally-renown centre for engineering studies with international
recognition.

Considering Professor Adrian Bejan’s highly relevant contributions to the
general progress of scientific knowledge with his academic pursuits of inspiring
and vast interdisciplinary repercussions, as well as his distinguishing this
University with his valuable collaboration, the Scientific Council and the
University Senate unanimously approved conferring to Professor Adrian Bejan an
Honorary Doctorate from this University of Évora. The honorary degree was
awarded in solemn academic plenary section on the 1st November 2003. The award
took place in the main academic hall, which dates back to the foundation of the
University of Évora in
1559.

In
addition to that academic tribute to Prof. Adrian Bejan as an eminent scientist
and world expert in mechanical and thermal engineering, the University of Évora organized a one day Symposium on “Bejan’s Constructal
Theory of Shape and Structure” which was held on October the 31st 2003,
with invited papers by Sylvie Lorente (INSA, France), Ibrahim Dincer (University
of Ontario, Canada), Ventsislav Dzimparov (Gabrovo Technical University,
Bulgaria), Vitor Costa (University of Aveiro, Portugal), Rui Rosa,
António F. Miguel and A. Heitor Reis (University of Évora,
Portugal) and Adrian Bejan himself. That was a memorable occasion for those
who participated or attended the overview, albeit partial, of applications
of the Constructal Principle.

In the
present volume we collected the papers offered by our speakers. We feel that
in bringing them to a wider audience, the University of Évora is also
contributing to make the Constructual Theory more widely known so that it might
inspire further researchers in a growing number of fields of
knowledge.