Angèle Kremer Marietti

Université de Picardie Jules Verne, Amiens, France Groupe d'Études et de Recherches Épistémologiques, La Maison d'Auguste Comte, Paris, France

The constructal principle


With Adrian Bejan’s constructal theory, the perfect plan of the machines rests on a new scientific method, derived from the knowledge of diverse imperfections of the beings and the things from which are drawn and released the adequate drawings. Adrian Bejan’s book, Shape and Structure, from Engineering to Nature [7], underlines the relations between the forms and the structures of nature and the forms and the structures that the engineer has to render concrete. My presentation highlights the four Aristotelian causes which found the realization of constructions in engineering of which realization depends on the efficient cause (the sculptor), of the material cause (marble), of the formal cause (configuration of the statue), and, above all, of the final cause, i.e. the destination for which the statue is built. By applying the constructal method, the engineer eliminates the imperfections or, at least, makes them profitable in the accomplished realization of the scientific design. Among the philosophical implications of the constructal theory, we can count optimism.
Keywords: scientific design, circulation of fluids, forms and structures, causes, constructal theory.

1. What is constructal theory ?

The search for optimal systems has lead Professor Adrian Bejan to establish that flow structures morph in such a time that they provide greater flow access, for example, less resistance to fluid flow.  Resistances can be distributed optimally, and from this process comes the design of the river basin, power plant, human lung.  Flow systems acquire configuration and achieve high performance : the optimal distribution of imperfection is obtained through a balancing act with the surrounding flows. The general design of this process concerns cities, geography, economics. It concerns also medicine, because constructal geometry could answer to this question : why some people can resist to some virus and not to others ? [9]

The constructal principle is applied as follows. The elemental structure is defined, determined and optimised. The first problematic question is to defining and determining the ideal proportions adapted to the available elemental system, which has been defined and determined. The next step consists in uniting several elemental structures in a network. It is the way by which the perfect plan of the machines rests on a new scientific method, entirely deterministic, derived from the knowledge of diverse imperfections of the beings and the things from which the adequate drawings are drawn and released.

2. Scientific origins of the constructal theory

Mechanical engineering began with the possibility to applicate algebra to geometry. In the seventeeth century, Pierre Fermat (1601-1665) made major contributions by emphasing the sketching of loci. Then René Descartes (1596-1650) in Geometry (1637) succeeded to interpret algebra geometrically and reciprocally to realize geometrical constructions of positive solutions of determinate algebraic equations. For the negative coordinates, we must wait until the mid-eighteenth century and for higher dimensions until the nineteenth century.

Denis Papin (1647-1712), leading physicists to study the transformation of heat into mechanical energy, inaugurated the industrial era with the consideration of the work of steam engines. Steam engineering influenced thermodynamics. In his book on the correlation of figures in geometry (1801), Lazare Nicolas Carnot (1753-1823) used the notion of algebraic measure and explained a principle according to which mechanical efficiency decreases as the difference in velocity of moving parts brought into contact increases. Nicolas Léonard Sadi Carnot (1796-1832) translated his father’s principle into a prohibition of direct flow of heat through contact of engine parts at different temperatures ; he gave his Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (1824), in which he brought the second principle of thermodynamics, the “Carnot’s principle” according to which it is impossible to realize a thermal engine working with only one source of heat. Rudolf Clausius (1822-1888) succeeded in a mathematical theory with an analytic expression for entropy (1859). Thanks to Ludwig Boltzmann (1844-1906) it appeared that the second law of thermodynamics is nothing else than a statement of relative probabilities (1877); hence the definition of a gas entropy is proportionnal to the logarithm of the number of microscopic states which define its macroscopic condition

Joseph Fourier (1768-1830) gave an influential mathematical theory of heat conduction independent of the caloric hypothesis (1811), therefore he was in favor with the kinetic theory (against the caloric theory). Through conciliating heat and motion, the success of constructal method has been confirmed by the observation of fluid fluss in natural systems as in constructed systems. On that subject we can recall that Justus Liebig (1803-1873) et Johannes Müller (1801-1858) establish through studies on animal economy that a correlation exist between heat, chimical reactions and muscular work. Other physicists agreed on the equivalence of all kinds of  force.

Now, the last two decades have marked important changes in how thermodynamics is taught, researched and practiced. We can quote the methods of exergy analysis, entropy generation minimization or thermodynamic optimization, and thermoeconomics: they are the most established methods that have emerged. And more recently, we can add thermodynamic optimization developed extensions in physics, where it also acquired new names (endo-reversible, finite-time).

Energetics, mechanics of fluids, mechanics of solids are the fields the most known and developed by Adrian Bejan, who is overall a specialist in thermodynamics [15] with a great interest in architecture and biology. Bejan’s research covers a wide range of topics in thermal engineering: that is to say, entropy generation minimization, exergy analysis, natural convection, combined heat and mass transfer, convection in porous media, transition to turbulence, melting, solidification, condensation, fouling, solar energy conversion, cryogenics, applied superconductivity and tribology. Bejan is dealing with resistances, frictions, circulation, always referring to the decreasing of thermodynamic imperfection.

3. Philosophical background of the constructal theory

The philosophical implications of the constructal theory are numerous. By envisioning the subsystems we see that they are small parts; envisioning the entire network, we see that every subsystem is linked with each individual in the whole. Like the Aristotelian use of the concept of actuality as an activity that by definition does not attain any completed state, the perfect network can be presented in a new more complicated way, if necessary. By Aristotle (384-22 BC), the notion of the world as appearance implied a notion of the world's essence : and vice versa [1]. Through both of these notions, a notion of the world's actuality includes its essence and its appearance, which refer simultaneously to a determinacy of being and a determinacy of thought. If we consider with Aristotle a statue as an individual substance, then we can distinguish within it matter and form (two of its causes). The form of the statue has its existence in the matter of which it is made : for instance, bronze or marble. Bronze is cast in a certain shape: for instance, that of a god. But the statue ceases to exist when the matter, as bronze, is melt down in some other god or thing, or, as marble, is cut up into some other representation. But neither bronze nor marble are substances; the shape is not belonging either to another category. Composites of form and matter are perishable.

Then the design of engineering planned and realized by the constructal theory has a similar interpretation between all and part or, concretely, the whole and its parts. It is why we recognize in the constructal process the four Aristotelian causes which found the realization of constructions in engineering of which realization depends on the efficient cause (the sculptor), on the material cause (bronze or marble), on the formal cause (configuration of the statue), and, above all, on the final cause, i.e. the destination for which the statue is built. After having envisioned the final intended cause the designer’s formal part of the work is the part of decision. Without the final cause there is no design; and the design obeys to the formal cause, which is mathematically thought by the designer. He combines the formal data with the final wished product, which came to him at first in his thought. He is able to know how to organize his thought in order to accomplish what he wants to create. Bodies and minds are dependent on each other, but reason governs action and theory of action. Every new situation must have been reflected and analysed, with the purpose of what should be aimed at, and that for the common good of human beings.
Beyond Aristotelian philosophy, I find even more philosophy in the constructal theory. Descartes and Leibniz (1646-1716) have printed out their tracks on it. Besides Cartesian co-ordinates and analytic geometry, Descartes [5] conceived a philosophy which aimed to establish a basis for certainty. He compared particularly ancient cities and new cities to conclude that ancient cities are badly laid out. And, in the Discourse on Method (1637) , he suggested that old cities are not laid out by a designer, because they extended themselves progressively in such a way that they have become with time large towns without any precise design. These remarks fit very well with the constructal designer’s purposes : every time, he knows how to join what would be useful and what would be perfect in any constructal project. Descartes’ theories are exposed like the story of a dramatic voyage of discovery, going from doubt to certainty. Thinking that the senses are deceiving, Descartes suspected all existential claims through two arguments : the not conclusive “dreaming argument” (I cannot prove that I am awake) and the argument of a malignant demon: “this demon cannot cause me to be nothing”, but something exists which is “I”(“I think, therefore I am”). Meanwhile Descartes underlines strongly human will. Like for Aristotle, for Descartes mind and body were interdependent, even if Descartes gave the representation of an incorporeal mind lodged in a mechanical body. The metaphor of the tree sums up Descartes’ philosophy with metaphysics as roots, physics as trunk, and as branches : morals, medicine, and mechanics. We must note that the representation of the tree is important in the constructal theory : for example, tree-shaped architectures are generated by optimising every geometric design. Each feature of the tree is deterministic and the result of a single principle of optimization, a pattern of cooperation which is responsible for the formation too of societal trees, from bacterial colonies to urban growth.

As to Leibniz [3], like Descartes he discovered a new trend of mathematics, the differential calculus, also separately invented by Newton (1642-1727). Leibniz is another “continental rationalist”; he had in his Monadology (1721) the original idea that in the whole of reality there were independent monads or unities which did not influence each other, because they had no windows, but were nevertheless working themselves through according to a “pre-established harmony”. Each one of all elements was expressing all others together with the whole universe : the part-whole relationships were implicated by an a priori harmony established by God. We can see an analogy between this leibnizian idea with that of the contructal theory for which the designer has adapted every subsystem to all the others in such a way within the network that the whole has the perfect shape with the perfect efficiency. Thanks to the pre-established harmony, for Leibniz windowless substances were independent from each other but each of them owned a “complete concept” like a law-governed whole : that is, every part was itself a whole. The complete concept of a substance was presented as being sufficient to identify a substance : hence, the leibnizian “principle of indiscernibles”. The result was the best possible world, the parts of which were indispensable parts. However human will was seen by Leibniz as free : even if man must follow the strongest motive, human inclination to the motive is not logically necessary, only hypothetically necessary. Following the “principle of the best”, God has created a world that is the richest possible, according to which the “principle of sufficient reason” concerns only logical explanation for everything and for human action, and which means that nothing can be definitively so without a reason.

Mechanism and teleonomy are absolutely not opposites; Kant [2] did not think them as opposites because they can explain each other. The constructal theory establishes that shapes and structures are governed by a teleonomy, so that there is a mechanical finality which proceeds in such a way of growth that the whole transcends the parts which get organized themselves into a sufficient determination of the whole. In the quest for the perfect design, constructal theory has a design of the world which is resolutely optimistic. There is an original constructal perception of the world.

4. The open future of constructal theory

The best of constructal structures is realized in the equilibrium structures for which the performance level is the highest and which do not change even though the flow architecture continues to change with maximum freedom. At equilibrium the flow structure achieves the most that its freedom to morph has to offer.

In the constructal perspective, the designer works essentially on geometric basis with an important effect played by the notion of surface.Thanks to geometry [15], all the system imperfections can be shared out among the diverse elements and a particular shape can contribute better to this sharing : for heat rejection, for instance, a rectangular shape is better, more efficient than a square one. Elementary surfaces are organized in one network of collection. In the progression from the smallest to the greatest, one can reach a very high degree of complexity in functioning together with a high degree of freedom.

With the book, Shape and Structure, from Engineering to Nature [7], Adrian Bejan shows perfectly the deterministic principle generating geometry form in natural systems. Natural flow structures result from an optimization process of constructing. The objective and constraints principle used in engineering is identified as the mechanistic principle from which the geometry (shape and structure) in natural flow systems emerges. Therefore, one can provide the morphology of tree flows in many sectors: heat transfer, microchannel networks, electronic cooling, fluids engineering, urban design, geophysics, physiology, transportation.

Heitor Reis [12] has seen the constructal view of global circulation and climate : he has defined models promising to be useful in the study of Earth’s climate. Adrian Bejan and Sylvie Lorente [13] have expressed the thermodynamic formulation of the constructal law, in particular in the case of maximization of flow access in systems with heat and fluid flow irreversibilities, but also with freedom to change configuration. The constructal analysis of dendritic growth has been explained by Antonio F. Miguel [14] around the question : What shape is optimal for survival ? Roots provide the best structure to acced to most nutrients in the shortest time.

A team of researchers at Duke University’s Pratt School of Engineering and Pennsylvania State University have found reports that all animals bear the same stamp of physics in their design. Basic characteristics of locomotion for every creature are explained by the constructal theory that is a confirmed analytical approach to describing movement or flows in nature : because animal locomotion is not different than other flows, animate and inanimate. After their findings they have implications for understanding factors that guide evolution by suggesting that many important functional characteristics of animal shape and locomotion are predictable from physics [17].

5. Constructal comparisons in art and science

The constructal theory can contradict C. P. Snow’s denunciation [6] concerning the absence of mutual intermingling between the two fields of art and science, between human and natural science. But, In Shape and Structure, from Engineering to Nature [7], Adrian Bejan has alluded to Beauty.

Nature offers perfect shapes : a tree has a spacious top because it needs to expose its body to the sunshine, and divergent roots because it needs to obtain the maximum nutritive matter from the earth. Friction and gravity influences the flow of its nutritive matter [16]. Blood circulates in the human body. The ramifications are typical of wood, vascular system and human lungs.

There is a relationship between artistic activity and technical activity; and “design” is a borderline concept in the middle of making a thing and giving it an appropriate shape. Here is a tradition coming back from the Renaissance searching inspiration in the shapes observed in Nature, natural laws being considered as perfection rules [11].

From engineering to nature : that is the flow of Bejan’s book [7]. Form is generated by the principle of the optimal distribution of imperfection. In the constructal perception, the engineered and the natural worlds tend to be united. The generation of geometric forms in natural systems obey to a deterministic principle.


[1] Aristotle, Complete Works of Aristotle, Vol. 1, Vol. 2, J. Barnes Ed., Princeton /Bollingen, 1971, 1995.
[2] I. Kant, Critique of Pure Reason, New York: Prometheus Books, 1990.
[3] G. W. Leibniz’s Monadology. An edition for Students by Nicholas Rescher, Pittsburgh, Pa: University of Pittsburgh Press, 1991.
[4] A. Bejan, Advanced Engineering Thermodynamics, second edition, New York: Wiley, 1997.
[5] R. Descartes, Discourse on Method. Meditations on First Philosophy, Indiana: Hackett Publishing Company, 1998.
[6] C. P. Snow, The two Cultures, Cambridge University Press, 1998.
[7] A. Bejan, Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, UK, 2000.
[8] H. Poirier, “Une théorie explique l’intelligence de la nature”, Science et Vie, n°1034 , Paris, November 2003, pp. 44-63.
[9] Jean-Paul Basquiat, « Discussion. Mémétique et théorie constructale », in Automates intelligents 2003 :
[10] R. N. Rosa, A. H. Reis and A. F. Miguel, Eds., Bejan’s Constructal Theory of Shape and Structure, Évora Geophysics Center, University of Évora, Portugal, 2004.
[11] R. N. Rosa, “A brief appraisal of Professor Adrian bejan work », in Bejan’s Constructal Theory of Shape and Structure (op. cit.), Évora, Portugal, 2004, pp. 5-14.
[12] A. H. Reis, “Constructal view of global circulation and climate”, in Bejan’s Constructal Theory of Shape and Structure (op. cit.), Évora, Portugal, 2004, pp.171-189.
[13] A. Bejan, S. Lorente, “Thermodynamic Formulation of the Constructal Law”, in Bejan’s Constructal Theory of Shape and Structure (op. cit.), Évora, Portugal, 2004, pp. 95-119.
[14] A. F. Miguel,  “Dendritic Growth : Classical Models and Denditric Analysis”, in Bejan’s Constructal Theory of Shape and Structure (op. cit.), Évora, Portugal, 2004, pp. 75- 93.
[15] A. Bejan, S. Lorente, La loi constructale, Avant-propos d’Angèle Kremer-Marietti, Paris, L’Harmattan, 2005.
[16] M. Giboda, “Interaction – Science and Art. Divergence and convergence”, Czech-Argentine Biennale, “e-Golems”, July 2-5, 2005.
[17] Bejan and J. H. Marden, “Unifying constructal theory for scale effects in running, swimming and flying”, Jan.3, 2006, Journal of Experimental Biology 209, 238-248 (2006).

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