Branching out: Engineering professor and theory conceiver Adrian Bejan. Chris Hildreth
Branching out: Engineering professor and theory conceiver Adrian Bejan. Chris Hildreth

Going With the Flow

Can a "commonsense, concise, and useful" theory predict the shape of things that are and the shape of things to come?
October 1, 2007

Manhattan is all about movement. This summer the Museum of Modern Art mounted a retrospective of the work of Richard Serra, who crafts enormous plates of steel into sculptures. When you enter one of the spiraling shapes, you're led to no visible destination; the tilted walls sometimes open up invitingly, sometimes close in claustrophobically. This is a museum encounter that demands not just observation but experience—the experience of merging into a crowd as it flows through a work of art.

A couple of blocks up from the exhibit, along Columbus Circle, the preoccupation is with the art of efficient customer flow. There, Whole Foods, the gourmet supermarket, has abandoned separate checkout lines, including those that turn out to be inevitably and annoyingly slow-moving, in favor of a single, serpentine line. As soon as a cash register becomes available, the next customer is summoned. In a front-page article, The New York Times called the process an emblem of “queue management.”

Wherever a flow system is involved, Adrian Bejan has something to say about it. A decade ago, Bejan, J.A. Jones Professor of mechanical engineering, coined the term “constructal theory,” originally as an idea applied to thermodynamics. How might heat be dispersed in small electrical devices, he wondered, so they wouldn't burn up? He found that in many cases, the answer is to let the heat spread out like a tree's limbs and leaves or like a river's tributaries. In a book published by Cambridge University Press in 2000, Shape and Structure, From Engineering to Nature, Bejan found parallels between engineering principles and mechanisms in natural flow systems. An optimal engineering system, he argued, hinges on the ability to minimize all the resistances to internal flows—whether those are the flows of heat, fluid, or electricity.

patterns in nature

Writing in the January 2006 International Journal of Heat and Mass Transfer, Bejan and geophysicist A. Heitor Reiss of the University of Évora in Portugal turned to constructal thinking in an audacious application of the theory. They wanted to predict the climate, the large-scale movement of air that distributes heat on the surface of the Earth—the grandest of all flow systems. “Nothing flows ideally,” Bejan says. “Every flow system is destined to remain imperfect. The struggle of nature is to be the least imperfect it can be.”

From thermodynamics, constructal theory has morphed into a theory of pretty much everything, natural or manmade. Bejan says it unites physics with Darwinian evolution. According to the theory, if free to do so, a flow system—a river basin, a blood stream, or city traffic—will evince a pattern that allows for optimal movement.

That's also true of things that fly, run, and swim. To fly at optimal speed is to strike a balance between the vertical and horizontal loss of energy, says Bejan. “The bird is basically a falling body, a rock. In every time interval that the bird falls, the bird has two jobs. One is to lift itself vertically back to where it was. But it also has to advance horizontally, which means it has to overcome drag. When the cruising is fast, it takes a little work to lift itself up. But it takes a lot of work to go forward. Once you put your finger on that, you know the optimal flapping frequency for the wings.” Larger birds, then, fly faster and flap their wings less frequently, though with greater force.

Vascularized design: Rectilinear representation of optimal flow in areas as various as biological tissues and microelectronics. Red and blue channels alternately collect from and discharge into the yellow interstices

Vascularized design: Rectilinear representation of optimal flow in areas as various as biological tissues and microelectronics. Red and blue channels alternately collect from and discharge into the yellow interstices


The same tasks are demanded of a running or a swimming animal. And the same mechanisms that produce flying efficiency produce efficiency in runners and swimmers. “The runner has to get off the ground, which is vertical work. And then the runner has to advance against the horizontal ground and air friction; at higher speeds, the gazelle or the cheetah struggles mightily against drag.” In Bejan's view, all forms of locomotion, managing as they do to surmount obstacles in physics through a balancing act of good design, are essentially identical. And that fact illustrates the presence of a universal principle.

“People didn't copy a bird to make an airplane. They tried all sorts of shapes, and are still trying. And, as it turns out, the ones that are better and better look more and more birdlike.”

At a constructal-theory conference held at Duke this past spring, a mathematician gave a presentation that considered a dog with a seeming capacity to calculate in constructal terms. That capacity actually is an aspect of survival: When a lion is going to chase down an antelope, or when that dog is going to retrieve a stick thrown from a lakeside beach into the water, it has to calculate precisely the most efficient way to perform the feat, given its properties of locomotion. So the dog doesn't make a direct-line approach; at some point, it dashes into the water and swims at an angle toward the stick. The dog finds the optimal path to retrieving its object.

In a book published this year, Constructal Theory of Social Dynamics (Springer Science+
Business Media), Bejan and his co-editor, Gilbert W. Merkx, a sociologist and the vice provost for international affairs at Duke, take the theory across new intellectual boundaries. In the book's preface, Bejan and Merkx declare that the theory is so “commonsense, concise, and useful” that it applies to the social sciences, as well as to engineering science. Merkx contributed a chapter on “Constructal Models in Social Processes.” Duke colleagues Kenneth Manton A.M. '71, Ph.D. '74, Kenneth Land, and Eric Stallard wrote on “Human Aging and Mortality”; Edward Tiryakian on “Sociology Theory, Constructal Theory, and Globalization”; and John Staddon on “Is Animal Learning Optimal?”

Merkx, in his chapter, scrutinizes Mexican migration to the United States. The migration stream began to develop rapidly with World War II, he points out, when the U.S. responded to labor shortages by recruiting Mexicans. Originally, labor would flow north over the border for planting and harvest, and flow back over the border in the off seasons. Merkx argues that as the U.S. clamped down on the natural flow system—that is, the move across borders, including the easy flow of Mexicans back to their home country—the system lost its efficiency.

“There will be less flow through big channels,” he says. “But the flow of people will still leak through in other ways, and they won't go back, because the transaction costs are so high. That means they want to bring their families over, too, so you're actually moving more people and creating more and more of a one-way flow by closing the border. Another effect is, once the border is sealed, you begin to see this sort of washing-through effect: The population spreads out and disperses beyond border areas to places like New England and North Carolina.”

Having grown up in Venezuela, Merkx came with his family to the U.S. when he was eleven. His childhood across borders, he says, “gave me a lifelong interest in viewing things from a comparative perspective.” As a Harvard undergraduate, he studied both sociology and cultural anthropology, which whetted his appetite for understanding social processes. The sociology department at Harvard at the time was led by Talcott Parsons, who was committed to uncovering “pattern variables,” the basic attributes or properties that characterize all social systems.

Merkx: studying human migration as a natural flow system

Merkx: studying human migration as a natural flow system. Megan Morr

Merkx first met his constructal-theory co-editor in the fall of 2003, when Bejan was appointed to the Provost's International Advisory Committee, which Merkx chairs. Bejan later asked Merkx for travel support for a research trip to Eastern Europe. After the trip, the two came together for lunch, and Bejan talked about his theory, sketching for Merkx trees and rivers with their patterns of channels extending from delta shapes. Those images, Merkx recalls, mirrored familiar patterns in the social sciences. “I've always had a kind of system approach,” he says. “I see countries as systems. And sometimes if the system is not functioning well, then it breaks down, the same way a flow system can break down when it has too many obstacles.”

The theory's inventor, Bejan, has faced his own obstacles as an engineer with a theoretical bent. “To me, engineering is a science, and on this I fight with everybody,” he says. “I want respect for engineering science. Engineering is not something that you tinker with in a shop, sell, and make money. That is technician work. Engineering is a mental viewing, it is ideas, it is rare and noble, just like frontier physics or biology.”

Bejan has long been enamored of patterns—and freedom of movement. After all, he says, science seeks deterministic principles in a world seemingly ruled by chance events. And scientists insist on the freedom to question, overturn, and invent those principles.

He grew up enduring the restricted patterns of Communist Romania. His father, a veterinarian, had been imprisoned with the beginning of Communist rule in 1948, and his mother, a pharmacist, similarly “disappeared” for a time ten years later. Beginning in third grade, Bejan took afterschool drawing and painting lessons at a fine-arts school; there, as he puts it, he “learned the language of patterns.”

But basketball competed with art for his attention. He was exposed to the sport first as a ball boy at a local sports arena. In high school, he was a starting player in a professional league: He jokes, “Moi, LeBron James!” After graduating from high school, he enrolled at the Galati Polytechnic University, in part to continue playing basketball in national competition at the highest level: Galati had a team in the top league.

Fish in motion: Swimming, like flying and running, involves striking a balance between the vertical and horizontal loss of energy

Fish in motion: Swimming, like flying and running, involves striking a balance between the vertical and horizontal loss of energy

Basketball, a flow system in its own way, is always with him, Bejan says; at Duke's faculty club, he can be seen practicing the flow of the game, dribbling, pivoting, and shooting. When, in his schoolboy days, his basketball coach was asked about producing a great shooter, he would reply that his interest was in producing a great passer. The game fundamentally is about moving the ball, and that imperative involves, moment by moment, choosing the more efficient scoring path. “My coach taught that when you see a good opening, pass the ball. Or, if you don't see the opening, give it to a guy who knows how to dribble.

“The playing field is like vascularized muscle and arterial blood flow. The players are milling around in order to create pores for all these possible paths. And a good team puts the ball in the right channels—the right channels over space and time.”

Then, in a reaction to the Prague Spring of 1968, a current of liberalization that swept briefly through Eastern Europe, Romania's Education Ministry offered scholarships for study in the West. On the basis of a competitive exam, Bejan earned one of the half-dozen places.

He enrolled at the Massachusetts Institute of Technology; it was the only school that Romanian authorities allowed him to apply to. As a freshman he took a strength-of-materials course, in which he learned how to calculate the maximum stress that a beam could accommodate. “I said to myself, this is amazing,” he recalls. “I know what will happen to something without having to build it and test it.” He had discovered the power of theory. The next year, he took a class quiz that required predicting how a machining process would produce chips off a chunk of metal. It was the first time as a college student, he says, that he was encouraged to be an original thinker—a quality, he adds, that he encourages in his own classes.

Bejan refused to return to Romania after graduation, instead accepting an offer to continue work at MIT toward his doctorate. In the eyes of the government, he wasn't just a defector but a traitor as well. At his old high school, posters went up identifying him as an example to be avoided.

“Constructal theory shows that freedom is good for design,” he says. “Also freedom to morph is good for design.” A political system —like an engineering system or a natural system—has to be self-correcting to endure. Freedom, in the realms of politics and economics, nurtures networks that are efficient, including networks for encouraging creativity and for maximizing profits. That's why democracy has staying power, he says.

Outside his office in the Pratt School of Engineering, Bejan displays a quote from Plato: “Let no one untrained in geometry enter my house.” Inside, on one wall, he has certificates grouped according to a strict pattern: on the left, awards from professional societies; on the right, fifteen honorary degrees, clustered geographically, from Western to Eastern Europe. On another wall is a seascape he did in grade school and his intricately rendered, multiple-perspective projections of a kite, from his polytechnic days.

Within Pratt, Bejan is considered an iconoclast—and a maverick. In the preface to his earlier book, he repeats a lesson about academic colleagues he learned from one of his former MIT professors. The lesson came in the form of an insight from Sancho Panza, loyal servant to Don Quixote: The windmills hit his master just as hard as he hit them.

Beyond the campus, too, Bejan is regarded as an unconventional thinker—though he's succeeded at working with other unconventional thinkers, including Sylvie Lorente, professor of civil engineering at the National Institute of Applied Sciences in Toulouse, France. Lorente, with Bejan, helped develop a Duke mechanical-engineering course on constructal theory.

This summer, Bejan traveled to Portugal for an international constructal-theory conference, which drew more than 100 physicists, biophysicists, and engineers. On a Paris stopover, he met with other groups of constructal-theory enthusiasts. Some specialized scientific journals, he laments, haven't been quick to publish his work because engineers don't often have status outside the engineering profession—even as, to some engineers, theory-powered thinking doesn't do much to confer status. Still, a decade ago, he was awarded the Worcester Reed Warner Medal from the American Society of Mechanical Engineering International. The medal goes to one individual each year in recognition of “outstanding contribution to the permanent literature of engineering.” In Bejan's case, it honored “his originality, challenges to orthodoxy, and impact on engineering thermodynamics and heat transfer.”

Today, he says he maintains the drive to create that comes from being an outsider. 
“I feel I have to prove myself. I feel that every day.”

Bejan is quick with the constructal quip, referring to animals, in constructal terms, as “walking trees,” including terms like “svelteness” in the technical diagrams he shares with visitors, and declaring that “the future belongs to the vascularized” (a saying he borrowed from his collaborator Lorente). For all his enthusiasm, he has confronted questions about the novelty of a concept linking the shapes of systems with their other properties. An anonymous posting on a physics website, for example, declared that “The idea of deriving outcomes of (biological, astronomical, other) systems based on the simple laws that govern them is gorgeous, necessary, and very, very old.” But Bejan says that what makes the theory “dangerous,” or the observations embedded in the theory seemingly old, is the harking back to engineering as a scientific pursuit. What a theory can do, he adds, is to tie together seemingly random observations into a grand package, or to reveal the pattern that's not apparent.

City systems: Over time, streets become wider and longer as evolving transportation technologies make it possible to go farther, faster.  Here, aerial view of London, 1851

City systems: Over time, streets become wider and longer as evolving transportation technologies make it possible to go farther, faster. Here, aerial view of London, 1851. Bridgeman Art Library

And those patterns are everywhere, including human creations like street blocks and airports. Merkx, Bejan's co-editor on the new book, observes that a well-laid-out block or a well-planned airport, like Atlanta's, minimizes the average time, effort, and energy expended for a traveler between each mode of travel. “Different block lengths and house heights and sizes also evolve to minimize the time required for the average traveler from point to point,” he says. If walking is the primary mode, as it would have been in the time before cars ruled the roadways, streets can be relatively narrow, and houses will not be too deep nor have many stories.

“Let's say you add the horse and buggy, plus walking. Then streets must be wider and blocks longer, but houses will remain not too large or high. If you add a slow elevator to the mix, then houses can efficiently rise in height without sacrificing time. If you add cars and high-speed elevators instead of buggies, blocks can get longer and buildings much taller, with the same time efficiency. But the speed of walking does not change. Therefore, the buildings tend to go higher more rapidly than they get wider or deeper, because the time required for walking limits the horizontal distances.”

As Merkx sees it, the Pentagon, with its endless horizontal corridors and modest vertical scale, is a case study of inefficient flow. (According to a new history of the Pentagon, the five-sided plan conformed to the shape of the plot of land, while the low height of the building was meant to keep it in harmony with the low Washington skyline.) “If you want to have a building that is really huge in terms of the horizontal dimensions, like an airport terminal, then you have to put in high-speed people-movers, like trams, to minimize the time or maximize the efficiency.”

Constructal theory in practice: Atlanta's airport is designed to minimize passenger time in getting from point to point

Constructal theory in practice: Atlanta's airport is designed to minimize passenger time in getting from point to point. Ron Sherman

In constructal terms, is language a high-speed ideas-mover? That was the research starting point for Cyrus Amoozegar, a former student of Bejan's who is a Duke senior majoring in biomedical engineering and mechanical engineering and with minors in chemistry and Chinese. In a chapter he contributed to the book, he examines the flow paths of modern languages and two of the earliest languages, ancient Egyptian and Chinese.

“Through time, written language develops a set of pieces from which the most basic ideas are constructed,” he says. “In English, these pieces are the alphabet. The forms and uses of these pieces change through time so that they are easy enough to remember but complex enough to be distinguishable from one another and numerous enough so that ideas can be conveyed easily.”

According to constructal theory, a written language evolves to “connect” better to the masses, even as it's able to provide a more accurate description of the world. If the elements that constitute a language are complicated, the language will take too long to write and will be more difficult to remember. The global resistance will increase. On the other hand, if the language elements are too simple, the users of the language will lack precision. The meaning of words will be misconstrued. The natural evolution of written language, then, must head for a balance between the complicated and the simple.

With interests in history and engineering, another former student, Gideon Weinerth '07, wrote a term paper in Bejan and Lorente's course applying constructal theory to ancient warfare. Weinerth says that warfare can be understood, after all, in terms of flow systems. The Greek phalanx, for example, would maximize its effectiveness by taking on the same shape that Bejan noticed in riverbeds, that is, a semi-circle. A deeper phalanx of soldiers offered more pushing power than a narrow formation. But in making its flanks wider and thinner, a phalanx could build a strong defense. Those two actions would be at cross-purposes, so the idea was to find the perfect geometric balance. By the Roman period, the phalanx had been reorganized into an independent, highly mobile, and rapidly adjustable unit. “This is simply a validation of the freedom of design providing advantages in efficiency,” Weinerth writes.

In class Bejan compared the optimization of the material in a cantilevered beam designed by Galileo with how the Roman army maximized the strength of all of its soldiers. By that account, Galileo was unconsciously a constructal theorist. The class discussion “began to tip me off to possible avenues for investigation,” Weinerth recalls. He says he was surprised to find that studies of military strategy have been largely devoid of references to math or physics.

Today Bejan is interested in linking constructal theory and another sort of global phenomenon, higher education. Universities always have been a morphing flow system, he says. Through the centuries, ideas, and the people who generate them, have moved through channels from centers of learning in Bologna, then Padua, then Paris, then the United Kingdom, now the U.S. Those channels may swell or shrink, and the nodules—the learning centers—along the channels may grow or diminish in importance. But, as in any effective flow system, the hierarchy remains essentially fixed and recognizable.

Bejan worries that engineering itself may be too fixed and recognizable for its own good; and part of his crusade is to get the profession to think in grander terms. In his earlier book, he observes that engineering “ranks either low or not at all on the ladder of respect.” He adds, “Biologists and physicists are describing what nature is and how it works. What do engineers bring to this apparently full table? Engineers describe how a system changes its configuration in time so that its global performance improves.”

With figures like Gustave Eiffel and Leonardo da Vinci as his models, he suggests that engineers can blur the lines between the natural and the artificial, that they can define the theoretical agenda for the life sciences. It's just a matter of going with the flow of good ideas —or against the flow of conventional thinking.