To optimize the arrangement of heat-generating components, plate spacing and number of columns can vary. If spacing is too large or too small, the hot-spot temperature is high (in red). The optimal spacing is shown in the middle frame, where the hot spots are the coolest

Thermodynamics represents a powerful tool that engineers can use to gain a more profound understanding of how naturally organized systems arise and evolve. Since the dramatic rise in the cost of fuel in the 1970s, engineers have applied thermodynamics to improve the performance, increase the efficiency, and lower the cost of man-made systems. In the process, engineers have extended the theory that underlies thermodynamics—which was originally conceived to define engine performance in terms of heat and work—so that it also applies to the operation and optimization of highly complex systems. Thermoeconomics and the analysis of exergy (the loss of available energy) are examples of such applications. Even more important, engineers have simultaneously 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 a system—natural or man-made—and its components.

This development, known as entropy-generation minimization, is an aspect of thermodynamic optimization that sheds considerable light on 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 physical parameters, including size, shapes, and materials. After gaining this understanding, engineers—at least in theory—can go on to optimize the system's performance in terms of its constraints, which are responsible for its irreversible operation.

For example, engineers can use the energy-minimization method to optimize the "rhythm" of intermittent processes in which irreversibility is caused by time-dependent diffusion—the growth of a layer of ice on a cooled surface, for instance. Thickness of the ice increases with the square root of the duration of the freezing process. The rate of ice production decreases, however, as the freezing process continues. If the purpose of the system is to maximize the production of ice (or refrigeration or exergy storage), engineers can "optimize" the freezing process by periodically interrupting it, scraping the surface clean, and restarting the process. The time-averaged rate of ice production is proportional to the square root of the duration of freezing divided by the freezing and surface-cleaning times. Using this knowledge, engineers can maximize ice production by fine-tuning the on-and-off freezing process.

Such rhythms can help engineers understand a wide variety of man-made and natural phenomena. For example, engineers could proceed in this way to predict the existence of unique, finely tuned frequencies for heartbeats and breathing, which decrease as body size increases. Such allometric laws, which have been observed empirically in biology for quite some time, can now be anticipated theoretically according to engineering thermodynamics and entropy-generation minimization.

The shape-optimization principle applies at larger volume scales: On the left is the optimized shape for a first construct; and on the right, for a second construct

Of course, the larger issue here is not just to observe and describe such processes but also to understand the purpose of the optimization that rules a naturally occurring structure. After all, when engineers design a device or system, they must first understand its purpose. The device must function—fulfill its purpose—subject to certain constraints. Merely analyzing the device is not sufficient; the real objective is to optimize it, construct it, and make it work. The unique understanding that engineers offer in the search for the origins and evolution of naturally occurring structures is that many "designs" for such structures—just like those for man-made ones—have nearly the same overall performance as the optimal design, even though they differ in their finer details.

Put another way, this engineering insight helps account for the evolution of naturally occurring systems—at least those subject to flow and size constraints—from the simple to the complex. Understanding how nature is "engineered"—in other words, grasping the thermodynamic principles underlying a naturally occurring structure's finite size, constraints, optimization, and construction—can help engineers determine how a shape occurs and how a structure develops as it moves from one scale to the next larger scale.

Perhaps the best way to illustrate this concept is to consider the topology of a structure and the way its components interact. Using thermodynamic-optimization methods, engineers can determine the optimal dimensions of components. For example, in the most elementary passage of a heat exchanger, the generation of entropy is due to both heat transfer and fluid friction, which compete against one another. The hydraulic diameter of the passage can be selected such that the sum of the two irreversibilities is minimal. The dimensions of bodies immersed in external convection can be selected in a similar way. Even simpler is the sizing of a system in which a single transport mechanism causes the irreversibility, such as heat transfer. When the heat current is imposed, minimizing the entropy generation means minimizing the resistance to heat flow.

To cool electronic packages, for example, both the volume and the heat-generation rate that is distributed uniformly over that volume are fixed. The heat current is removed by a single-phase stream with natural or forced convection. The geometric arrangement of heat-generating components can then be optimized such that the hot-spot temperature is minimal. In attempting such an arrangement, the plate-to-plate spacing (or number of columns) is free to vary. If the spacing is too large, however, there is not enough heat-transfer area and the hot-spot temperature becomes high; when the spacing is too small, the coolant flow rate decreases and the hot-spot temperature is again high. Between the two lies an optimal spacing—an optimal package architecture—that minimizes the thermal resistance between the system and the environment. This geometric principle is applied in many man-made and natural systems. A swarm of bees, for example, regulates its maximum internal temperature by constructing similar cooling channels; the bees rearrange themselves to create vertical (internal-chimney) channels through the swarm.

Researchers recently discovered that, when minimizing the thermal resistance between a fixed heat-generating volume and one point, every portion—every subsystem of the given volume—can have its shape optimized. This principle can be illustrated at the smallest volume scale, where a single high-conductivity fiber removes the heat generated by the low-conductivity material from the system. Ultimately, an optimal rectangular shape can be found to minimize the thermal resistance between the element and the exit end of its high-conductivity fiber.

The same geometric-optimization principle applies at larger scales. The next volume is an assembly of optimized volume elements of the smallest size. This construct can also have its shape (or number of constituents) optimized. The process of construction and shape optimization continues toward stepwise larger scales until the given volume is covered. The end result is shape and structure—the optimized architecture of the composite (high- and low-conductivity materials) that connects the sink point to the finite-size volume. The high-conductivity paths form a tree, and the low-conductivity paths reach the infinity of points of the given volume.

Tree networks abound in nature, in both animate and inanimate flow systems. We find them everywhere: plants, leaves, roots, lungs, vascular tissues, neural dendrites, river drainage basins, lightning, and dendritic crystals. Every detail of every natural tree, for example, can be anticipated through the construction and optimization shown for the heat tree already discussed. In fluid trees, the small-scale volumetric flow is by slow viscous diffusion (such as Darcy flow in the wet banks of the smallest rivulet), while the larger scale flow is organized into faster conduits. The high-conductivity channels form a tree. More important, each feature of the tree is deterministic, the result of a single principle of optimization.

This conclusion runs counter to currently accepted doctrine that natural structures are nondeterministic, the result of chance and necessity. In fractal geometry, any tree can be simulated by repeating an assumed algorithm and truncating this operation at an arbitrary, small (finite) scale. But fractal geometry is descriptive, not predictive. The discovery, then, is the mechanism that generates this structure, from one scale to the next.

Volume-to-point constructs have a definite time direction: from small to large, and from shapelessness (diffusion) to structure (channels and streams). Determinism results only if this time arrow is respected. If the time direction is reversed, such as from large to small, through the repeated fracturing of a postulated network into smaller and smaller pieces (as in fractal geometry), then it is impossible to predict the optimal volume-to-point flow architecture.

The optimized geometry formed by low- and high-conductivity flow regimes unites all the volume-to-point flows. Think of how oxygen flows through a mammal: The low-conductivity flow is volumetric mass diffusion through tissues, while the high-conductivity flow is stream flow through blood vessels and bronchial passages. Also consider turbulent flow: Diffusion in the smallest volume elements is accompanied by the structure of faster currents known as eddies. Artificial constructs, such as the internal arrangement of components in a computer, require the same cooperation between slow and fast heat transfer, with the slow mode placed at the smallest scale.

This pattern of cooperation is also responsible for the formation of societal trees, from bacterial colonies to urban growth. Every member of the living group has a place in the structure, in such a way that every member benefits. The urge to organize is thus an expression of selfish behavior, i.e., actions for the good of the individual rather than of the group.

The geometric principle that generates tree networks in living groups is even clearer in the context of minimizing the travel time between one point and a finite-size area with an infinity of points. Travelers in this hypothetical area have access to more than one mode of locomotion, starting with the slowest speed (walking) and proceeding toward faster modes. 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 theory predicts urban growth, from backyards and alleys to streets, avenues, and highways. Along the way, we discover that tree networks are results of a unique geometric optimization principle. The principle represents the physics behind fractals, the reason why natural tree networks happen to look like the truncated fractal images postulated by mathematicians.

In the past, the minimization of travel time has been invoked to account for the shape of light rays. Research as far back as Heron of Alexandria has shown that when the ray strikes a mirror, the optimal angle of incidence equals the angle of reflection. We were also taught that the ray is bent to an optimal angle as it passes from one medium into another.

This travel time, or the principle of minimizing resistance, suggests a generalization: For a finite-size system to persist in time—that is, to live—it must evolve 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 (such as shapes and structures) through constrained open systems. It also accounts for the evolution (improvement) of these paths, which occurs in an identifiable direction that can be aligned with time.

This generalization accounts for the "choices" that are made by natural and man-made macroscopic open systems subject to flow and size constraints. It bridges the gaps not only between physics, biology, economics, and many related fields but also between physics and engineering. Moreover, it introduces engineering insights into the current debate on natural order—and rightfully so. Nature is the supreme form of engineering design and organization. Who is better qualified than an engineer to extend the deterministic powers of thermodynamics so that they help account for the grand design of the natural world?

Adrian Bejan, a Fellow of ASME, is the J.A. Jones Professor of Mechanical Engineering at Duke University in Durham, N.C.

This research was supported by the National Science Foundation.

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