S. Lorente

Constructal theory of generation of configuration in nature and engineering

Constructal theory is the view that the generation of flow configuration is a physics phenomenon that can be based on a physics principle (the constructal law): “For a finite-size flow system to persist in time (to survive) its configuration must evolve in such a way that it provides an easier access to the currents that flow through it” [A. Bejan, Advanced Engineering Thermodynamics, 2nd ed. (Wiley, New York, 1997); Int. J. Heat Mass Transfer, 40, 799 (1997)]. This principle predicts natural configuration across the board: river basins, turbulence, animal design (allometry, vascularization, locomotion), cracks in solids, dendritic solidification, Earth climate, droplet impact configuration, etc. The same principle yields new designs for electronics, fuel cells, and tree networks for transport of people, goods, and information. This review describes a paradigm that is universally applicable in natural sciences, engineering and social sciences.

The constructal law and the evolution of design in nature.

The constructal law accounts for the universal phenomenon of generation and evolution of design (configuration, shape, structure, pattern, rhythm). This phenomenon is observed across the board, in animate, inanimate and human systems. The constructal law states the time direction of the evolutionary design phenomenon. It defines the concept of design evolution in physics. Along with the first and second law, the constructal law elevates thermodynamics to a science of systems with configuration. In this article we review the more recent work of our group, with emphasis on the advances made with the constructal law in the natural sciences. Highlighted are the oneness of animate and inanimate designs, the origin of finite-size organs on animals and vehicles, the flow of stresses as the generator of design in solid structures (skeletons, vegetation), the universality and rigidity of hierarchy in all flow systems, and the global design of human flows. Noteworthy is the tapestry of distributed energy systems, which balances nodes of production with networks of distribution on the landscape, and serves as key to energy sustainability and empowerment. At the global level, the constructal law accounts for the geography and design of human movement, wealth and communications.

Constructal law of design and evolution: Physics, biology, technology, and society

This is a review of the theoretical and applied progress made based on the Constructal law of design and evolution in nature, with emphasis on the last decade. The Constructal law is the law of physics that accounts for the natural tendency of all flow systems (animate and inanimate) to change into configurations that offer progressively greater flow access over time. The progress made with the Constructal law covers the broadest range of science, from heat and fluid flow and geophysics, to animal design, technology evolution, and social organization (economics, government). This review presents the state of this fast growing field, and draws attention to newly opened directions for original research. The Constructal law places the concepts of life, design, and evolution in physics.

The constructal law of design and evolution in nature.

Constructal theory is the view that (i) the generation of images of design (pattern, rhythm) in nature is a phenomenon of physics and (ii) this phenomenon is covered by a principle (the constructal law): ‘for a finite-size flow system to persist in time (to live) it must evolve such that it provides greater and greater access to the currents that flow through it’. This law is about the necessity of design to occur, and about the time direction of the phenomenon: the tape of the design evolution ‘movie’ runs such that existing configurations are replaced by globally easier flowing configurations. The constructal law has two useful sides: the prediction of natural phenomena and the strategic engineering of novel architectures, based on the constructal law, i.e. not by mimicking nature. We show that the emergence of scaling laws in inanimate (geophysical) flow systems is the same phenomenon as the emergence of allometric laws in animate (biological) flow systems. Examples are lung design, animal locomotion, vegetation, river basins, turbulent flow structure, self-lubrication and natural multi-scale porous media. This article outlines the place of the constructal law as a self-standing law in physics, which covers all the ad hoc (and contradictory) statements of optimality such as minimum entropy generation, maximum entropy generation, minimum flow resistance, maximum flow resistance, minimum time, minimum weight, uniform maximum stresses and characteristic organ sizes. Nature is configured to flow and move as a conglomerate of ‘engine and brake’ designs.

Optimal tree-shaped networks for fluid flow in a disc-shaped body

In this paper we consider the fundamental problem of how to design a flow path with minimum overall resistance between one point (O) and many points situated equidistantly on a circle centered at O. The flow may proceed in either direction, from the center to the perimeter, or from the perimeter to the center. This problem is an integral component of the electronics cooling problem of how to bathe and cool with a single stream of coolant a disc-shaped area or volume that generates heat at every point. The smallest length scale of the flow structure is fixed (d), and represents the distance between two flow ports on the circular perimeter. The paper documents a large number of optimized dendritic flow structures that occupy a disc-shaped area of radius R. The flow is laminar and fully developed in every tube. The complexity of each structure is indicated by the number of ducts (n0) that reach the central point, the number of levels of confluence or branching between the center and the perimeter, and the number of branches or tributaries (e.g., doubling vs. tripling) at each level. The results show that as R/d increases and the overall size of the structure grows, the best performance is provided by increasingly more complex structures. The transition from one level of complexity to the next, higher one is abrupt. Generally, the use of fewer channels is better, e.g., using two branches at one point is better than using three branches. As the best designs become more complex, the difference between optimized competitors becomes small. These results emphasize the robustness of optimized tree-shaped networks for fluid flow.

Tree-shaped flow structures designed by minimizing path lengths

Abstract This paper outlines a direct route to the construction of effective tree-shaped flow structures. Dendritic flow structures dominate the design of natural and engineered flow systems, especially in thermal and fluid systems. The starting point is the optimization of the shape of each elemental area or volume, such that the length of the flow path housed by the element is minimized. Proceeding toward larger and more complex structures – from elements, to first constructs, second constructs, etc. – the paper develops tree-shaped flow structures between one point and a straight line, one point and a plane, a circle and its center, and a point and many points distributed uniformly over an area. In the latter, the construction method is applied to a fluid flow configuration with laminar fully developed flow. The constructions reveal several features that are supported by empirical observations of natural tree-shaped flows: asymmetry, flow rate imbalance, pairing or bifurcation, angles between branches, and Y-shaped constructs that lie in a plane. It is shown that these basic features are necessary because of “packing”, i.e., assembling optimized elements into a fixed space, and filling the space completely. For the flow between an area and one point, the best elemental shape is the regular hexagon. It is shown that the emergence of string-shaped links that connect two or more elements are necessary features, which are also required by packing. Strings cover some of the inner zones of the tree network, particularly the inner zones of large and complex trees. Dichotomous Y-shaped constructs dominate the tree structure, especially the peripheral zones of the tree canopy. The practical importance of the simplified design method is discussed.

Constructal design for cooling a disc-shaped area by conduction

Abstract This paper describes a hierarchical strategy to developing the optimal internal structure of a round heat-generating body cooled at its center with the help of optimally distributed inserts of high-conductivity material. The sequence begins with optimizing the geometry of the smallest heat generating entity – a sector-shaped elemental volume with the smallest dimension, and a single high-conductivity insert. Many such elements are assembled into disc-shaped constructs, or into sector-shaped constructs in which the elemental volumes are grouped into a formation shaped as a fan. When several sector-shaped constructs are assembled into a disc, they constitute a quasi-radial heat-flow structure in which each high-conductivity insert exhibits one branching. Every geometric detail of the optimized two-material conductive structures is determined based on principle – the minimization of global resistance subject to global constraints (total volume, total volume of high-conductivity material). The inserts of high-conductivity material form structures shape as trees. The global thermal resistance of each tree-shaped construct is reported. The minimization of global thermal resistance is the criterion for choosing between a design with radial inserts and one with branched inserts.

Networks of channels for self-healing composite materials

This is a fundamental study of how to vascularize a self-healing composite material so that healing fluid reaches all the crack sites that may occur randomly through the material. The network of channels is built into the material and is filled with pressurized healing fluid. When a crack forms, the pressure drops at the crack site and fluid flows from the network into the crack. The objective is to discover the network configuration that is capable of delivering fluid to all the cracks the fastest. The crack site dimension and the total volume of the channels are fixed. It is argued that the network must be configured as a grid and not as a tree. Two classes of grids are considered and optimized: (i) grids with one channel diameter and regular polygonal loops (square, triangle, hexagon) and (ii) grids with two channel sizes. The best architecture of type (i) is the grid with triangular loops. The best architecture of type (ii) has a particular (optimal) ratio of diameters that departs from 1 as the crack...

Thermodynamic optimization of flow geometry in mechanical and civil engineering

Abstract Recent developments in thermodynamic optimization are reviewed by focusing on the generation of optimal geometric form (shape, structure, topology) in flow systems. The flow configuration is free to vary. The principle that generates geometric form is the pursuit of maximum global performance (e.g., minimum flow resistance, minimum irreversibility) subject to global finiteness constraints (volume, weight, time). The resulting structures constructed in this manner have been named constructal designs. The thought that the same objective and constraints principle accounts for the optimally shaped flow paths that occur in natural systems (animate and inanimate) has been named constructal theory. Examples of large classes of applications are drawn from various sectors of mechanical and civil engineering: the distribution of heat transfer area in power plants, optimal sizing and shaping of flow channels and fins, optimal aspect ratios of heat exchanger core structures, aerodynamic and hydrodynamic shapes, tree-shaped assemblies of convective fins, treeshaped networks for fluid flow and other currents, optimal configurations for streams that undergo bifurcation or pairing, insulated pipe networks for the distribution of hot water and exergy over a fixed territory, and distribution networks for virtually everything that moves in society (goods, currency, information). The principle-based generation of flow geometry unites the thermodynamic optimization developments known in mechanical engineering with lesser known applications in civil engineering and social organization. This review extends thermodynamics, because it shows how thermodynamic principles of design optimization account for the development of optimal configurations in civil engineering and social organization.

Vascularized materials: Tree-shaped flow architectures matched canopy to canopy

In this paper we develop flow architectures for “vascularizing” smart materials that have self-healing capabilities. The flow architectures are configured as two trees matched canopy to canopy. A single stream flows through both trees and bathes every subvolume (crack site) of the material. Several types of tree-tree configurations are optimized. Trees that have only one level of branching and bathe a rectangular domain have optimal external shapes that are nearly square. They also have optimal ratios of channel sizes before and after branching. Trees optimized on square domains perform nearly as well as trees on freely morphing rectangular domains. The minimized global flow resistance decreases slowly as the number of subvolumes increases. It is more beneficial to bathe the entire volume with a single (optimized) one-stream architecture than to bathe it with several streams that serve small clusters of volume elements. These conclusions are reinforced by an analytical optimization of the same class of architectures in the limit of a large number of assembled subvolumes. We also show that the freedom to morph the design and to increase its performance can be enhanced by using tree-tree architectures with more than one level of branching.