From heat transfer principles to shape and structure in nature: Constructal theory
This lecture reviews a relatively recent body of heat transfer work that bases on a deterministic (constructal) principle the occurrence of geometric form in systems with internal flows. The same principle of global optimization subject to constraints allow us to anticipate the natural (animate and inanimate) flow architectures that surround us. The lecture starts with the example of the optimal spatial distribution of material (e.g., heat exchanger equipment) in power plants. Similarly, void space can be allocated optimally to construct flow channels in the volume occupied by a heat generating system. The lecture continues with the optimization of the path for heat flow between a volume and one point. It shows that when the heat flow can choose between at least two paths, low conductivity versus high conductivity, the optimal flow structure for minimal global resistance in steady flow is a tree. Nearly the same tree is deduced by minimizing the time of discharge in the flow from a volume to one point. Analogous tree-shaped flows are constructed in pure fluid flows, and in flow through a heterogeneous porous medium. The optimization of trees that combine heat transfer and fluid flow is illustrated by means of two-dimensional trees of plate fins. The method is extended to the superposition of two fluid trees in counterflow, as in vascularized tissues under the skin. The two trees in counterflow are one tree of convective heat currents that effect the loss of body heat. It is shown that the optimized geometry of the tree is responsible for the proportionalities between body heat loss and body size raised to the power 3/4, and between breathing time and body size raised to the power 1/4. The optimized structures are robust with respect to changes in some of the externally specified parameters. When more degrees-of-freedom are allowed, the optimized structure looks more natural. The lecture outlines a unique opportunity for engineers to venture beyond their discipline, and to construct an engineering theory on the origin and workings of naturally organized systems.
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