“Constructal Tree Network for Fluid Flow between a Finite-Size Volume and One Source or Sink”, 1997-09 ():
[cf. et al 2000] The ‘constructal theory’ of formation of structure in nature is extended to fluid-flow systems. The fluid flow path between one point and a finite-size volume (an infinite number of points) is optimized by minimizing the overall flow resistance when the flow rate and the duct volume are fixed. The solution is constructed as a sequence of optimization and organization steps.
The sequence has a definite time direction: it begins with the smallest building block (elemental system, with flow by volumetric diffusion), and proceeds toward larger building blocks (assemblies, with flow collected in ducts). Optimized at each level are the shape of the assembly, the number of constituents (ie. smaller assemblies), and the distribution of the duct volume.
It is shown that the ducts of the optimized assemblies form a tree-like structure, in which every architectural detail is deterministic. It is also shown that the structure cannot be determined when the time direction is reversed, from large elements toward smaller elements.
The general importance of the constructal law (access-optimization principle) in physics, biology and economics is discussed.
[Keywords: constructal theory, optimization and organization steps, tree-like structure / reversed time direction]
Hanson: …Will this focus on cooling limit city sizes? After all, the surface area of a city, where cooling fluids can go in and out, goes as the square of city scale, while the volume to be cooled goes as the cube of city scale. The ratio of volume to surface area is thus linear in city scale. So does our ability to cool cities fall inversely with city scale?
Actually, no. We have good fractal pipe designs to efficiently import fluids like air or water from outside a city to near every point in that city, and to then export hot fluids from near every point to outside the city. These fractal designs require cost overheads that are only logarithmic in the total size of the city. That is, when you double the city size, such overheads increase by only a constant amount, instead of doubling.
For example, there is a fractal design for piping both smoothly flowing and turbulent cooling fluids where, holding constant the fluid temperature and pressure as well as the cooling required per unit volume, the fraction of city volume devoted to cooling pipes goes as the logarithm of the city’s volume. That is, every time the total city volume doubles, the same additional fraction of that volume must be devoted to a new kind of pipe to handle the larger scale. The pressure drop across such pipes also goes as the logarithm of city volume.
The economic value produced in a city is often modeled as a low power (greater than one) of the economic activity enclosed in that city. Since mathematically, for a large enough volume a power of volume will grow faster than the logarithm of volume, the greater value produced in larger cities can easily pay for their larger costs of cooling. Cooling does not seem to limit feasible city size. At least when there are big reservoirs of cool fluids like air or water around.
I don’t know if the future is still plastics. But I do know that a big chunk of it will be pipes.