Murray's law

In the paper published in PNAS in 1926, based on consideration of how to ensure nutrition transfer with full coverage and fluency as a precondition, Murray deduced that the cost of operation of physiological systems tends to be a minimum for optimum substance transfer networks and formulated what is now known as Murray’s law. Murray derived his law for optimal cardiovascular design that defines the sizes of blood vessels from the aorta through progressive branch points to the capillaries. Like the laws of Poiseuille and Fick, which were also formulated from a biological context, Murray’s law is a basic physical principle for transfer networks. However, since its discovery, little attention has been paid to exploit this law for designing advanced materials, reactors and industrial processes for maximizing mass or energy transfer to improve material performance and process efficiency. Moreover, the special Murray’s law is only applicable to flow processes involving no mass variations. In the paper published in Nat.Commun.in 2017, the generalized Murray's law deduced by Zheng et al can be applicable for optimizing mass transfer involving mass variations and chemical reactions involving flow proceses, molecule or ion diffusion, etc. Murray's law is a powerful biomimetic design tool in engineering—for example it has recently been applied in the design of minimum mass vascular networks carrying a liquid healing agent to areas of damage in a self-healing material and the expression developed could readily be applied to minimum mass or energy transfer systems in other engineering applications, such as in lithium-ion battery electrodes, photocatalysis and gas sensing.

The special Murray's law deduced by Murray and sherman is a formula for relating the radii of daughter branches to the radii of the parent branch of a lumen-based system. The branches classically refer to the branching of the circulatory system or the respiratory system, but have been shown to also hold true for the branchings of xylem, the water transport system in plants.

Murray's original analysis was intended to determine the vessel radius that required minimum expenditure of energy by the organism. Larger vessels lower the energy expended in pumping blood because the pressure drop in the vessels reduces with increasing diameter according to the Hagen-Poiseuille equation. However, larger vessels increase the overall volume of blood in the system; blood being a living fluid requires metabolic support. Murray's law is therefore an optimisation exercise to balance these factors.

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