Murray's law

The Murray's law is a rule that predicts the thickness of branches in a hierarchical porous networks, in a way that it minimises the cost for both the transport of substances and the maintenance of the transport medium. This law is observed in the vascular and respiratory systems of animals, xylem in plants, and the respiratory system of insects.^[1] In the paper published in PNAS in 1926,^[2]^[3] based on consideration of how to ensure nutrition transfer with full coverage and fluency as a precondition, Murray deduced that the operation cost 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 Hagen-Poiseuille Law and Fick's Law, which were also formulated from a biological context, Murray’s law is a basic physical principle for transfer networks.^[4]^[5] 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.^[6] Murray's law is a powerful biomimetics design tool in engineering. It has been applied in the design self-healing materials, batteries, photocatalysts, and gas sensors.

Moreover, the special Murray’s law is only applicable to mass-conservative transport in the branching network. A generalized Murray's Law for non-mass conservative networks was derived, which describes effects such as chemical reactions and diffusion through the walls.^[1].

The special Murray’s law

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.^[2]^[3] The branches classically refer to the branching of the circulatory system or the respiratory system,^[4] but have been shown to also hold true for the branchings of xylem, the water transport system in plants.^[5]

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.

For $n$ daughter branches arising from a common parent branch, the formula of special Murray's law for laminar flow is:

$r_{p}^{3}=r_{{d_{1}}}^{3}+r_{{d_{2}}}^{3}+r_{{d_{3}}}^{3}+...+r_{{d_{n}}}^{3}$

where $r_{p}$ is the radius of the parent branch, and $r_{{d_{1}}}$ , $r_{{d_{2}}}$ , $r_{{d_{3}}}$ ... $r_{{d_{n}}}$ are the radii of the respective daughter branches.

However, the special Murray's law deduced by Murray and sherman is only applicable to laminar flow.

Williams et al. deduced the formula for turbulent flow:^[6]

$r_{p}^{{(7/3)}}=r_{{d_{1}}}^{{(7/3)}}+r_{{d_{2}}}^{{(7/3)}}+r_{{d_{3}}}^{{(7/3)}}+...+r_{{d_{n}}}^{{(7/3)}}$

where $r_{p}$ is the radius of the parent branch, and $r_{{d_{1}}}$ , $r_{{d_{2}}}$ , $r_{{d_{3}}}$ ... $r_{{d_{n}}}$ are the radii of the respective daughter branches.

Derivation

Murray's Law derives from the minimisation of the energy spent to transport quantities constrained by the energy spent to maintain the transport medium (i.e. the fluid, such as blood).

Laminar Flow

The power (energy per time) in case of laminar flow is

$P=Q\,\Delta p=\left({\frac {\pi \,r^{4}}{8\,\mu }}{\frac {\Delta p}{l}}\right)\Delta p$

where $Q$ is the laminar flow rate given by Hagen-Poiseuille Law, $\mu$ is the dynamic viscosity of the fluid, $\Delta p$ is the pressure difference between the entry and exit of a tube of radius $r$ and length $l$ . The objective function also requires the power spent to maintain the transport medium, which is given by the multiplication of the volume of the cylinder $V=\pi \,r^{2}\,l$ to a Lagrangian multiplier $\lambda$ . Hence, the objective function is minimum where

${\frac {d}{dr}}(P-\lambda \,V)={\frac {\pi \,r^{3}}{2\,\mu }}{\frac {\Delta p^{2}}{l}}-\lambda 2\pi \,r\,l=0\Leftrightarrow {\frac {\Delta p}{l}}={\frac {\sqrt {4\,\mu \lambda }}{r}}$

where ${\frac {\Delta p}{l}}$ is the pressure gradient, which is substituted into the Hagen-Poiseuille equation to obtain the flow rate

$Q={\frac {\pi \,r^{4}}{8\,\mu }}{\frac {\Delta p}{l}}={\frac {\pi \,r^{4}}{8\,\mu }}{\frac {\sqrt {4\,\mu \lambda }}{r}}=\pi {\sqrt {\frac {\lambda }{16\,\mu }}}r^{3}=k\,r^{3}$

where $k$ is an arbitrary constant, since $\lambda$ is also arbitrary. Thus, in a branching systems where mass is conserved, the flow rate of the parent branch is the sum of the flow rate in the children branches

$Q_{p}=\sum _{i=1}^{N}Q_{i}\Rightarrow k\,r_{p}^{3}=\sum _{i=1}^{N}k\,r_{i}^{3}\Leftrightarrow r_{p}^{3}=\sum _{i=1}^{N}r_{i}^{3}$

Diffusion

The power spent by diffusion is given by

$P=Q\,\Delta C=\left(\pi \,D\,r^{2}{\frac {\Delta C}{l}}\right)\Delta C$

where the flow rate is given by Fick's Law, whose $D$ is the diffusivity constant and $\Delta C$ is the difference of concentration between the ends of the cylinder. Similarly to the case of laminar flow, the minimisation of the objective function results in

${\frac {\Delta C}{l}}={\sqrt {\frac {\lambda }{D}}}\Rightarrow Q=\pi \,D\,r^{2}{\sqrt {\frac {\lambda }{D}}}\Leftrightarrow Q=k\,r^{2}$

Hence,

$Q_{p}=\sum _{i=1}^{N}Q_{i}\Rightarrow k\,r_{p}^{2}=\sum _{i=1}^{N}k\,r_{i}^{2}\Leftrightarrow r_{p}^{2}=\sum _{i=1}^{N}r_{i}^{2}$

The generalized Murray’s law

However, the special Murray’s law is only applicable to flow processes involving no mass variations. Significant theoretical advances need to be made for more broadly applicable in the fields of chemistry, applied materials, and industrial reactions.

Murray networks. (Zheng, CC-BY-SA)

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.^[1]

For connecting a parent pipe with radius of r₀ to many children pipes with radius of r_i , the formula of generalized Murray's law is: $r_{o}^{a}={1 \over 1-X}\sum _{i=1}^{N}r_{i}^{a}$ , where the X is the ratio of mass variation during mass transfer in the parent pore, the exponent α is dependent on the type of the transfer. For laminar flow α =3; for turbulent flow α =7/3; for molecule or ionic diffusion α =2; etc.

I t is applicable to an enormous range of porous materials and has a broad scope in functional ceramics and nano-metals for energy and environmental applications.

Murray materials

The generalized Murray’s law defines the basic geometric features for porous materials with optimum transfer properties. The generalized Murray’s law can be used to design and optimize the structures of an enormous range of porous materials. This concept has led to materials, termed as the Murray materials, whose pore-sizes are multiscale and are designed with diameter-ratios obeying the generalized Murray’s law.^[1]

A diagram of Murray materials with macro-meso-micropores built by nanopaticles as building blocks. (Zheng, CC-BY-SA)

As lithium battery electrodes, the Murray materials can reduce the stresses in these electrodes during the charge/discharge processes, improving their structural stability and resulting in a longer life time for energy storage devices. This material could also be used for boosting the performance of a gas sensor and a photocatalysis process that broke down a dye using light.

Murray materials in leaf and insect. (Zheng, CC-BY-SA)

To achieve substances or energy transfer with extremely high efficiency, evolution by natural selection has endowed many classes of organisms with Murray materials, in which the pore-sizes regularly decrease across multiple scales and finally terminate in size-invariant units. For example, in plant stems and leaf veins, the sum of the radii cubed remains constant across every branch point to maximize the flow conductance, which is proportional to the rate of photosynthesis. For insects relying upon gas diffusion for breathing, the sum of radii squared of tracheal pores remains constant along the diffusion pathway, to maximize gases diffusion. From plants, animals and materials to industrial processes, the introduction of Murray material concept to industrial reactions can revolutionize the design of reactors with highly enhanced efficiency, minimum energy, time, and raw material consumption for a sustainable future.

References

1 2 3 4 Zheng, Xianfeng; Shen, Guofang; Wang, Chao; Li, Yu; Dunphy, Darren; Hasan, Tawfique; Brinker, C. Jeffrey; Su, Bao-Lian (2017-04-06). "Bio-inspired Murray materials for mass transfer and activity". Nature Communications. 8. ISSN 2041-1723. doi:10.1038/ncomms14921.
1 2 Murray, Cecil D. (1926). "The Physiological Principle of Minimum Work: I. The Vascular System and the Cost of Blood Volume". Proceedings of the National Academy of Sciences of the United States of America. 12 (3): 207–214. PMC 1084489 . PMID 16576980. doi:10.1073/pnas.12.3.207.
1 2 Murray, Cecil D. (1926). "The Physiological Principle of Minimum Work: II. Oxygen Exchange in Capillaries". Proceedings of the National Academy of Sciences of the United States of America. 12 (5): 299–304. PMC 1084544 . PMID 16587082. doi:10.1073/pnas.12.5.299.
1 2 Sherman, Thomas F. (1981). "On connecting large vessels to small. The meaning of Murray's law" (pdf). The Journal of General Physiology. 78 (4): a 431–453. PMC 2228620 . PMID 7288393. doi:10.1085/jgp.78.4.431.
1 2 McCulloh, Katherine A.; John S. Sperry; Frederick R. Adler (2003). "Water transport in plants obeys Murray's law". Nature. 421 (6926): 939–942. PMID 12607000. doi:10.1038/nature01444.
1 2 Williams, Hugo R.; Trask, Richard S.; Weaver, Paul M.; Bond, Ian P. (2008). "Minimum mass vascular networks in multifunctional materials". Journal of the Royal Society Interface. 5 (18): 55–65. PMC 2605499 . PMID 17426011. doi:10.1098/rsif.2007.1022.

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