World Library  
Flag as Inappropriate
Email this Article

Murray's law

Article Id: WHEBN0021461479
Reproduction Date:

Title: Murray's law  
Author: World Heritage Encyclopedia
Language: English
Subject: Xylem, Botany, Biomimetics, Equations, Developmental biology
Collection: Botany, Developmental Biology, Equations
Publisher: World Heritage Encyclopedia

Murray's law

Murray's law, or Murray's principle is a formula for relating the radii of daughter branches to the radii of the parent branch of a lumen-based system.[1][2] The branches classically refer to the branching of the circulatory system or the respiratory system,[3] but have been shown to also hold true for the branchings of xylem, the water transport system in plants.[4]

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 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.

Murray's law is seeing increasing use as a 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[5] and the expression developed could readily be applied to minimum mass fluid systems in other engineering applications. The trade-off is directly analogous—larger diameter tubes are heavier because of both the tubing and the additional volume of enclosed fluid, but the pressure losses incurred are reduced and so the mass of the pumping system required is lower. The (inner) tube diameter d_i which minimizes the total mass (tube + fluid + pump), is given by the following equation in laminar flow:[5]

d_i^6 = \frac{1024 Q^2 \mu }{ \pi^2 k [ \rho_{\text{tube}} (c^2+2c) + \rho_{\text{fluid}} ] }

where Q is the volume flow rate, \mu the fluid viscosity, k the power-to-weight ratio of the pump, \rho_{\text{tube}} the density of the tubing material, c a constant of proportionality linking vessel wall thickness with internal diameter and \rho_{\text{fluid}} the density of the fluid.

For turbulent flow the equivalent relation (derived from the Darcy-Weisbach equation) is:[5]

d_i^7 = \frac{80 Q^3 \rho_{\text{fluid}} f }{ \pi^3 k [ \rho_{\text{tube}} (c^2+2c) + \rho_{\text{fluid}} ] }

where f is the Darcy friction factor. The junction relations above can therefore be applied in the following form in turbulent flow:

r_p^{(7/3)} = r_{d_1}^{(7/3)} + r_{d_2}^{(7/3)} + r_{d_3}^{(7/3)} + ... + r_{d_n}^{(7/3)}


  1. ^ 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.  
  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.  
  3. ^ 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.  
  4. ^ McCulloh, Katherine A.; John S. Sperry and Frederick R. Adler (2003). "Water transport in plants obeys Murray's law". Nature 421 (6926): 939–942.  
  5. ^ a b c Williams, Hugo R.; Trask, Richard S., Weaver, Paul M. and Bond, Ian P. (2008). "Minimum mass vascular networks in multifunctional materials". Journal of the Royal Society Interface 5 (18): 55–65.  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.