A Critical Reassessment of the Hess-Murray Law

The Hess–Murray law is a correlation between the radii of successive branchings in bi/trifurcated vessels in biological tissues. First proposed by the Swiss physiologist and Nobel laureate Walter Rudolf Hess in his 1914 doctoral thesis and published in 1917, the law was “rediscovered” by the American physiologist Cecil Dunmore Murray in 1926. The law is based on the assumption that blood or lymph circulation in living organisms is governed by a “work minimization” principle that—under a certain set of specified conditions—leads to an “optimal branching ratio” of ri`1 ri “ 1 3 2 “ 0.7937. This “cubic root of 2” correlation underwent extensive theoretical and experimental reassessment in the second half of the 20th century, and the results indicate that—under a well-defined series of conditions—the law is sufficiently accurate for the smallest vessels (r of the order of fractions of millimeter) but fails for the larger ones; moreover, it cannot be successfully extended to turbulent flows. Recent comparisons with numerical investigations of branched flows led to similar conclusions. More recently, the Hess–Murray law came back into the limelight when it was taken as a founding paradigm of the Constructal Law, a theory that employs physical intuition and mathematical reasoning to derive “optimal paths” for the transport of matter and energy between a source and a sink, regardless of the mode of transportation (continuous, like in convection and conduction, or discrete, like in the transportation of goods and people). This paper examines the foundation of the law and argues that both for natural flows and for engineering designs, a minimization of the irreversibility under physically sound boundary conditions leads to somewhat different results. It is also shown that, in the light of an exergy-based resource analysis, an amended version of the Hess–Murray law may still hold an important position in engineering and biological sciences.