Theory and ExamplesThe best branching angles for blood

Chapter 4, Problem 35AAE

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Problem 35AAE

Theory and Examples

The best branching angles for blood vessels and pipes When a smaller pipe branches off from a larger one in a flow system, we may want it to run off at an angle that is best from some energy saving point of view. We might require, for instance, that energy loss due to friction be minimized along the section AOB shown in the accompanying figure. In this diagram, B is a given point to be reached by the smaller pipe, A is a point in the larger pipe upstream from B, and O is the point where the branching occurs. A law due to Poiseuille states that the loss of energy due to friction in non-turbulent flow is proportional to the length of the path and inversely proportional to the fourth power of the radius. Thus, the loss along AO is and along OB is where k is a constant, d1 is the length of AO, d2 is the length of OB, R is the radius of the larger pipe, and r is the radius of the smaller pipe. The angle is to be chosen to minimize the sum of these two losses:

In our model, we assume that AC = a and BC = bare fixed. Thus we have the relations

so that

We can express the total loss L as a function of :

a. Show that the critical value of for which equals zero is

b. If the ratio of the pipe radii is r/R = 5/6, estimate to the nearest degree the optimal branching angle given in part (a).