On the Nusselt number in heat transfer between multiple parallel blood vessels

Arrays of two or more parallel blood vessels in a tissue matrix have been studied extensively in the context of bioheat transfer. The average vessel Nusselt number (based on the difference between the mixed-mean blood temperature and the average vessel surface temperature) is a crucial parameter in such studies. Various workers have noted tht in particular cases the average Nusselt number is identical to that for fully developed flow in a single vessel in an infinite medium. In other words, the Nusselt number is unaffected by the presence of other vessels. It is proven here that this surprising result holds true for arbitrary number, size, flow direction, and velocity profile in the blood vessels, and for very general boundary conditions on the outer tissue boundary. A useful corollary is that the average wall temperature in a particular vessel may be found by evaluating the temperature fields due to the other vessels and the tissue boundaries at a single point, the center of the vessels in question.