Consider the rat tail model described in Problem 10.9. Assume that the base of the tail is at the artery supply temperature and that the tip is insulated. Show that the axial temperature distribution and total heat transfer rate from the tail are given by Construct a plot of the axial temperature distribution and calculate the heat transfer rate for the following data:

Problem 10.9

Fin approximation can be applied in modeling organs such as the elephant ear, rat tail, chicken legs, duck beak and human digits. Temperature distribution in these organs is three-dimensional. However, the problem can be significantly simplified using fin approximation. As an example, consider the rat tail. Anatomical studies have shown that it consists of three layers: bone, tendon and cutaneous layer. There are three major axial artery-vein pairs: one ventral and two lateral. These pairs are located in the tendon near the cutaneous layer as shown. The ventral vein is small compared to the lateral veins, and the lateral arteries are small compared to the ventral artery. Blood perfusion from the arteries to the veins takes place mostly in the cutaneous layer through a network of small vessels. Assume that blood is supplied to the cutaneous layer at uniform temperature all along the tail. Blood equilibrates at the local cutaneous temperature Ta0 before returning to the veins. Assume further that (1) cutaneous layer, tendon and bone have the same conductivity, (2) negligible angular variation, (3) uniform blood perfusion along the tail, (4) negligible metabolic heat, (5) steady state, (6) uniform outer radius and (7) negligible temperature variation in the radial direction (fin approximation is valid, Surface heat exchange is by convection. The heat transfer coefficient is h and the ambient temperature is Using Pennes model for the cutaneous layer, show that the heat equation for the rat tail is given by