Chapter 11, Problem 11.46P

The human brain is especially sensitive to elevatedtemperatures. The cool blood in the veins leaving theface and neck and returning to the heart may contributeto thermal regulation of the brain by cooling the arterial blood flowing to the brain. Consider a vein andartery running between the chest and the base of the skull for a distance L=250 mm, with mass flow ratesof $3\times {10}^{-3}\text{ kg/s}$ in opposite directions in the two vessels. The vessels are of diameter D=5 mm and areseparated by a distance w = 7 mm. The thermal conductivity of the surrounding tissue is $k_t=0.5\text{ W/m}\cdot \text{K}$ .If the arterial blood enters at $37°C$ and the venousblood enters at $27°C$ , at what temperature will the arterial blood exit? If the arterial blood becomes overheated, and the body responds by halving the bloodflow rate, how much hotter can the entering arterialblood be and still maintain its exit temperature below $37°C$ ? Hint: If we assume that all the heat leavingthe artery enters the vein, then heat transfer between the two vessels can be modeled using a relationship foundin Table 4.1. Approximate the blood properties asthose of water.