The human brain is especially sensitive to elevated temperatures. The cool blood in the veins leaving the face and neck and returning to the heart may contribute to thermal regulation of the brain by cooling the arterial blood flowing to the brain. Consider a vein and artery running between the chest and the base of the skull for a distance L = 250 mm, with mass flow rates of 3 x 10^-3 kg/s in opposite directions in the two vessels. The vessels are of diameter D = 5 mm and are separated by a distance w = 7 mm. The thermal conductivity of the surrounding tissue is k, = 0.5 W/m ∙ K. If the arterial blood enters at 37°C and the venous blood enters at 27°C, at what temperature will the arterial blood exit? If the arterial blood becomes over-heated, and the body responds by halving the blood flow rate, how much hotter can the entering arterial blood be and still maintain its exit temperature below 37°C?