Serway & Jewett — The First Law of Thermodynamics: Problem 26.

The human body must maintain its core temperature inside a rather narrow range around \(37^\circ\text{C}\). Metabolic processes, notably muscular exertion, convert potential energy into internal energy deep in the interior. From the interior, energy must flow out to the skin or lungs to be expelled to the environment. During moderate exercise, an 80-kg man can metabolize food energy at the rate \(300\text{ kcal/h}\), do \(60\text{ kcal/h}\) of mechanical work, and put out the remaining \(240\text{ kcal/h}\) of energy by heat. Most of the energy is carried from the body interior out to the skin by forced convection, whereby blood is warmed in the interior and then cooled at the skin, which is a few degrees cooler than the body core. Without blood flow, living tissue is a good thermal insulator, with thermal conductivity about \(0.210\text{ W/m}\cdot^\circ\text{C}\). Show that blood flow is essential to cool the man’s body by calculating the rate of energy conduction in kcal/h through the tissue layer under his skin. Assume that its area is \(1.40\text{ m}^2\), its thickness is \(2.50\text{ cm}\), and it is maintained at \(37.0^\circ\text{C}\) on one side and at \(34.0^\circ\text{C}\) on the other side.