A heat pipe is basically a hollow fin that is partially filled with fluid. The fluid evaporates at the hot end, travels the length of the fin and condenses at the cold end. Capillary forces return the liquid to the hot end. We can model the heat pipe by modifying the fin equation using an internal heat transfer coefficient that describes whether heat is gained or lost by the fin via evaporation whose driving force is given by the temperature difference between the fin wall temperature, \(T\), and the fluid's boiling temperature, \(T_{v}\).

a. Derive the differential equation describing the operation of the heat pipe including an Out term defined by heat transfer to the surroundings at \(T_{\infty}\) and an internal convection term describing the evaporation and condensation process.

b. Solve the differential equation, assuming that the fin base temperature is \(T_{b}\) and the fin tip is held at the temperature of the surroundings. (Hint: It may help to separate the constant terms and use the trial solution, \(T=A \exp (r x)\) ).