The other way to operate a heat pipe is to fix the tip temperature as we did before and to use the internal integral to define the "operating temperature" of the pipe \(\left(T_{v}\right)\).

a. Solve the heat pipe equation, assuming that the tip is held at the temperature of the surroundings and use the internal integral to obtain the operating temperature.

b. For an \(\mathrm{Al}\) cylindrical heat pipe that is \(3 \mathrm{~mm}\) in diameter and \(40 \mathrm{~mm}\) long, using pentane as an operating fluid, the external and internal heat transfer coefficients are \(15 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\) and \(300 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\). Assuming a base temperature of \(100^{\circ} \mathrm{C}\) and a free-stream air temperature of \(20^{\circ} \mathrm{C}\), what is the operating temperature of the heat pipe?

c. Since the heat pipe is filled with a pure fluid, use the Antoine equation and coefficients for pentane to determine the internal operating pressure of the heat pipe based on your results from part (b).