A miniature heat pipe is a heat transfer device that operates via an evaporation/condensation mechanism. The heat source evaporates fluid at one end of the device. The fluid travels the length of the device condensing on the walls and at the opposite end to reject heat. The situation is shown in Figure P9.8. Consider such a device with a square cross-section. The temperature profile within the wall of the device near the evaporating end is of interest. We can treat the evaporating section as a slab of length, \(L\), and width, \(w\), and thickness, \(t_{h}\). At the start of the device a known amount of heat, \(q_{o}\), is injected. At the end of the evaporator, a constant temperature, \(T_{a}\), is reached so that heat flow in the \(x\)-direction is zero. Convective boundary conditions exist on the inside and outside of the device. Determine an expression for the temperature distribution within the wall in the \(x\) - and \(y\)-directions.

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FIGURE P9.8 Region of Interest ho Tx Liquid X Vapor Evaporating h; Ty Adiabatic Condensing