Consider the fuel cell stack of Problem 1.58. The t = 0.42 rum thick membranes have a nominal thermal conductivity of k = 0.79 W/m ∙ K that can be increased to keff,x = 15.1 W/m ∙ K by loading 10%, by volume, carbon nanotubes into the catalyst layers (see Problem 3.12). The membrane experiences uniform volumetric energy generation at a rate of q = 10 X 106 W/m3. Air at Ta = 80oC provides a convection coefficient of ha = 35 W/m2 ∙ K on one side of the membrane, while hydrogen at Th = 80°C, hh = 235 W/m2 ∙ K flows on the opposite side of the membrane. The flow channels are 2L = 3 mm wide. The membrane is clamped between bipolar plates, each of which is at a temperature Tbp = 80°C.

(a) Derive the differential equation that governs the temperature distribution T(x) in the membrane.

(b) Obtain a solution to the differential equation, assuming the membrane is at the bipolar plate temperature at x = 0 and x = 2L.

(c) Plot the temperature distribution T(x) from x = 0 to x = L for carbon nanotubes loadings of 0% and 10% by volume. Comment on the ability of the carbon nanotubes to keep the membrane below its softening temperature of 85°e.