The electrolytic membrane of the fuel cell in Example 1.4 is a thin composite structure consisting of sandwiched layers of delicate materials, as shown in the sketch. The thickness of the polymer core is t_pc = 0.20 mm, while the thickness of each of the catalyst layers is t_el = 0.01 mm. The gas diffusion layers are each t_gdl = 0.1 mm thick. Since the membrane can soften and lose durability at temperatures exceeding 85°C, the materials engineer decides to strengthen the membrane by implanting long carbon nanotubes (diameter D_cn = 14 nm, k_cn = 3000 W/m ∙ K) lengthwise within both catalyst layers. Determine the value of the effective thermal conductivity, k_eff.x. of the membrane assembly that is defined by the relation q_x = k_eff ... wt ∆T/L, where L, w, and t are the length, width, and total thickness of the membrane assembly respectively; q, is the heat transfer rate along the assembly; and 6.T is the temperature drop along a section of the assembly of length L. Determine the value of keff.x for carbon nanotubes loadings of f = 0, 10%, 20%, and 30%, where f is the volume fraction of carbon nanotubes in the catalyst layers. The thermal conductivity of the polymer core is k_pc = 0.25 W/m ∙ K and the thermal conductivities of the gas diffusion layers and catalyst layers are k_gdl = 1.3 W/m ∙ K and k_cl = 1 W/m ∙ K, respectively.

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Air Catalyst layers Gas diffusion layers Polymer Core "pc Carbon nanotubes Hydrogen Den