A novel process has been proposed to create a composite palladium tube for use as a hydrogen separation membrane in order to produce high-purity hydrogen. To fabricate the composite palladium tube, a gas containing palladium (species A) flows through a porous-walled tube, and the palladium deposits into the pores of the tube wall. The gas mass flow rate is m and the inlet mass concentration of palladium is P_A,m,i, The mass transfer coefficient for transfer of palladium between the gas and the surface is h_m1, and the deposition rate is proportional to the mass concentration of palladium at the tube surface, that is, - n''_A,s = k_1PA,s' The palladium is a dilute species, so the total mass flow rate is approximately constant down the length of the tube.

(a) Building upon the convection mass transfer analysis presented in Section 8.9, derive an expression for the variation of the mean species density of palladium with distance from the tube entrance, and determine an expression for the local deposition rate (kg/m² ∙ s) for a tube of diameter D. Neglect any leakage of gas through the porous tube walls.

(b) If the tube is too long, the variation in deposit thickness will be unacceptably large. What is the ratio of the deposition rates at x = L and x = O?