Concentration Profile in Catalytic Pores

This problem involves the derivation and solution of a differential equation that gives the concentration of a species A in a binary mixture while undergoing a first order reaction in a cylindrical, catalytic pore. The system to be described is given below:

To enhance the effective surface, and hence the chemical reaction rate, catalytic surfaces often take the form of porous solids. One such solid may be visualized as consisting of a larger number of cylindrical pores, each of diameter D and length L. Consider conditions involving a gaseous mixture of A and B for which species A is chemically consumed at the catalytic surface. The reaction is known to be first order k₁C_A. Under steady state, flow over the porous solid is known to maintain a fixed value of the molar concentration C_A0 at the pore mouth. Beginning from fundamentals, obtain the differential equation that governs the variation of C_A with distance x along the pore. Applying appropriate boundary conditions, solve the equation to obtain an expression for C_A(x).

This equation is obtained by application of fundamental principles of material balances, reaction kinetics, mass transfer, and calculus. The derivation is given in the image below: