A cylindrical catalyst pellet of radius, \(r_{o}\), and length, \(L\), has a chemical reaction occurring in it. In the free stream of the fluid the concentration of reactant is \(c_{a \infty}\) and the flux of reactant to the pellet's surface is characterized by a mass transfer coefficient, \(k_{c}\). The partition coefficient, defined as \(K_{e q}=c_{a \propto} / c_{a, p e l l e t}=10\). The length of the pellet is much larger than its diameter, so the mass transfer is essentially one-dimensional, in the radial direction.

a. Derive the differential equation for the concentration profile inside the catalyst assuming a zero-order reaction with rate constant, \(k_{o}\), and diffusivity, \(D_{o}\).

b. Derive the differential equation for the concentration profile inside the catalyst assuming a first-order reaction with rate constant \(k^{\prime \prime}\).

c. What are the boundary conditions?

d. Solve the equations for the concentration profile.

e. Calculate the flux at the pellet's surface.

f. How does the external mass transfer affect the solution?
g. How do I know if the process is:

1. Reaction rate controlled? 

2. Internal mass transfer controlled?

3. External mass transfer controlled?