Reaction$-$diffusion in a cylindrical catalyst pellet. Catalyst pellets come in various sizes

and shapes. They are porous bodies, into which reactants diffuse and get converted into

new species on surfaces. See the figure below. Cylindrical catalyst pellets with a large

aspect ratio $(\frac{L}{R}1)$ can be modeled as infinite cylinders.

a$.$ For an infinite cylinder, derive the following differential equation that describes

reaction and diffusion inside the catalyst at steady$-$state, assuming that the reaction

can be approximated as a homogeneous first$-$order reaction.

$\frac{d}{dr}(r\frac{dC}{dr})-\frac{k^{\text{'}\text{'}}}{D_e}rC=0$

$D_e$ is effective diffusivity inside the porous pellet.

b$.$ The concentration in the gas stream around the pellet and the concentration on the

outer surface of the pellet are the same: $C_0.$ Solve the differential equation above

and show that the concentration profile is given by:

$C=C_0\frac{I_0(\sqrt[\phantom{2}]{\frac{k^{\text{'}\text{'}}}{D_e}}r)}{I_0(\sqrt[\phantom{2}]{\frac{k^{\text{'}\text{'}}}{D_e}}R)}$