$($Reaction$-$Diffusion in a spherical catalyst pellet$)$

Consider a spherical catalyst pellet. Assume that transport of reactant is by diffusion,

with diffusivity $D_{\text{e}\text{f}\text{f}}.$ We assume that the transport of reactant within the pellet is

decoupled from the transport of product. Mass transfer in the gas is sufficiently rapid

so that the reactant concentration at the catalyst surface is maintained at $c_{}.$ Suppose

that $D_{\text{e}\text{f}\text{f}}=D_0($independent of position$)$ and the reaction is the $1$ st order reaction, i$.$e$.,$

$R=-k_1^{\text{e}\text{f}\text{f}}C.$

$($a$)$ Define Damkholer number for this reaction rate.

$($b$)$ To determine the concentration distribution of the reactant inside the pellet

assuming that the Damk$$hler number is small,

$Da1,$

Obtain both the first approximation $($i$.$e$.O(l)$ and the $2^{nd}$ approximation $($i$.$e$.O(Da))$

for $C(r)$ in this limit using an appropriate scaling.

Compare with the numerical solution with $R=1um,D=1u\frac{m^2}{s},k=1se{c}^{-1}.($Draw a

graph for the first approximation solution, the $2^{nd}$ approximation solution, and

numerical solution.$)$ Does it make sense and why?

$($c$)$ Now solve it for $Da1$ up to $O((\frac{1}{D}a{)}^{\frac{1}{2}})$ and compare with numerical solution

as done in $($b$).($Draw a graph for the first approximation solution, the $2^{nd}$

approximation solution, and numerical solution.$)$