One argument may be the feasibility providing additional area for the reactant to enter the pellet. Since the reagent can enter the hollow region and then enter the pellet, the locations closer to the central axis can be richer in terms of the reactant this way, especially in the case of a fast reaction.

a$.$ By modeling the pellets shown in the photo as infinite cylinders, obtain the dimensionless concentration distribution, $\backslash$theta $=$ CA$/$CAR$,$ in terms of modified Bessel

functions I and K$,$ the Thiele modulus, $=($k

$$R

$2$

$/$De$)$

$1/2,$ and dimensionless radial distance, $\backslash$xi $=$ r$/$R$.$ Investigate the case where the supply of A to both the outer and inner pellet surfaces is not hindered by mass transfer limitations$.$

b$.$ Investigate your solution when $\backslash$kappa $->0.$

c$.$ Show that the effectiveness factor for this geometry can be represented with the

following formula:

$()$

$()2$

$2$

H $\backslash$kappa $,$

$1\backslash$kappa

$$

$$

$=$

$$

where H$(\backslash$kappa $,)$ is a complicated function that contains modified Bessel functions of

order zero and one.

d$.$ Plot the effectiveness factor as a function of $.$ Keep the x$-$axis logarithmic. Include

curves for $\backslash$kappa $=0\mathrm{.}1,0\mathrm{.}35$ and $0\mathrm{.}6.$ On the same graph, plot the effectiveness factor of

a full cylinder and comment on the results.

e$.$ Note that for a hollow cylindrical pellet with the same outer radius and length with

a full cylinder, the volume taken up by a single catalyst pellet is the same. Since the

hollow pellet contains less catalyst, a counter argument to using such pellets could be the fact that there is less $$effective volume$$ for reaction to occur within the

volume taken up by the pellet. Compare the volumetric rate of consumption, i$.$e$.$the total reaction rate per volume taken up in the reactor, of both the full and the

hollow pellets. To do this, take the ratio of rates with the rate for the solid pellet on the denominator. Again, plot this ratio as a function of $$ and keep the x$-$axis

logarithmic. Include curves for $\backslash$kappa $=0\mathrm{.}1,0\mathrm{.}35$ and $0\mathrm{.}6.$ Comment on your results.

f$.$ How realistic are the boundary conditions for the hollow cylinder? Based on the photos available on the internet, what do you roughly see the upper limit for $\backslash$kappa $?$

Based on your answers to these questions and your results in part $($e$),$ provide your thoughts on why one would consider a hollow pellet instead of a solid pellet.