Exercise $7\mathrm{.}9$: A hollow$-$cylinder catalyst pellet in a fixed$-$bed reactor

Estimate the mass of catalyst required in an isothermal fixed$-$bed reactor for the second$-$

order reaction.

$A{k}_{1}B$

$r=k_1{c}_A^2,k_1=1\mathrm{.}21{10}^5\frac{c{m}^3}{(mol)*s}$

The feed is pure A at a molar flowrate of $15mo\frac{l}{s},$ the total pressure is $3\mathrm{.}0$atm and

the temperature is $550K.$ The reactor achieves $90\%$ conversion of $A.$ The catalyst is a

hollow cylinder with an outer diameter of $1\mathrm{.}3cm,$ an inner diameter of $0\mathrm{.}5cm,$ and a

length of $0\mathrm{.}7cm.$ The pellet density ${}_p=0\mathrm{.}73\frac{g}{c}{m}^3$ and the bed density ${}_B=0\mathrm{.}58$

$\frac{g}{c}{m}^3.$ The effective diffusivity of $A$ is $0\mathrm{.}008c\frac{m^2}{s}$ and may be assumed constant.

You may assume the bulk and pellet surface concentrations are equal.