An irreversible exothermic gas$-$phase reaction:

A $->3$ B

$-$ra$=$ K$($CA$^2)$

and is carried out isothermally in a plug flow reactor $($negligible $\backslash$Delta P$).$ The feed is a $1$:$8($by mol$)$ mixture of species A and an inert $($I$)$ at $250$ kPa and $600$ K$.$ The $800$ L reactor is operating at a conversion of $70\%.$ The rate constant for the gas$-$phase reaction is $0\mathrm{.}050$ L mol$1$ s$1.$

a$)$ What is the volumetric flow rate into the PFR$?$

b$)$ PEOPLE would like to use to reduce the size of reactor necessary for the reaction. The catalyst can also perform the reaction at a lower temperature, saving energy costs. They plan to operate the catalyzed unit at $500$ K$,$ and the rate constant for the catalyzed reaction is $5$ L$2$ mol$1$s$1$g$1.$ They want to validate

their predictions of the catalyst performance so $10\%$ of your PFR feed is split off to a packed bed reactor. The pressure$-$drop parameter at these conditions is $0\mathrm{.}15$ g$1.$

How much catalyst do they need to load into the PBR to maintain the $80\%$ conversion?