Problem $3.$ Integral Catalytic Fixed$-$bed Reactor Sizing $(40$ points$)$

A pilot$-$scale fixed$-$bed reactor was operated to approximate the behavior of an isothermal

plug$-$flow reactor $($PFR$)$ system. The heterogeneous reaction kinetics for the isomerization of a

particular light hydrocarbon $($specie A$)$ were previously shown to follow first$-$order, irreversible

behavior. In one series of experiments using $1kg$ of $\frac{1}{8}-$in$.$ spherical catalyst in a fixed$-$bed

reactor, the fractional conversion of the hydrocarbon was $0\mathrm{.}80.$ The molar flow rate of the

hydrocarbon at the reactor inlet was $2$kmol$\frac{A}{m}in$ and the inlet feed composition was $0\mathrm{.}1$

kmo$\frac{l}{m^3}.$

The project manager for the process scale$-$up group has asked you to use this data to predict

how much catalyst $($in kilograms$)$ would be required to achieve a fractional conversion of $0\mathrm{.}99$

for the same hydrocarbon if the feed molar flow rate of the hydrocarbon in the larger reactor

was $1200$kmol$\frac{A}{h}r$ and the inlet feed composition was now $0\mathrm{.}2$kmo$\frac{l}{m^3}$ under otherwise

identical process conditions.

Show all the equation$($s$),$ variables and their units, and your result$($s$).$