Substance $A($gas$)$ is converted to $P($isomerization reaction$)$ at $373K$ in the presence of a catalyst. The catalyst is $3-$mm diameter nonporous spheres, and these catalyst beads are packed in a $2-$cm ID metal tube. The reaction occurs on the outside surface of the sphere so fast that the surface concentration of A can be assumed to be zero. Thus the apparent rate of reaction is entirely controlled by the rate of mass transfer of A to the surface of the catalyst.

Pure A $($molecular weight $=50)$ is forced at $1$atm and $373K$ through the packed bed of catalyst spheres at a velocity which gives a Reynolds number of $100.$ For the gas mixture, the Schmidt number is $5,$ and the viscosity is $0\mathrm{.}02{10}^{-3}$Pas. The spheres form a bed that has a void fraction of $0\mathrm{.}4.$

$($a$)(20$ points$)$ Predict $k{G}^{\text{'}}($mass transfer coefficient$)$ for substance A$.$

$($b$)(20$ points$)$ What catalyst bed length is required to obtain a $99$ percent conversion of $A$ to $P?$

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