A second order irreversible gas solid catalytic reaction $2AR$ is carried out

isothermally in a packed bed reactor of height $3m$ and diameter $0\mathrm{.}3m.$ The packed

bed reactor can be treated as a CSTR$.$ The reactor is filled with porous, $50mm$

diameter spherical catalyst pellets to give a bed voidage of $46\%.$ In two trial runs,

at the same temperature, the following data were obtained:

$($a$)$ Show that external mass transfer does not present a significant resistance

under these reactor conditions.

$($b$)$ Determine the effectiveness factor and determine the reaction rate constant

per unit area of catalyst surface.

Data:

Specific surface area

$2\mathrm{.}0{10}^5{m}^2$ cat$-$surf$/$kgcat

Pellet density

$1\mathrm{.}2{10}^{3}k\frac{g}{m^3}$cat

Effective diffusivity of $A$ in catalyst pellet

$8\mathrm{.}0{10}^{-7}{m}^3$ gas$/($mcat $*s)$

For a $n^{th}$ order GS catalytic reaction with sphere catalysts, $=\frac{R}{3}\sqrt[\phantom{2}]{\frac{(n+1)k_{es}{C}_{AS}^{E-1}}{2D_{el}}}$