The decomposition of cyclohexane to benzene and hydrogen, $C_6{H}_{2}longrightarrow{C}_6{H}_6+H_2$ is mass transfer$-$limited at high temperatures. The reaction is carried out in a packed bed having inside diameter of $5cm.$ The reactor was packed with plate pellets, $6mm2mm1mm.$ The pellets are coated with the catalyst only on the outside. The bed porosity is $40\%$ and the entering volumetric flow rate is $60{10}^{5}c\frac{m^3}{m}in.$

$($a$)$ Show that the relationship of conversion with reactor length is given by:

$\frac{d{x}_A}{dz}=\frac{a_c{k}_c}{U}(1-x_A),$ where: $U=U_0(1+{}_A{x}_A),{}_B={}_c(1-)$

$($b$)$ Calculate the reactor length necessary to achieve $99\%$ conversion of cyclohexane from an entering gas stream of $10\%$ cyclohexane at $2$atm and $450C.$

$($c$)$ Find the required catalyst weight.

Additional information:

${}_g=1\mathrm{.}9{10}^{-4}\frac{g}{c}{m}^3,=1\mathrm{.}7{10}^{-4}\frac{g}{c}m-s,{}_c=1\mathrm{.}922\frac{g}{c}{m}^3{D}_{AB}(at25C$ and $1(atm))=0\mathrm{.}32c\frac{m^2}{s}$