$4\mathrm{.}9$ The following catalytic reaction is carried out in an isothermal circulating fluidized bed reactor:

$A_{(g)}>B_{(g)}$

For a surface$-$reaction limited mechanism. in which both A and $B$ are absorbed on the surface of the catalyst, the rate law is

$-r_A\frac{k_1{C}_A}{1+k_2{C}_A+k_3{C}_B}$

where $r_A$ is the rate of the reaction in kmo$\frac{l}{m^3}.s,C_1$ and $C_B$ are concentrations of A and $B,$ respectively, in kmo$\frac{l}{m^3},$ and $k_1,k_2,$ and $k_3$ are constants.

Assume that the solids move in plug flow at the same velocity of the gas $(U).$ Evaluate the height of the reactor at which the conversion of $A$ is $60\%.$ Additional data are as follows:

$C_{10}=0\mathrm{.}2$kmo$\frac{l}{m^3}$

$C_{B0}=0$

$U=7\mathrm{.}5\frac{m}{s}$

$k_1=8s^{-1}$

$k_2=3\frac{m^3}{k}$mol

$k_3=0\mathrm{.}01\frac{m^3}{k}$mol