$(50$ marks$)$ A multistage fixed bed catalytic reactor was to be designed for the dehydrogenation of ethylbenzene $(50$ marks$)$ A multistage fixed bed catalytic reactor was to be designed for the dehydrogenation

of ethylbenzene $(C_8{H}_{10}=A)$ into styrene $(C_8{H}_8=S),$ using excess steam to control the reaction

temperature. The goal is to reach $70$ percent conversion of ethylbenzene, using a multi$-$stage

system if necessary. A new catalyst has been developed that allows operation at temperatures as

high as $950K.$ Assume that the pseudohomogeneous one$-$dimensional plug flow model is valid,

and that each stage operates isobarically.

a$.$ If the reactor operates adiabatically, how many stages would be required in order to achieve

the target conversion with the new catalyst? Assume that the outlet temperature from each

stage is $25C$ greater than the equilibrium temperature. List the outlet fractional conversion

and operating temperature for each stage. The graph provided on the next page may be

helpful.

b$.$ Assuming $30\%$ conversion is obtained in the first stage of this process, how much catalyst

would be required for this stage? Use a step size of $0\mathrm{.}10$ for fractional conversion for your

numerical integration.

c$.$ If the Thiele modulus $(\{$:${}_G)$ for the catalyst particles in part $($b$)$ was equal to $1\mathrm{.}5,$ how much

extra catalyst $($in $\%)$ would be needed relative to your answer in part $($b$)?$

Data:

$)-1$