$(50$ marks$)$ A multistage fixed bed catalytic reactor was to be designed for the dehydrogenation of ethylbenzene $)=(A$ into styrene $)=(S,$ in the presence of steam. The Thiele modulus for the catalyst is $1\mathrm{.}2,$ assumed to be constant. Eighty percent conversion of ethylbenzene is desired. The maximum operating temperature is $1075K,$ dictated by sintering of the catalyst.

a$)$ Assuming adiabatic operation, how many stages will be required to achieve $80\%$ conversion of ethylbenzene, given that the outlet temperature from each stage differs from the equilibrium temperature by $25K?$ Show the operating lines on the diagram on the next page. Provide the inlet and outlet conversion and temperature for each stage.

b$)$ If $30$ percent conversion is achieved in the first stage of this system, how much catalyst is required, assuming adiabatic, isobaric operation and that the pseudohomogeneous onedimensional plug flow model is valid. Use a step size of $0\mathrm{.}10$ for the fractional conversion.

c$)$ Identify and justify your preferred mode of operation if your goal is to minimize the catalyst requirement.

Data: $F_{Ao}=100mo\frac{l}{s}$;$F_{H2O}=1200mo\frac{l}{s}$;$P_o=\frac{2\mathrm{.}0}{b}ar$ ;$H_{RA}=126k\frac{J}{m}ol$;$c_P=2\mathrm{.}4J{g}^{-1}{K}^{-1}$; ;

Molar masses: $C=12$;$H=1$;$O=16$

Graph provided on the next page.

Answers:

$($a$)2$ stages; Stage $1$ outlet conditions: $f_A=0\mathrm{.}70$;$T=960K$

$($b$)W=3\mathrm{.}5$ tonnes; $T_f=1026K$;