the elemental gas phase reaction, A$+$BC$+$D$,$ takes place at a constant temperature in a Straight$-$Through Transport Reactor $($SSTR$)$ with a diameter of $0\mathrm{.}5$ m$.$ k $=1\mathrm{.}0$ dm$3/$mol$.$s and the volumetric flow rate is v$0=10$ dm$3/$s$.$ The feed consists of A and B in a stoichiometric ratio. The feed concentration of A is $0\mathrm{.}5$ mol$/$dm$3.$ The catalyst decay law follows first order. kd$=0\mathrm{.}2$ s$-1.$

a$)$ Find the conversion would be obtained in a $10$ m high reactor. Assume that the catalyst moves at the same speed as the gas in the reactor. Plot X$-$Z curve.

b$)$ What is the activity of the catalyst at the exit of the reactor? Plot a$-$Z curve

c$)$ What reactor size would you choose, considering the fixed and operating costs of the reactor? Explain Why?