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 $0\mathrm{.}5m.k=1\mathrm{.}0d\frac{m^3}{m}ol.$s and the volume flow rate is $v_0=10d\frac{m^3}{s}.$ The feed consists of A and $B$ in a stoichiometric ratio. The feed concentration of $A$ is $0\mathrm{.}2mo\frac{l}{d}{m}^3.100kg$ of catalyst is used in the reactor. The catalyst decay law follows second order. $k_d=0\mathrm{.}2s^{-1}.$

a$)$ Find the conversion would be obtained in a $10m$ 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 if the reactor? Explain Why?