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The first$-$order irreversible gas$-$phase reaction

n$-$pentane $$ i$-$pentane is currently carried out in a packed bed reactor with $1000kg$ of reforming catalyst packed in a $4cm$ diameter pipe. $14\mathrm{.}1\%$ conversion is achieved. The catalyst particles are $0\mathrm{.}5cm$ in diameter, and the bulk density of the packed catalyst is $1000k\frac{g}{m^3}.$ The entering pressure is $20$atm, and the pressure at the exit of the reactor is $9\mathrm{.}0$atm.

a$)$ What conversion could be achieved in a CSTR with the same catalyst weight and no pressure drop? Evaluate analytically. Hint: use CSTR mole balance with $W$ instead of $V$ and $r_A\text{'}$ instead of $r_A.$

b$)$ Calculate the maximum value of the pressure drop parameter, $,$ that you can have and still maintain an exit pressure of $1$atm.