P$10-20$A The vapor$-$phase cracking of gas$-$oil in Example $10-6$ is carried out over a different catalyst, for which the rate law is

$-r_A^{\text{'}}=k^{\text{'}}{P}_A^2,$ with $k^{\text{'}}=5{10}^{-5}\frac{\text{k}\text{m}\text{o}\text{l}}{kg-\text{c}\text{a}\text{t}*s*at{m}^2}$

$($a$)$ Assuming that you can vary the entering pressure and gas velocity, what operating conditions would you recommend?

$($b$)$ What could go wrong with the conditions you chose?

Now assume the decay law is

$\frac{da}{dt}=k_{D}a{C}_{\text{c}\text{o}\text{k}\text{e}},$ with $k_D=100\frac{d{m}^3}{(mol)*s}at400C$

where the concentration, $C_{\text{c}\text{o}\text{k}\text{e}\text{,}in}mo\frac{l}{d}{m}^3,$ can be determined from a stoichiometric table.

$($c$)$ For a temperature of $400C$ and a reactor height of $15m,$ what gas velocity do you recommend? Explain. What is the corresponding conversion?

$($d$)$ The reaction is now to be carried in an STIR $15m$ high and $1\mathrm{.}5m$ in diameter. The gas velocity is $2\mathrm{.}5\frac{m}{s}.$ You can operate in the temperature range between $100$ and $500C.$ What temperature do you choose, and what is the corresponding conversion?

$($e$)$ What would the temperature$-$time trajectory look like for a CSTR$?$