Problem $1.$ Diisopropyl ether $($DIPE$)$ can be synthesized by the dehydration of isopropyl alcohol $($IPA$)$:

$2C_3{H}_{7}O{H}_{\text{I}\text{P}\text{A}}{C}_3{H}_{7}O{C}_3{H}_7{?}_{\text{D}\text{I}\text{P}\text{E}}+H_{2}O$

Due to a complex mechanism, the rate of formation of DIPE over $-$alumina catalyst is zero$-$order with

respect to IPA $[1],$

$r_{\text{D}\text{I}\text{P}\text{E}}=\frac{3\mathrm{.}08*{10}^{-3}\frac{\text{m}\text{o}\text{l}}{k}\frac{g_{\text{c}\text{a}\text{t}}}{s}}{1+\frac{0\mathrm{.}403}{k}Pa{P}_W}$

where $P_W$ is the partial pressure of water and the kinetic parameters are reported for $488K.$ Water

adsorption onto the catalytic sites inhibits the reaction.

a$)$ Assuming there is no pressure drop, estimate the PBR catalyst mass necessary for a reactor feed

$F_{\text{I}\text{P}\text{A}\text{o}}=12\mathrm{.}5$ tonne $\frac{?}{hr}$ at $P=1$atm with a single$-$pass conversion $x_{\text{I}\text{P}\text{A}}=0\mathrm{.}8.$