At $5C$ the reaction of ketene to diketene

$2(C{H}_2=CO)(C{H}_2=CO{)}_2$

occurs rather slowly in the liquid phase, with second order kinetics

$-$rket $=k{C}_{\text{k}\text{e}\text{t}}^2.$ We wish to design a flow process to treat a ketene feed $(v_0=0\mathrm{.}8\frac{m^3}{hr})$

to $98\%$ conversion in $2$ horizontal tanks in series, each mechanically agitated and

having $3$ well mixed compartments. In a batch reactor this conversion would re$-$

quire a $12$ hour reaction time.

$($a$)$ In theory, what volume would be required for $1$ PFR$?$

$($b$)$ What tank volume would be needed to do this is $1$ CSTR$?$

$($c$)$ What about $2$ CSTRs and what is the total of the two?

$($d$)$ Now for the design objective of $6$ total well$-$mixed flow compartments, and

what is the total of the $6?$ Hint: do not try to solve this one analytically. Instead

develop an expression that relates $x_i,$ the conversion leaving compartment $i,$

to $x_{i-1}.$ Then try to find a volume that will give the correct $x_6$ and $x_0.$

$($e$)$ What is the size of each of the two tanks? Assume the internals $($i$.$e$.$ the ag$-$

itators, baffles, cooling coils, etc.$)$ occupy $10\%$ of the tank volume. State any

assumptions, such as constant liquid density. Draw or create a picture of your

reactors with you standing next to it for scale.