Problem $3$: CSTR vs$.$ PBR for a catalyzed reaction

You are given the rate equation, $-$rA$=$ kA$$ cA for the reaction A$->$ B$.$ The inlet flow rate

is $200$ mol$/$s of pure A$.$ Pure A has a density of $500$ kg$/$m$3$ and a molar mass of $500$ g$/$

mol. The rate constant is $10-6$ m$/$s$.$ The reactor has a catalyst of surface area Sg of $5$ m$2/$

g$,$ and the catalyst density in the reactor is $30$ kg$/$m$3.$

a$.$ To achieve $40\%$ conversion in an isothermal, steady$-$state, liquid phase CSTR$,$

what is the volume required? $[0\mathrm{.}9$ m$3]$

b$.$ What volume is required for the same conversion in an isothermal, steady$-$state,

liquid phase PBR$?[0\mathrm{.}7$ m$3]$

c$.$ What is the residence time in a CSTR vs$.$ a PBR$?$ Calculate a residence time with

units of catalyst weight$/$volumetric flow rate as well as the usual reactor

volume$/$volumetric flow rate. The former does not have the convenient units of

time, but physically expresses the residents time accurately in a reaction catalyzed

by a solid catalyst in a reactor. Explain in a single sentence, why the residence

time for the CSTR is different from the PBR$.[$CSTR $=4\mathrm{.}5$ s$,$PBR $=3\mathrm{.}5$ s$,$w$,$CSTR $=130$

kg$*$s$/$m$3,$w$,$PBR $=100$ kg$*$s$/$m$3]$

d$.$ It is decided to decrease the residence time by a factor of $2$ by doubling the flow

rates. What happens to the conversion in a PBR and CSTR$.$ Support your

conclusions with calculations. $[$X $=0\mathrm{.}25]$