The irreversible elementary gas phase reaction $2A2B$ is carried out isothermally at $488k$

in a packed bed reactor containing $50kg$ catalyst. Currently $30\%$ conversion is realized for

pure A entering at $20$atm. The pressure at the exit of the first reactor is $15$atm. It is proposed

to put a second packed bed reactor containing the double catalyst weight in series,

downstream the first packed bed reactor. The particle size in the second reactor is half that

of the first reactor $(D_{P2}=0\mathrm{.}5D_{P1}),$ the catalyst porosity is same in both reactor at $40\%.$ Viscosity

and density of the gas stream is same as that of air at $200C.$ The flow rate through the

reactor bed is very high and the flow can be assumed turbulent.

Assume variable density case where

$\frac{P}{P_0}=(1-W{)}^{\frac{1}{2}}$

a$)$ What are the pressure and conversion exiting the last reactor?

b$)$ Without doing detailed calculations, what will be the exit conversion in the second reactor

if the flow becomes laminar with no other changes to the catalyst?