P$14-9$ B OEQ $($Old Exam Question$).$ In a certain chemical plant, a reversible fluid$-$phase isomerization

$AB$

is carried out over a solid catalyst in a tubular packed$-$bed reactor. If the reaction is so rapid that mass transfer between the catalyst surface and the bulk fluid is rate$-$limiting$,$ show that the kinetics are described in terms of the bulk concentrations $C_A$ and $C_B$ by

$-r_A^{\text{'}\text{'}}=\frac{k_B[C_A-(\frac{1}{K})C_B]}{\frac{1}{K}+\frac{k_B}{k_A}}$

where $-r{\text{'}}_A=$moles of A reacting per unit area catalyst per

$k_A,k_B=$ transfer coefficients for A and B

$K=$ equilibrium constant

It is desired to double the capacity of the existing plant by processing twice the feed of reactant A while maintaining the same fractional conversion of A to B in the reactor. How much larger a reactor, in terms of catalyst weight, would be required if all other operating variables are held constant? You may use the Thoenes$$Kramers correlation for mass transfer coefficients in a packed bed.