P$11-6B_B$ 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 reac$-$

tion 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 $,-r_A^{\text{'}\text{'}}=$ moles of $A^{\text{'}}$ reacting per unit area catalyst per unit time

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

$K=$ reaction 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.

Describe the effects of the flow rate, temperature, particle size at conversion.