Problem $4.$ Size of Catalyst Particles $(10$ points$)$

Our lab$-$scale packed bed reactor is conducting a gas phase reaction $AP$ at a total pressure of

$10$atm and $336C$ using our latest spherically$-$shaped catalyst developed internally by our

research team. A conversion of $90\%$ was achieved using a feed that consists of pure reactant A$.$

The catalyst salesman who handles our account states that in the absence of any pore diffusion

resistance, their technology group has shown that our reaction will proceed on their new catalyst

$(D_{\text{e}\text{f}\text{f}}=2{10}^{-6}\frac{m^2}{s})$ with a rate given by $-r_A=k{C}_A$mol$\frac{A}{m^3}$ where $k$ is $0\mathrm{.}08s^{-1},$ which is much

better than our current catalyst.

What size of spherical catalyst should we order to take advantage of their new catalyst product?

$($Hint: Pore diffusional resistances can be safely neglected for all particle shapes if the Thiele

modulus $0\mathrm{.}4.$