The heterogeneous liquid phase reaction, the alkylation of phenylethane $($PE$)$ with light olefinssuch as ethylene and propylene was executed over a commercial zeolite $($solid$)$ catalyst in a fixed$-$bed reactor. This liquid phase reaction was conducted in excess PE at $25$ bar pressure and $190C$ temperature. Assume that this reaction is pseudo$-$first$-$order with respect to olefin. The porosity of the catalyst, the tortuosity and the density was $0\mathrm{.}5,5\mathrm{.}0,$ and $1000kg{m}^{-3},$ respectively. The observed rate $(r_{\text{o}\text{b}\text{s}})$ and rate constant $(k_{\text{o}\text{b}\text{s}})$ were measured for two different catalyst pellet sizes. The results are given below:

$\backslash$table$[[$R$\_($p$)($mm$),$k$\_($e$)($ms$^(-1)),\backslash$table$[[$External$],[$Surfacearea of$],[$Catalyst$,],[$S$\_($p$)($m$^(2)$kg$^(-1))]],\backslash$table$[[$r$\_($obs $)],[($kmol$(($kgcats$))^(-1))]],\backslash$table$[[$K$\_($obs $)],[($m$^(2)(($kgcats$))^(-1))]]],[0\mathrm{.}63,,4\mathrm{.}62,,],[0\mathrm{.}17,5\mathrm{.}69\backslash$times $10^(-4),17\mathrm{.}13,8\mathrm{.}64\backslash$times $10^(-6),0\mathrm{.}33\backslash$times $10^(-3)]]$

a$)$ Determine whether or not external and internal mass transfer limitations are significant for each case. Assume the diffusivity of olefins in phenylethane is $D_{AB}=1\mathrm{.}9{10}^{-4}c{m}^2{s}^{-1}.$

b$)$ Calculate the Thiele modulus, $,$ and the internal effectiveness factor, $,$ for each case.

c$)$ Determine the overall effectiveness factor for each case.