The gas phase reaction $($A$->$ B$+$C$)$ must be carried out in a fixed$-$bed tubular reactor using a solid catalyst in a spherical shape of $0\mathrm{.}4$ cm in diameter. The reactor must operate isothermally at $573$ K and an inlet pressure of $1$ atm. The reaction is first order and the reactor is fed with pure A at a molar flow of FA$0=1$ g$-$mol$/$s$.$ The internal diameter of the reactor must be $2\mathrm{.}54$ cm$.$

It is known: $-$The reaction on the surface of the catalyst is of the type rA$=$kCA.; $-$The particle density gp$=2$ g$/$cm$3$; $-$The porosity of the bed gl$=0\mathrm{.}40.$

$-$This reaction, when carried out using the catalyst in powder form, obtained a speed of RA$=1\mathrm{.}10^-3$ g$-$mol$/$g$-$cat.s$,$ at a temperature of $573$ K and pure reagent A at a pressure of $1$ atm $.$

$-$Consider that the reactor is being operated under conditions free from resistance to external mass transfer to the particle.

$-$The effective molecular diffusivity in the gas phase $($DAB$)$ to solve this exercise must also be considered constant with the temperature and composition DAB$=2\mathrm{.}010^-4$ cm$2/$s$.$

We ask:

a$)$ Perform the steady$-$state differential mass balance for one of the reactants within an infinite porous occurrence plate. Dimensionlessize the resulting resolution, so that it applies the Thiele differential modulus.

b$)$ Sketch an effectiveness factor curve versus Thiele$$s modulus. Show, on this curve, the regions where the diffusional resistance in the pores is negligible

$($which is the controlling stage in this case?$)$ and where this resistance is the controlling stage $($the asymptotic region$).$

Calcule: $1)$ O valor do modulo de Thiele; $2)$O valor do fator de efetividade; $3)$ O comprimento do reator para se obter a convers$$o na sa$$da do reator XA$=0,8.$