The system of parallel reactions of the figure $($ A$,$ B and C gases$)$ was studied in a non$-$porous

catalytic granule with a diameter of $1\mathrm{.}4cm.$ The catalytic granule was exposed to a gas stream

containing $0\mathrm{.}9\%$molA in air, under a pressure of $1$atm and a temperature of $620K($Kelvin$).$

$($a$)$ Find the required minimum velocity of the gas stream so that the temperature at the surface

of the grain does not exceed $630K.$

Hint: To simplify the calculations, you could assume that the value of the Reynolds number

corresponding to the required minimum velocity of the gas stream is large enough so that, for

this system, it can be assumed that $Sh=0\mathrm{.}6R{e}^{\frac{1}{2}}S{c}^{\frac{1}{3}}$ and $=0\mathrm{.}6R{e}^{\frac{1}{2}}P{r}^{\frac{1}{3}},$ where $Re$ :

Reynolds number, Sh: Sherwood number, Sc: Schmidt number, Nu: Nusselt number, and Pr:

Prandl number. However, you should check the correctness of this assumption after calculating

the minimum velocity.

$($b$)$ Calculate the point selectivity with respect to B and compare its value with that calculated

under main current conditions $($bulk$).$Is the result of the comparison expected? Explain briefly.

$($c$)$ Calculate the external efficiency factor for each of the two reactions.

Additional data:

Activation energies of the two reactions: $E_1=165kJmo{l}^{-1},E_2=108kJmo{l}^{-1}$

enthalpy of the two reactions: $H_{R,1}=-337kJmo{l}^{-1},H_{R,2}=-477kJmo{l}^{-1}$

The two reactions are of order $1$ with respect to A with $k_1=1\mathrm{.}81cm{s}^{-1}$ and $k^2=1\mathrm{.}42cm$

$s^{-1}$ at $620K.$

Under the conditions of the problem, the diffusivities of A$,$ B and C in the reactant mixture

can be considered constant and equal to $0\mathrm{.}175(c{m}^2**s^{-1}),$ the thermal conductivity of the

reactant mixture is equal to $4,{85}^{**}{10}^{-2}((W)m^{-1}{K}^{-1}),$ the specific heat of the reactant

mixture is equal to $1,162(k{J}^{-1}**K^{-1}),$ the kinematic viscosity of the reactant mixture is

equal to $5,6**{10}^{-5}(m^2**s^{-1})$ and the density of the reactant mixture is equal to $0,565kg$

$m^{-3}$

A$,$B$,$C behave as ideal gases. Please full answer with calculations and final answers.