Using Interfacial Mass Transfer Phenomena formulas please calculate

Tutorial $12$: Diffusion$-$reaction in spherical coordinates

A chemical dimerization reaction in the gas$-$phase takes place on the surface of metal$-$coated

spherical beads $(5mm$ in diameter$),$ acting as a catalyst in a fluidised bed reactor:

$2A{A}_2$

Under the operating conditions the reaction is controlled by both diffusion of A to the catalyst

surface and the reaction rate of $A$ at the surface which follows first order kinetics:

$r_A=-k{c}_A$ with $k=0\mathrm{.}01\frac{m}{s}$

a$)$ Make a sketch to indicate the system with all flows and boundaries

b$)$ If pure A enters in to the reactor $($which can be taken as a CSTR$)$ and a per$-$pass

conversion of $80\%$ is achieved, show that the exit mol fraction of $A$ is $0\mathrm{.}33$

c$)$ Using the appropriate formulation of Fick's Law and the continuity equation, show

that the system can be described by the following expression

$N_{A(R)}\frac{\text{d}\text{e}\text{l}\text{r}}{r^2}=-\frac{D_{A-A2}c}{R^2}\frac{del{y}_A}{(1-0\mathrm{.}5y_A)}$

d$)$ Integrate this term between appropriate limits$,$ simplify appropriately and determine

the rate of production of $A_2$ per unit surface area of the catalyst

Data

Binary diffusivity of $A$ in $A_2$ :

$D_{A-A2}=2\mathrm{.}3*{10}^{-5}\frac{m^2}{s}$

$T=160C$;

$p=\frac{10}{b}ar$

Temperature:

Pressure: