$(30$ pts$)$ A spherical catalyst particle with a diameter $R$ is suspended in a fluidized bed.

$C_2{H}_6$ diffuses, through a stationary gas layer, to the catalyst surfac

which the following reaction occurs:

$C_2{H}_6{C}_2{H}_4+H_2$

$($i$)$ Express the ethane flux $(N_A)$ in terms of diffusion coefficient

total gas concentration $C,$ and mole fraction of $C_2{H}_6(x_A)$ along

radial $(r)$ direction in its most simplified form. State your

assumptions;

$($ii$)$ Write down the governing equation with respect to $x_A{?}_{Ny}$

performing shell mass balance $($do NOT solve the equation$)$;

$($iii$)$ If the rate constant on the catalyst surface is very large relative to the diffusion coefficient, i$.$e$.\frac{k}{?}$

$D,$ and boundary layer thickness $$ is very small relative to the catalyst radius $R,$ i$.$e$.\frac{}{R}0,$

express $N_A$ in $C,C_{AO},D,$ and $.$ Assume first order reaction rate equation with $A$

being ethane$),$ and bulk phase ethane concentration being $C_{A0}.$ Hint: Simplify the problem by

considering the effect of very large $\frac{k}{D}$ ratio on ethane concentration at the catalyst surface and small

$\frac{}{R}$ on surface curvature.