Using Interfacial Mass Transfer Phenomena formulas please calculate

Heterogenous Diffusion Reaction Spherical Dimensions

A carbon particle is burnt in an environment of air at a temperature of $200C$ and a pressure

of $6$ bar according to the following reaction stoichiometry.

$4C_{(s)}+3O_{2(g)}2C{O}_{2(g)}+2C{O}_{(g)}$

It can be assumed that the particle is spherical with a diameter of $8mm.$ It is uncertain whether

diffusion or reaction is controlling. It is known however that the reaction follows first$-$order

kinetics with respect to the concentration of oxygen at the surface such that:

$r_{s,02}=-k{c}_A$

with

$k=1\mathrm{.}5{10}^{-3}(\frac{m}{s})$

$c_A=$ total molar concentration $of$ Oxygen $at$ the particle surface $(mo\frac{l}{m^3})$

$($a$)$ Make a schematic drawing of this mass transfer system, showing your choice of

dimensional axes and the direction of the fluxes of each species. Indicate which, if any,

of the fluxes are zero and why. $[$choose $"$A$"$: $O_2,"B"$: $C{O}_2,"C"$: $CO,"D"$: $N_2]$

$($b$)$ With the appropriate form of the continuity equation and Fick's $1^{st}$ Law show that the

differential equation for the molar flux of oxygen is given by the following

expression:

$N_{Ar}=-\frac{D_{AM}}{(1+\frac{y_B}{3})}c\frac{del{y}_A}{\text{d}\text{e}\text{l}\text{r}}$

$($c$)$ Assuming that the system is "diffusion controlled" and that the concentration of oxygen

at the surface is zero, show that the molar flux of oxygen can be given by the following

expression after integrating the expression in $($b$)$:

$N_{A-R1}=-\frac{3c{D}_{AM}}{R_1}ln(1+\frac{y_{A,b}}{3})$

$($d$)$ Now assuming that the system is controlled by the reaction at the surface, show that the

molar flux of oxygen can be given by the following expression after integrating the

expression in $($b$)$:

$N_{A-R1}=-\frac{3c{D}_{AM}}{(R_1+\frac{D_{AM}}{k})}ln(1+\frac{y_{A,b}}{3})$

$($e$)$ How long does it take for the carbon particle to burn completely in the case that

diffusion is controlling? $[$Ans: $0\mathrm{.}5h]$

$($f$)$ How long does it take for for the carbon particle to burn completely in the case that the

reaction rate is controlling? $[$Ans: $1\mathrm{.}89h]$

$D_{AM}$:$1\mathrm{.}98{10}^{-5}\frac{m^2}{s}$