Using Interfacial Mass Transfer Phenomena formulas please calculate

Diffusion$-$Reaction cylindrical dimensions

A gas stream of pure $\text{'}B\text{'}$ at pressure $P=\frac{2\mathrm{.}0}{b}ar(\frac{?}{abs})$ and at temperature $T=80C$ is contacted

with liquid $\text{'}A\text{'}$ through a cylindrical membrane of $1mm$ thickness. The liquid $A$ is flowing

through the $1-$meter cylindrical membrane hose at a flowrate of $100m\frac{L}{h},$ with the inner

diameter of the hose at $7mm.$ Gas $\text{'}B\text{'}$ diffuses through the membrane and reacts with liquid

$\text{'}A\text{'}$ at the inner membrane wall $($i$.$e$.$ on the liquid side$),$ according to the following reaction

stoichiometry:

$A_{(l)}+3B_{(g)}2C(g)+D_{(l)}$

Reagent $\text{'}A\text{'}$ and reaction product $\text{'}D\text{'}$ remain in the liquid phase. Gases $\text{'}B\text{'}$ and $\text{'}C\text{'}$ are not

soluble in the liquid. The concentration of $\text{'}C\text{'}$ outside the membrane is negligible. The molar

rate of surface reaction, $r_{s,B},$ at the inner wall membrane follows first$-$order kinetics in terms of

the molar concentration of gaseous $\text{'}B\text{'}$ at the inner membrane wall:

$r_{s,B}=-k{c}_B$

with

$k=5{10}^{-5}(\frac{m}{s})$

$c_B=$ total molar concentration $of$ gaseous $Bat$ membrane surface $(mo\frac{l}{m^3})$

Assuming isothermal steady$-$state conditions:

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 through the membrane.

Indicate which, if any, of the fluxes are zero and why. State any assumptions you

consider are appropriate to assist with quantitative analysis of the mass transfer.

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 $\text{'}B\text{'}$ through the membrane at its inner

surface

$N_B=-\frac{D_{BC}}{(1-\frac{y_B}{3})}c\frac{del{y}_B}{\text{d}\text{e}\text{l}\text{r}}$

c$)$ By integration of the equation in b$)$ between appropriate limits$,$ calculate the rate of

production of $\text{'}D\text{'}$ per $m^2$ of membrane. The diffusivity of $\text{'}B\text{'}$ through $\text{'}C\text{'}$ within

the porous membrane wall is: $D_{BC}=1\mathrm{.}75{10}^{-9}\frac{m^2}{s}.$ Ans $5\mathrm{.}47{10}^{-5}mo\frac{l}{m^2.s}]$

d$)$ What is the exit concentration of $D$ in the liquid in $mo\frac{l}{m^3}$ assuming reagent A enters

the cylindrical hose free of the product D$?$ Ans $43\mathrm{.}3\frac{\text{m}\text{o}\text{l}}{m^3}$