Helium is a nonrenewable resource currently obtained from natural gas. We want to examine the

practicality of separating He from methane using a membrane. Assume $100,000mo\frac{l}{m}in$ of gas

containing $1\%$ Helium and $99\%$ methane is introduced at $50$ bar and $25C.$ The membrane skin is

$0\mathrm{.}0002mm$ thick, and the permeabilities of helium and methane are $100$ barrer and $1$ barrer,

respectively. The resulting permeate has a pressure of $1$ bar.

a$.$ Calculate the material balance as a function of the cut and mole fractions of the permeate,

which is defined as

$=$ cut $=$ fraction $of$ feed permeated $=\frac{n_P}{n_F}$

Hint: you should solve for mole fraction of each component in the raffinate $(x_{R,He}$ and $x_{R,M}).$

b$.$ What is the separation factor ${}_{H\frac{e}{M}}$ in terms of cut and $y,He?$

c$.$ Use the separation factor relationship below to derive an equation containing only cut and

$y_{P,He}$ as variables. You do not need to simplify this equation. Calculate the fraction of

Helium in the permeate $(y_{P,He})$ for when cut is equal to $0\mathrm{.}0001,0\mathrm{.}001,0\mathrm{.}01,$ and $0\mathrm{.}1.$

${}_{\frac{A}{B}}=\frac{P_{M,A}}{P_{M*B}}**[\frac{x_{R,A}**({}_{\frac{A}{B}}-1)+1-r{}_{\frac{A}{B}}}{x_{R,A}**({}_{\frac{A}{B}}-1)+1-r}]$

Where $r=\frac{\text{p}\text{r}\text{e}\text{s}\text{s}\text{u}\text{r}\text{e}of\text{p}\text{e}\text{r}\text{m}\text{e}\text{a}\text{t}\text{e}}{\text{p}\text{r}\text{e}\text{s}\text{s}\text{u}\text{r}\text{e}of\text{f}\text{e}\text{e}\text{d}}$

d$.$ Use flux to derive an equation for area of the membrane and calculate the area required

for a cut of $0\mathrm{.}0001,0\mathrm{.}001,0\mathrm{.}01,$ and $0\mathrm{.}1.$ Assume total pressure of the raffinate stream is

approximately equal to the pressure of the feed.

e$.$ Calculate the molar flux of Helium in the permeate for a cut of $0\mathrm{.}0001,0\mathrm{.}001,0\mathrm{.}01,$ and

$0\mathrm{.}1.$