QUESTION $4$

a$)$ Natural gas with $8\%$ hydrogen content is transported in an above ground pipe line of inner diameter $1\mathrm{.}5$ meter and a thickness of $0\mathrm{.}01m.$ The pipe has a length of $100m$ and is maintained at a pressure of $500$kPa and a temperature of $273K.$ The highest rate of hydrogen loss through the pipe at steady state conditions is to be determined. The binary diffusion coefficient of hydrogen in the steel pipe at the operation temperature is given to be $2\mathrm{.}9{10}^{-13}\frac{m^2}{s}.$ The molar masses of $H_2$ and $C{H}_4$ are $2$ and $16k\frac{g}{k}$mol, respectively. The solubility of hydrogen gas in steel is given as $w_{H_2}=2\mathrm{.}09{10}^{-4}$exp$(-\frac{3950}{(T)})P_{H_2}^{0\mathrm{.}5}.$ The density of steel pipe is $7854k\frac{g}{m^3}.$

Assume: Mass diffusion is steady and one one$-$dimensional as the hydrogen concentration inside the pipe is constant and the atmosphere is negligible. There are no chemical reactions in the pipe that results in the generation or depletion of hydrogen. Both $H_2$ and $C{H}_4$ are ideal gases.

$[16$ Marks$]$