Question $1-$ Compression of a $C\frac{H_4}{C}{O}_2$ mixture $($van der Waals EoS$)$

A mixture of methane $(C{H}_4)$ and carbon dioxide $(C{O}_2)$ is produced as off$-$gas in a

composting process, comprising of $85$ mol$\%$ methane and $15$ mol$\%$ carbon dioxide. This

gas is to be stored in compressed form to be used as fuel gas for a thermal process in the

operation. The off$-$gas is available at the temperature and pressure of $293K$ and $\frac{1}{b}ar,$

respectively, and needs to be compressed to the pressure of $10$ bar in a steady$-$state

process. The efficiency of the available adiabatic compressor is $85\%.$ The gas mixture

can be described using the van der Waals equation of state:

$(p+\frac{a_{\text{m}\text{i}\text{x}}}{V_{{?}^{?}}(2)})*(V_{-}{b}_{\text{m}\text{i}\text{x}})=RT$ with $a_{\text{m}\text{i}\text{x}}=0\mathrm{.}2378*\frac{Pa*m^6}{mo{l}^2}$ and $b_{\text{m}\text{i}\text{x}}=4\mathrm{.}311*{10}^{-5}*\frac{m^3}{\text{m}\text{o}\text{l}}$

The rest of required data are available in Table $1.$

$35\mathrm{.}699$ I

Table $1.$ Properties of methane and carbon dioxide

$($a$)$ Make a drawing of the process and define your system by the relevant balances.

$($b$)$ Show that the final temperature upon compressing the gas from $1$ bar to $10$ bar is

$474K,$ if the gas obeys the Ideal Gas Law.

$($c$)$ Show that the gas mixture is close to ideal at both the inlet and the outlet

conditions of the compressor.

$($d$)$ Determine the amount of work required if the flow rate of the gas mixture is $1\mathrm{.}0$

$mo\frac{l}{s}.$

$($e$)$ What is the actual final temperature of the gas leaving the compressor

considering its efficiency?

$17\mathrm{.}6493+3\mathrm{.}8637+0\mathrm{.}048435T$