PROBLEM $2(70$ POINTS$)$

The Hampson$-$Linde cycle is the key step in most gas liquefaction processes, invented by William Hampson and Carl von Linde in $1895.$ The cycle uses successive pressurization and expansion through a Joule$-$Thomson orifice $($essentially a throttling valve$)$ to achieve cooling until the gas is cold enough to condense. In this problem, we will study a simplified version of the process.

Argon gas at $300K$ and $1$ bar in a steady flow of $1mo\frac{l}{s}$ is first compressed isothermally to $60$ bar, and then passed through a throttling valve until the pressure goes back to $1$ bar. The throttling valve can be assumed to be adiabatic.

$($a$)$ If we treat argon as an ideal gas, what would be the final temperature after passing through the valve $(10$ points$)?$

$($b$)$ The ideal gas assumption does not work in our case, because it fails to produce the cooling effect when a real gas is passed through a valve $($called the Joule$-$Thomson effect$).$ Instead of using the ideal gas assumption, we will use the Pitzer B correlation for argon:

where:

$Z=\frac{PV}{RT}=1+B^0\frac{P_r}{T_r}$

$B^0=0\mathrm{.}083-\frac{0\mathrm{.}422}{T_r^{1\mathrm{.}6}}$

Here, $T_r=\frac{T}{T_c}$ is the reduced temperature and $P_r=\frac{P}{P_c}$ is the reduced pressure. Note that for argon, the acentric factor $=0,$ so we can leave out the $B^1$ term. The critical temperature is $T_c=150\mathrm{.}9K$ and the critical pressure is $P_c=48\mathrm{.}98$ bar.

For this equation of state, derive expressions for $(\frac{\text{d}\text{e}\text{l}\text{Z}}{\text{d}\text{e}\text{l}\text{T}}{)}_P,(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P,(\frac{\text{d}\text{e}\text{l}\text{S}}{\text{d}\text{e}\text{l}\text{P}}{)}_T$ and $(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{P}}{)}_T.$ The expressions should be in terms of constants $($including $P_c$ and $T_c),T_r$ and $P_r$ only $(20$ points$).$

$($c$)$ Assuming the compression step is reversible, calculate the heat rate and the shaft power of the compressor $(15$ points$).$

$($d$)$ Calculate the final temperature of the argon gas after passing through the valve. For argon gas at $1$ bar, $C_P=2\mathrm{.}5R$ and can be assumed to be independent of temperature $(10$ points$).$

$($e$)$ Using residual properties, verify that the molar enthalpy change of the argon gas across the throttling valve is approximately zero $(15$ points$).$

Note: In the real Hampson$-$Linde process, the isothermal compression is often achieved by a real compressor $($which is far from isothermal$),$ and cooling with the cold product stream using a heat exchanger. One step is usually not enough to liquefy the gas, so the process is repeated.