At modest pressures the equation of state of a gas can be...

At modest pressures the equation of state of a gas can be written as $p V=R T+B p$, where $B$ is a function of $T$ only. (i) Find an expression for the Joule-Kelvin inversion temperature in terms of $B$ and $\mathrm{d} B / \mathrm{d} T$. (ii) For helium gas between $5 \mathrm{~K}$ and $60 \mathrm{~K}, B$ can be represented as $B=m-(n / T)$, where $m=15.3 \times 10^{-6} \mathrm{~m}^3 \mathrm{~mole}^{-1}$ and $n=352 \times 10^{-6} \mathrm{~m}^3$ $\mathrm{K} \mathrm{mole}{ }^{-1}$. From this information, estimate the inversion temperature for helium. For an expansion starting below the inversion temperature, comment on the efficiency of the process (i.e. cooling per unit pressure change) as the temperature falls. (iii) In a helium liquifier, compressed helium gas at $14 \mathrm{~K}$ is fed to the expansion valve, where a fraction $x$ liquifies and the remaining fraction $(1-x)$ is rejected as gas at $14 \mathrm{~K}$ and atmospheric pressure. If the specific enthalpy of helium gas at $14 \mathrm{~K}$ is given by $$ h=a+b\left(p-p_0\right)^2 $$ where $a=71.5 \mathrm{~kJ} \mathrm{~kg}^{-1}, b=1.3 \times 10^{-12} \mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~Pa}^{-2}$, and $p_0=3.34 \mathrm{MPa}$ $(=33.4 \mathrm{~atm})$, and the specific enthalpy of liquid helium at atmospheric pressure is $10 \mathrm{~kJ} \mathrm{~kg}^{-1}$, determine what input pressure will allow $x$ to achieve a maximum value, and show that this maximum value is approximately 0.18 .

At modest pressures the equation of state of a gas can be written as $p V=R T+B p$, where $B$ is a function of $T$ only. (i) Find an expression for the Joule-Kelvin inversion temperature in terms of $B$ and $\mathrm{d} B / \mathrm{d} T$. (ii) For helium gas between $5 \mathrm{~K}$ and $60 \mathrm{~K}, B$ can be represented as $B=m-(n / T)$, where $m=15.3 \times 10^{-6} \mathrm{~m}^3 \mathrm{~mole}^{-1}$ and $n=352 \times 10^{-6} \mathrm{~m}^3$ $\mathrm{K} \mathrm{mole}{ }^{-1}$. From this information, estimate the inversion temperature for helium. For an expansion starting below the inversion temperature, comment on the efficiency of the process (i.e. cooling per unit pressure change) as the temperature falls. (iii) In a helium liquifier, compressed helium gas at $14 \mathrm{~K}$ is fed to the expansion valve, where a fraction $x$ liquifies and the remaining fraction $(1-x)$ is rejected as gas at $14 \mathrm{~K}$ and atmospheric pressure. If the specific enthalpy of helium gas at $14 \mathrm{~K}$ is given by
$$
h=a+b\left(p-p_0\right)^2
$$
where $a=71.5 \mathrm{~kJ} \mathrm{~kg}^{-1}, b=1.3 \times 10^{-12} \mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~Pa}^{-2}$, and $p_0=3.34 \mathrm{MPa}$ $(=33.4 \mathrm{~atm})$, and the specific enthalpy of liquid helium at atmospheric pressure is $10 \mathrm{~kJ} \mathrm{~kg}^{-1}$, determine what input pressure will allow $x$ to achieve a maximum value, and show that this maximum value is approximately 0.18 .

Key Concepts

Enthalpy Conservation in Throttling Processes

In a throttling process, the specific enthalpy of the gas remains constant during the expansion. This conservation is used to relate the initial state values to those after expansion, even when phase change (i.e. partial liquefaction) occurs. By analyzing the enthalpy before and after the expansion, one can derive relations that determine key performance metrics such as the fraction of the gas liquefied.

Joule?Thomson Effect

The Joule?Thomson effect describes the temperature change that occurs in a real gas when it is allowed to expand adiabatically through a throttling process, without performing work. Whether the gas cools or heats during this process depends on its initial temperature relative to the inversion temperature. This effect is central to cryogenic engineering and gas liquefaction, and it underscores the importance of non-ideal gas behavior in practical applications.

Process Optimization in Cryogenic Liquefaction

Engineering applications such as helium liquefiers require optimizing the operating conditions to maximize the conversion of gas to liquid. This involves balancing the input pressure, the enthalpy changes, and the Joule?Thomson effect to achieve the highest possible liquefaction efficiency. The analysis often leads to conditions that maximize the liquefied fraction, which is critical to the design and performance of cryogenic systems.

Joule?Thomson Inversion Temperature

This is the specific temperature at which the sign of the Joule?Thomson coefficient changes, meaning that a gas will neither cool nor heat upon throttling at this point. It is determined through the balance of intermolecular interactions as embodied in the equation of state and its temperature dependence. The inversion temperature is characterized by the derivative of the non-ideal correction term (in this case B) with respect to temperature, highlighting its dependence on molecular properties and interparticle forces.

Non?Ideal Gas Equation of State

Real gases deviate from ideal behavior, especially under conditions of high pressure or low temperature. The modified equation of state, pV = RT + Bp, introduces a temperature-dependent correction term to account for intermolecular attractions and repulsions. This correction is crucial for accurately predicting gas behavior and for the analysis of thermodynamic processes, such as the Joule?Thomson effect.

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