The only known sources of helium are the atmosphere (mole...

The only known sources of helium are the atmosphere (mole fraction approximately $5 \times 10^{-6}$ ) and natural gas. A large unit is being constructed to separate $100 \mathrm{~m}^{3} / \mathrm{s}$ of natural gas, assumed to be 0.001 helium mole fraction and 0.999 methane. The gas enters the unit at $150 \mathrm{kPa}, 10^{\circ} \mathrm{C}$. Pure helium exits at $100 \mathrm{kPa}, 20^{\circ} \mathrm{C}$ and pure methane exits at $150 \mathrm{kPa}, 30^{\circ} \mathrm{C}$. Any heat transfer is with the surroundings at $20^{\circ} \mathrm{C}$. Is an electrical power input of $3000 \mathrm{~kW}$ sufficient to drive this unit?

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Key Concepts

Thermodynamic Minimum Work of Separation

This concept refers to the minimum amount of work, calculated from the change in Gibbs free energy, that is theoretically required to separate a mixture into its pure components using a reversible process. It sets an ideal lower bound on energy consumption, providing a benchmark against which the energy input and actual performance of a separation process can be compared.

Energy Balance and Second Law Analysis

The energy balance in separation processes involves applying both the first and second laws of thermodynamics to quantify the energy inputs and outputs. The second law, in particular, introduces the concept of irreversibility and exergy destruction, which helps in understanding why real processes require more energy than the theoretical minimum and in evaluating whether a given energy input is sufficient to drive the separation process.

Mole Fraction and Partial Pressure in Gas Mixtures

Mole fraction is a measure of the concentration of a component within a mixture and is directly related to its partial pressure in an ideal gas mixture. This concept is critical in thermodynamic calculations, such as those for the Gibbs free energy change of mixing, and plays a fundamental role in determining the conditions and energy requirements for separating components in gas mixtures.

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