A rigid tank is filled initially with 5.0 kg of air at a pressure of 0.5 MPa and a temperature

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A rigid tank is filled initially with 5.0 kg of air at a...

A rigid tank is filled initially with $5.0 \mathrm{~kg}$ of air at a pressure of $0.5 \mathrm{MPa}$ and a temperature of $500 \mathrm{~K}$. The air is allowed to discharge through a turbine into the atmosphere, developing work until the pressure in the tank has fallen to the atmospheric level of $0.1 \mathrm{MPa}$. Employing the ideal gas model for the air, determine the maximum theoretical amount of work that could be developed, in $\mathrm{kJ}$. Ignore heat transfer with the atmosphere and changes in kinetic and potential energy.

Key Concepts

First Law of Thermodynamics

This principle governs energy conservation within thermodynamic systems. In any process, the change in internal energy equals the net heat transferred into the system minus the work done by the system. For problems involving work extraction, applying the first law correctly allows one to relate energy changes in the system to the work output, even when mass is being discharged or when volume is held constant.

Ideal Gas Model

The ideal gas assumption simplifies the analysis by establishing that the gas follows the ideal gas law (PV = mRT), and that its internal energy depends solely on temperature. This assumption is crucial for calculating thermodynamic state changes such as internal energy and entropy variations during the process, thereby aiding in the determination of maximum work outputs.

Reversible Process

A reversible process is an idealization in which the system changes state in a quasi-static manner without generating entropy. This concept is important because maximum theoretical work is obtained under reversible conditions; any irreversibilities, such as friction or turbulence, would reduce the actual work output compared to this ideal.

Exergy Analysis (Availability)

Exergy analysis quantifies the maximum work obtainable from a system as it comes into equilibrium with its environment. It accounts not only for energy but also for entropy production and the quality of energy. This approach is useful in determining the theoretical work that can be extracted from a system by considering the decrease in the system's ‘available energy’ relative to a given reference state.

Closed System (Control Mass) Analysis

In a closed system, such as a rigid tank, the mass remains fixed even though energy may be exchanged with the surroundings or work may be extracted. Analyzing such systems involves tracking changes in internal energy and other properties as mass is allowed to exit through controlled processes, thereby enabling the calculation of work extracted under constant volume conditions.

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