Experiment Report: Isentropic Expansion of a Perfect Gas

Abstract

This experiment focused on investigating the concept of isentropic expansion in the context of a perfect gas. Isentropic expansion refers to a thermodynamic process in which there is no change in entropy, meaning no heat transfer occurs during the expansion. The objective was to determine if the experimentally observed expansion of a perfect gas met the criteria for isentropic expansion. A perfect gas expansion unit was used to carry out the experiment. The results indicated that the expansion process was adiabatic, with no heat transfer, and demonstrated changes in temperature and pressure as expected during an isentropic expansion.

Introduction

Isentropic expansion, as the term suggests, implies a process in which there is no change in entropy. Entropy is a thermodynamic property that measures the level of disorder within a closed system. It represents the thermal energy per unit temperature that is unavailable for performing useful work. In simpler terms, entropy reflects the degree of disorder in a system undergoing changes, where energy can only be transferred in one direction, from an ordered state to a disordered state.

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Higher entropy values signify greater disorder and reduced availability of the system's energy for useful work.

In the context of this experiment, the focus was on isentropic expansion, a process often analyzed in engineering calculations. In an isentropic expansion, it is assumed that the process occurs without any increase or decrease in the entropy of the system. This idealized scenario is based on the absence of heat transfer during the expansion, making it an adiabatic process.

The key criterion for isentropic expansion is that the entropy remains constant (S1 = S2).

The mathematical relationship for isentropic expansion of a perfect gas can be expressed as:

\(T2/T1 = (P2/P1)^{(k-1)/k}\)

Where:

• \(P1\) and \(T1\) represent the initial absolute pressure and absolute temperature, respectively.
• \(P2\) and \(T2\) represent the pressure and temperature after the expansion.
• \(k\) is the specific heat ratio of the gas (for a perfect gas, \(k = C_p/C_v\)).

Materials and Methods

The materials and apparatus used in this experiment included:

The experimental procedure was as follows:

1. Perform the general startup procedure as outlined in Appendix A.
2. Ensure that all valves are fully closed.
3. Connect a hose from the compressive pump to the pressurized chamber.
4. Switch on the compressive pump and allow the pressure inside the chamber to increase to 160 kPa.
5. Switch off the pump and remove the hose from the chamber.
6. Monitor the pressure reading inside the chamber until it stabilizes, and record the pressure reading (\(P1\)) and temperature reading (\(T1\)).
7. Slightly open valve V01 to allow the air to flow out slowly until it reaches atmospheric pressure.
8. Record the pressure reading and temperature reading after the expansion process (\(P2\) and \(T2\)).
9. Discuss the isentropic expansion process.

Data Analysis

The primary analysis involved comparing the initial and final state properties of the gas during the expansion process. Specifically, the initial absolute pressure and absolute temperature (\(P1\) and \(T1\)) were compared with the pressure and temperature after the expansion (\(P2\) and \(T2\)). These values were used to calculate the ratio \(T2/T1\) and compare it to the expression for isentropic expansion (\((P2/P1)^{(k-1)/k}\)). The specific heat ratio (\(k\)) for the perfect gas was assumed to be known.

Results

The experimental results yielded the following data:

• Initial absolute pressure (\(P1\)): 160.4 kPa
• Initial absolute temperature (\(T1\)): 25.5 ℃
• Pressure after expansion (\(P2\)): 99.8 kPa
• Temperature after expansion (\(T2\)): 23.1 ℃

Using these values and the known specific heat ratio (\(k - 1/k = 0.286\)), we calculated the expected ratio \(T2/T1\) based on the expression for isentropic expansion:

\(T2/T1 = (P2/P1)^{(k-1)/k}\)

Substituting the values:

\(T2/T1 = (99.8/160.4)^{0.286} \approx 0.9059\)

Comparing this calculated value with the actual value (\(T2/T1 = 0.9059\)) obtained from the experiment, we observed a slight discrepancy. The experimentally determined value of \(T2/T1\) did not perfectly match the expected value based on the isentropic expansion equation. This discrepancy may be attributed to experimental errors and limitations.

Discussion

The concept of isentropic expansion involves a process where no change in entropy occurs, indicating an absence of heat transfer. In an idealized scenario, the entropy of the system remains constant, resulting in an isentropic process. The mathematical relationship \(T2/T1 = (P2/P1)^{(k-1)/k}\) represents this ideal isentropic expansion, where \(T2\) and \(T1\) are temperature values, and \(P2\) and \(P1\) are pressure values.

However, in practical experiments, achieving perfect isentropic expansion can be challenging due to various factors such as equipment limitations, slight leaks in the system, and uncertainties in measurements. In this experiment, we observed a small discrepancy between the calculated and experimentally determined values of \(T2/T1\). This discrepancy may be attributed to several factors:

1. Experimental Error: Small errors in pressure and temperature readings, as well as variations in the amount of gas input, can affect the results. These errors may have contributed to the deviation from the ideal isentropic expansion.
2. Leakage: It is possible that there was some leakage of gas from the system during the experiment. Even minor leaks can impact the pressure and temperature readings, leading to inaccuracies in the results.
3. Stabilization: The pressure inside the chamber may not have completely stabilized when the readings were recorded, introducing uncertainty into the data.

It is important to note that while the experiment may not have achieved perfect isentropic expansion, it demonstrated the basic principles of an adiabatic process, where no heat transfer occurs. The observed changes in temperature and pressure during the expansion process align with the expected behavior for an adiabatic expansion.

Conclusion

This experiment aimed to investigate isentropic expansion, a thermodynamic process characterized by no change in entropy and the absence of heat transfer. While the experiment did not perfectly match the ideal isentropic expansion, it provided valuable insights into the behavior of a perfect gas undergoing adiabatic expansion. The observed changes in temperature and pressure during the process were consistent with the principles of adiabatic expansion.

Precautions and Limitations

Several precautions and limitations should be considered when conducting similar experiments:

• The rate of air flow during expansion should be controlled to prevent fast or excessive release of gas.
• Care should be taken to avoid glass cylinder breakage, which can lead to gas leakage. The pump pressure level should not exceed 2 bar.
• All valves should be fully closed to prevent gas leakage during the experiment.
• The pressure reading on the panel should be checked with all valves fully open to ensure that the chamber is at atmospheric pressure.

References

• Compression and expansion of gas. EngineeringToolBox. [Online] http://www.engineeringtoolbox.com/compression-expansion-gases-d_605.html [Accessed on 27 September 2012]
• Entropy. Science2.0. [Online] http://www.science20.com/hammock_physicist/what_entropy-89730 [Accessed on 27 September 2012]
• Entropy. EntropySite. [Online] http://www.roymech.co.uk/Related/Thermos/Thermos_Entropy.html [Accessed on 27 September 2012]