# Isenteropic expansion of ideal gas

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**ME 2140**

**Isentropic Expansion of An Ideal Gas**

**REFERENCE:** Cengel, Y.A. and Boles, M.A., *Thermodynamics, An Engineering Approach, 6th Ed.*, McGraw-Hill, Inc., 2008, Sections 7-3, 7-5, 7-7, 7-9.

**OBJECTIVE:** Evaluate and assess a laboratory procedure for measuring the specific heat of a gas at constant pressure (C_{p}).

**THEORY OF OPERATION:**

We shall consider the system shown in Figure 1 which consists of a vessel and the associated tubing, valves and a pressure gauge for measuring the pressure in the vessel. We shall pressurize the vessel to some pressure above atmospheric pressure and allow the vessel to come to thermal equilibrium (room temperature). We will term this equilibrium condition, State 1.

**Figure 1** – Specific heat apparatus.

Subsequently, we shall open the valve allowing the vessel to be exhausted to atmospheric pressure and a lower temperature than the starting temperature. This equilibrium condition will be termed State 2. Upon closing the valve, the remaining contents of the vessel will then warm up, with a corresponding increase in pressure. This final equilibrium condition will be termed State 3.

**ANALYSIS:**

** **

Let us first consider the process 1-2 and let the system be only that gas contained in the vessel at the end of the expansion. In differential form, the first law of thermodynamics for a closed system (fixed mass) is:

. **(1)**

If we assume the expansion process is adiabatic, then q = 0 or

. **(2)**

If we assume the work done by the expansion of the remaining gas is done reversibly then:

. **(3)**

If, in addition, we assume that the working substance is an ideal gas with constant specific heats, then we can show from Gibbs’ equations that:

. **(4)**

During the process 2-3, the gas warms up at constant volume. Using the equation of state, we can show that:

, **(5)**

where it may be assumed that: T3 = T1.

**PROCEDURE:**

- Record atmospheric pressure, p2.

- Record room temperature, T1 = T3.

- Slowly pressurize the vessel to approximately 5 psig. Turn off the air supply and close the valve that isolates the compressed air inside the vessel. Allow the system to come to thermal equilibrium (i.e. no pressure change for at least one minute) and record the steady-state pressure, p
_{1}.

- Depressurize the vessel by prying the stopper loose with your thumbs. Allow the discharge to stop and reseat the stopper.

- Allow the system again to reach thermal equilibrium while the pressure rises. Record the steady-state pressure, p
_{3}.

- Repeat steps 3-5 to obtain a total of three sets of readings for p1 and p3.

**REPORT:**

This exercise is an examination of a laboratory procedure with the claimed potential to measure the specific heat of an unknown gas. The following items should be included in your evaluation report.

- 1. Include all pertinent information from the pre-laboratory.

- Calculate the following for each test run: T2 ,Cp, Cv.

- Plot all three system states on both T-s and P-v diagrams. Tabulate the results also. Compare your results with those from the gas tables.

- Calculate the work done by the system as it expands.

- Discuss the assumptions needed to model the process 1-2 as an isentropic expansion.

- Evaluate the potential use of this technique for measuring specific heats of unknown gas mixtures: would you recommend the use of the technique ?

ME/ESE 2160 – Fluids Lab Intro & Uncertainty‐ 1 ME/ESE 2160 Introduction and Uncertainty Analysis ME/ESE 2160 – Fluids Lab Intro & Uncertainty‐ 2 Introduction and Uncertainty Analysis A. Introduction 1. Overview a) This course will provide an introduction to experimental techniques used in the laboratory as well as laboratory and teamwork skills. In the process, fundamental concepts from Thermodynamics, Fluid Dynamics, and Heat Transfer will be demonstrated. Laboratory write‐ups and presentations are a primary focus of the course. 2. Approach a) Most of the technical material in this course is not new, but an application of what you have already learned in previous courses. As such, there will be limited material provided other than the handouts and the student is expected to use the referenced material to learn more about what they will be doing. The exception is the heat transfer material that some students may not yet have seen. Lectures on this material will be more comprehensive. ME/ESE 2160 – Fluids Lab Intro & Uncertainty‐ 3 Introduction and Uncertainty Analysis A. Introduction 3. Course materials are all on‐line • WyoCourses • Syllabus • Schedule • Lab Assignments • Lab Partners • Resources • ME Class References • Curve Fitting • Documenting Sources • Experimental Errors • Laboratory Report Formats • Propagation of Errors • Curve Fitting and Regressions • Oral Presentations • Common Problems in Long Lab Reports • Example Presentations • Equations, Plots, Tables, Drawings Reference ME/ESE 2160 – Fluids Lab Intro & Uncertainty‐ 4 Introduction and Uncertainty Analysis B. Uncertainty Analysis 1. Consists of 2 parts a) Determine the uncertainty of measured quantities • Precision uncertainties (Random, Statistical) • Bias uncertainties (Systematic) • Reference: Experimental Errors – ME References b) Determine the uncertainty of quantities determined from the measured quantities • Propagation of Error • Reference: Propagation of Errors ME/ESE 2160 – Fluids Lab Intro & Uncertainty‐ 5 Introduction and Uncertainty Analysis B. Uncertainty Analysis 2. Precision Errors ‐ Px a) Errors that exist due to random variations in a measurement b) Characterized by the standard deviation • It is not equal to the standard deviation • Two measurements of a quantity, one with 10 samples and one with 100 samples, may have the same standard deviation. Which one is a better estimate? c) Introduce the 95% confidence interval • Bounds between which 95% of the measurements will fall Student’s t Distribution ME/ESE 2160 – Fluids Lab Intro & Uncertainty‐ 6 Introduction and Uncertainty Analysis B. Uncertainty Analysis 3. Bias Errors – Bx a) Errors that exist due to systematic variations in a measurement b) Arise from different sources • Smallest measurement unit • Calibration bias – your own or factory calibration • Digitization error – rounding due to data collection c) Burden is on the experimenter to come up with good estimates of the Bias 4. Total Uncertainty – a) Combination of Biases and Precision b) For this class, we’ll use • Bx2 = Bx,12 +Bx,22 +Bx,32 + … • Ux2 = Px2 +Bx2 ME/ESE 2160 – Fluids Lab Intro & Uncertainty‐ 7 Introduction and Uncertainty Analysis A. Uncertainty Analysis 1. Propagation of Errors a) Calculated quantities often depend on several measured quantities. How do we calculate the uncertainty of the calculated quantity? b) The impact of a measured quantity on a calculated quantity depends on the sensitivity of the calculated quantity to the measured quantity • Measured quantities X and Y, calculated quantity R • Sensitivities are • Uncertainty of calculated quantity R

lab 1 | ||||||||

T_{1} = T_{3} = 293 K |
293.00 | |||||||

p_{2} = 603 mm Hg |
||||||||

80.40 | Kpa | |||||||

R= | 0.29 | |||||||

S. No | (Kpa) | (Kpa) | (Kpa) | (K) | (K) | (K) | (kJ/kg.K) | (kJ/kg.K) |

Pressure | Pressure | Pressure | Temperature | Temperature | Temperature | Specific Heat | Specific Heat | |

p_{1} |
p_{2} |
P_{3} |
T_{1} |
T_{2} |
T_{3} |
(Cp) | (Cv) | |

1.00 | 114.18 | 80.40 | 87.30 | 293.00 | 269.86 | 293.00 | 1.22 | 0.94 |

2.00 | 114.53 | 80.40 | 87.30 | 293.00 | 269.86 | 293.00 | 1.23 | 0.95 |

3.00 | 112.12 | 80.40 | 87.30 | 293.00 | 269.86 | 293.00 | 1.16 | 0.87 |

(m^{3}) |
(m^{3}) |
(m^{3}) |
||||||

volume | volume | volume | ||||||

V_{1} |
V_{2} |
V_{3} |
||||||

0.74 | 0.96 | 0.31 | ||||||

0.73 | 0.96 | 0.31 | ||||||

0.75 | 0.96 | 0.31 |

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