Determination of the Charge to Mass Ratio of the Electron

Richard Plews

Determination of the Efficiency of the Stirling Engine Cycle

Student: Richard Plews

Tutor: A.Aruliah

Course 1B40

22nd March 2006

Abstract

The Stirling engine was invented by Robert Stirling in 1816. . At that time, Stirling engines were recognized as a safe engine that could not explode like steam engines of that era often did. It could also use any form of energy, including solar energy and heat energy.

The variations in temperature, pressure and volume of the Stirling engine with time were used to calculate the efficiency with three different methods of increasing complexity, producing three values of the efficiency, the most accurate of which being (1.83±0.89)%. The precision of the result is the lowest, but takes errors into account better than the other two methods

Introduction

In the early 1800s, the widely used steam engines were dangerous, harming many in unpredictable explosions. The ‘hot air engine’ developed by Scottish minister Robert Stirling was designed to be a safer replacement.Stirling’s design also had the potential to be the most efficient engine, with up to 50% of the energy put into operating the engine being returned as useful work. Calculating the efficiency of engines is a useful way to determine their usefulness in comparison to other types of engine available.

The Stirlingengine operates as a heat pump, using energy from a warm reservoir, converting some of this energy into a different form to perform work.

A gas absorbs heat energy QA at a high temperature TH, and releases energy QB to a cold reservoir at lower temperature TL, and converts some of this energy into work, W. This cyclic process is repeated. The efficiency of a heat pump is given by

(1)

If there was no energy released to the cold reservoir, the efficiency, η would be equal to 1, indicating 100% efficiency. However no heat engine can accomplish this. The theoretical efficiency of the Stirling engine is nearly equal to their theoretical maximum efficiency, known as the Carnot Cycle efficiency,

(2)

Or alternatively, the efficiency can be calculated from a Pressure against Volume graph showing the Stirling engine cycle

The area of a Pressure against Volume graph is equal to the Work done; this is true because of the formula

(3)

Therefore the total work put into the cycle is the area underneath the curve A (QA), and the useful work out of the cycle is the area underneath curve B subtracted from the area underneath curve A (W). From which the efficiency is calculated with equation (1).

Equipment

The Stirling engine

This Stirling engine is driven by a motor connected to the flywheel by a band. The flywheel drives two cams operating 90º out of phase, each causing the working and displacement piston to oscillate. The working piston increases and decreases the volume of the gas in the cycle, heating and cooling the gas respectively, the displacement piston then moves the cooled gas to one end of the displacement chamber, and the heated gas to the other. This is demonstrated in figure

Method

The apparatus is set up as shown

The Temperature module of the Stirling engine was plugged into the CASSY sensor interface, ensuring that the probes were in thermal contact with the measuring points on the displacement chamber. The software is configured to take readings of the two temperatures at 15 second intervals after the engine was turned on, and did so until the temperatures stabilized. The engine must be turned off after recording has finished to prevent overheating. A graph was then plotted of temperature T against time t. Using averages of the flat regions of the graph, TL and THwere calculated.These two temperatureswere entered into equation (2) to give the maximum value that the efficiency can be equal to (due to the ≤ sign).

The temperature module was then removed and replaced by the Volume and Pressure modules. The pressure sensing module works by comparing the pressure inside the chamber with atmospheric pressure. As such the software requires a reading from a nearby barometer to use as an offset, which can then be used to determine the pressure changes inside the engine.

This time, after the Stirling engine had been running for 10 minutes, the software was used to record pressure and volume changes over 4 cycles, which takes a total of ~1 second (1 cycle ≈ 250ms). Readings were taken every millisecond to give a high resolution of data from which integrations can be made. The engine is then turned off.

The CASSY lab software’s integration tool can be used to calculate the area underneath parts of a single cycle, values of QA and W were taken using this method, and a more accurate value of the efficiency determined using equation (1).

If the Pressure and Volume data are plotted against time (separate graphs of P against t and V against t) the results should give sinusoidal waves oscillating about Po (atmospheric pressure) and Vo (the midpoint of the volume inside the system – i.e. When the working piston is at it’s centre point). The Sigmaplot software package can then be used to fit sinusoidal curves to the data and output a general formula for a sin curve with the correct parameters. The general form of a sinusoidal wave is as follows,

(4)

Where yo is the value around which the oscillations occur, a is the amplitude, (2π/b) is the Eachgraph (Volume and Pressure) will have it’s own fitted sine wave, and will share the same value of b as the two waves share a constant phase difference, therefore the angular frequency, ω, is the same.

Customizing equation (4) for each curve,

(5)

(6)

The parameters a and bare the amplitudes for each curve. Φis the phase difference between the two waves and is only required in one of the equations. Now that the formulae are set up, equation (6) can be rearranged to make ωthe subject of the formula, and can then be substituted into (5) to show the relation between P and V. Equation (7) is also the equation of the PV diagram.

(7)

To calculate the Work from this formula, it must be integrated with respect to the volume, as in equation (3). The result of this complex integration is shown in equation (8). The ± sign is used to determine whether the area determined is that of under the top or bottom of the PV cycle.

(8)

(9)

Results

Using the first method and equation (2),the values of TL and TH were calculated and used to obtain the first value of the efficiency, η. Because the Stirling engine can operate as both a heat pump and a refrigerator depending on the direction of rotation of the flywheel, a value of η is given for each operational mode.

Despite these values of η seeming incredibly low, the values are acceptable and can be attributed to design flaws in the particularStirling engine that was used. The errors were calculated by calculating the fluctuations from the average value over the flat region of the line from which the values of TL and TH were taken. These errors are very small, indicating that the results were very precise.

The next two methods will be conducted with the Stirling engine operating in heat pump mode, ie the flywheel will be rotating clockwise. Not only is this mode slightly more efficient, but to take more readings of the engine in refridgeration mode, the engine would have to be run for at least 10 minutes in the opposite direction, in addition to the time needed to reverse the temperatures at each end.

The values received byusing the CASSY lab software integration tool were used next, it was not possible to propagate an error for this value as the software had no way of returning an error value, the accuracy to which the sensor interface records data could have been considered, but this would give a negligable value due to the high precision of the sensors.

Four cycles were measured in the 1 second recording period. One of these cycles was used in the CASSY lab software to calculate the efficiency.

Figure 7 shows the Stirling cycle after averages of each phase point were taken. The graph of one cycle contained more noise on the points in the x-axis, indicating that the volume readings were less precise. This is due to the nature of the volume sensor relying on a small potentiometer, consisting of a carbon track over which a wire oscillates. The noise is caused as the wire ‘scratches’ over the carbon track, jumping slightly.

Area under top curve=11335J

Area under bottom curve=10968J

η=3.2%

As expected, this value of η is lower than the first method’s result, which gave the maximum value that the efficiency could have been. The Pressure and Volume waves then had sinusoidal waves fit using Sigmaplot and the parameters returned were entered into formula (9). The value of the phase change between the pressure and volume theoretical sine waves, Φ was calculated by finding the difference between the phase differences of each wave. From this theoretical data, the efficiency was evaluated to be η = (1.83 ± 0.89)% The error was determined by using error propagation theories with equation (9) The result of this error propagation is shown below.

(10)

Conclusion

All three methods of calculating the efficiency of the Stirling engine returned different results. Method one gave a maximum value for the efficiency, the other two methods agree with this as they are both lower values.

Method 1 η= (3.52±.0.01)%

Method 2 η= 3.2%
Method 3 η= (1.83 ± 0.89)%

Method 3 is the most accurate value. It uses methods to eliminate noise from the readings by creating a theoretical Stirling engine cycle. However method 3 also has a large error calculated with error propagation. The individual errors for each parameter used in equation (9) each contributed to this error, the largest of which was ΔP0. This means Sigmaplot found the greatest inaccuracy in the fitting of the sine wave to be establishing the y0 value in equation (4). To decrease the error in the atmospheric pressure, more cycles could have been recorded, from which the sinusoidal fit would then be taken.

More repeat sets of data could have been obtained, should more time have been available. It was observed that there was a large difference in the rotational speeds of different Stirling engines in the laboratory; this has been attributed to the volume sensors, which begin causing a frictional effect on the oscillations with age. Overcoming this problem by redesigning the sensor could yield more efficient results for all three methods. A more effective improvement on the method would be to attempt the same experiment with different sets of similar equipment, to detect any possible anomalies in any of the several devices used.

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