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
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Richard Plews
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 Stirling engine operates as a heat pump, using energy from a warm reservoir, converting
some of this energy into a different form to perform work.
Warm Reservoir
QA
Heat
Pump
External Work, W
QB
Cold Reservoir
Figure 1. the Heat Pump
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

W
QA
(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,
  1
(2)
TL
TH
Or alternatively, the efficiency can be calculated from a Pressure against Volume graph
showing the Stirling engine cycle
Pressure
A
B
Volume
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Figure 2. Stirling PV diagram
Richard Plews
The area of a Pressure against Volume graph is equal to the Work done; this is true because of
the formula
(3)
WD  P.dV

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
Temperature sensor B
Temperature sensor A
Flywheel
Volume sensor
Displacement Piston
Pressure sensor
Working Piston
Figure 3. 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
90º phase difference between cranks
Flywheel
Displacement Piston
Different temperatures
Working Piston
Figure 4. Stirling engine operation
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Richard Plews
Method
The apparatus is set up as shown
To computer
CASSY sensor interface
Stirling engine
To Temperature module
Figure 5. Apparatus
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 TH were calculated. These two temperatures were 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,
 2

y  yo  a sin 
x  c
 b

Where yo is the value around which the oscillations occur, a is the amplitude, (2π/b) is the
Each graph (Volume and Pressure) will have it’s own fitted sine wave, and will share the
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(4)
Richard Plews
same value of b as the two waves share a constant phase difference, therefore the angular
frequency, ω, is the same.
P/V
Pressure
t
Volume
Figure 6. graph of P and V against time t
Customizing equation (4) for each curve,
P  P0  a sin( t   )
(5)
V  V0  b sin( t )
(6)
The parameters a and b are the amplitudes for each curve. Φ is 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 ω the 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.
(V  V0 ) 2
 V  Vo 
P  P o  a cos  
  a sin  1 
b2
 b 
(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)
1
WD  2 P0 b  ab sin 
2
W
2a sin 


Q A 4 P0  a sin 
(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.
Anticlockwise (refrigerator)
Clockwise (Heat pump)
TL /K
289.42
294.93
5
TH /K
299.82
305.70
Max value of η
(3.47±.0.02)%
(3.52±.0.01)%
Richard Plews
Despite these values of η seeming incredibly low, the values are acceptable and can be
attributed to design flaws in the particular Stirling 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 by using 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.
Pressure (Pa)
Volume (cm3)
Figure 7. Graph of PV results for average of four cycles
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
Area under bottom curve
η
=11335J
=10968J
=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
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Richard Plews
(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.
  
2
 a  2    2 

2a 2 sin 2 
  P0 2
 1 
   
2 
a
tan

  4 P0  a sin   
  
(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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