The and current outputs of the power supply

The Effect of the Rate of Heat Flow into the Stirling Engine on its

Student Number: 179061361

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Department of Physics,
University of Bath, UK, BA2 7AY

Abstract. The efficiency of the
Stirling Engine in converting heat energy into work is investigated by changing
the rate of heat flow into the machine. The power input into the engine is
varied by adjusting the voltage and current outputs of the power supply
connected to the heater. These values are used to determine the heat energy
input into the engine in one cycle. The area encapsulated by the plot of
the pressure and volume of the gas is used to determine the work done in one
cycle. At a rate of heat flow of (189±2) W, the engine shows an
efficiency of (8.7±0.1) %, whereas at a rate of heat flow of (97±1) W, the
efficiency is (7.6±0.1) %. The results therefore suggest that with an increased
rate of heat flow, there is an increase in the efficiency of the engine.



The Stirling engine has the
capacity to produce a zero or low level of pollution 1. This makes it a
viable candidate for the eco-friendly production of kinetic energy from all
sources of heat energy, and so there is an interest in increasing the
efficiency of such an engine so as to make it a feasible alternative.

The experiment tests one
method into increasing the efficiency, by increasing the rate of heat flow into
the engine. The ideal Stirling cycle can only be achieved with infinite rates
of heat transfer 2, so in theory, running the engine with a higher rate of
heat flow should produce a cycle closer to that of ideality, and so should
increase efficiency of the engine.

Prior work into increasing
the efficiency of the Stirling engine includes the changing of geometric
parameters of heat exchangers 3. This looked into the various ways of getting
heat energy into the engine. There was also the heat transfer study of the
cavity receiver for the optimization of solar energy with minimum heat loss for
the Stirling engine 4.



The Stirling Engine used
(Figure 1) consists of a heater, a regenerator, two pistons, and a flywheel,
all encapsulated in two glass containers, with cool water running in-between
the two. There is also a pressure sensor connected to the main vessel, and a
piston displacement sensor attached to the lower piston. The upper piston has
the regenerator built into it, and both of the pistons are connected to the
flywheel by rods of differing lengths. The heater is connected to a power
supply that provides constant voltage and current. The pressure sensor records
the relative pressure of the gas inside the vessel, and the piston displacement
sensor records the displacement of the piston from a set position.

Figure 1: Apparatus used in the experiment

The engine is started by
firstly setting the voltage and current to be supplied to the heater to the
desired values on the power supply. After waiting five seconds, the flywheel is
then pulled so as to start the engine.  


Results and Specific Discussion

The piston displacement
sensor and the pressure sensor were used in conjunction with a computer
program, in this case CASSY LAB 2, to plot the relative pressure of the gas
against the volume of the container, the values obtained by the program had to
be modified so as to represent the real pressure and volume of the gas at each


The area encapsulated by the plot was measured using the
‘calculate integral’ function in CASSY LAB 2. This is then used to calculate
the work done by the engine, using the equation,




where W is the work
done in joules, and A is the area
obtained from the graph. It is evident that the engine with the 189W heat flow
has a larger area, and so did more work per cycle than the 97W heat flow, with
specific values being (3.2558±0.008) J and (2.3048±0.0009) J respectively. The
uncertainty came from taking multiple measurements of the area, finding the
mean, and then finding the standard error of said mean. This standard error is
then used as the uncertainty.

            The time interval between each
measurement was set to be 2 ms in the software, and so the volume of the
container is plotted as a function of time, as shown in figure 3.


These plots are used to
determine the period, T, of one cycle
of the engine, by finding time difference between two identical points on the
sinusoidal waves. The electrical energy input into the machine in one cycle can
then be calculated using,




Where QA is the electrical energy used in the heater, I is the current supplied to the heater,
V is the voltage over the heater, and
T is the period of one cycle. For the
97W heat flow, this came out to be (30.2±0.4) J; for the 189W heat flow, QA came out to be (37.4±0.4)
J. The uncertainties come from statistical methods of propagating errors using
partial differentiation, using the resolution error of I and V, and as an
estimate for the degree of accuracy in reading values from the graph on the plots
in figure 3 for T.

The efficiency of the engine
in converting the heat energy supplied to it into work is calculated by,




Where ? is the efficiency of the engine, and W and QA are
as previously stated. The equation gives values of (0.076±0.001) and
(0.087±0.001) for the 97W heat flow and 189W heat flow, respectively. These
values can then be converted into percentages so as to better represent what
they mean;    (7.6±0.1) % and (8.7±0.1) %.
The uncertainties come from the same statistical methods as mentioned

The values obtained from
equations (1, 2, 3) are presented in table 1 for both engines.

1: The efficiency, ?, heat energy input, QA, and the work done by the
engine, W,
for one cycle of the Stirling Engine when powered by a heater of power, P.


P / W

QA / J

W / J

? / %















The results show that the
189W heat flow engine was more efficient at transferring electrical energy into
work per cycle than the 97W heat flow engine. This implies that with a higher
rate of heat flow, there is a higher efficiency of transfer of energy per
cycle, however there is not enough experimental evidence to state this


General Discussion

The final results appear to agree with the theory mentioned previously,
albeit with an effect less than expected.

There are many limitations to the experiment. For example, the filament
can only get so hot before burning out, and so the maximum rate of heat flow is
limited by the apparatus. The system is also not frictionless, and so work must
be done against this whilst the pistons are moving, taking away from the measured
work done. There is also thermal loss throughout the whole process, with heat
energy being transferred to the surroundings.

Further testing could be done with the same apparatus at different rates
of heat flow into the engine to further consolidate the effect on efficiency,
as well as performing in an evacuated environment to lesser the loss of heat
energy to the surroundings.



To conclude, the experimental results show a link between efficiency and
rate of heat flow into the engine, indicating a higher rate of heat flow leads
to a larger efficiency. There is no overlap with the uncertainties between the
two final values, so it can be said that they are reliable.


1          M.J. Collie, 1979, Stirling Engine Design and Feasibility
for Automotive Use, New Jersey,        Noyes
Data Corporation, p6.

2          G. Walker, 1980, Stirling Engines, Oxford, Oxford
University Press, p203.

3          G. Xiao, U. Sultan, M. Ni, H. Peng, X. Zhou, S. Wang, Z.
Lou, Design optimization with computational fluid dynamic analysis of ?-type
Stirling engine, 2017, Applied Thermal Engineering, Vol. 113, p87.

4          T. Hussain, M.D. Islam, I. Kubo, T. Watanabe, Study of heat
transfer through a cavity receiver for a solar powered advanced Stirling engine
generator, 2016, Applied Thermal Engineering, Vol. 104, p751.