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

Efficiency

Student Number: 179061361

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.

Introduction

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.

Experimental

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

reading.

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,

(1)

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,

(2)

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,

(3)

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

previously.

The values obtained from

equations (1, 2, 3) are presented in table 1 for both engines.

Table

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

? / %

97±1

30.2±0.4

2.3048±0.0009

7.6±0.1

189±2

37.4±0.4

3.2558±0.008

8.7±0.1

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

outright.

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.

Conclusion

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.

References

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.