UMEÅ UNIVERSITY Department of Physics

Leif Hassmyr

Updated versions 2019-11-19: Vladimir Miranda

Andreas Nordenström

Hot Air Engine, Type Stirling

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Hot Air Engine, type Stirling

The object with this experiment is to make you familiar with cyclic processes, pV-diagrams,

efficiency, refrigerators, heat pumps, hot air engines, etc. The following tasks have to be

completed during the laboratory and included in the written report.

pV-diagram and piston movements

Study the pistons’ movement and correlate them to the different points in the theoretical pV-

diagram.

Draw at page 11-13 (Appendix 1 - Appendix 3) the positions and movement of the pistons in

Figure 9, when the gas is at the points A, B, C, D as Figure 8. Also, show with arrows in both

Figures 8 and 9 how heat is transported away and delivered to the system in the three cases:

See example at page 13 (Appendix 3).

1. Refrigerator.

2. Heat pump.

3. Hot air engine.

Experiment I

Study the Stirling Engine as a refrigerator and heat pump when the engine is driven by an

electric motor (See experiment I, page 6-8). Depending on the direction of rotation of the

flywheel, the Stirling Engine will either emit heat (i.e. work as a heat pump), or absorb heat

(i.e. work as a refrigerator). Answer the questions about the cooling and heating processes:

a) Record the temperature of 1.5 cm3 water in a test tube.

b) How much water freezes instantaneously after super cooling? (Tip: Use the temperature-

jump during this process)

c) Explain the difference when you compare the slopes of the curve just before the freezing

with the slope just after the freezing.

d) Why is the time it takes for water to freeze different from the time it takes to melt ice?

Experiment II

Study the Stirling Engine as a Hot Air Engine and record a pV-diagram.

a) Use A3 paper with mm2 squares to measure the pV diagram. Remember to calibrate the

units of the recorded pV diagram, i.e. determine what 1 mm2 on the paper corresponds to in

terms of pressure and volume.

b) Determine both the thermodynamic efficiency and the temperature TH from the

pV-diagram.

Experiment III

Study the Stirling Engine’s useful efficiency by applying different loads.

a) Make a diagram of the useful efficiency as a function of the number of revolutions/sec.

b) Draw a power distribution scheme (see page 10).

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1. Description of the Processes

The idealized Stirling process is described in the pV-diagram in Figure 1 below.

Figure 1 pV-diagram of the idealized Stirling cycle. In this diagram, Vb > Va and Th > Tc.

The ideal Stirling cycle consists of two isothermal and two isochoric processes. For an ideal

Stirling process where the working fluid is an ideal gas (𝑝𝑉 = 𝑛𝑟𝑇) we can describe the

changes to the system (which in this case is the ideal gas), under the assumptions that we have

quasistatic, fricionless conditions. We can also determine the thermodynamic efficiency.

The work on the gas as it changes from state A to B is: 𝑊𝐴−𝐵 = −∫ 𝑝𝑑𝑉𝐵

𝐴

In an ideal gas, where the internal energy 𝑈 = 𝑓(𝑇), we know according to the first law of

thermodynamics that ∆𝑈 = 𝑄 +𝑊 = 0 in an isothermal process.

The first law of thermodynamics then becomes 𝑄𝐴−𝐵 = −𝑊𝐴−𝐵. Throughout this lab we use

the sign convention that Q and W are positive when energy is going into the gas.

According to Figure 1:

1 2 : The gas is isothermally compressed at the temperature 𝑇𝑐 and the work

𝑊1−2 = −𝑛𝑅𝑇𝑐𝑙𝑛𝑉𝑎

𝑉𝑏= 𝑛𝑅𝑇𝑐𝑙𝑛

𝑉𝑏

𝑉𝑎> 0 is done on the gas.

The heat 𝑄1−2 = −𝑊1−2 < 0 is leaving the gas.

2 3 : The gas is heated at constant volume (isochoric) to the temperature 𝑇ℎ by supplying

the heat 𝑄2−3 = 𝑄𝑅 > 0. No work is done

3 4 : The gas expands isothermally at the temperature 𝑇ℎ and does work on the

environment, which mean work done on the gas is negative.

𝑊3−4 = −𝑛𝑅𝑇ℎ𝑙𝑛𝑉𝑏

𝑉𝑎< 0 .

The heat 𝑄3−4 = −𝑊3−4 > 0 is absorbed in the gas during this process.

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4 1 : The gas is cooled at constant volume to the temperature 𝑇𝑐. No work is done and the

heat is 𝑄4−1 = −𝑄𝑅 < 0, which means heat leaves the gas.

To calculate the thermodynamic efficiency of the Stirling cycle we define the thermal energy

added to the gas 𝑄ℎ and the waste heat 𝑄𝑐 as

𝑄ℎ = 𝑄3−4 = 𝑛𝑅𝑇ℎ𝑙𝑛𝑉𝑏

𝑉𝑎,

and

𝑄𝑐 = −𝑄1−2 = 𝑛𝑅𝑇𝑐𝑙𝑛𝑉𝑏

𝑉𝑎.

For processes 2-3 and 4-1 we note that 𝑉 is constant and since no work is done (dV = 0 W

= 0) we have 𝑑𝑈 = 𝑄. Furthermore we know that

(𝜕𝑈

𝜕𝑇)𝑉= 𝐶𝑉.

The total change in internal energy for these isochoric processes will then be 𝑑𝑈 = 𝐶𝑉𝑑𝑇.

More specifically

𝑄4−1 = 𝐶𝑉(𝑇𝑐 − 𝑇ℎ) and 𝑄2−3 = 𝐶𝑉(𝑇ℎ − 𝑇𝑐) respectively.

That is,

𝑄4−1 = −𝑄2−3 which gives 𝑄4−1+𝑄2−3= 0 (assuming no heat is lost).

The thermodynamic efficiency of Stirling cycle can be written as the produced energy, 𝑊𝑒

(work) divided by the input energy 𝑄ℎ:

𝜂𝑡−𝑑 =𝑊𝑒

𝑄ℎ=𝑄ℎ − 𝑄𝑐𝑄ℎ

= 1 −𝑄𝑐𝑄ℎ

= 1 −𝑛𝑅𝑇𝑐𝑙𝑛 𝑉𝑏 𝑉𝑎⁄

𝑛𝑅𝑇ℎ𝑙𝑛 𝑉𝑏 𝑉𝑎⁄

or

𝜂𝑡−𝑑 = 1 −𝑇𝑐

𝑇ℎ.

Note that the total work, 𝑊𝑒, corresponds to the enclosed area of the pV diagram.

We see that the efficiency for the cycle is the same as for the Carnot cycle. This is possible if

we can keep the heat 𝑄4−1 stored in the machine. This heat should later be returned to the gas

as the heat 𝑄2−3. In practice this is achieved with the so called regenerator.

If the heat 𝑄4−1 is transported away with the cooling water one would have to supply the heat

𝑄2−3 externally and then the efficiency would not be the same as for the 'Carnot-machine'.

The efficiency can also be improved by lowering the low temperature 𝑇𝑐 or by raising the

high temperature 𝑇ℎ, but it is impossible to reach an efficiency equal to 1.

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Note that this is not caused by mechanical shortcomings as friction. This can be derived for

all reversible processes, in which there are no losses.

2. Description of the Machine

A sketch of the machine is found in Figure 2. The main parts are a precision cut glass

cylinder (1) with two movable pistons (2) and (3) attached to a flywheel (4). In the upper part

of the cylinder there is a heating arrangement (heated tungsten spiral (5)) and the lower part is

surrounded by a plastic cooling jacket (6) with inlet and outlet for the cooling water (7). The

displacing piston (2) transports the gas from the warm to the cold part of the cylinder (and the

other way around).

The working piston (3), which moves with a 90 degrees phase difference relative to the

displacing piston, compresses the gas and thus controls the volume. The working piston

isolates the gas from the surroundings and work is taken away or delivered via this piston.

Figure 2 Schematics of different parts of the Stirling engine used in the lab

The displacing piston is made of a heat resisting glass and the bottom of it is sealed with a

water-cooled metal plate with radial slots that allows air to pass during heat exchange. This

piston has been given a special shape with an axial cavity filled with copper wool as the

regenerator (8). The purpose of the copper wool is to absorb heat when the gas passes to the

colder part of the cylinder and to emit heat when the gas passes in the opposite direction. In

this way heat is conserved and the efficiency increases.

The pistons are connected with piston rods to a heavy flywheel (4) to give the machine a

smooth running. At the rod of the working piston (9) there is an outlet (10) for measuring the

pressure in the cylinder via a channel in the piston rod. The outlet is connected to a pV-

indicator for producing a pV-diagram of the process.

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The flywheel has a key groove for connections to other machines (for example an electric

motor). A handle can temporarily be attached to the flywheel. Then one can turn the flywheel

around manually and make a detailed study of the process.

The pV-indicator (Figure 3) consists of a mirror assemblage (11) which is possible to rotate

in both horizontal and vertical directions. The volume variations of the working gas are

transferred via a string (12) to the horizontal movement of the mirror holder (13). The

pressure variations are transferred via a thin PVC-tube (14) to the vertical movement. By

lighting the mirror in an appropriate way one can observe simultaneous variations in pressure

and volume of the gas, (i.e. we have a pV-diagram for the process).

Figure 3 A schematic of the pV-indicator

3. Experimental preparations - pV diagram and piston

movements

(Has to be completed before the experimental procedure)

Turn the flywheel manually and check that the movable piston moves freely.

Study the pistons’ movement and correlate them to the different points in the theoretical pV-

diagram.

Draw at page 11-13 (Appendix 1-Appendix 3) the positions and movement of the pistons in

Figure 9 when the gas is at the points A, B, C, D in Figure 8. Also, show with arrows in both

Figures 8-9 how heat is transported away and delivered to the system in the three cases: See

example at page 13 (Appendix 3). Think about where the hot and cold reservoirs are in each

case and how it would affect the pV-diagram.

1. Refrigerator.

2. Heat pump.

3. Hot air engine.

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4. The Experimental Procedure

For practical reasons it is best to do the experiments in the following order.

I. Refrigerator and Heat pump.

II. Hot air engine – Thermodynamic efficiency.

III. Hot air engine - Useful efficiency

General instructions

Turn on the cooling water and check that it flows.

Lubricate the machine if neccessary according to the supervisors instructions. Note: only

silicon oil! Always check that the machine runs without any part touching other parts by turning the

flywheel manually.

If the cooling water is lost: Turn off the filament current within three seconds.

Put the protection cover over the heated filament when it is removed.

Always position the displacement piston at the lowest configuration when the machine is

turned off.

Never leave the machine running unattended!!!!

4.1. Experiment I: Refrigerator and heat pump

In these two experiments one uses the arrangement shown in Figure 4 and Figure 5.

Figure 4 Arrangement of the Stirling engine, used as a refrigerator and heat pump. Note the external motor connected to the

flywheel.

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Figure 5 An image of the arrangement in experiment I and II. It is important to use the Plexiglass protection.

The flange with the heating filament is changed to a flange to which you can attach a test-

tube. The machine is driven by an electric motor and the flywheel can rotate both clockwise

and counter clockwise. In this experiment you should analyse and explain what's happening

with help of the Stirling cycle's pV-diagram. Further, you should demonstrate and investigate

the machine's use as a refrigerator and heat pump.

4.1.1. Cooling of Water (Refrigerator)

Fill the test tube with 1.5 cm3 of water. Measure the temperature in the test-tube with a

thermo-couple (type K, 40µV/K). (Important: The thermocouple should not touch the

glass!).

Note: The Plexiglas protection must be mounted.

Turn on the machine as a refrigerator.

Study how the temperature depends on time with the help of a t/y-printer (set to measure

temperatures from -25 ºC to +100 ºC and the timescale: 0.5 mm/sec). Wait until the

temperature reaches -20 ºC.

Plexiglass

protection

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4.1.2. Heating of Water (Heat pump)

When the temperature in the test tube is around -20 °C, change the direction of revolution for

the flywheel. Heat to about +40 ºC.

Observe and compare the cooling and heating processes.

4.2. Experiment II: The hot Air Engine – Thermodynamic efficiency

Note: Turn the flywheel so that the displacing piston ends up in its lowest position when the

motor has stopped. Otherwise there is a risk for overheating and thereby cracking.

For a demonstration how thermal energy is converted to mechanical energy the machine is set

up as in the Figure 6. (Note: 12.5 A is enough)

Figure 6 A schematic of the Stirling engine setup during the brake test.

Preparing the setup

Mount the flange with the heating filament so that the heating filament never touches

the displacing piston.

Check that the cooling water flows.

Connect the pV-indicator

Check that the displacing piston is in its lowest position. Important!

Start the heat engine

Connect the heating filament (1Ω) to the source and increase the voltage.

A suitable filament current is 12 A. Be careful!! The filament gets really hot and

starting to glow red.

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Start the machine by manually turning the flywheel in clockwise direction. (The

filament will cool down)

Observations

Record the pV-diagram by the help of the pV-indicator

One problem is to determine the scale of the pressure. It must be determined by static

measurements. The instructor gives the necessary instructions. The volume scale can

be determined as one knows that Vmin= 130 cm3 and Vmax = 270 cm3.

When the motor does not do any work, the thermodynamical efficiency (𝜂𝑡−𝑑) can be

determined from calculations in the pV-diagram.

Determine Th if Tc = 20 ºC.

4.3. Experiment III: The hot Air Engine – Useful efficiency

The brake test is done by putting a friction band of copper over the wheel attached to the

outgoing axis. In the end a suitable weight is added and the frictional force is measured with

help of a dynamometer.

The number of revolutions per second is measured with the help of a stroboscope.

The power Pout is determined from the relation Pout , where is the angular velocity and

is the torque.

Useful efficiency:

outout

in

in

P

PP U I ,

Follow the instructions on how to turn on the heat engine. (Use the same Pin)

Let the engine run and stabilize

Determine for at least 5 different loads, and plot out as a function of the number of

revolutions/sec.

Be careful! Be careful when adding loads. If too much loads are added, the gas will get

extremely hot and the glass cylinder can break.

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How to draw a power distribution scheme

Draw a power distribution scheme like the one in Figure 7 below, for the maximum Pout –

value when you apply loads on the Stirling engine. Give the powers in Watt and draw the

width of the arrows in proportion to the power.

Figure 7 Power distribution scheme of the Stirling engine with important energy losses included.

Heat losses to the surroundings = Pin – Qh n

Losses in the Stirling cycle = Qc n

Frictional losses = We n - Pout

(n = number of revolutions/sec) Pout

Pin

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Appendix 1

Refrigerator

Figure 8 Theoretical pV-diagram for the Stirling cycle

Figure 9 Glass cylinders for studying the piston movements

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Appendix 2

Heat Pump

Figure 8 Theoretical pV-diagram for the Stirling cycle

Figure 9 Glass cylinders for studying the piston movements

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Appendix 3

Hot Air Engine

Figure 8 Theoretical pV-diagram for the Stirling cycle

Figure 9 Glass cylinders for studying the piston movements

The working piston

is not moving.

The displacing piston

moves up.

The heat QR is given

off by the gas

to the regenerator in

the displacement piston.

QR

D

Working

piston

Displacement

piston