Finalreport

1.0 Introduction

The Mechanical Engineering department of Dalhousie University has contracted the

development and construction of a solar powered Stirling engine. The design team

selected for this endeavor consists of Paula Cook, Dale DeMings, Susan Foster, Jonathan

Fraser, and Charles Harrison. The design team is supervised by Dr. Murat Koksal.

The Stirling engine is to be used in thermodynamics and energy conversion classroom

demonstrations. For this reason, the engine is designed to best demonstrate the principles

of these courses. Another design parameter was that the final product is be powered

solely by solar energy.

2.0 Requirements

The final project was to consist of a constructed engine to be easily transported for

classroom demonstrations. The engine was to be simple and safe to use. The engine was

to be able to operate using only the energy supplied from the solar collector. Extra

thermal input may be utilized for demonstration purposes in place of, or in addition to,

solar energy. The operation of the engine was to be visible through transparent

components. Various sensors were to be included to enhance the effectiveness of

classroom demonstration. The engine was designed to heat quickly for a fast startup time.

3.0 Theory

Stirling engines are very different from the common internal combustion engines found

in most present day vehicles. Stirling engines do not require the use of fossil fuels and

therefore can be used without producing harmful waste products. They can use solar

energy or waste energy from other sources to produce power. This capability makes the

Stirling engine a very environmentally friendly power source.

The Stirling engine creates work as a result of temperature and pressure differentials. To

understand the project, it is important to first understand the Stirling cycle.

The Stirling cycle is a heat addition and heat dissipation process just like the well-known

Carnot cycle. Heat addition comes from the high temperature reservoir, TH, and then

later in the cycle, heat is rejected to the low temperature reservoir, TL. In our Stirling

engine, the high temperature reservoir is provided by the sun’s solar energy. During the

heat addition and rejection stages, the ideal Stirling cycle is a constant temperature

process. During the other two stages of the cycle, a regenerator causes an increase in

temperature while volume remains constant within the system.

Figure 1: P-v and T-s Diagram for the Ideal Stirling Cycle.

Figure 1 shows the P-v and T-s diagrams of an ideal Stirling Cycle with regeneration.

The four steps are summarized as follows:

1-2 T = constant → expansion (heat addition from external source)

2-3 ν = constant → regeneration (internal heat transfer from the working fluid to the

regenerator)

3-4 T = constant → compression (heat rejection to external sink)

4-1 ν = constant → regeneration (internal heat transfer from regenerator back to the

working fluid)

Because it is impossible to attain an ideal cycle, the P-v and T-s diagrams will most likely

have more rounded edges and therefore the four stages will mesh into one another. That

is, during the first stage (expansion), T will not exactly be constant, but it will remain

increasing through the first part of that stage.

The cycle we predict for our Stirling engine is the four step process shown in Figure 2.

For simplicity, the regeneration is left out of this diagram.

Figure 2: The Sirling Cycle Stages.

4.0 Design Selection

The following section describes the designs that were considered for our Stirling engine

and solar collector. Pros and cons of these ideas are discussed and followed by a

weighted chart that aided our final design selection. From this, our final design of the

displacer regenerator engine using a parabolic solar collector was chosen.

4.1 Displacer Piston

A half-disk displacer is contained in a shallow cylinder filled with gas. As the gas is

heated it expands and is forced into the piston. The movement of the piston pushes the

displacer disk to the hot side, allowing the remaining air to cool and contract. This

contraction will pull the piston, and force the displacer from the hot side.

Figure 3: Displacer Piston.

4.2 Dynamic Heat Sleeve

A heated metal sleeve is mounted concentrically to the piston. This sleeve is raised up to

surround the cylinder to heat and expand gas inside. When the gas inside is expanded, the

piston raises and causes the heat sleeve to lower. This allows the hot gas inside the

cylinder to cool, bringing the piston down and raising the sleeve. This design would

likely use two pistons.

Figure 4: Dynamic Heat Sleeve.

4.3 Rotary Chamber

A shaft is eccentrically mounted in a cylinder with four perpendicular telescopic arms.

Each arm creates a seal with the sides of the cylinder, isolating four distinct chambers. As

each of the four chambers reaches smallest volume, it is exposed to an outside heat

source, which causes the gas to expand and forces the compartment to a larger volume

and into the next stage of the cycle. As each chamber expands, it causes the shaft to

rotate, and aids in the contraction of the other three chambers. As a chamber rotates away

from the heat source, it is cooled by the ambient air and contracts, aiding in the shaft

rotation.

Figure 5: Rotary Chamber Design.

4.4 Large Piston

Rather than using a coupled displacer-piston device, a large piston is used to act as its

own displacer. The air and piston are heated at the bottom, causing the air to expand and

driving the piston to the cooled area. The piston is cooled, cooling the air below it and

causing contraction. This pulls the piston back to the heated area to begin the cycle again.

A hollow piston could be used to increase the speed of temperature change.

Figure 6: Large Piston.

4.5 Regenerator in Piston

A displacer piston and a power piston are connected by a drive shaft. The displacer piston

is insulated and loosely-fitted in its chamber. The displacer isolates the gas, causing it to

be alternately heated and cooled. A conduit connects the displacer and power pistons, and

as hot gas is transferred to the power piston the movement is converted to power.

Figure 7: Regenerator in Piston.

4.6 Bellows

Two flexible-walled chambers are connected by a conduit, and their movement is

constrained by a drive shaft and cams. Both chambers start at the top. As the air in one

chamber is heated, expansion occurs and the bottom of the chamber is driven downwards,

rotating the shaft. Because of the CAM, the second chamber remains at the top. As

rotation continues, the cam on the heated chamber reaches maximum height, the pistons

then move the gas from the hot side to the cold side maintaining a constant volume of

gas. The air in this chamber is cooled and contracted, and as its cam reaches maximum

height, the air is transferred back to the first chamber, where it is heated again. This

design could incorporate a regenerator in the transfer conduit to improve efficiency.

Figure 8: Bellows.

4.7 Weighted Chart for Engine Selection

Table 1: Engine Selection.

Eas

e of

Con

stru

ctio

n

Eas

e of

Dem

onst

ratio

n

Dur

abili

ty

Eff

icie

ncy

Cos

t

Port

abili

ty

Wei

ght

Wor

kabi

lity

Sim

plic

ity

Tot

al

Weighting 8 9 7 6 5 8 6 10 9

1 Rotary Displacer 6 4 5 7 8 8 9 6 7 442

2 Dynamic Heat

Sleeve 7 6 7 4 6 7 6 4 4

341

3 Rotary Chamber 3 5 6 5 8 8 9 5 6 353

4 Large Piston 10 8 7 4 7 8 7 2 8 438

5 Regenerator in

Piston 8 10 8 8 8 8 7 9 7

467

6 Bellows 7 8 3 5 7 6 8 5 8 382

4.8 Selected Design

We chose a displacer design which incorporates the use of a regenerator that will improve

the overall engine efficiency. This is a unique design as displacer engines do not

normally incorporate regenerators. The displacer design uses one cylinder to expose the

contained gas to either a hot or a cold source and a second cylinder to convert the hot gas

expansion to power. The cylinders are connected by a conduit to allow the gas to be

transferred. Some of the components were to be constructed from transparent materials

to facilitate the demonstration of thermal principles acting on the mechanical

components. Refer to Figure 9 for a conceptual view of our selected design.

Figure 9: Selected Design

Dominant factors that were considered when selecting the design were:

- Simple – good demonstration tool

- Uses a regenerator – better efficiency

- Ease of construction

- Closed system allows use of gases other than air, i.e. helium

- Durable

- Parabolic solar collector – reaches high temperatures quickly, easily positioned

and inexpensive to manufacture

5.0 Parts

The main components of our engine are: a solar collector, two pistons, a regenerator, a

flywheel and a drive shaft. These components will be discussed later on in this report.

5.1 Solar Collector

A parabolic solar collector was purchased to concentrate the solar rays. The concentrated

thermal energy could then be transferred to heat the air inside the displacer chamber.

5.1.1 Parabolic Collector Theory

The parabolic shape of the collector reflects and concentrates the parallel solar rays to a

focal point. The focus is given by

p = x2

4y

0

2

4

6

8

1 0

-1 0 -8 -6 -4 -2 0 2 4 6 8 1

x

y

y = 0 .1 x 2

0

F o c a l P o in t

Figure 10: Focal Point of a Parabola

The parabola above (Figure 10) has the equation y = 0.1x2, and has a focal point at p =

x2/0.4x2 = 2.75, as shown on the figure. On a solar collector, the focus represents the

point to which all parallel solar rays will be reflected.

The collector was purchased from Edmund Scientifics, and has the following

specifications (Table 2).

Table 2: Solar Collector Specifications

Material Aluminum

Thickness 0.04 inch

Aperture (top opening) 24 inch diameter

Depth 6 inch

Centre Hole 1.5 inch diameter

The geometry of the collector is further described by

ρ = 2f

(1 + cosθ)

where ρ = distance from focal point to mirror surface

f = focal length (= 6”)

θ = angle between optical axis and ρ See Figure 11

Figure 11: Solar Collector Geometry

0

1

2

3

4

5

6

7

- 1 2 - 8 - 4 0 4 8

I n c h e s

I n c h e s

ρ θ

1 2

Taking the focal length as 6”, as specified by the manufacturer, the equation yields a ρ of

12” at the rim of the collector (θ = 90º), as anticipated from the specified 24” diameter.

5.1.2 Theory of Solar Collection

The aperture size of the collector determines the amount of solar energy that can be

collected. Our collector will be tilted so that the top opening is always perpendicular to

the solar rays. This means that the solar incident area is given by the circular area of the

top of the collector, an area of 3.14 ft2, or 0.292 m2. At our latitude, the sun provides

600 W/m2 of energy to the earth. We therefore estimate collecting energy at a rate of

~175 W.

5.1.3 Transmission of Energy to the Engine

To transmit the energy collected by the solar collector to the engine a rod assembly was

constructed (Figure 12). The insulation theory will be discussed later. The basic principle

employed in the rod design was the conduction of heat through a highly conductive

medium (copper). The collector focuses heat energy to a focal point near the top of the

copper rod. This rod is attached to the solar collector, passing through the hole in its base.

The bottom of the rod is threaded into the copper top of the displacer chamber. Heat is

conducted down the rod and into the copper top, which heats the enclosed air by

radiation.

Steel Tube

Copper Block

Bisque Ceramic Tile

Bisque Ceramic Tile

Copper Collecting Rod

Figure 12: Conducting Rod Assembly

5.2 Insulation

Insulation was needed to ensure effective transfer of heat from the focal point of the

collector to the displacer chamber. The insulation had to minimize heat loss at two major

locations: to the air surrounding the collecting rod and to the ambient air above the

displacer top.

Initial testing of the solar collector and collector rod was carried out in January by

attaching a thermocouple to the rod at the focal point. A temperature of 550ºC was

achieved in 40 seconds, at which point the thermocouple burnt off (Figure 13).

Figure 13: Solar Collector Testing - Thermocouple at Focal Point

This experimentation led us to use 500ºC as a probable rod temperature to design around.

Most conventional insulation is not effective to this extreme a temperature, so insulation

selection was difficult. A ceramic wrap insulation was located which was effective to

2300ºF (~900ºC). This product was intended for use inside walls, and is dangerous to

work with (inhalation hazard), so we decided not to use this to insulate the rod.

5.2.1 Air as an Insulator

On further research, we determined that a thin film of air could be an effective means of

insulating the rod. An enclosed air space of 1/8” has an insulation value of 0.0263 W/mK.

By enclosing a thin air space around the rod, the losses to the ambient would be reduced.

5.2.2 Mechanism of Enclosing Air

The air was enclosed around the copper rod by using an insulated steel tube, separated by

a ceramic spacer (Figure 12). The steel is less conductive than the copper rod, and

Aluminum-vinyl pipe insulation provides further insulation value. The insulating air

reduces the overall temperature of the steel tube so that the pipe wrap can be used; the

Aluminum-vinyl insulation is not effective on a 500ºC rod.

The ceramic spacer is used to reduce direct heat conduction from the rod to the steel tube.

A hole was drilled in a small ceramic tile, which was then slid onto the rod. The ceramic

has an insulation value of 0.1 W/ºC, to reduce direct conduction from the hot copper to

the steel.

5.2.3 Reducing Heat Loss from the Displacer

A second larger tile was placed over the copper top of the displacer casing to prevent heat

loss to the ambient air from the exposed top. The goal of the inclusion of all the

insulation materials was to direct as much of the collected heat into the displacer chamber

as possible.

5.2.4 Testing of the Collector and Rod

The first tests of the solar collector were carried out in January, as mentioned above.

Tests were also completed on the rod assembly, and on the rod attached to the displacer

chamber. Four series of tests were performed. A summary of the results appears below

(Table 3). The testing locations are found in Figure 14.

Table 3: Testing Results

Test 1 Test 2 Test 3 Test 4

Collector and

Rod

Collector on

Engine

Collector and

Rod

Collector and

Rod

Day March 31, 2004 March 31, 2004 April 1, 2004 April 1, 2004

Time 3:20 pm 4:30 pm 11:30 am 11:45 am

Weather Intermittent

Clouds

Intermittent

Clouds

Sunny Sunny

Ambient Air

Temperature (ºC)

10 8 12 12

Temperatures (ºC)

(1) Focal Point 200 250 330 360

(2) Ceramic Spacer 76 - 150 170

(3) Top of Insulation 49 - 90 125

(4) Middle of Insulation 44 - 55 80

(5) Bottom of Insulation 40 - 44 68

(6) Nut Below Collector 44 50 - -

(7) Bottom of Rod 85 N/A 150 170

(8) Side of Displacer

(top)

N/A 38 N/A N/A

(9) Side of Displacer

(bottom)

N/A 18 N/A N/A

The testing results demonstrate that the

insulation is doing its job, since the

temperature at the bottom of the rod is

consistently higher than the temperature

along the insulation. The majority of the

heat is being transferred into the displacer

chamber.

The heat values on the outside of the

insulation are higher than desired, however.

For safety, the insulation should be cool

enough to touch, and temperatures in excess

of 100ºC reveal that heat energy is being

lost as it travels down the copper rod.

(4)

(6)

(5)

(8)

(7)

(9)

(2)(3)

(1)

Figure 14: Testing Locations

5.3 Piston Sizing

The power piston casing was designed to be well sealed to prevent air losses and to allow

maximum work to be obtained from the volume change. The power piston should be as

small and light as possible, while still capable of transferring work. The size of the power

piston was determined by the desired power output and the volume of the displacer

casing. The shafts of both the displacer and power piston are lubricated for ease of

sliding.

5.3.1 Calculations

The following calculations were made to estimate the size of the power and displacer

cylinders needed as well as the work output of the engine. Calculations were based on

the ideal Stirling cycle, the ideal gas law, and the following assumptions corresponding to

the ideal Stirling cycle:

rpmNairKkgJR

WsJQ

KCTKCTkPaPP

in

H

L

1)(/287

400/400

47320029320325.10142

=⋅===

=°==°=

==

Ideal efficiency of the cycle can be calculated immediately from the reservoir

temperatures.

%38%1004732931%1001 =×⎟

⎠⎞

⎜⎝⎛ −=×⎟

⎠⎞

⎜⎝⎛ −=

KK

THTLη

Step 4 to 1 is a constant volume process so the following formula can be used to find : 1P

( ) ( ) kPaK

KkPaT

TPT

TPP

L

H 164300

475325.1014

4

141 =

×=

×=

×=

The same thing can be done to find : 3P

( ) ( ) kPaK

KkPaT

TPT

TPP

H

L 63475

300325.1012

2

323 =

×=

×=

×=

The ideal gas law can also be used to find specific volumes, 1ν and 3ν . Based on the

ideal Stirling cycle, we can also assume that 41 νν = and 23 νν = .

( ) ( ) kgmkPa

KKkgkJP

TR /83.0160

475/287.0 3

1

141 =

×⋅=

×==νν

( ) ( ) kgmkPa

KKkgkJP

TR/34.1

64300/287.0 3

3

323 =

×⋅=

×==νν

The qin required per kilogram of gas per cycle can be determined by the following

formula (note: T2=T1 so that term becomes zero):

( ) kgkJkPa

kPaKPP

RTT

CTsTq PHin /65164

325.101ln287.0473lnln1

2

1

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=∆=

A similar calculation can also be made for qout:

( ) kgkJkPa

kPaKPP

RTT

CTsTq PLout /4063

325.101ln287.0300lnln3

4

3

4 =⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=∆=

Since the cycle happens once per second and the only lasts for half of the cycle, it can

be said that only 200 of the 400 J are transferred to the system. The following calculation

determines the mass of air capable of running in this ideal cycle.

inQ

kgkgkJkJ

qQ

min

in 0031.0/65

200.0===

We can now calculate the actual volumes of air at every stage:

( )( )( )( ) LmkgkgmmVV

LmkgkgmmVV

1.40041.00031.0/34.1

6.20025.00031.0/83.033

232

33141

===×==

===×==

ν

ν

Total work generated, Wout, by the cycle may be calculated now. Since 1rpm was

assumed, this value is also our output wattage.

( )( ) kJkgkJkgkJkgqqmW outinout 076.0/40/650031.0)( =−=−=

To check to see if our calculations are correct, we can check our efficiency using heat

transfer.

( )( ) %38%100076.0

0031.0/65%100 =×=××

=kJ

kgkgkJW

mq

out

inη

This efficiency agrees with the efficiency calculated via temperatures. Finally, now that

we have the upper and lower volume limits, we can determine the size of the displacer

cylinder and the power cylinder. Since the power cylinder should not contain any

volume at minimum, V1 and V4 is equal to the displacer cylinder volume, 2.6L. The

difference between V2=V3 and V1=V4 is therefore the power cylinder volume, 1.5L.

From these volumes we can determine ideal sizes of pistons. If we were to assume a

power piston diameter of 10cm and displacer piston width of 10cm, the heights of the

power cylinder and displacer cylinder would then be 20cm and 26cm, respectively. See

Appendix A for the Microsoft Excel spreadsheet of these calculations and the generated

P-v diagram.

Subsequent to making these calculations, we received our working solar collector. We

began testing of the collector to see realistically, how well it would perform as a source

of heat for the hot side of our Stirling Engine. As is discussed already, the solar collector

performed well and led us to change our preliminary assumptions and consequently the

calculated size of our engine. Firstly, we increased our high temperature reservoir

temperature to 300ºC instead of the 200ºC we originally had. However, we felt that our

actual power input from the collector may have been optimistic at 400W so we reduced

this value to 300W based on an assumed 600W/m2 solar output on a sunny day. By

completing the same calculations as above with the new assumptions, we found an

optimal size of 1.13L for the displacer casing, 1.08L for the power cylinder and an actual

work output of 73kJ as compared to our 76kJ found previously. These calculations are

also completed in a Microsoft Excel spreadsheet and attached in Appendix A.

5.3.2 Displacer Casing

With these volumes in mind, we had to decide on actual dimensions of the square

displacer casing as well as the power cylinder. Because we were concerned with

conduction down the metal sides of the displacer casing, we decided that it would be a

good idea to make the sides fairly long compared to the cross section of the casing. This

would mean that the cold end would not be influenced by the extremely hot end as

quickly and therefore maintain a temperature differential and run the engine longer. In

addition to these long sides, we chose 1/8” stainless steel as our material for the three

metal sides for its relatively low conduction rate compared to other metals. The top and

bottom ends of the displacer casing were to be made of highly conductive metal to ensure

that the heat and cold reached the air appropriately. Copper is the ideal metal for these

ends, however a reasonably thick piece was needed to act as a thermal capacitor and such

a piece of copper was found to be scarce. We located enough copper for one end, we

chose that to be the hot end, and used 3/4” thick plate to hold our heat with. On the cold

end, we used the same size piece of aluminum as it was the next best conducting metal

that was readily available. Ultimately, our displacer casing had internal dimensions of

3.25” by 3.25” square and 7.5” high. This came very close to meeting our calculated size

of 1.13L. The constructed displacer casing is seen in figure 15.

Figure 15 - Displacer Casing

5.3.3 Power Piston Casing

The power cylinder was going to be approximately the same size as discussed above;

however, it was to have a cylindrical shape. We were not particularly concerned with

conduction in the power cylinder so we chose steel as our working metal because it was

fairly inexpensive. To allow for the air duct to plug into the top of the power cylinder,

we wanted its height to be not as large as that of the displacer casing. Therefore we

constrained it vertically and found the appropriate diameter. We decided on a piston

throw of 5” and a diameter of 4”. This gave us our desired volume change of

approximately 1.08L and still gave us room to place the cylinder on the engine stand and

connect via a duct to the displacer casing side (near the hot side). The piston itself was

also machined from steel to allow for smooth operation in the steel cylinder, and also to

have a comparable thermal expansion coefficient in the event that this side of the engine

became hot. The sides of the piston were built long to reduce binding, but the inside was

machined out to reduce as much weight as possible and effectively reducing efficiency

loss. Figure 16 shows a picture of our initial power piston and cylinder.

Figure 16 - Power Piston and Cylinder

5.3.4 Testing and Modification

Testing on the current design began at this time and instead of using the solar collector,

we felt it would be more efficient use of time to use a propane torch for ease of

experimentation. It was found that after disconnecting the drive shaft and allowing the

displacer piston to be maneuvered manually, the power piston yielded very little

movement as a result of displacer actuation. After this unsuccessful experimentation, we

concluded that changes needed to be made to our design. Specifically, two main issues

concerning the thermal workings of the engine were found. The first was constrained

flow within the air duct, and second and more importantly, it seemed that the engine

required too large of a volume change in the power piston. Initially, we shortened the

throw of the power piston from 5” to 2” by modifying drive shaft linkages, in effect

reducing the expansion volume by 60%. After doing this, we began testing and yet again

were unsuccessful. We then decided that our next step would be to increase the air duct

size to allow easier flow. At that time, we also felt that the power piston was too large,

heavy and caused excessive friction so we decided to replace this with a smaller version

of the same concept.

In determining the new power piston size, we decided that a drastic size drop was

necessary so we reduced its size from a 4” to a 1” diameter as this was most likely our

last chance given the time constraints. Furthermore, we increased our duct size from 1/2”

inner diameter to 7/8” inner diameter in an attempt to eliminate the majority of the

efficiency losses. We introduced labyrinth seals on the power piston to maintain

lubrication within the cylinder and to reduce pressure blowback past the piston as air

leakage seemed to be a problem as well. The new power cylinder is seen in Figure 17.

Figure 17 - Power Cylinder

During this modification process, the stainless steel displacer casing sides were replaced

with aluminum sides and the duct connection location was moved from the hot side of the

displacer casing to the middle. This choice of location is understood within the Stirling

Engine community as an ideal location for maximum efficiency.

Future recommendations to the power piston would be to ensure an excellent seal to

prevent any air leakage around the piston through to the bottom of the cylinder. This

leakage issue plagues the displacer casing as well and in the future, a square casing would

not be advisable. Ideally, a cylindrical casing would be the most effective, and to allow

for viewing of the displacer, an entirely Pyrex cylinder could be used. This would also

reduce internal conduction from the hot to the cold end of the cylinder.

5.4 Regenerator

The main purpose of the regenerator is to improve the efficiency of the engine. A

possible regenerator design involves using a series of wire mesh layers, using enclosed

air spaces as insulators to trap the heat energy. This type of regenerator is illustrated in

Figure 18.

Figure 18: Wire Mesh Regenerator.

A regenerator works by removing heat from the working fluid during the cooling process

(steps 2-3 as seen on the P-v diagram) and storing it. This stored heat is then transferred

back to the working fluid during the heating process (steps 4-1 as seen on P-v diagram).

Through this method, energy that would normally be lost to the environment is used to

reheat the gas, thus improving efficiency by requiring less outside energy to heat the gas.

5.4.1 Calculations

There are some important considerations involved when designing a regenerator. The

first consideration is that the regenerator should not directly conduct heat from the hot

side to the cold side of the regenerator. The second consideration is that in order to

increase the effectiveness of the regenerator a certain amount of surface area must be

present based on the speed of the working fluid. And finally, in our case we must also

consider the weight of the material.

To ensure minimal heat conduction in the direction of heat flow, consider the equation of

conduction:

qcond = -kA dT/dx

where:

qcond = heat rate (W)

k = thermal conductivity (W/mK)

dT/dx = the change of temperature over a distance x (K/m)

Since the overall temperature change is fixed, changes in the thermal conductivity,

determined by the choice of material, must be considered. Plain carbon steel is a poor

choice because its thermal conductivity is 60.5 W/m°K. Stainless steel is a better choice,

since its conductivity is about 15 W/m°K. Preferred choices are Pyrex, with a

conductivity of only 1.5 W/m°K, or ceramics, which can achieve even lower conductivity

based on their composition. One of the best insulators available is air having a

conductivity of only 0.0263 W/mK. The problem with air is that its fluid composition

makes it prone to convection losses, which eliminate the benefits of its low conduction.

To stop this problem the air can be held in small volumes, which restrict its movement.

The second consideration is the amount of surface area present. The more surface area

available, the more convection can occur. Since convection is the main method for

transferring heat from our system to the regenerator and back to the system, the system

should incorporate the maximum possible surface area for the available volume.

The rate of heat flow from convection is defined by the equation:

qconv = hA(Ts –Tinf)

where:

q = heat flow

h = convection coefficient (typically between 25-250W/m2K)

This depends on both air speed and temperature of the surface and air.

A = Surface area (m2)

From this formula it is seen that the surface area is the only value that can be easily

manipulated. The downside of having a high surface area is that it restricts the flow of the

gas, resulting in more force needed to pass the gas through the regenerator.

To calculate the size of the spacing required the following equation is used:

δ = (2k/ωCpρ)-1/2

where:

δ = optimal spacing (m)

k = conduction coefficient

Cp = specific heat at constant pressure (J/kgK)

ρ = density (kg/m3)

ω = 2πf where f is the frequency of the gas moving through the regenerator in cycles/sec

This equation will give us the optimum spacing required, and hence surface area.

Based on the background information and manufacturing availability. It was chosen to

use a modular regenerator in the Stirling engine. This allows for testing of different

regenerator designs, and provides a method of demonstrating the benefits of the different

regenerators by showing the efficiency change of the engine.

5.4.2 Chosen Regenerator Design

Figure 19 – Regenerator

The current regenerator is composed of 10 aluminum sheets with an offset pattern of

holes. These are equally spaced to produce the regenerator (Figure 19). One benefit of

this design is that spacing the aluminum sheets allows air to be used as an insulator. This

air will insure the proper working of the regenerator by greatly limiting the amount of

conductive heat transfer from the hot to the cold side during the engines operation. The

second benefit is the pattern of holes in the sheets. These holes are 1/4” in diameter and

are offset so that there is no straight path from one side of the regenerator to the other. If

these holes were not present the air would simply flow around the sides and very little

area would be contacted, reducing the efficiency of the regenerator. Also, if the holes

were all in line with each other the air would flow straight through the regenerator and

not be forced to circulate within each of the air spaces in the sheets.

5.4.3 Improvements

Possible improvements to the selected regenerator design are to replace the aluminum

sheets with stainless steel and to change the size of the holes in the sheets. Replacing the

aluminum sheets with stainless steel would be done since aluminum has a high

conductivity (237 W/m°K compared to stainless steel at 15 W/m°K), since conductivity

is not desired, the stainless steel is a better choice. The stainless steel plates were the first

material proposed for sheet construction, but stainless steel is more difficult to machine

than aluminum. Since time is a consideration in this project, and recognizing that the

sheets are spaced apart to minimize the actual effects of conduction within the

regenerator, it was decided that it would be sufficient to construct the sheets of

aluminum.

Using smaller holes in the sheets has both advantages and disadvantages. The obvious

advantage is that by reducing the holes size, the amount of surface area in the displacer is

increased. The disadvantage is that by reducing the holes size, the flow rate of air that can

flow through the displacer is reduced. For this reason a balance must be found between

the amount of surface area and the flow rate of air. The optimal hole size is based on the

speed of the engine during operation; the faster the engine runs, the larger the holes in the

sheets need to be, and conversely the slower it runs, the smaller the holes.

Besides the chosen regenerator design other regenerator possibilities include using a wire

mesh between two plates; this has the advantage of a very large surface area, the

disadvantage is greater conduction. Ceramic is also a possibility; its advantage is a very

low thermal conductivity, but it has the problem of being brittle and difficult to machine.

5.5 Connecting Rods

The connecting rods are used to connect the displacer and power pistons to the drive

shaft. The original connecting rods were made of two 1/4” diameter steel shafts

connected with a pin joint to 1/8” thick flat bars. The pin joint allows the top and the

bottom of the rods to move independently of one another and is required so that the

engine can rotate. The top halves (steel shafts) of the rods move vertically up and down

with the pistons while the lower halves (steel bars) move in a circular pattern with the

drive shaft. The 1/4” diameter shafts and 1/8” bars were used to keep the overall weight

of the engine down. The two rods are different lengths to accommodate the different

throws of the pistons. The displacer piston connecting rod also has to travel through the

bottom of the displacer casing while the power piston connecting rod is suspended in the

air.

5.5.1 Modifications

After preliminary testing it was found that that the connecting rods needed to be

modified. The displacer piston connecting rod was too flexible due to its length and was

binding against the bottom of the displacer casing. The power piston connecting rod

needed to be modified to account for the changes in throw that were decided upon from

the testing results. To fix these issues the displacer connecting rod was changed to a 1/2”

diameter steel shaft and an oilite bushing was added under the displacer casing to allow

the shaft to run without binding. The throw of this piston stayed the same and therefore

no changes were made to either the lengths of the top or bottom link. To adjust the

power piston connecting rod the top link was reduced by two-thirds its original length

and the bottom link was doubled in length. These changes to the power piston

connecting rod reduced the throw of the piston and therefore reduced the volume change

required to rotate the drive shaft.

After testing the engine thermally, it was realized that further modifications were

required to get the engine to work properly. These modifications required changes to the

connecting rods. The top and bottom links and the pin joint needed to be remade to a

higher tolerance. The final connecting rods have the same overall dimensions as the

previous ones, but are made to a higher tolerance. The rods are more rigid and have fewer

mechanical losses then the previous rods. The final connecting rods are as light as the

previous rods and allow the engine to run mechanically sound when manually cranked.

The bushing under the displacer casing was lengthened to provide more support to the

connecting rod and to further reduce the chance of it binding.

The final connecting rods are well built and suitable for further use with this engine.

Although the rods are well built, it will be difficult to make any future changes to the

throw. If modifications to the engine are needed that require a throw change in either

piston a new connecting rod will need to be fabricated.

5.6 Drive Shaft

The drive shaft is an integral part of the Stirling engine. It ties the engine components

together and transfers the generated power from the engine to the output device. There is

a 90o bend in the shaft to force the displacer and power pistons to be 90o out of phase.

The phase difference means that if one piston is at the top dead center position

(completely up) the other piston is in the half way up position and vise versa. This phase

difference is used to control the amount of air exposed to the heat source at a given time

and also to prevent the engine from reaching equilibrium. The phase difference prevents

equilibrium from occurring because when the displacer piston is covering the heat source

(top dead center), the air starts to cool and will approach its minimum volume. When the

air does reach its minimum volume the displacer piston will have already moved to the

half down position allowing the air to start to reheat. Due to this motion, the power

piston (being 90o out of phase with the displacer piston) will always be chasing the

equilibrium position, and therefore will keep the engine rotating.

The preliminary drive shaft was constructed using 1/4” diameter steel threaded shafts

bolted to 1/8” thick steel bars. This design was chosen to keep the weight of the drive

shaft to a minimum and also to keep the fabrication simple. Refer to Figure 20 for a

photograph of the preliminary drive shaft.

Figure 20: Preliminary Drive Shaft

After preliminary testing, the shaft proved to be too flexible and the shaft would not run

properly when manually cranked.

5.6.1 Modifications

At this point a second drive shaft needed to be designed to solve the issues which arose

from testing of the preliminary shaft. The new shaft would need to be rigid and yet

remain lightweight. To accomplish this, the thickness of the steel bars was increased to

1/2” and the shaft diameter was increased to 1/2”. To keep the weight of the shaft down

aluminum was used for the bar sections. The shaft sections were also to be made of

aluminum to keep the weight to a minimum; but due to time constraints and poor contact

resistance of aluminum on aluminum, threaded steel rod was used. The threaded steel

rod increased the mechanical loses in the system but it was the best option available.

Once the shaft was together it resolved the issues with the preliminary drive shaft.

Although it was slightly heavier it was much more rigid and ran mechanically sound

when manually cranked. Refer to Figure 21 for a photograph of the second drive shaft.

Figure 21: Second Drive Shaft

After testing the engine thermally it was realized that further modifications were required

to get the engine to work properly. These modifications required changes to the second

drive shaft. The threaded rod needed to be replaced and the entire engine had to be made

to a higher tolerance.

The final drive shaft has the same overall dimensions as the second drive shaft, but is

made to a higher tolerance. The threaded steel rod was replaced with a steel rod and

bushings were incorporated at each end to reduce friction losses and play in the shaft. In

the previous two designs the shaft simply rotated in the mounts attached to the stand.

The shaft was also pinned and brazed together, instead of being bolted together. This

process made the drive shaft more rigid then the others. Refer to Figure 22 for a

photograph of the final drive shaft.

Figure 22: Final Drive Shaft

The final drive shaft is the most rigid and has fewer mechanical losses than the two

previous shafts. It is also lighter then the second drive shaft and runs mechanically sound

when manually cranked.

The final drive shaft is well built and suitable for further use with this engine. However,

future modifications probably will be required to get the engine to work. Although well

built, it will be difficult to make any changes to the throw or the phase angle of the drive

shaft. If the modifications to the engine require that either one of these parameters be

changed, a new drive shaft will need to be fabricated.

5.7 Flywheel

Experimentation with the constructed Stirling engine demonstrated that a flywheel is

necessary to maintain the rotation through all stages of the piston motion. A flywheel acts

as a reservoir to absorb energy during the points of rotation where the turning moment is

greater than the resisting moment, and restores energy when the turning moment is less

than the resisting moment. The absorbing of energy must be accompanied by an increase

in speed, while restoring energy necessitates a decrease in speed. These speed

fluctuations are small, but the flywheel must be properly proportioned so that these

changes of speed do not exceed permissible limits. The kinetic energy of the flywheel is

given by

IKsω2 = ½ Ef

where I = mass moment of inertia of the flywheel = mass*(radius of gyration)2 = mk2

Ks = speed coefficient

ω = mean angular speed

Ef = energy fluctuation = area under torque vs. rotation angle diagram

For optimal flywheel performance, the effective weight must be as far from the centre of

the shaft as possible (maximal radius of gyration).

Figure 23: Flywheel #1

The first flywheel constructed was a 7” round disk that was 1/8” thick (figure 23). This

design was constructed of steel and had material removed from the inner portion to

maximize the performance of the flywheel with respect to weight. This flywheel was not

intended to be the final design. The final design would only be determined after the

engine was constructed and running; this is due to the fact that the size and weight of the

flywheel is dependant on both the running speed of the engine and the amount of friction

that exists in the drive train while running at the operating speed.

After the engine was constructed, a large amount of friction was observed within the

system, so a larger flywheel was constructed. This second flywheel had dimensions of 6”

diameter and 3/4” thickness and was made of steel. Once the second flywheel was

installed on the second drive shaft, testing was done to ensure it was the proper size. This

was done by manually moving the power piston at approximately 60 RPM, which is the

projected running speed of the engine. It was then observed that at the top and bottom of

the power pistons cycle that the flywheel proved sufficient to provide the required force

to maintain rotation in the drive shaft. This is important because the power piston is

unable to provide power in these locations.

Figure 24: Flywheel #3

The third and final flywheel (Figure 24) was constructed to accompany the third drive

shaft. It is composed of an aluminum disk measuring 1.5cm by 12.5cm diameter. The

aluminum was chosen because its reduced density reduces the overall weight without

affecting the flywheel’s efficiency. The weight was reduced in order to minimize the

bending in the drive shaft, which could cause misalignment and adversely affect the

running of the engine. After testing the flywheel, it was found to be slightly undersized

for the amount of friction in the system. This conclusion was reached from moving the

power piston by hand; the flywheel will sometimes propel the drive shaft through the

trouble areas but not consistently. In order to fix this problem it is recommended to return

to the second flywheel design.

A future improvement of the flywheel would be to optimize its size based on the

equations above, once the engine’s running speed is known. For demonstrational

purposes of the engine the second flywheel design should easily meet this requirement.

5.8 Transparent Side

One of the design requirements was that the displacer piston be visible while in

operation. To accomplish this, a transparent material suitable to withstand approximately

500 oC was required. The first materials researched were Plexiglas and Pyrex products.

These products were the first choice due the machineability of the materials and also their

transparent properties. The melting point of Plexiglas is approximately 70oC and the

Pyrex was more then our budget would allow.

The second option available was to use a glass product. Although glass can withstand

high temperatures, it is very difficult to machine and is very brittle. A glass supplier was

contacted who was able to supply and machine a piece of glass to fit our engine. This

product is commonly used in wood stoves. The glass, Neoceram, has a melting

temperature of 2500oC, which more than exceeds our requirements. A rubber gasket was

made and the glass was bolted to the displacer piston to allow for engine thermal testing.

After the testing, the displacer was disassembled and the Neoceram cracked due to an

unnoticed alignment issue. A slight leak was also detected during the initial testing

between the glass and the displacer. Refer to Figure 25 for a photograph of the Cracked

Neoceram Glass.

Figure 25: Broken Neoceram Glass

A redesign of the glass mounting system is required. The redesign will need to both

eliminate the original alignment issue that caused the crack and also eliminate the sealing

problem. To accomplish this, a piece of the Neoceram glass should be pressed and sealed

between two sheets of stainless steel. The steel could then be bolted to the existing

displacer casing and sealed. Unfortunately this modification will need to be completed in

the future. Refer to Figure 26 for a sketch of the proposed mounting system.

Figure 26: Proposed Redesigned Glass Mounting

5.9 Rotating Engine Stand

The main purpose of the stand is to support the engine. The displacer and power pistons

sit on top of a horizontal surface. This surface is pivoted to permit swiveling from the

full vertical position to a full horizontal position. This swivel is necessary to keep the

solar collector focused at the sun. Two mounting brackets are attached to the underside

of the flat surface to hold the drive shaft in position. The table is supported on either end

by a set of legs.

The stand proved fairly stable through the preliminary testing and it functioned well.

However, there were several issues with the stand that needed to be resolved. The flat

top itself was bowed in the middle causing the two piston casings to be on a slight angle

away from each other. This potentially could cause more mechanical loss than necessary

in the drive shaft. The mounting brackets that supported the drive shaft were flexible and

the drive shaft was set in holes cut in either bracket. This also proved to increase the

mechanical loses in the drive shaft. The stands legs moved independently of one another

making it awkward to carry.

No changes were made to the stand until the power piston, drive shaft and connecting

rods were redesigned for the final time. When the stand was modified the flat table was

replaced with a flatter piece of steel. The swivel and mounting brackets were bolted on

instead of welded on. This measure provided a more accurate mounting system for the

drive shaft. The mounting brackets were replaced with more rigid ones to eliminate the

flexing issue. Instead of the drive shaft running in holes in the mounting brackets,

bushings were added to the end of each bracket for the shaft to travel through. The

bushings reduced the mechanical losses encountered in the original stand. Finally the

feet of the stand were tied together with two lateral bars to make the stand more stable

when being transported. The final stand can be seen in Figure 27.

Figure 27: Tilted Stand With Engine

6.0 Testing

6.1 Temperature Measurement

To measure the temperature drops across the engine we purchased a digital thermometer

from Omega (Figure 28). This handheld thermometer was chosen with the intention of

mounting it directly to the engine. The two thermocouple inputs are useful to read the

difference in temperature between two points instantaneously.

Figure 28: Omega Digital Two-Input Thermometer

Using the solar collector, we achieved the following values:

Thermocouple Position Temperature Reading (˚C)

Focal Point of Collector 360

Top of Displacer Casing 230

Bottom of Displacer Casing 26

∆T = 204˚C for the displacer.

The large temperature difference between the focal point and the top of the displacer

casing does not correlate with our finite element analysis for the heat loss of the copper

rod (Figure 29). The rod lost much more heat than anticipated for the insulation

surrounding it. It is possible that the gaps at the insulation seams may have been a

contributing factor to these losses. There were also sections of the copper rod that could

not be easily insulated because other parts of the engine were mounted to it; where the

collector was positioned and where the rod threaded into the copper plate on the top of

the displacer were difficult areas to incorporate insulation. We also believe there was

some contact resistance between the threaded copper rod and the threads in the copper

plate. This could also account for some of this temperature disparity. Due to the scarcity

of available insulators able to withstand the anticipated rod temperatures, and due to

space and safety constraints, the results are in an acceptable range. The temperature at the

top of the displacer casing is still sufficiently hot to nullify the impact of these losses.

Figure 29: ANSYS Prediction of Rod Heat Conduction

A large temperature drop in the displacer casing is desired to optimize the performance of

our engine. However, we don’t want the heat to be lost before adequately heating the air

in the hot side of the displacer. The material initially chosen for the displacer casing was

stainless steel, however, because of availability and time, we used aluminum. Aluminum

is more conductive than ideally desired for the displacer casing walls; we would prefer

conduction from the outside to the inside but not in the vertical direction of the walls.

6.1.1 Recommendations

Stainless steel is a better suited material for the displacer due to its lower conduction

value. We would suggest that the final displacer casing be constructed from stainless

steel.

Since conduction is not desired in the displacer piston casing, we further recommend that

the casing around the middle of the displacer be constructed of a material with very low

conduction, such as ceramic. This would minimize the conduction of heat from the hot

side to the cold side and vice versa.

A very useful addition to the displacer design would be attaching fins to the inside walls

of the displacer casing. Fins on the inner walls of the hot side would increase start-up

time by transferring the heat from the copper to the air in the hot side of the piston more

quickly than the current assembly. Fins would be useful on the inner and outer walls of

the cold side allowing it to more rapidly transfer the heat from the chamber. We would

have liked to add fins, but they were not included on the current design primarily due to

time constraints.

6.2 Force Measurements

Using a force meter and pushing on the drive shaft, we measured a maximum required

force of 4lb. This was the maximum force because it was the force required to begin

rotating the drive shaft or push the power piston upwards. This converted to a required

torque of 0.5 ft-lb by using the 1.5 inch link attaching the power piston connecting rod to

the shaft. These forces are reasonable for the size of the engine and its components.

6.2.1 Recommendations

Further reduction of the frictional losses is desired. Reducing the throw would also

increase the rigidity of our links and could improve the performance of the drive shaft.

We also recommend replacing the current bushings in the drive shaft with ball bearings to

remove some of the friction from the shaft.

By completely sealing our displacer piston, the forces calculated above would be easily

achievable with air pressure changes. This will be expanded in the next section.

6.3 Pressure Measurements

To measure the pressure our engine was capable of holding, we used a vacuum pump to

drop the pressure in the piston casing. We read the pressure at which our seal gave way

with a pressure gage. We saw that our pressure was only 0.725 psi below atmospheric

pressure when the seal failed. We approximated this pressure drop as the equivalent

pressure increase our engine could withstand during operation. Using the force calculated

above, and the cross-sectional area of our power piston, we were able to estimate a

required pressure drop/increase as 5 psi. The displacer piston needs alterations to

withstand this pressure change. We initially underestimated the difficulty of sealing our

square piston.

6.3.1 Recommendations

The team decided on a displacer piston casing made from fewer pieces. One solution to

consider is to make the sides of the displacer casing from square tubing. Then we could

seal the holes to the connections and secure two end-caps. A cylindrical casing would be

ideal for better sealing. A better solution might be to use steel rod and machine the casing

out of one piece of stock.

7.0 Final Budget

Based on a current design, the following costs have been accrued:

Solar collector mirror: $178.92

Digital Thermometer: $132.25

Metal: $260.00

Neoceram Glass: $46.00

Ceramic: $30.00

Miscellaneous: $62.21

Total: $709.38

8.0 Future Recommendations

Sealing is the major problem with our Stirling Engine design. In order for the engine to

work properly no air should be able to escape from the engine once it is sealed. Therefore

our first recommendation is to replace the current displacer piston casing with a square or

cylindrical stainless steel tube. This would prevent the air from leaking out at the seams

as it does in our current design.

To further improve sealing a cap could be manufactured to go over the hot end of the

displacer. This would allow the cap to be sealed to the rest of the displacer casing at a

cooler location further from the top. This would keep the temperature of the sealed region

within the allowable limits of more readily available sealants.

Fins should be incorporated in the displacer piston on the inside of the hot side and on

both the outside and the inside of the cold side. These fins would increase the rate at

which the air in the system is heated and cooled.

We also propose that the displacer walls be separated in the center by an insulating

material such as ceramic. This would help prevent heat propagation from the hot side to

the cold side of the displacer piston. A possible problem that may develop from this

modification is additional leaks in the displacer casing.

Since the engine is to be used as a demonstrational tool the glass face of the displacer

casing should be reintegrated. To accomplish this, a redesign of the glass mounting

system is required. The redesign will need to both eliminate the original alignment issue

that caused the glass to crack and also eliminate the sealing problem. To accomplish this,

a piece of the Neoceram glass could be pressed and sealed between two sheets of

stainless steel. Then the steel plates could be attached to the displacer casing and sealed.

If a round displacer casing is incorporated it could be made entirely from Pyrex. This

would allow the displacer to be visible and provide minimal seams where air leakage

could occur.

To make the drive shaft run true, counterweights could be added to balance the shaft.

With the addition of counterweights, less force would be required to make the shaft

complete a full rotation. The flywheel could also be made lighter with the addition of

counterweights because it would have to overcome less force to keep the shaft rotating.

Another recommendation would be to further reduce the mechanical losses of the system.

Two main ways are proposed to accomplish this: (1) replacing the bushings with ball

bearings to reduce the friction on the drive shaft, and (2) reduce the piston connecting rod

lengths to make them more rigid and lighter. This could be done to allow the engine to

run with a smaller pressure/temperature differential.

It may be beneficial to try to incorporate an overhead drive shaft design with the solar

collector. With an overhead design, gravity would pull the pistons down and the

generated pressure would push the pistons back up. Currently gravity pulls the pistons

down and the generated pressure has to try to pull the pistons back up.

The regenerator could not be adequately tested without a fully operational engine.

Therefore various regenerator combinations should be tested to determine the best

arrangement for this application. Variables in the regenerator design could include the

material used, the volume, the hole pattern, size of holes/mesh, and others.

9.0 Conclusion

The selected Stirling engine design has not yet met the specifications of our client. With a

sealed displacer piston, adequate pressure will be developed in the displacer chamber to

drive the action. By incorporating the recommendations outlined above, we believe that

the engine will meet the required design criteria described in the design requirements

memo. Although the engine, in its anticipated future configuration, will not be able to

produce the 50 Watts of power initially envisioned, it will produce a visible power

output, be an asset to classroom demonstrations, be portable and run from a solar heat

source.

10.0 References

Bevel, T. (1971). The Theory of Machines (3rd ed.). Great Britain: William Clowes and Sons. Çengel, Y.A. & Boles, M.A. (1998). Thermodynamics. An Engineering Approach (3rd ed). New Jersey: McGraw-Hill. Daniels, F. (1964). Direct Use of the Sun’s Energy. New Haven and London: Yale University Press. Diel Ltd. (2001). The Stirling Hot Air Engine. Retrieved September 9, 2003, from http://www.stirlinghotairengine.com Incropera, F.P. & DeWitt, D.P. (2002). Introduction to Heat Transfer (4th ed.). New York: John Wiley & Sons, Inc. Lewitt, E.H. (1965). Thermodynamics Applied to Heat Engines (6th ed.). London: Sir Isaac Pitman & Sons. Montgomery, R.H. (1978). The Solar Decision Book. New York: John Wiley. Mueller Solartechnik. (1998). Solar Collectors. Retrieved September 24, 2003, from http://www.mueller-solartechnik.de/koch_eng.htm Nice, K. (n.d.). How Stirling Engines Work. In How Stuff Works Inc. Retrieved September 9, 2003, from http://www.howstuffworks.com/stirling-engine.htm Ross, A. (1977). Stirling Cycle Engines. Phoenix: Solar Engines. Schmidt, F.W. and A.J. Willmott. (1981). Thermal Energy Storage and Regeneration. New York: McGraw-Hill. The Solar Server. (n.d.). Solar Collectors: Different Types and Fields of Application. Retrieved September 16, 2003, from http://www.solarserver.de/wissen/sonnenkollektoren-e.html Zarem, A.M. (1963). Introduction to the Utilization of Solar Energy. New York: McGraw-Hill.