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Stirling Engine Optimization

1

Optimization of Stirling Engine Power Output Through Variation of Choke Point

Diameter and Expansion Space Volume

Anna Brill

Massachusetts Academy of Math and Science

Abstract

Fixed choke points of different diameters have been tested in a Stirling engine. The

resultant power output was measured and the diameter which produces the most power was

determined. Additionally, for each choke point diameter the optimum volume of the expansion

space was determined. Adapting a quarter turn ball valve to adjust the choke point while the

engine was running was deemed improbable because changing the size of the passageway also

changes the direction of the air flow, preventing the engine from running. It was determined that

the optimal choke point diameter was 0.67469 cm and the optimal expansion space volume was

47.665 mL.

Introduction

A Stirling engine is an external combustion engine based on the Stirling Cycle.

Developed first in 1816 by Robert Stirling, this engine produces power from differences in

temperature. The working fluid inside the engine, typically air, hydrogen or helium, is heated on

one end and cooled on the other, consequently causing the gas to expand and compress,

respectively. In addition, the expansion and compression of the working fluid moves two pistons

within the engine cylinder which in turn are, depending on the configuration, coupled in some

manner with a drive mechanism to produce a net power output (Energy Conversion 2010). For nearly two hundred years Stirling engines have been constantly improved upon with

regards to efficiency. Gas combustion engines far outrun Stirling engines because they are

inexpensive, but lately, with rising fossil fuel prices, Stirling engines have come back into the

picture as a green, inexpensive alternative to the gas combustion engine. Stirling engines can

achieve efficiencies of 65% to 75% that of the Carnot efficiency (Stirling Engines, 2010). However, efficiency can be easily lost. Good heat transfer devices are crucial to achieving any

kind of useful power output. Without the proper material, heat transfers can be ineffective and as

a result not create a large enough temperature difference for the engine to have a net power

output. Too much focus in research has been on regenerators and pistons while the heat transfers

have been neglected.

Stirling engines lack a throttling method inherently built into the engine. One method of

changing power output is to change the diameter of the choke point of the engine, which is the

point at which the hot air flows from the heat transfer to the engine cylinder. Choke points are

often specifically designed to work optimally at one setting. A choke point that can be varied is

advantageous because power output can be altered on the fly. The goal of this project is to

develop a simple variable choke point for use in a small Stirling engine.


2

Definitions and Terminology

A great amount of literature on the subject of Stirling engines exists, but no standard

terminology does. Most of the literature was published sporadically over the past 200 years and

there exists no standardization or clarification of terminology.

Originally, types of Stirling engines were classified into three groups according to the

Kirkley-Walker classification system: Alpha, Beta, and Gamma. Now these terms only describe

the cylinder couplings of a Stirling engine. Cylinder coupling identifies the way in which the

displacer piston and the power piston are connected, with respect to the connection of the

variable volume working spaces. These are the spaces inside the engine cylinder where the

working fluid is heated and cooled, respectively (Sandfort, 1962).

An Alpha arrangement uses two separate cylinders that each has a sealed piston, either

the displacer or the power piston. Power output is produced by the separate motion of the

individual pistons. The term Beta covers the group of Stirling engines that use a single cylinder

arrangement where the displacer and power piston are in tandem and power is produced by the

action of the pistons together. In simple engines the piston and the displacer can often be just one

piston. A Gamma arrangement is more or less a hybrid of the Alpha and Beta arrangements.

Gamma engines have two separate cylinders like the Alpha, but power output is produced in the

same manner as in Beta engines (Urieli, 2010).

Figure 1. Three types of Stirling engines are: Alpha Twin Piston,

Beta Piston-Displacer, and Gamma Piston-Displacer (Hooper & Reader, 1983).

Lately this type of classification has become insufficient because the categorization does

not identify the mode of operation, the form of crank-drive, or the power take-off mechanism of

a Stirling engine. In order to classify Stirling engines correctly, the mode of operation, the form

of cylinder coupling and the form of piston coupling should be identified.

The mode of operation of an engine denotes how the components of a Stirling engine

work together. There are six terms used to describe the mode of operation: double-acting, single-

acting, single phase, multiphase, resonant, and non-resonant. A double-acting or differential

Stirling engine has multiple cylinders that contain two pistons, the power piston and the

displacer, whereas a single-acting Stirling engine has cylinders that separately house the power

piston and the displacer. In a single-acting Stirling engine the multiple cylinders have to work


3

together in order to produce a net power output, and so a single-acting engine must have at least

two cylinders. Single-phase and multiphase refer to the state of the working fluid inside a Stirling

engine. The working fluid in a single-phase engine stays at a constant pressure while the working

fluid in a multiphase engine varies. The terms resonant and non-resonant refer to the properties

of the cylinders themselves and how they behave under the stress of a working Stirling engine.

Piston coupling is very important in Stirling engines. There are three basic forms with

many further subdivisions. A rigidly coupled Stirling engine uses a solid mechanical linkage that

connects the reciprocating elements to each other and the power take-off mechanism. Typical

types of rigid coupling include: slider-crank, rhombic drive, swashplate, Scotch Yoke, crank-

rocker, and Ross rocker. In gas coupled machines the pistons are coupled by gas dynamic effects

rather than solid mechanical linkages. A few examples of this type of coupling are: free-piston,

free-displacer, and free-cylinder. Liquid coupled Stirling engines use liquid to connect the

pistons. There are at least three ways Stirling engines can be liquid coupled: jet-stream, rocking

beam, and pressure feedback. The classification of liquid coupling can also be used to describe

hybrid Stirling engines that have the power piston rigidly coupled with the output shaft but the

displacer gas coupled to the power piston (Hooper & Reader, 1983).

Stirling Engine Function and Design

A Stirling Engine is a heat engine that operates on a closed regenerative cycle. Energy

transfer occurs through the cylinder walls of the heat exchanger. The modern Stirling engine is

based on the hot-air engine invented by Robert Stirling in 1816. These engines cool the working

fluid and compress it, heat the fluid, and then expand it in the basic heat engine cycle. What

makes a Stirling Engine different is that heat energy is transferred into and out of the engine

through the cylinder walls or heat exchanger. Moreover, the gas stays permanently inside. In

order for a Stirling engine to run, a necessary temperature difference must occur between the hot

end and the cold end. To obtain this, the engine is divided into a cold and hot space between

which the fluid can be moved by pistons. The volume variations that occur when the working

fluid is moved by the pistons must be out of phase with each other if power is to be produced.

Because the heating (expansion) process occurs at a greater working pressure than the cooling

(compression) process, a net work output is produced. A displacer is a piston that just fits inside

the working chamber with a small distance, the annulus, between it and the walls of the chamber

that allows the working fluid to pass through. The displacer moves the working fluid between the

hot space and the cold space, and so the fluid is alternatively heated and cooled, and work can be

produced. Because the power piston and the displacer also move out of phase with each other,

the Stirling engine requires an unconventional drive system.

Stirling engines lose efficiency due to large differences in the temperature of the working

fluid and the heating and cooling spaces. As the fluid passes around the displacer constant heat is

being added, and the fluid reaches the cold space at a higher temperature than necessary. This

occurs again in reverse when the fluid passes around the displacer on the way back, arriving at

the hot space colder than necessary. A regenerator, or economizer as Robert Stirling called it,

was developed to increase the efficiency of a Stirling engine. The design was originally a mass

of steel wire located in the annulus that absorbed excess energy as the working fluid passed

through it. A regenerator is essentially a pre-cooler, reducing the thermal load on the main


4

cooler, as well as a pre-heater, reducing the energy required by the main heater to heat the

working fluid.

Figure 2. A regenerator in the annulus of simple Beta configuration

Stirling engine (Hooper & Reader, 1983).

Philips developed a more efficient Stirling engine that eliminates the problem of rapid

energy transfers. Instead of placing the regenerator directly in the annulus, Philips researchers

made use of tubular heat exchangers seen in Figure 3 (Hargreaves, 1991).

Figure 3. The complete cycle of a Beta Stirling engine with tubular regenerators (Hooper &

Reader, 1983).

Regenerators can improve the performance of a Stirling Engine by alternatively storing

and returning heat energy so that the heat input can be kept to a minimum while still producing a

useful power output.

As the working fluid flows between the ends of the engine, it passes through a choke

point, or a narrow hole. The choke point allows for regulation of the speed and amount of

working fluid that passes through the engine cylinder.


5

Engine Efficiency

The efficiency of a Stirling engine is dependent on several different parameters. They can

achieve efficiencies of 65%-70% of the Carnot efficiency. However, the engine efficiency is

reduced up to 0.5% for each degree rise in coolant temperature.

Dead space in a Stirling engine, which is required to accommodate the necessary heat

exchangers, accounts for up to 50% of the total internal gas volume of the engine. Depending on

the location of the dead space it can have differing effects on the efficiency of the engine.

Altering the location can also provide a means to control the power output.

The diameter of the choke point, which regulates airflow from the expansion space to the

main engine cylinder, also controls the power output. Torbjorn Bergstrom, a professor at

Worcester Polytechnic Institute, noted that in professional applications, the choke point is often

altered as a means of throttling the engine (personal communication).

Efficiency is also dependent on the speed at which the engine is running. As the speed is

increased, aerodynamic drag becomes a predominant factor because it is proportional to the

square of the speed. To reduce these losses, light working fluids, such as helium and hydrogen,

are used. However, these gasses are difficult to contain, especially hydrogen due to its ability to

diffuse through solid material. Therefore, Stirling engines that use hydrogen or helium for their

working fluid are often expensive and bulky.

Stirling engines can be very quiet depending on their construction. Because they have no

exhaust valves, they are significantly quieter than internal combustion engines. However, free

piston Stirling engines can potentially be very noisy, depending on their operational mode. And

these engines can get noisy due to the timing gears and combustion blower, but they are still

relatively quiet compared to internal combustion engines.

In addition, Stirling engines can utilize energy sources that do not pollute the atmosphere.

Even when fossil fuels are used to power Stirling engines, the inherent steady flow combustion

process reduces the amount of pollutants released (Walker, 1980).

The fraction of cyclic energy rejected by the cooler in a Stirling engine is between 60%

and 250% more than a conventional reciprocating engine. As a result, large radiators are needed

to handle this large thermal loading.

The theory behind a Stirling engine requires the heat transfer process to occur reversibly.

For this to occur, several conditions must be met:

1. Reversible processes can only occur when the process is at all times in thermodynamic equilibrium. This means that the regenerator system needs to be quasi-static (passing through

a series of equilibrium states) during the flow periods. However, this cannot happen in

practice because the process needs to occur at an infinitely slow pace. Stirling engines often

have high shaft speeds and thus have extremely high flow rates. Because this requirement

can never be satisfied, the remaining requirements must be satisfied as well as possible.

2. Because the amount of excess energy given up by the working fluid as well as the rate of rejection are both extremely high, the bulk heat transfer must be infinite in order to balance

the system. Yet again, this is not practically obtainable in the real world. Therefore, Stirling

engine must obtain the highest bulk heat transfer possible within design constraints.

3. The heat transfer area (surface area) must be infinite in order to enable the ideal conditions under which Stirling engines operate optimally. Clearly this is not physically possible, and so

certain steps must be taken to make the heat transfer area as large as possible within design


6

constraints. Typical practical solutions include the use of wires or small particles to

maximize the surface area of a regenerator.

4. The heat capacity of the regenerator must be zero or infinite. To ensure this, the ratio of the heat capacities of the working fluid to that of the regenerator material must be kept to a

minimum. Additionally there should be no axial heat conduction and there should be

maximum conduction perpendicular to the flow.

As evident from the above requirements, a practical Stirling engine is very different from

an idealized one. The working fluid properties of density, velocity, viscosity, and pressure will

change within the engine, the heat transfer areas will not be infinite, there will be axial

conduction, and the conduction perpendicular to the flow will be imperfect, and so on. Practical

Stirling engine designers must therefore strive to fulfill the above requirements to the best of

their ability.

Variations in Stirling Engines

Drive mechanisms pose a difficult problem for Stirling engines because discontinuous

motion is required to achieve the volumetric changes that result in a net power output. There are

four main drive mechanisms for Stirling engines: crank-rocker, rhombic drive, swashplate, and

slider-crank. Crank-rocker was the original drive mechanism in which a rocker connects to a

piston and displacer through two arms and the piston is driven off the crankshaft (Figure 4).

Figure 4. A crank-rocker drive mechanism in a Stirling engine (Hooper & Reader, 1983).

This set up is useful only in small engines because the crankcase has to be pressurized,

and there is no way to dynamically balance a single cylinder engine. Because larger Stirling

engines were needed, Phillips developed the rhombic drive in the 1950s (Hargreaves, 1991). This

drive mechanism is dynamically balanced and the crankcase does not need to be pressurized.

However, it has the disadvantage of being mechanically complicated. The swashplate

configuration is mainly used where space is tight. This drive mechanism is dynamically balanced

at a fixed swashplate angle, and the cylinders are easily sealed off so the entire crankcase does

not need to be pressurized. The swashplate also adds the function of varying power output by

changing the angle of the swashplate, but the engine is only dynamically balanced at one angle.


7

The slider-crank drive mechanism is very reliable and has been used widely in other

internal combustion engines; however, it is almost impossible to balance dynamically. This is the

mechanism generally used in twin-cylinder Stirling engines (Hooper & Reader, 1983).

Figure 5. This slider-crank Stirling engine has tubular regenerators (Hooper & Reader,

1983).

A free-piston Stirling engine is an engine where the pistons are not coupled mechanically.

William Beale first developed this concept in a practical device, and so the term Beale free-

piston engine is frequently used to describe free-piston Stirling engines (FPSE) (Walker, 1980).

Figure 6. The diagram shows a Free-Piston Stirling engine in an alpha configuration (Hooper &

Reader, 1983).

This configuration is the same as the most basic alpha configuration, but there is no

mechanical crank mechanism and the cylinder is fully sealed at both ends. As heat is applied to

the fluid inside the engine it expands, causing the pressure to increase and the power piston as

well as the displacer to move down the cylinder. To make sure the power piston and the displacer

are out of phase with each other, the displacer is made lighter so even though the pressure change

is approximately equal, the displacer has a smaller mass and therefore accelerates faster than the

piston. The working fluid is then forced through a connecting passage (which could contain a

regenerator) into the hot space where it is heated even more, causing an increase in pressure.

Eventually, the displacer comes in contact with the power piston, and no more fluid flows into


8

the hot space. At this point the pressure begins to decrease, but the inertia of the pistons in

tandem continues the expansion process. Then, because the displacer is lighter, it halts rapidly

and becomes separated from the power piston. Once the displacer has stopped, the working fluid

flows from the expansion space to the compression space, and the difference in pressure causes

the displacer to move up rapidly in the cylinder, forcing all the fluid into the compression space.

The power piston continues moving downward and then upward again, compressing the fluid. As

a consequence, the working fluid pressure increases and now there is a downward force on the

displacer. The displacer again comes in contact with the power piston, and the working cycle

repeats. In all types of FPSEs the working cycle is the same but the machine dynamics are

different.

Applications and Recent Developments in Stirling Engine Technology

Applications for Stirling engines are typically very specialized because of cheaper

alternatives. Stirling engines also have the disadvantage of needing time to warm-up. Most often

they are used in submarines and other marine applications because of their reduced engine noise

and vibration. The speeds required of a marine-based Stirling engine are also much less than

land-based ones and so helium, air, or nitrogen can be used as the working fluid instead of

hydrogen without compromising any engine performance. In addition, with marine-based

applications sea water can be efficiently used as a cooling method.

Stirling engines for use in automobiles have been the most extensively tested. During the

1970s and 1980s, Ford tried outfitting some of its standard models with Stirling engines, and

MTI built a car around a Stirling engine. Ultimately all of these ventures folded, either because

of a lack of popular appeal or failure to create a safe, reliable, and efficient engine in an

automobile (Hooper & Reader, 1983).

The most popular mechanical (non-propulsive) application of Stirling engines is in

pumping systems. If crank-type Stirling engines are used, however, the idiosyncrasies of the

separate pumping system would still be intrinsic. Free-piston Stirling engines are therefore used

in pumping systems because the pump then becomes an integral part of the engine.

While Stirling engines have no specific application yet, their potential for a better

alternative to current popular engines is great.

For the past several years there has been renewed interest in Stirling engines and

increasing their performance. These modifications have been mostly in regards to larger Stirling

engines. Kroliczek, Nikitkin, and Wolf developed a different type of heat transfer where Loop

Heat Pipes and Capillary Pumped Loops are used to transfer heat more efficiently (2010).

Another modification of a Stirling engine increased power output by using a coaxial power

mechanism (Lin, 2010). Much research has also been done involving regenerators in Stirling

engines such as Abdulrahmans work in designing and testing low cost, efficient materials for engines (2011). Pistons, displacers, and types of working fluid have also been the object of

several studies and patents in the past ten years. However, not as much investigation has been

made into increasing the efficiency of small, tabletop Stirling engines. While they are not the

most useful for applications that need large power outputs, the technology used in them can be

scaled up to larger applications.


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Research Plan

Engineering problem being addressed:

Problems with Stirling engine design, specifically the volume of the expansion space and

choke point, result in decreased efficiency and power output.

Hypothesis/Engineering Goal:

The goal of this project is to develop a variable choke point for use in a small Stirling

engine.

Description in detail of methods or procedures:

A Beta configuration Stirling engine similar to the one below in Figure 8 will be built.

Figure 8. This diagram shows a Beta Stirling engine similar to the one used in this project

("Solar-13 stirling engine," 2010).

To test the choke point the engine will first be run with the typical diameter choke and

then the power output will be determined based on the speed and friction of the engine. To power

the engine, a standard alcohol burner will be used. This allows the amount of heat applied to the

engine to stay constant throughout all tests. Once the power output of the engine with the

standard choke has been determined, the same process will be used to test with eight other

different diameters (nine diameters will be tested in all).

Testing of the expansion space volume will commence in a similar manner. The

temperature applied to the engine will stay constant for all tests, and the standard volume will

first be tested. Then, as the engine is running, the volume will be reduced by 2.835 mL. Two

magnets will be used to move a ball bearing (radius of 1.905 cm) inside the test tube to reduce

the volume. The temperature of the engine will never exceed 200 degrees Celsius. The testing

will occur in a controlled environment when conditions will be kept the same throughout the

different experiments.

Methodology

The Stirling engine was built from a kit (a modified version of the one manufactured by

PM Research, Solar-13 model) used in the ME 1800 course at Worcester Polytechnic Institute in

Massachusetts. The engine was powered by a standard alcohol burner with denatured alcohol

used as the fuel. The heat transfer used was steel wool (4 g) and a ball bearing very close in

diameter to the test tube (1.905 cm) was placed in the end of the tube. After the engine was

heated for three minutes, it was kick-started. Using a tachometer (Monarch Instrument, Pocket


10

Laser Tach 200) the speed of the crankshaft was measured and recorded. The following

procedure was used to determine the optimal expansion space volume for multiple choke point

diameters. Once the initial speed was recorded, the ball bearing was moved 1 cm by placing a

magnet on the glass where the ball bearing was and sliding it over. The volume of the test tube

was changed in this manner. The speed was recorded, and again the ball bearing was moved over

again. This process was repeated 5 times. The engine was then allowed to cool down. The choke

point of 0.3175 cm in diameter was then taken out and replaced by one of 0.47625 cm in

diameter. The process for measuring the speed of the crankshaft was then repeated for each

movement of the ball bearing. Then seven more choke points of diameters 0.555625, 0.5953125,

0.635, 0.6746875, 0.714375, 0.79375, and 0.9921875 cm were tested in the same manner. These

choke points were chosen based on drill bits which were in English units and then converted to

metric.

To calculate power, standard physics equations were used (see appendix).

Results

Table 1. Average power output for each choke point diameter and expansion space volume.


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Choke Diameter Volume speed 1 speed 2 speed 3 speed 4 vavg St. Dev. aavg Favg Pavg

(cm) (mL) (rps) (rps) (rps) (rps) (m/s) (m/s2) (N) (W)

0.3175000 50.500 33.236 33.287 32.687 32.345 0.329 0.453 0.065778 0.0006315 0.062413

0.3175000 47.665 31.880 32.028 32.037 32.020 0.320 0.074 0.063983 0.6142320 0.196500

choke 1 0.3175000 44.829 30.670 30.276 30.056 30.442 0.304 0.260 0.060722 0.5829312 0.176984

0.3175000 41.994 31.935 31.955 31.972 31.914 0.319 0.025 0.063888 0.6133248 0.195920

0.3175000 39.159 32.052 32.353 32.456 32.407 0.323 0.182 0.064634 0.6204864 0.200523

0.4762500 50.500 32.412 32.330 32.561 32.635 0.162 0.139 0.032485 0.3118512 0.050652

0.4762500 47.665 31.978 31.887 31.967 31.899 0.160 0.046 0.031933 0.3065544 0.048946

choke 2 0.4762500 44.829 32.123 35.446 36.231 35.443 0.174 1.830 0.034811 0.3341832 0.058166

0.4762500 41.994 - - - - - - - - -

0.4762500 39.159 - - - - - - - - -

0.5556250 50.500 34.512 34.875 34.622 34.362 0.346 0.216 0.069186 0.6641808 0.229758

0.5556250 47.665 34.950 34.739 34.880 34.744 0.348 0.104 0.069657 0.6687024 0.232897

choke 3 0.5556250 44.829 34.342 34.214 34.154 34.198 0.342 0.081 0.068454 0.6571584 0.224926

0.5556250 41.994 34.442 35.160 34.460 34.232 0.346 0.404 0.069147 0.6638112 0.229503

0.5556250 39.159 33.985 33.880 33.883 33.920 0.339 0.049 0.067834 0.6512064 0.220870

0.5953125 50.500 34.327 34.566 34.525 34.624 0.345 0.129 0.069021 0.6626016 0.228667

0.5953125 47.665 33.903 33.900 33.968 33.909 0.339 0.032 0.067840 0.6512640 0.220909

choke 4 0.5953125 44.829 34.116 34.060 34.125 34.190 0.341 0.053 0.068246 0.6551568 0.223558

0.5953125 41.994 33.530 33.496 33.642 33.575 0.336 0.063 0.067122 0.6443664 0.216254

0.5953125 39.159 33.752 33.770 33.690 33.642 0.337 0.059 0.067427 0.6472992 0.218227

0.6350000 50.500 34.020 34.100 34.060 34.003 0.340 0.043 0.068092 0.6536784 0.222550

0.6350000 47.665 32.680 32.700 32.890 32.640 0.327 0.111 0.065455 0.6283680 0.205649

choke 5 0.6350000 44.829 32.980 32.900 33.000 33.020 0.330 0.053 0.065950 0.6331200 0.208771

0.6350000 41.994 32.662 32.547 32.545 32.606 0.326 0.056 0.065180 0.6257280 0.203925

0.6350000 39.159 32.396 32.308 32.308 32.385 0.323 0.048 0.064699 0.6211056 0.200923

0.6746875 50.500 65.420 66.030 65.557 65.890 1.314 0.284 0.262897 2.5238112 3.317512

0.6746875 47.665 68.553 68.399 66.544 68.425 1.360 0.960 0.271921 2.6104416 3.549169

choke 6 0.6746875 44.829 67.865 68.010 67.483 66.725 1.350 0.575 0.270083 2.5927968 3.501352

0.6746875 41.994 66.950 67.031 66.872 66.580 1.337 0.197 0.267433 2.5673568 3.432980

0.6746875 39.159 42.330 45.565 39.980 47.781 0.878 3.448 0.175656 1.6862976 1.481041

0.7143750 50.500 67.011 66.465 66.587 66.732 1.334 0.235 0.266795 2.5612320 3.416619

0.7143750 47.665 66.191 66.020 65.810 65.955 1.320 0.158 0.263976 2.5341696 3.344800

choke 7 0.7143750 44.829 66.547 66.495 66.517 66.488 1.330 0.027 0.266047 2.5540512 3.397488

0.7143750 41.994 64.375 64.501 63.897 64.245 1.285 0.260 0.257018 2.4673728 3.170796

0.7143750 39.159 31.058 31.120 31.110 31.109 0.622 0.028 0.124397 1.1942112 0.742781

0.7937500 50.500 55.730 55.465 55.839 55.599 0.557 0.162 0.111317 1.0686384 0.594785

0.7937500 47.665 63.049 62.945 62.751 63.033 0.629 0.137 0.125889 1.2085344 0.760706

choke 8 0.7937500 44.829 62.701 62.611 62.715 62.561 0.626 0.074 0.125294 1.2028224 0.753532

0.7937500 41.994 55.110 55.090 55.087 55.131 0.551 0.020 0.110209 1.0580064 0.583009

0.7937500 39.159 35.837 36.101 35.844 35.903 0.359 0.123 0.071843 0.6896880 0.247745

0.9921875 50.500 68.581 68.049 68.271 68.568 0.684 0.256 0.136735 1.3126512 0.897424

0.9921875 47.665 67.684 67.702 67.665 67.682 0.677 0.015 0.135367 1.2995184 0.879556

choke 9 0.9921875 44.829 68.525 67.569 68.533 68.503 0.683 0.476 0.136565 1.3110240 0.895200

0.9921875 41.994 33.275 33.263 33.198 33.301 0.333 0.044 0.066519 0.6385776 0.212386

0.9921875 39.159 32.268 32.304 32.279 32.265 0.323 0.018 0.064558 0.6197568 0.200051


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Figure 9. Power outputs for the Stirling engine with nine different choke point diameters and five

different expansion space volumes. The maximum power output is 3.5492 Watts at a choke

diameter of 0.6746875 cm and a volume of 47.665 mL.

Table 2. Average power outputs arranged to create the three-dimensional graph in Figure 9. The

maximum power output is inside the bolded cell.

50.500

47.665

44.829

41.994

39.159

0.0

1.0

2.0

3.0

4.0

0.318 0.476 0.556 0.595 0.635 0.675 0.714 0.794 0.992

Vo

lum

e (

mL)

Ave

rage

Po

we

r (W

)

Choke size (cm)

Average Power 3.0-4.0 2.0-3.0 1.0-2.0 0.0-1.0

Average Power (W)

Choke sizes (cm)

0.31750 0.47625 0.55563 0.59531 0.63500 0.67469 0.71438 0.79375 0.99219

50.500 0.062413 0.050652 0.229758 0.228667 0.222550 3.317512 3.416619 0.594785 0.897424

Volume 47.665 0.196500 0.048946 0.232897 0.220909 0.205649 3.549169 3.344800 0.760706 0.879556 Pavg(mL) 44.829 0.176984 0.058166 0.224926 0.223558 0.208771 3.501352 3.397488 0.753532 0.895200 (W)

41.994 0.195920 - 0.229503 0.216254 0.203925 3.432980 3.170796 0.583009 0.212386

39.159 0.200523 - 0.220870 0.218227 0.200923 1.481041 0.742781 0.247745 0.200051


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Data Analysis and Discussion

The data reveal a specific point at which the power output of the engine is at its

maximum. With a choke point diameter of 0.6746875 cm and a volume of 47.665 mL, the

greatest power output is 3.5492 Watts. The goal of this engineering project was to find the

optimal choke diameter and expansion space volume to produce the greatest power output.

A trend in the data is that as the choke point diameter increases so does the power output,

to a certain extent. If the choke becomes too large, the power output drastically drops off. A

similar trend occurs with the expansion space volume. The percent increase from the weakest

power output to the strongest power output was 7151%. From the standard choke point diameter

(0.625 cm) and standard volume (50.5 mL) to the largest power output there was a percent

increase of 1495%. The percent decrease from the largest power output to the largest choke point

tested was 94%. There was a huge increase from the weakest to the strongest and there was also

a large increase from the standard choke and volume used to the setting with the most power

output. The drop off after the strongest power output was slight in comparison with the other

numbers. For both the choke diameters and the expansion space volume, the optimal setting

tends to be in the middle. Too large, and there is too much air, too small and there is not enough

air to heat up.

An anomaly occurred with choke point 2 (0.47625 cm) as the engine would not run with

the smallest two volumes. A possible explanation for this is that at the small choke point size

more hot air was needed to power the engine. However, this does not explain why the smallest

choke point size (0.3175 cm) still ran at all volumes.

Sources of error in this project include slight changes in the temperature of the engine

over time, though this issue was minimized as much as possible by keeping the same level of

alcohol in the burner, and allowing the engine to cool off before testing the next choke point.

Additionally, at certain choke point diameters the speed was erratic, and significant standard

deviations of 1.829 to 0.453 were achieved. This occurred in the two smallest chokepoints

(0.3175 and 0.47625 cm).

With the adapted quarter turn ball valve the engine did not run on any of the choke or

volume settings. In preliminary analysis the quarter turn ball valve was determined to be the best

choice for this project. Possible reasons why the ball valve was not successful in varying the

choke diameter while the engine was running include the change in direction of the airflow due

to the mechanics of the valve. Rather than channeling the air straight through, the valve changed

the direction and eliminated the laminar flow necessary for the engine to run. Additionally, the

copper material of the valve absorbed the heat from the working fluid and so the working fluid

compressed before reaching the piston.

Conclusions

It was concluded that the optimal setting of the Stirling engine was a choke point of

diameter 0.6746875 cm and a volume of 47.665 mL and the greatest power output was 3.5492

Watts. The engineering goal was to develop a variable choke point for a beta type Stirling

engine. While testing a variable choke point was not accomplished, the results of this project

indicate that continuing research might be valuable. Nine choke points have been tested and the


14

differences in power output are large enough to encourage pursuing a choke point that can be

varied while the engine is running.

Despite the variable choke point being inoperative, several points can be taken from this

experiment. It is of utmost importance to preserve the laminar flow of air through the engine in

order to have it run. Rather than having a choke which diverts the flow of air when its diameter is

changed, a choke which can expand and contract in a circle (like the iris of a camera) would be

desired. However it was beyond the scope of this project to obtain or manufacture a choke

similar to the iris of a camera. Moreover, the material the choke and cylinder are made of should

not be a material with a high heat transfer to withstand the high heat of the engine.

Limitations and Assumptions

The current device is limited in the power it produces. The engine is on a small scale and

does not generate a useful power output. By itself, the engine is not designed to produce much

power, but is designed to only be a testing ground for the variable choke point. The focus of this

project was not to use the power outputted. Instead, the focus was on developing a variable

choke point for use in a Stirling engine.

Another limit of this device is that it does not work for Stirling engines that use working

fluids other than air. Most Stirling engines in commercial applications use hydrogen or helium as

the working fluid. Using the current design would result in engine failure because the materials

used would allow a leakage of the working fluid.

The largest limitation of this device is that the variable choke point changes the direction

of the air flow and so the engine cannot run. A solution to this problem would be to develop a

valve like the iris of a camera that would not change the direction of the airflow, but only the

diameter of the choke point.

An assumption in this investigation is that the simplified Stirling engine used in the

development of the variable choke point models the larger, more complex ones accurately. With

this assumption the choke point can be easily scaled up in order to work with larger, more

powerful engines where it would be of more use. The current engine is not one used in many

applications and so using the device as it stands would not result in a useable amount of power.

Additionally, this project assumes that the engine functions in the same way every time.

Applications and Future Experiments

The variable choke point can be enlarged for use in larger commercial engines. The

current size of the engine seriously limits the power output and so a larger, more powerful engine

would use the choke point more effectively. The Stirling engines used in marine applications,

such as submarines, can utilize the variable choke point to adjust the power output of the engine

on the fly. This can be very useful because it allows the engine to, based on its needs, adjust the

power output and energy needed to run it. Consequently, when the engine does not need the full

power output, the choke point can be changed and the amount of heat needed to power the

engine can be lessened.

Possible extensions of research include modifying the choke point to work with different

types of working fluids. In most applications, Stirling engines use helium or hydrogen as a

working fluid and to safely contain these light molecules the choke point would need to be made

of different materials to prevent leakage of the working fluid.


15

Additional experiments can be performed to determine more precisely the best volume of

the expansion space for each choke point diameter. The experiments performed in this project

only found the power output for seven different choke point diameters and five different volumes

for each diameter. Further research could determine the power output for more choke point

diameters and expansion space volumes.

In the current engine the differences between the choke point sizes of the three highest

power outputs are so close that precise machinery would be needed to determine if there is a

choke point diameter that produces more power than the current one. In a commercial sized

engine however, the small differences between the settings would increase. More testing should

be done when the choke points are scaled up to determine if there is another choke diameter

which produces more power in between the choke diameters tested in this project.

Another extension of this project would be to develop a variable choke point that does

not alter the direction of the flow of the working fluid. This would allow changing the diameter

to be done while the engine was running. Having the choke point mimic the iris of a camera

might be one way to achieve this.

Literature Cited

Abdulrahman, A. S. (2011). Selection and experimental evaluation of low-cost porous materials

for regenerator applications in thermoacoustic engines. Materials & Design, 32(1), doi:

10.1016/j.matdes.2010.06.012

Energy Conversion. (2010.) In Encyclopedia Britannica. Retrieved September 22, 2010, From

Encyclopedia Britannica Online: www.britannica.com

Hargreaves, C., M., (1991). The Philips Stirling engine. New York: Elsevier Science Publishers.

Hooper, C., Reader, G. T., (1983). Stirling engines. Cambridge: University Press.

Kroliczek, E. J., Nikitkin, M., Wolf, D. A. (2010) U.S. Patent No. 7,708,053. Washington D.C.:

U.S. Patent and Trademark Office.

Lin, P. (2010). U.S. Patent No. 7,712,310. Washington D.C.: U.S. Patent and Trademark Office.

Sandfort, J. F. (1962) Heat engines. New York: Anchor Books Doubleday & Company, INC.

Solar-13 Stirling engine model. (2010). Retrieved from http://www.pmresearchinc.com/store/

product.php?productid=3101&cat=5&page=1

Stirling Engines. (2010). In Access Science. Retrieved September 22, 2010, Access Science

online: www.accessscience.com

Urieli, I. (2010, September). Stirling cycle machine analysis. Retrieved from

http://people.ohio.edu/urieli/stirling/me422.html


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Walker, G. (1980) Stirling engines. Oxford, England: Clarendon Press

Appendix

Standard Physics Equations Used:

Acknowledgements

The author wishes to thank several mentors who assisted in various aspects of this

project. Mr. Torbjorn Bergstrom kindly provided ongoing guidance and use of his lab and

materials. He willingly gave materials and advice regarding construction of the engine. Mr.

William Ellis additionally gave guidance and support in experiment design. The author also

wishes to thank her parents for being supportive, no matter how many times she forgot her

camera batteries.

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