Stirling Eng

Q: How do Stirling Engines work?A: Stirling engines can be hard to understand. Here are the key points. EveryStirling engine has a sealed cylinder with one part hot and the other cold. Theworking gas inside the engine (which is often air, helium, or hydrogen) is movedby a mechanism from the hot side to the cold side. When the gas is on the hot sideit expands and pushes up on a piston. When it moves back to the cold side itcontracts. Properly designed Stirling engines have two power pulses perrevolution, which can make them very smooth running. Two of the more commontypes are two piston Stirling engines and displacer-type Stirling engines. The twopiston type Stirling engine has two power pistons. The displacer type Stirlingengine has one power piston and a displacer piston.

Displacer Type:The displacer type Stirling engine is shown here. The space below the displacerpiston is continuously heated by a heat source. The space above the displacerpiston is continuously cooled. The displacer piston moves the air (displaces theair) from the hot side to the cold side.

Displacer Engine Detail:

Click here for animation...

Gas expands when heated, and contracts when cooled. Stirling engines move the gas from thehot side of the engine, where it expands, to the cold side, where it contracts.

DISPLACER PISTONWhen there is a temperature difference between upper displacer space and lower displacerspace, the engine pressure is changed by the movement of the displacer. The pressure increaseswhen the displacer is located in the upper part of the cylinder (and most of the air is on the hotlower side). The pressure decreases when the displacer is moved to the lower part of the cylinder.The displacer only moves the air back and forth from the hot side to the cold side. It does notoperate the crankshaft and the engine. In other words, the connecting rod to the displacer couldbe a string in this engine and it would still work.

POWER PISTONWhen the engine pressure reaches its maximum because of the motion of the displacer, a powerpiston is pushed by the expanding gas adding energy to the crankshaft. The power piston shouldideally be 90 degrees out of phase with the displacer piston. The displacer type Stirling engine isoperated by the power of the power piston.

A special thanks to Koichi Hirata for the excellent illustrations!

Two Piston Type:The two piston type Stirling engine is shown here. The space above the hot piston iscontinuously heated by a heat source. The space above the cold piston is continuouslycooled.

Two-Piston Engine Detail:

Click here for animation...

HEATINGLet's start from top dead center of the hot piston. The hot piston moves to the upper part of thecylinder and the cold piston moves to the lower part of the cylinder during the first 90 degrees ofrevolution. The working air is moved from the cold space to the hot space. And the pressure in theengine is increased.

EXPANSIONDuring the next 90 degrees of revolution, the two pistons both move the lower part accepting theair pressure. The engine gets its power during this portion of its cycle.

COOLINGThe crankshaft revolves by power stored in the flywheel for the next 90 degrees. The hot pistonmoves to the lower part and the cold piston moves to the upper part. The air is moved from thehot space to the cold space. And the pressure in the engine is decreased.

CONTRACTIONThe two pistons are moved to upper part by the contraction of the air during the next 90 degrees.The engine also gets power during this portion of its cycle. The two piston type Stirling engine thenrepeats this cycle.

A special thanks to Koichi Hirata for the excellent illustrations!

1. Q: Are Stirling engines really the most efficient engines possible?

A: In the mid 1800's a very bright Frenchman named Sadi Carnot figured out themaximum efficiency possible with any heat engine. It is a formula like this(Temperature of the hot side - Temperature of the cold side)/Temp of hot side x100 equals the max theoretical efficiency. Of course the temperatures must bemeasured in degrees Kelvin or Rankine. Stirling engines (with perfectregeneration) match this cycle. Real Stirling engines can reach 50 percent of themaximum theoretical value. That is an incredibly high percentage!

1. Q: If Stirling engines are so efficient, why don't I have one in my car?

A: The best answer for that is to pick the MM-1 engine up after it gets up tospeed. Notice that it keeps running for a minute or so. While it's very easy to builda Stirling engine that will stop instantly, there is not one thing in the world anyonecan do to make one start instantly. When I get in my car I want it to startimmediately (if not sooner) and be able to burn rubber off the tires as I leave theparking lot! Stirling engines can't do that. In spite of these limitations, Ford, GM,and American Motors Corp. spent millions of dollars developing Stirling enginesfor cars, back in the 1970's. Ford even built a Stirling that could drive away fromthe curb (with relatively low power) twenty seconds after you turned the start key!Many prototypes were built and tested. Then oil prices came down in the 1980's,and people started to buy bigger cars. Suddenly there was no compelling reason tobuild an engine that was substantially more efficient than internal combustionengines, but wouldn't start instantly. Here is a picture of a 1979 AMC Spirit. Itwas equipped with an experimental Stirling engine powerplant called the "P-40".The Spirit was capable of burning gasoline, diesel, or gasohol. The P-40 Stirlingengine promised less pollution, 30% better mileage, and the same level ofperformance as the car's standard internal combustion engine. [From "AnIntroduction to Stirling Engines"] The French Research Sub Saga is Stirlingengine powered. Stirling engines also work exceptionally well as auxiliary powergenerators/heaters on yachts (see Victron Energy.), where their silence is valued

and good cooling water is available. They would also work very well in airplaneswhere the air gets colder as the plane climbs to altitude. There is no aircraft powerplant (jets included) that gets any improvement in any operating conditions fromclimbing. Stirling engines won't lose as much power as they climb as do eitherpiston engines or jets. Also wouldn't you like to have silent airplanes with veryefficient engines that also have exceedingly low vibration levels?

1. Q: What are Stirling engines being used for today?

A: The modern uses of Stirling engines are invisible to almost everyone. Therehave been many research engines built in recent years but there are only threeareas where Stirling engines have made a dramatic impact. There are Stirlingengines in Submarines, stirling machines used as cryocoolers, and Stirling enginesin classrooms. Cryogenics is the science of things that are exceedingly cold andStirling engines are one tool that can be used to make things exceedingly cold. It'snot obvious but a Stirling engine is a reversible device. If you heat one end andcool the other, you get mechanical work out, but if you put mechanical work in,by connecting an electric motor, one end will get hot and the other end will getcold. If you design the machine correctly, the cold end will get extremely cold. Infact, Stirling coolers have been made that will cool below 10 degrees Kelvin.Micro Stirling coolers have been produced in large numbers for cooling infraredchips down to 80 degrees Kelvin for use in night vision devices.

1. A good general guideline is that if the hot side of the engine is not at least 500deg. F. (260 deg. C) the engine will be too bulky for the amount of power it putsout.

1. I don't think there is a theoretical upper limit on power in a Stirling engine. 2. Thebigger the temperature difference the easier it is to get power out of a smallengine. In other words to put out any significant amount of power an enginerunning on small temperature differences would have to be physically very large.3. The place where metals are critical is in the hot side of the engine. If you aregoing to build an engine that puts out a significant amount of power you willprobably want to build the heater head out of at least a good grade of stainlesssteel and perhaps a more exotic metal like Inconnel or Hasteloy.

Modern Stirling Engine Development

Today, there are many companies developing Stirling devices for niche markets, such as cogenerationunits and power generation using alternative fuels. Stirling engines have come a long way from the large andheavy engines of the 19th century, thanks to advancements in materials, manufacturing processes, theory andanalysis methods. This page contains a handful of links to some of these companies. Click on the images to learn moreabout these organizations and the engines they produce.

All images and information related to these devices are property of and are assumed to be copyrighted bytheir respective owners.

STM Corporation

SOLO Kleinmotoren GmbH

Stirling Energy Systems, Inc.

Kockums Sweden.

Sunpower, Inc.

Infinia Corporation

Tamin Enterprises

NASA Glenn Research Center

The Stirling Engine's most basic configuration consists of two pistons each in itsown cylinder. (Sometimes it is easier to envision these two cylinders as one longtube with the piston heads facing each other inside the tube (see the figurebelow)). Note that between these two pistons heads are the heater, cooler andregenerator. The regenerator (usually a block of woven wire) is in the center ofthis tube and the heater is between the regenerator and one piston (in red) whilethe cooler is between the regenerator and the other piston (in Blue). The volume

attached to the 'heater' is the 'expansion space' where the hot gas pushesagainst the 'expansion piston'. The volume attached to the 'cooler' is the'compression space'. The regenerator is where the excess heat of the gas is stored in theregenerator matrix on the way to the compression space from the expansionspace and then the heat is recovered on the way back from the compressionspace to the expansion space.

Graphic courtesy of Dr. Israel Urieli of Ohio University.

Stirling Engine operation can be explained in a non technical way that applies tomany but not to all engines that may be called Striling Engines.

The working gas trapped between the two piston heads is pushed by theCompression Piston through the regenerator where it is heated by the energy inthe regenerator heated to its hottest in the Heater and expands in the ExpansionSpace. This increased pressure pushes on the Expansion Piston so that itmoves away from the regenerator pushing on a mechanism which changes thelinear movement of the piston to a rotary motion. This continues until all the gasthat will expand has been pushed into the heater area and expanded. Themechanism also pushes the Compression Piston further toward the Regeneratorpushing all the gas out of the Compression Space in to the gas circuit (heater,cooler, regenerator).

Then the mechanism, to which both pistons are connected (but 90 degreesapart), begins to move the Expansion piston back the other way pushing the hotgas through the Heater backwards and then on to the Regenerator and finallyinto the Cooler where it begins to Cool and contract (the pressure starts todrop). The Compression Piston is also moving away from the regenerator whilethe Expansion piston comes toward the regenerator moving the gas through theregenerator into the compression space without compressing the gas.

The linkage continues to move the pistons until the Compression Piston is all theway back and the Expansion piston is all the way forward. At this point the

mechanical arrangement moves the pistons together but because of the way thepiston moves up and down in the cylinder and the mechanism is moving in acircle, the Expansion Piston does not move very far but the Compression Pistonmoves toward the regenerator actually compressing the gas and begining topush the gas through the regenerator. (That is why it is called the CompressionPiston.)

This brings us to the first line of this explanation to complete the cycle and beginagain.

Stirling Engineering (Technical Explanation)

First Approximation of the power of a Stirling Engine (kinematic or free piston)Power = (Beale.number) x (pressure(mean)) x (Volume Exp) x (frequency)Watts = 0.116 x Pascals(10E-6) x (Cm^3) x (Hz)

A Stirling "Air" Engine is a mechanical device which operates on a closedregenerative thermodynamic cycle with cyclic compression and expansion of theworking fluid (air) at different temperature levels. The flow of the working fluid iscontrolled by changes in the volume of the hot and cold spaces, eliminating theneed for valves. The Stirling Engine is reversible, meaning that an input of heatenergy (burning fuel, for example) will produce an output of mechanical energy,and an input of mechanical energy (electric motor, etc.) will produce an output ofheat energy. In this manner, the Stirling Engine can be used as a heat pump inmuch the same way as traditional refrigeration units, only without theenvironmentally harmful refrigerants.

The most basic engine consists of a set of pistons, heat exchangers, and adevice called a 'regenerator'. The engine is filled with a working fluid (gas) whichis commonly Air, but some more advanced engines may use Nitrogen, Helium orHydrogen. The pistons are arranged such that they create both a change involume of the working fluid and create a net flow of the fluid through the heatexchangers. In this manner, heat is absorbed from an external source in the 'hot'end, creating mechanical energy, and rejected in the 'cold' end to theenvironment.

In a Stirling engine, the working fluid is completely contained inside the engine atall times, meaning the cycle is closed, As opposed to a typical gasoline engine,which takes in 'fresh air' for each new cycle. This enables a Stirling Engine tooperate cleanly and quietly as there are no combustion products coming into

contact with any of the engine's working components and no release ofhigh-pressure gasses.

An important feature in Stirling Engines is the regenerator. On the most basiclevel, a regenerator is a device that absorbs heat from the working fluid as itenters the 'hot' end, and re-heats the fluid as it enters the 'cold' end. This internalrecycling of energy allows for much higher efficiencies, and better performanceoverall. The regenerator is such a critical component that most Stirling Enginescannot operate efficiently without one!

Stirling Engineering ( Deeper understanding)This link is a much deeper look into the theory. Click Here for a look at theDetailed Theory of Operation.This information is from Dr. Israel Urieli of Ohio University. Caution: Containscalculus, partial differential equations and thus requires a knowledge of thecalculus. Also contains source code modules for a second order simulator (In'C').

Stirling Engines - Mechanical ConfigurationsThe mechanical configurations of Stirling engines are generally divided into three groupsknown as the Alpha, Beta, and Gamma arrangements. Alpha engines have two pistons inseparate cylinders which are connected in series by a heater, regenerator and cooler. BothBeta and Gamma engines use displacer-piston arrangements, the Beta engine having boththe displacer and the piston in an in-line cylinder system, whilst the Gamma engine usesseparate cylinders.

The Alpha engine is conceptually the simplest Stirling engine configuration,however suffers from the disadvantage that both pistons need to have seals tocontain the working gas. Andy Ross of Columbus, Ohio has been developingsmall air engines with extremely innovative Alpha designs, including the classicalRoss-Yoke drive and more recently a balanced "Rocker-V" mechanism, asshown below.

The Alpha engine can also be compounded into a compact multiple cylinderconfiguration, enabling an extremely high specific power output, as is required ofan automotive engine. A schematic diagram of this configuration is shown below.Notice that the four cylinders are interconnected, so that the expansion space ofone cylinder is connected to the compression space of the adjacent cylinder viaa series connected heater, regenerator and cooler. The pistons are typicallydriven by a swashplate, resulting in a pure sinusoidal reciprocating motionhaving a 90 degree phase difference between the adjacent pistons.

Beta Type Stirling EnginesThe Beta configuration is the classic Stirling engine configuration and hasenjoyed popularity from its inception until today. Stirling's original engine from hispatent drawing of 1816 shows a Beta arrangement. A photograph of RobertStirling, the original patent drawing, as well as an animated model of Stirling'sengine is clearly shown in an interesting website by Bob Sier. Another importantearly Beta engine is Lehmann's machine on which Gusav Schmidt did the firstreasonable analysis of Stirling engines in 1871.

From the figure we see that unlike the Alpha machine, the Beta engine has asingle power piston and a displacer, whose purpose is to "displace" the workinggas at constant volume, and shuttle it between the expansion and thecompression spaces through the series arrangement cooler, regenerator, andheater.

Rolf Meijer of Philips, Holland, derived his famous vibrationless rhombic drive forBeta engines in the early 1960s.Probably the most ingenious Stirling engines yet devised are the free-pistonengines invented and developed by William Beale at Ohio University in the late1960s. He later formed the company Sunpower, Inc., which has been the leaderin the development of free-piston Stirling engines and cryocoolers to this day. Allof Sunpower's engines are Beta arrangements and employ no mechanicallinkage system. The main aspect of the free piston machine is that the outputpower can be obtained through a linear alternator, allowing the entire system tobe hermatically sealed. Sunpower have recently begun to manufacture Stirlingcycle croygenic coolers for liquifying oxygen. Over the years Sunpower hastransformed Athens, Ohio into a hotbed of Stirling cycle machine activity, whichnow includes four R&D and manufacturing companies as well as oneinternationally recognized consultant in the area of Stirling cycle computeranalysis.Stirling Technology Inc. is a spinoff of Sunpower, and was formed in order tocontinue the development and manufacture of the 5 kW ST-5 Air engine. Thislarge Beta type engine burns biomass fuel (such as sawdust pellets or ricehusks) and can function as a cogeneration unit in rural areas. It is not afree-piston engine, and uses a bell crank mechanism to obtain the correctdisplacer phasing.Global Cooling is a licencee of Sunpower, mainly in order to develop free-pistonStirling cycle coolers for home refrigerator applications. These systems, apartfrom being significantly more efficient than regular vapor-compressionrefrigerators, have the addad advantage of being compact, portable units usinghelium as the working fluid (and not the Ozone destroying CFCs).External Power is a very recent licencee of Sunpower, and was formed tomanufacture biomass fueled (sawdust pellets) free-piston cogeneration units forhome use.

Gamma Type Stirling EnginesGamma type engines have a displacer and power piston, similar to Betamachines, however in different cylinders. This allows a convenient completeseparation between the heat exchangers associated with the displacer cylinderand the compression and expansion work space associated with the piston.Thus they tend to have somewhat larger dead (or unswept) volumes than eitherthe Alpha or Beta engines.

Furthermore during the expansion process some of the expansion must takeplace in the compression space leading to a reduction of specific power. Gammaengines are therefor used when the advantages of having separate cylindersoutweigh the specific power disadvantage.Because of the convenience of two cylinders in which only the piston has to besealed, the gamma configuration is a favorite among modellers and hobbyists.

Ideal Isothermal AnalysisThe invention of the Stirling engine in 1826 was well in advance of all pertinentscientific knowledge of that time. The first attempt at an analysis of the cycle waspublished in 1871 by Gustav Schmidt. Much as the Otto cycle has become theclassic Air standard cycle to describe the spark ignition engine, the cycledescribed by Schmidt has become the classic ideal Stirling cycle. This isunfortunately mainly because the Schmidt analysis yields a closed form solutionrather than its ability to predict the real cycle, however we use it as a startingpoint to guide us ultimately to a more realistic approach.

Consider the Ideal Isothermal model of a Stirling engine as shown below.

The principal assumption of the analysis is that the gas in the expansion spaceand the heater is at the constant upper source temperature and the gas in thecompression space and the cooler is at the constant lower sink temperature.This isothermal assumption makes it possible to generate a simple expressionfor the working gas pressure as a function of the volume variations. Thisexpression may then be used to investigate how different drive mechanismsaffect the output power. To obtain closed form solutions, Schmidt assumed thatthe volumes of the working spaces vary sinusoidally.The assumption of isothermal working spaces and heat exchangers implies thatthe heat exchangers (including the regenerator) are perfectly effective, with aspacial temperature distribution as indicated in the figure above. The engine isconsidered as a five component serially connected model, consistingrespectively of a compression space c, cooler k, regenerator r, heater h andexpansion space e. Each component is considered as a homogeneous entity orcell, the gas therein being represented by its instantaneous mass m, absolutetemperature T, volume V and pressure p, with the suffix c, k, r, h, and eidentifying the specific cell.The starting point of the analysis is that the total mass of gas in the machine isconstant, thus:

M = mc + mk + mr + mh + me

Substituting the ideal gas law given by

m = p V / R T

we obtain

M = p (Vc / Tk + Vk / Tk + Vr / Tr + Vh / Th + Ve / Th) / R

For the assumed linear temperature in the regenerator we can show that theeffective regenerator temperature Tr is given by

Tr = (Th - Tk) / ln(Th / Tk)

Thus given the volume variations Vc and Ve we can solve the above equation forpressure p as a function of Vc and Ve.

The work done by the system over a complete cycle is given respectively by thecyclic integral of p dV

On evaluating the heat transferred over a complete cycle to the various cells wefind remarkably that the cyclic heat transferred to all three heat exchanger cellsis zero! Thus:

Qc = WcQe = WeQk = 0Qh = 0Qr = 0

This rather startling result implies that all the heat exchangers in the ideal Stirlingengine are redundant since all the external heat transfer occurs across theboundaries of the compression and expansion spaces. This apparent paradox isa direct result of the definition of the Ideal Isothermal model in which thecompression and expansion spaces are maintained at the respective cooler andheater temperatures. Obviously this cannot be correct, since the cylinder wallsare not designed for heat transfer. In real machines the compression andexpansion spaces will tend to be adiabatic rather than isothermal, which impliesthat the net heat transferred over the cycle must be provided by the heatexchangers. This will be resolved when we consider the Ideal Adiabatic model inthe next section.

The set of pertinent equations is shown in the following table.

In order to solve these equations we need to specify the working space volumevariations Vc and Ve as well as the respective volume derivatives dVc and dVewith respect to crankangle . One of the case studies of this course is the RossYoke-drive engine for which we have analized the volume variations, thus theabove equation set can be solved by numerical integration. In 1871 GustavSchmidt published an analysis in which he obtained closed form solutions for theabove equation set for the special case of sinusoidal volume variations. Wecontinue now with the Schmidt analysis.

The Schmidt AnalysisIn the previous section we derived the basic set of equations which describe theIdeal Isothermal model, as shown in the following table.

Gustav Schmidt of the German Polytechnic Institute of Prague Published ananalysis in 1871 in which he obtained closed form solutions of these equationsfor the special case of sinusoidal volume variations of the working spaces withrespect to the cycle angle . Consider the following diagram showing the volumevariations of the compression and expansion spaces (Vc and Ve) over a singlecycle. Notice the phase advance angle of the expansion space volumevariation with respect to the compression space volume variation:

The sinusoidal volume variations of the compression and expansion spaces arerespectively as follows:

Vc = Vclc + Vswc (1 + cos ) / 2

Ve = Vcle + Vswe (1 + cos( + )) / 2

where Vcl and Vsw represent respectively clearence and swept volumes, andis the cycle angle. Substituting for Vc and Ve in the pressure equation above andsimplifying we obtain

In order to simplify the pressure equation we now consider a trigonometricsubstitution of and c as defined by the following right-angled triangle

Substituting for and c in the pressure equation above and simplifying

where

= +

b = c / s

The maximum and minimum values of pressure can now be evaluated for theextreme values of cos

The average pressure over the cycle is given by

From tables of integrals, this reduces to

This equation is the most convenient way of relating the total mass of workinggas in the cycle to the more conveniently specified mean operating pressure.

The net work done by the engine is the sum of the work done by thecompression and expansion spaces. Over a complete cycle

W = Wc + We

The volume derivatives are obtained by differentiating Vc and Ve above

Substituting these and the pressure equation into the equations for Wc and We

The solution of these integrals requires the judicious use of tables of integralsand is done in the book by Urieli & Berchowitz, "Stirling Cycle Machine Analysis",Adam Hilger 1984. The book itself is out of print, however the relevant appendixin this book that deals with the Schmidt analysis has been placed on the web byGlobal Cooling, and can be downloaded in Acrobat pdf format. Finally we obtain

Ideal Adiabatic AnalysisIn the previous section we considered an ideal Stirling engine model in which thecompression and expansion spaces were maintained at the respective coolerand heater temperatures. This led to the paradoxical situation that neither theheater nor the cooler contributed any net heat transfer over the cycle and hencewere redundant. All the required heat transfer occurred across the boundaries ofthe isothermal working spaces. Obviously this cannot be correct, since thecylinder walls are not designed for heat transfer. In real machines the workingspaces will tend to be adiabatic rather than isothermal, which implies that the net

heat transferred over the cycle must be provided by the heat exchangers. Wethus consider an alternative ideal model for Stirling cycle engines, the IdealAdiabatic model.

As before the engine is configured as a five component serially connected modelhaving perfectly effective heat exchangers (including the regenerator) and in thisrespect is similar to the Ideal Isothermal model defined previously. However boththe compression and expansion spaces are adiabatic, in which no heat istransferred to the surroundings. In the following diagram we define the IdealAdiabatic model nomenclature. Thus we have a single suffix (c, k, r, h, e)representing the five cells, and a double suffix (ck, kr, rh, he) representing thefour interfaces between the cells. Enthalpy is transported across the interfaces interms of a mass flow rate m' and an upstream temperature T. The arrows on theinterfaces represent the positive direction of flow, arbitrarily defined from thecompression space to the expansion space.

Notice from the temperature distribution diagram that the temperature in thecompression and expansion spaces (Tc and Te) are not constant, but vary overthe cycle in accordance with the adiabatic compression and expansion occurringin the working spaces. Thus the enthalpies flowing across the interfaces ck andhe carry the respective adjacent upstream cell temperatures, hence

temperatures Tck and The are conditional on the direction of flow and aredefined algorithmically as follows:

if mck' > 0 then Tck = Tc else Tck = Tkif mhe' > 0 then The = Th else The = Te

In the ideal model there is no gas leakage, the total mass of gas M in the systemis constant, and there is no pressure drop, hence p is not suffixed andrepresents the instantaneous pressure throughout the system.Work W is done on the surroundings by virtue of the varying volumes of theworking spaces Vc and Ve, and heat Qk and Qh is transferred from the externalenvironment to the working gas in the cooler and heater cells, respectively. Theregenerator is externally adiabatic, heat Qr being transferred internally from theregemerator matrix to the gas flowing through the regenerator void volume Vr.

Development of the equation setThe general approach for deriving the equation set is to apply the equations ofenergy and state to each of the cells. The resulting equations are linked byapplying the continuity equation across the entire system. Consider first theenergy equation applied to a generalised cell which may either be reduced to aworking space cell or a heat exchanger cell. Enthalpy is transported into the cellby means of mass flow mi' and temperature Ti, and out of the cell by means ofmass flow mo' and temperature To. The derivative operator is denoted by d, thusfor example dm refers to the mass derivative dm/d , where is the cycle angle.

The word statement of the energy equation for the working gas in thegeneralised cell is

Mathematically, this word statement becomes

dQ + (cp Ti mi' - cp To mo') = dW + cv d(m T)

where cp and cv are the specific heat capacities of the gas at constant pressureand constant volume respectively. This equation is the well known classical formof the energy equation for non steady flow in which kinetic and potential energyterms have been neglected.

We assume that the working gas is ideal. This is a reasonable assumption forStirling engines since the working gas processes are far removed from the gascritical point. The equation of state for each cell is presented in both its standardand differential form as follows

p V = m R T

dP / p + dV / V = dm / m + dT / T

The starting point of the analysis is that the total mass of gas in the machine isconstant, thus:

mc + mk + mr + mh + me = M

Substituting for the mass in each cell from the ideal gas law above

p (Vc / Tc + Vk / Tk + Vr / Tr + Vh / Th + Ve / Te) / R = M

where for the assumed linear temperature profile in the regenerator the meaneffective temperature Tr is equal to the log mean temperature difference Tr = (Th- Tk) / ln(Th / Tk). Solving the above equation for pressure

p = M R /(Vc / Tc + Vk / Tk + Vr / Tr + Vh / Th + Ve / Te)

Differentiating the equation for mass above

dmc + dmk + dmr + dmh + dme = 0

For all the heat exchanger cells, since the respective volumes and temperaturesare constant, the differential form of the equation of state reduces to

dm / m = dp / p

dm = dp m / p = (dp / R) V / T

Substituting in the mass equation above

dmc + dme + (dp / R) (Vk / Tk + Vr / Tr + Vh / Th) = 0

We wish to eliminate dmc and dme in the above equation so as to obtain anexplicit equation in dp. Consider the adiabatic compression space (dQc = 0).

Applying the above energy equation to this space we obtain

-cp Tck mck' = dWc + cv d(mc Tc)

From continuity considerations the rate of accumulation of gas dmc is equal tothe mass inflow of gas given by -mck', and the work donw dWc is given by pdVc, thus

cp Tck dmc = p dVc + cv d(mc Tc)

Substituting the ideal gas relations p Vc = mc R Tc, cp - cv = R, and cp / cv = ,and simplifying

dmc = (p dVc + Vc dp / ) / (R Tck)

Similarly for the expansion space

dme = (p dVe + Ve dp / ) / (R The)

Substituting for dmc and dme above and simplifying

From the differential form of the equation of state above we obtain relations dTcand dTe

dTc = Tc (dp / p + dVc / Vc - dmc / mc)

dTe = Te (dp / p + dVe / Ve - dme / me)

Applying the energy equation above to each of the heat exchanger cells (dW =0, T constant) and substituting for the equation of state for a heat exchanger cell(dm = dp m / p = (dp / R) V / T)

dQ + (cp Ti mi' - cp To mo') = cv T dm = V dp cv / R

Thus for the three heat exchanger cells we obtain

dQk = Vk dp cv / R - cp (Tck mck' - Tkr mkr')

dQr = Vr dp cv / R - cp (Tkr mkr' - Trh mrh')

dQh = Vh dp cv / R - cp (Trh mrh' - The mhe')

We note that since the heat exchangers are isothermal and the regenerator isideal, Tkr = Tk and Trh = Th.Finally the work done in the compression and expansion cells is given by

W = Wc + We

dW = dWc + dWe

dWc = p dVc

dWe = p dVe

Stirling Engine Simple AnalysisOnce we have done an Ideal Adiabatic analysis on a specific Stirling engine, wewould like to evaluate the heat transfer and flow-friction effects of the three heatexchangers on the performance of the engine. This will enable us to do aparametric sensitivity analysis as required for design optimization.Forced convection heat transfer is fundamental to Stirling engine operation. Heatis transferred from the external heat source to the working fluid in the heatersection, cyclicly stored and recovered in the regenerator, and rejected by theworking fluid to the external heat sink in the cooler section. All of this is done incompact heat exchangers (large wetted area to void volume ratio) so as to limitthe "dead space" an acceptable value and thus allow for a reasonable specificpower output of the engine. We find that effective heat exchange comes at aprice of increased flow friction, resulting in the so-called "pumping loss". This lossrefers to the mechanical power required to "pump" the working fluid through theheat exchangers, and thus reducing the net power output of the engine.The theory and analysis of these effects is extremely complex, and we find thatwe can only rely on the plethora of documented experimental and empiricalstudies ( e.g. Kays & London ,"Compact Heat Exchangers"). Almost all of thisvast body of work is based on steady flow conditions and is thus not directlyapplicable to the oscillating flow conditions that apply to Stirling engines. In thissection we adopt a "Quasi-Steady Flow" approach, in that we assume that ateach instant of the cycle the fluid behaves as though it is in steady flow. Thus wehave called this analysis a "Simple" analysis because it is a gross simplificationof an extremely complex process. At this stage there is still a major controversyabout this approach, and we need to treat the results of this analysis with ahealthy measure of scepticism. The only alternative for design is the recent"Similarity and Scaling" approach which has been developed by Allan Organ andis presented in his book "The Regenerator and the Stirling Engine".

Design Process To establish a starting point for designing the Idaho Stirling Engine, webegan by modifying an existing Stirling engine that was developed by TedBoyl-Davis, a graduate student at the University of Idaho.

Variables

The following independent design variables were manipulated during theiterative design process of the ISE and empirically evaluated. The variableswere evaluated based on the previous list of design parameters:

· Type of energy source (Incandescent vs. Halogen light bulb)· Type of pressurized container (Tin vs. Glass/Pyrex)· Use of Insulation· Size of energy source (50, 100, 150 or 200 W bulbs)· Diaphragm material (balloon vs. inner-tube rubber)· Ice water bath or air-cooling fins· Solid vs. split displacer· Base design for energy source (can vs. plaster/putty)· Use of brass bushings· Seal for pressurized container (rubber band vs. jar lid)

Assembling drawing - No.1

1: Cylinder Cover 2: Heater 3: Flywheel 4: Crank Disk 5: Piston Holder6: Cylinder 7: Hot Piston 8: Cold Piston 9: Joint Board 10: Frame 11: Base

12: Shaft 13: Connecting Rod 14: Bush 15: Gasket 16: Gasket

Assembling drawing - No.2

Suggestions to assemble the engine

A: Seal and fix between a cylinder cover (No.1) and a cylinder (No.6) with asilicone gulue.B: Fix between a piston holder (No.5) and a piston (No.7,8) with a quick dryingglue.C: Fix a connecting rod (No.13) to a piston holder (No.5) with a bolt (No.24) anda nut (No.26) to move light.D: Cut a top of a bolt (No.23).E: Fix the bolts (No.23) to a flywheel (No.3) and a crank disk (No.4) using doublenut type.F: Fix bolts (No.22) to a base (No.11) with double nut type.Q44: I am building a displacer type Stirling engine. I don't know how many size is thelength of the displacer piston. The displacer piston bore of my engine is 42 mm, and thestroke is 30mm. The engine is used hot water asthe heat source and have air cooling.21 May, 1997T. Ueno

A44: The length of the displacer piston must be decided by the type of theheatsource and the structure of heat transfer parts. Your engine has a longerstroke, 30mm, then I think that the length of the displacer piston must be decidedtwo or three times of the stroke.I explain about a heat conduction loss caused by the length of the displacer

cylinder. The heat conduction loss, Qcond (W) is calculated by next equation.Qcond=R(Twh-Twc)(A/L)

R: Heat conduction ratio of cylinder wall (W/m2K)Twh: Temperature of hot side cylinder wallTwc: Temperature of cold side cylinder wallA: Section area of displacer cylinderL: Length of displacer cylinder

In this equation, you see the better size of the length of displacer piston andcylinder.

"Vintage" - Stirling Cycle Engine

"Vintage" is a 90 degree engine with a horizontal displacer cylinder and averticle power cylinder. Connecting rods for both cylinders us a single crank

pin. This makes for easy construction and interesting rod motion. The engineframe (blue) is made from aluminum plate, most of the rest of the engine ismachined from brass or stainless steel bar stock. Most of the brass parts werenickel plated, but that is not needed in any way - just cosmetics to suit me!

Vintage was designed to be a power source for the Miser engine. As such, it iswater cooled and a belt powered water pump circulates the warm waterthrough a ring that Miser sits on and back through the engine again. Thispowers the Miser while the Miser becomes the cooling "radiator" for Vintage!Just about any other simple radiator can be used such as a 10 foot loop ofvinyl aquarium air line tubing, etc!

The plans include the entire engine as shown including 2 different water pumpsand piping, a heater ring to operate a Miser engine and an alcohol burner (notshown). The sole exception is a straight spoke flywheel machined from solidwhich is similar to the one on the "Vickie" engine. An optional curved spokezinc alloy flywheel casting shown on the above engine photos is availablebelow.

Vintage runs very easily on a tiny 1/4" diameter by 1/4" high alcohol or propanegas flame.

The plans set consists of 16 sheets of drawings and 2 sheets of constructionand assembly notes.

Specifications:Flywheel Dia.: 3.33"Cylinder Bore: .5"Piston Stroke: .7"Engine Length: 6.25"Height: 4.5"

"Vintage" Engine Plans Set - $18.00 Post Paid in the USA(Use the Project Plans Order Form to place an order)

HTF (Hard-to-Find) Materials KitContents of kit:(1) 5/8" diameter x 1-1/2" long graphite rod to make piston(1) 3/8" diameter x 1-1/2" long delrin rod for crosshead(1) 1/4" diameter x 2" long delrin rod for bushing & small rod ends

(2) .187" ID x .375" OD x .125" thick precision ball bearings(1) 4-40 x 1/8" socket head set screw(1) 4-40 x 1/4" socket head set screw(2) 2-56 x 1/4" stainless steel panhead screws(20) 2-56 x 1/4" stainless steel socket head screws

"Vintage" HTF Materials Kit - $18.00post paid in the USA

(Use the Kits & Parts Order Form to place an order)

"Vickie" - Victorian Stirling Cycle Engine

Stirling engines have no valves, carburetor, ignition system or boilers and theyrun almost ghostly silent. Properly made, they will run flawlessly every time asource of heat is applied!

"Vickie" is a stirling cycle engine of modified Heinrici type with elegant victorianstyling designed for pleasing looks as was applied to 18th and 19th centuryengines and machines. Three fluted columnar legs and two stylish crossheadsof differing style blend perfectly with the curved and angular lines of the engineframes.

The engine is primarily made of aluminum with accents of polished brass andstainless steel and trimmed in dark green and maroon paint. A belt drivenbrass cooling fan competes with the rod and crosshead action for attention.Vickie is powered by an attractive horizontal brass alcohol burner which sportsan integral fuel level sight glass.

Vickie is considered by many to be one of the most beautiful stirling enginesever designed. I hope that you'll agree too! She is a true heirloom engine whichwill surely be handed down from generation to generation.

The plans set consists of 16 sheets of drawings and 3 sheets of constructionand assembly notes.

Specifications:Flywheel Dia.: 4-5/8"Cylinder Bore: .600Piston Stroke: 1"Overall Length: 10"

"Vickie" Engine Plans Set - $18.00 Post Paid in the USA(Use the Project Plans Order Form to place an order)

Graphite & Ball Bearings Kit

(1) 5/8" dia. x 1.4" long graphite for piston(2) .250" x .500" x .187" thick ball bearings

$14.00 post paid in the USA(Use the Kits & Parts Order Form to place an order)

Stirling Engine Heat Absorbers (Hot Cap)

The heat input area of a stirling engine is generally called the "hot cap" or "hotend" of the engine. As the engine operates due to the temperature differencebetween the hot end or hot cap and the cool end or displacer cylinder of theengine, we want to do what we can to prevent heat from conducting directly tothe cool end without doing any work for us. In electrical terms that would becalled a "short circuit".

The material used to make the hot cap must conduct heat through the wall tothe air inside the engine while at the same time conducting a minimum of heatto the cooler (displacer cylinder) area of the engine. Historically, mild steel hasbeen the most widely used material in model engines. It is a fair conductor ofheat as metals go - that is, it is a poorer conductor than brass or aluminum andmost other common metals. Right about here you might be thinking that if weuse a metal that is a poor heat conductor then we won't get much heat to theinside of the hot cap. The heat will only slightly be hindered going through thethin wall, but will be greatly hindered traveling the length of the tube.

To further minimize conduction of heat toward the displacer cylinder the wall ismade as thin as practical. This really does help. An example would be thatfewer cars can cross a single lane bridge at a given speed than can cross abridge having say 4 lanes at the same given speed. The number of lanesequals the thickness of the hot cap wall.

Thin wall tubing is usually selected to make the hot cap. A top flange is weldedor brazed to the tube for mounting to the displacer cylinder and a thin plug issimilarly attached to close the bottom end. I machine my hot caps from solidrod. This eliminates the welding or brazing and I can make the ID any size Iwant. I leave the bottom and part of the side wall from .025" to .050" thick. Tominimize the conduction to the displacer I greatly reduce the thickness of theupper portion of the wall. This creates a narrow waist just below the flange.The length of the waist is from 1/3 to 1/2 of the length of the hot cap.

I use stainless steel because it is not as good a conductor as other commonmetals. Before beginning to machine a hot cap, I make a plug .001" smallerthan the inside diameter of the hot cap is to be and as long as the hot cap is

deep inside. One end is drilled for a center and the other end is chamfered.After the hot cap is bored and turned on the outside, but before the waist ismachined, I insert the plug and bring up the tail stock center as a support. Nowit is possible to reduce the waist to a very thin wall without danger of distortingor otherwise ruining the work. The plug also prevents the wall from collapsingfrom the force of the cutting tool. Use a truly sharp tool bit with a small radius atthe tip (around .010"). Take lighter cuts as the wall gets thinner and use finefeeds. I routinely produce hot caps with walls at the waist as thin as .006". Idon't try to get thinner than this as I want to leave some metal for strength tosurvive bumps etc.!

Titanium is even a poorer heat conductor than stainless steel. Since it is notmuch different to machine than stainless steel and the fact that it is beginningto become readily available, I have been experimenting with it for hot caps andI like it. It is a better hot cap material. If you can get titanium at a reasonableprice, use it because you will like it too. My "Beamer" and "Vintage" engineshave titanium hot caps and they are the coolest running flame poweredengines I have - the hot cap flange and displacer cylinders runs at LESS thanluke warm.

Power Cylinder - The cylinder must be true and straight, no taper, bellmouth or barrel shapes allowed! If it was accurately machined with a nicesurface finish then all that is needed is a nice polish. The closer to a mirrorfinish the better.

Power Piston - The piston must also be true with no taper, etc. as thecylinder above. The graphite piston must be within .0005" of the cylinderdiameter. Pistons over .750" can be a little smaller than that and pistonssmaller than .600" should be a little larger than that. The correct fit is whenthe piston will fall through the cylinder of it's own weight, but when the pistonis pushed into the cylinder with the bottom closed it feels like there is a springunder it. Both cylinder and piston must be clean, dry and absolutely oil free.

Mechanical Tightness or Binding - Model stirling engines produce littlepower. Because of that if they are to run properly, or at all, the mechanicalaspects must not rob power. If there is any tightness or binding it must betracked down and corrected. Even a small amount of tightness or binding willrob much more power than you would expect. Better a little loose that tootight.

Displacer Timing - For all practical purposes, the displacer movementshould be 1/4 crankshaft revolution (90 degrees) ahead of the power piston -ie crank pin to crank pin. This is not critical and can vary a few degrees oneway or the other. Not all engines will be at optimum performance at 90degrees but it is the best test setting for a new engine.

Engine Balance - If the engine is a "Miser" or other low temperaturedifference engine, balance is important. With compression relieved byloosening or removing the bottom plate, adjust the balance disk so that theengine will stop at random places when given a spin. If the engine can't bebalanced, gradually enlarge or plug the balance disk holes as needed. If theengine is unbalanced, it will require more heat and operate at a higher RPMthan it would if balanced.

Air Leakage - Other than minute leakage around the piston and displacerrod bushing, there should not be any other air leaks. When given a spin, theengine should exibit some compression by coming to a stop at about the 3:00o'clock or 9:00 o'clock power piston crankpin positions (vertical engineexample). If it exhibits no compression and all else above is well, there is anair leak somewhere that must be found and corrected. Don't overlook thedisplacer itself. It must be a sealed air tight can. If you put it in the freezer andget it very cold and then submerge it into near boiling hot water it should notshow any bubbles coming from it. Low temperature "Miser" type enginesshould have non pourus foam displacers and are exempt from this test.

If your engine is of sound basic design and it passes all the above tests it willbe nearly impossible for it NOT to run! One last caveat - be careful not to usetoo large a flame to operate your engine as small models can easily bedamaged by overheating. An alcohol or propane flame of 1/4" diameter andaround 1/2" high (or less) will easily operate any of my engine designs. Misershould NEVER be operated over any flame.