Q: How do Stirling Engines work?
A: Stirling engines can be hard to understand. Here are the key points. Every
Stirling engine has a sealed cylinder with one part hot and the other cold. The
working gas inside the engine (which is often air, helium, or hydrogen) is moved
by a mechanism from the hot side to the cold side. When the gas is on the hot side
it expands and pushes up on a piston. When it moves back to the cold side it
contracts. Properly designed Stirling engines have two power pulses per
revolution, which can make them very smooth running. Two of the more common
types are two piston Stirling engines and displacer-type Stirling engines. The two
piston type Stirling engine has two power pistons. The displacer type Stirling
engine has one power piston and a displacer piston.
Displacer Type:
The displacer type Stirling engine is shown here. The space below the displacer
piston is continuously heated by a heat source. The space above the displacer
piston is continuously cooled. The displacer piston moves the air (displaces the
air) 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 the
hot side of the engine, where it expands, to the cold side, where it contracts.
DISPLACER PISTON
When there is a temperature difference between upper displacer space and lower displacer
space, the engine pressure is changed by the movement of the displacer. The pressure increases
when the displacer is located in the upper part of the cylinder (and most of the air is on the hot
lower 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 not
operate the crankshaft and the engine. In other words, the connecting rod to the displacer could
be a string in this engine and it would still work.
POWER PISTON
When the engine pressure reaches its maximum because of the motion of the displacer, a power
piston is pushed by the expanding gas adding energy to the crankshaft. The power piston should
ideally be 90 degrees out of phase with the displacer piston. The displacer type Stirling engine is
operated 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 is
continuously heated by a heat source. The space above the cold piston is continuously
cooled.
Two-Piston Engine Detail:
Click here for animation...
HEATING
Let's start from top dead center of the hot piston. The hot piston moves to the upper part of the
cylinder and the cold piston moves to the lower part of the cylinder during the first 90 degrees of
revolution. The working air is moved from the cold space to the hot space. And the pressure in the
engine is increased.
EXPANSION
During the next 90 degrees of revolution, the two pistons both move the lower part accepting the
air pressure. The engine gets its power during this portion of its cycle.
COOLING
The crankshaft revolves by power stored in the flywheel for the next 90 degrees. The hot piston
moves to the lower part and the cold piston moves to the upper part. The air is moved from the
hot space to the cold space. And the pressure in the engine is decreased.
CONTRACTION
The 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 then
repeats 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 the
maximum 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 x
100 equals the max theoretical efficiency. Of course the temperatures must be
measured in degrees Kelvin or Rankine. Stirling engines (with perfect
regeneration) match this cycle. Real Stirling engines can reach 50 percent of the
maximum 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 to
speed. Notice that it keeps running for a minute or so. While it's very easy to build
a Stirling engine that will stop instantly, there is not one thing in the world anyone
can do to make one start instantly. When I get in my car I want it to start
immediately (if not sooner) and be able to burn rubber off the tires as I leave the
parking lot! Stirling engines can't do that. In spite of these limitations, Ford, GM,
and American Motors Corp. spent millions of dollars developing Stirling engines
for cars, back in the 1970's. Ford even built a Stirling that could drive away from
the 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 to
build an engine that was substantially more efficient than internal combustion
engines, but wouldn't start instantly. Here is a picture of a 1979 AMC Spirit. It
was equipped with an experimental Stirling engine powerplant called the "P-40".
The Spirit was capable of burning gasoline, diesel, or gasohol. The P-40 Stirling
engine promised less pollution, 30% better mileage, and the same level of
performance as the car's standard internal combustion engine. [From "An
Introduction to Stirling Engines"] The French Research Sub Saga is Stirling
engine powered. Stirling engines also work exceptionally well as auxiliary power
generators/heaters on yachts (see Victron Energy.), where their silence is valued
and good cooling water is available. They would also work very well in airplanes
where the air gets colder as the plane climbs to altitude. There is no aircraft power
plant (jets included) that gets any improvement in any operating conditions from
climbing. Stirling engines won't lose as much power as they climb as do either
piston engines or jets. Also wouldn't you like to have silent airplanes with very
efficient 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. There
have been many research engines built in recent years but there are only three
areas where Stirling engines have made a dramatic impact. There are Stirling
engines in Submarines, stirling machines used as cryocoolers, and Stirling engines
in classrooms. Cryogenics is the science of things that are exceedingly cold and
Stirling engines are one tool that can be used to make things exceedingly cold. It's
not obvious but a Stirling engine is a reversible device. If you heat one end and
cool 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 get
cold. If you design the machine correctly, the cold end will get extremely cold. In
fact, Stirling coolers have been made that will cool below 10 degrees Kelvin.
Micro Stirling coolers have been produced in large numbers for cooling infrared
chips 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 500
deg. F. (260 deg. C) the engine will be too bulky for the amount of power it puts
out.
1. I don't think there is a theoretical upper limit on power in a Stirling engine. 2. The
bigger the temperature difference the easier it is to get power out of a small
engine. In other words to put out any significant amount of power an engine
running 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 are
going to build an engine that puts out a significant amount of power you will
probably want to build the heater head out of at least a good grade of stainless
steel 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 cogeneration
units and power generation using alternative fuels. Stirling engines have come a long way from the large and
heavy engines of the 19th century, thanks to advancements in materials, manufacturing processes, theory and
analysis methods.
This page contains a handful of links to some of these companies. Click on the images to learn more
about these organizations and the engines they produce.
All images and information related to these devices are property of and are assumed to be copyrighted by
their respective owners.
SOLO Kleinmotoren GmbH
STM Corporation Stirling Energy Systems, Inc.
Sunpower, Inc.
Kockums Sweden.
Infinia Corporation
NASA Glenn Research Center
Tamin Enterprises
The Stirling Engine's most basic configuration consists of two pistons each in its
own cylinder. (Sometimes it is easier to envision these two cylinders as one long
tube with the piston heads facing each other inside the tube (see the figure
below)). Note that between these two pistons heads are the heater, cooler and
regenerator. The regenerator (usually a block of woven wire) is in the center of
this tube and the heater is between the regenerator and one piston (in red) while
the 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 pushes
against 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 the
regenerator matrix on the way to the compression space from the expansion
space and then the heat is recovered on the way back from the compression
space 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 to
many but not to all engines that may be called Striling Engines.
The working gas trapped between the two piston heads is pushed by the
Compression Piston through the regenerator where it is heated by the energy in
the regenerator heated to its hottest in the Heater and expands in the Expansion
Space. This increased pressure pushes on the Expansion Piston so that it
moves away from the regenerator pushing on a mechanism which changes the
linear movement of the piston to a rotary motion. This continues until all the gas
that will expand has been pushed into the heater area and expanded. The
mechanism also pushes the Compression Piston further toward the Regenerator
pushing 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 degrees
apart), begins to move the Expansion piston back the other way pushing the hot
gas through the Heater backwards and then on to the Regenerator and finally
into the Cooler where it begins to Cool and contract (the pressure starts to
drop). The Compression Piston is also moving away from the regenerator while
the Expansion piston comes toward the regenerator moving the gas through the
regenerator into the compression space without compressing the gas.
The linkage continues to move the pistons until the Compression Piston is all the
way 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 the
piston moves up and down in the cylinder and the mechanism is moving in a
circle, the Expansion Piston does not move very far but the Compression Piston
moves toward the regenerator actually compressing the gas and begining to
push the gas through the regenerator. (That is why it is called the Compression
Piston.)
This brings us to the first line of this explanation to complete the cycle and begin
again.
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 closed
regenerative thermodynamic cycle with cyclic compression and expansion of the
working fluid (air) at different temperature levels. The flow of the working fluid is
controlled by changes in the volume of the hot and cold spaces, eliminating the
need for valves. The Stirling Engine is reversible, meaning that an input of heat
energy (burning fuel, for example) will produce an output of mechanical energy,
and an input of mechanical energy (electric motor, etc.) will produce an output of
heat energy. In this manner, the Stirling Engine can be used as a heat pump in
much the same way as traditional refrigeration units, only without the
environmentally harmful refrigerants.
The most basic engine consists of a set of pistons, heat exchangers, and a
device called a 'regenerator'. The engine is filled with a working fluid (gas) which
is commonly Air, but some more advanced engines may use Nitrogen, Helium or
Hydrogen. The pistons are arranged such that they create both a change in
volume of the working fluid and create a net flow of the fluid through the heat
exchangers. In this manner, heat is absorbed from an external source in the 'hot'
end, creating mechanical energy, and rejected in the 'cold' end to the
environment.
In a Stirling engine, the working fluid is completely contained inside the engine at
all 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 to
operate cleanly and quietly as there are no combustion products coming into
contact with any of the engine's working components and no release of
high-pressure gasses.
An important feature in Stirling Engines is the regenerator. On the most basic
level, a regenerator is a device that absorbs heat from the working fluid as it
enters the 'hot' end, and re-heats the fluid as it enters the 'cold' end. This internal
recycling of energy allows for much higher efficiencies, and better performance
overall. The regenerator is such a critical component that most Stirling Engines
cannot operate efficiently without one!
Stirling Engineering ( Deeper understanding)
This link is a much deeper look into the theory. Click Here for a look at the
Detailed Theory of Operation.
This information is from Dr. Israel Urieli of Ohio University. Caution: Contains
calculus, partial differential equations and thus requires a knowledge of the
calculus. Also contains source code modules for a second order simulator (In
'C').
Stirling Engines - Mechanical Configurations
The mechanical configurations of Stirling engines are generally divided into three groups
known as the Alpha, Beta, and Gamma arrangements. Alpha engines have two pistons in
separate cylinders which are connected in series by a heater, regenerator and cooler. Both
Beta and Gamma engines use displacer-piston arrangements, the Beta engine having both
the displacer and the piston in an in-line cylinder system, whilst the Gamma engine uses
separate cylinders.
The Alpha engine is conceptually the simplest Stirling engine configuration,
however suffers from the disadvantage that both pistons need to have seals to
contain the working gas. Andy Ross of Columbus, Ohio has been developing
small air engines with extremely innovative Alpha designs, including the classical
Ross-Yoke drive and more recently a balanced "Rocker-V" mechanism, as
shown below.
The Alpha engine can also be compounded into a compact multiple cylinder
configuration, enabling an extremely high specific power output, as is required of
an automotive engine. A schematic diagram of this configuration is shown below.
Notice that the four cylinders are interconnected, so that the expansion space of
one cylinder is connected to the compression space of the adjacent cylinder via
a series connected heater, regenerator and cooler. The pistons are typically
driven by a swashplate, resulting in a pure sinusoidal reciprocating motion
having a 90 degree phase difference between the adjacent pistons.
Beta Type Stirling Engines
The Beta configuration is the classic Stirling engine configuration and has
enjoyed popularity from its inception until today. Stirling's original engine from his
patent drawing of 1816 shows a Beta arrangement. A photograph of Robert
Stirling, the original patent drawing, as well as an animated model of Stirling's
engine is clearly shown in an interesting website by Bob Sier. Another important
early Beta engine is Lehmann's machine on which Gusav Schmidt did the first
reasonable analysis of Stirling engines in 1871.
From the figure we see that unlike the Alpha machine, the Beta engine has a
single power piston and a displacer, whose purpose is to "displace" the working
gas at constant volume, and shuttle it between the expansion and the
compression spaces through the series arrangement cooler, regenerator, and
heater.
Rolf Meijer of Philips, Holland, derived his famous vibrationless rhombic drive for
Beta engines in the early 1960s.
Probably the most ingenious Stirling engines yet devised are the free-piston
engines invented and developed by William Beale at Ohio University in the late
1960s. He later formed the company Sunpower, Inc., which has been the leader
in the development of free-piston Stirling engines and cryocoolers to this day. All
of Sunpower's engines are Beta arrangements and employ no mechanical
linkage system. The main aspect of the free piston machine is that the output
power can be obtained through a linear alternator, allowing the entire system to
be hermatically sealed. Sunpower have recently begun to manufacture Stirling
cycle croygenic coolers for liquifying oxygen. Over the years Sunpower has
transformed Athens, Ohio into a hotbed of Stirling cycle machine activity, which
now includes four R&D and manufacturing companies as well as one
internationally recognized consultant in the area of Stirling cycle computer
analysis.
Stirling Technology Inc. is a spinoff of Sunpower, and was formed in order to
continue the development and manufacture of the 5 kW ST-5 Air engine. This
large Beta type engine burns biomass fuel (such as sawdust pellets or rice
husks) and can function as a cogeneration unit in rural areas. It is not a
free-piston engine, and uses a bell crank mechanism to obtain the correct
displacer phasing.
Global Cooling is a licencee of Sunpower, mainly in order to develop free-piston
Stirling cycle coolers for home refrigerator applications. These systems, apart
from being significantly more efficient than regular vapor-compression
refrigerators, have the addad advantage of being compact, portable units using
helium as the working fluid (and not the Ozone destroying CFCs).
External Power is a very recent licencee of Sunpower, and was formed to
manufacture biomass fueled (sawdust pellets) free-piston cogeneration units for
home use.
Gamma Type Stirling Engines
Gamma type engines have a displacer and power piston, similar to Beta
machines, however in different cylinders. This allows a convenient complete
separation between the heat exchangers associated with the displacer cylinder
and the compression and expansion work space associated with the piston.
Thus they tend to have somewhat larger dead (or unswept) volumes than either
the Alpha or Beta engines.
Furthermore during the expansion process some of the expansion must take
place in the compression space leading to a reduction of specific power. Gamma
engines are therefor used when the advantages of having separate cylinders
outweigh the specific power disadvantage.
Because of the convenience of two cylinders in which only the piston has to be
sealed, the gamma configuration is a favorite among modellers and hobbyists.
Ideal Isothermal Analysis
The invention of the Stirling engine in 1826 was well in advance of all pertinent
scientific knowledge of that time. The first attempt at an analysis of the cycle was
published in 1871 by Gustav Schmidt. Much as the Otto cycle has become the
classic Air standard cycle to describe the spark ignition engine, the cycle
described by Schmidt has become the classic ideal Stirling cycle. This is
unfortunately mainly because the Schmidt analysis yields a closed form solution
rather than its ability to predict the real cycle, however we use it as a starting
point 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 space
and the heater is at the constant upper source temperature and the gas in the
compression space and the cooler is at the constant lower sink temperature.
This isothermal assumption makes it possible to generate a simple expression
for the working gas pressure as a function of the volume variations. This
expression may then be used to investigate how different drive mechanisms
affect the output power. To obtain closed form solutions, Schmidt assumed that
the volumes of the working spaces vary sinusoidally.
The assumption of isothermal working spaces and heat exchangers implies that
the heat exchangers (including the regenerator) are perfectly effective, with a
spacial temperature distribution as indicated in the figure above. The engine is
considered as a five component serially connected model, consisting
respectively of a compression space c, cooler k, regenerator r, heater h and
expansion space e. Each component is considered as a homogeneous entity or
cell, the gas therein being represented by its instantaneous mass m, absolute
temperature T, volume V and pressure p, with the suffix c, k, r, h, and e
identifying the specific cell.
The starting point of the analysis is that the total mass of gas in the machine is
constant, 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 the
effective 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 for
pressure p as a function of Vc and Ve.
The work done by the system over a complete cycle is given respectively by the
cyclic integral of p dV
On evaluating the heat transferred over a complete cycle to the various cells we
find remarkably that the cyclic heat transferred to all three heat exchanger cells
is zero! Thus:
Qc = Wc
Qe = We
Qk = 0
Qh = 0
Qr = 0
This rather startling result implies that all the heat exchangers in the ideal Stirling
engine are redundant since all the external heat transfer occurs across the
boundaries of the compression and expansion spaces. This apparent paradox is
a direct result of the definition of the Ideal Isothermal model in which the
compression and expansion spaces are maintained at the respective cooler and
heater temperatures. Obviously this cannot be correct, since the cylinder walls
are not designed for heat transfer. In real machines the compression and
expansion spaces 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. This will be resolved when we consider the Ideal Adiabatic model in
the 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 volume
variations Vc and Ve as well as the respective volume derivatives dVc and dVe
with respect to crankangle . One of the case studies of this course is the Ross
Yoke-drive engine for which we have analized the volume variations, thus the
above equation set can be solved by numerical integration. In 1871 Gustav
Schmidt published an analysis in which he obtained closed form solutions for the
above equation set for the special case of sinusoidal volume variations. We
continue now with the Schmidt analysis.
The Schmidt Analysis
In the previous section we derived the basic set of equations which describe the
Ideal Isothermal model, as shown in the following table.
Gustav Schmidt of the German Polytechnic Institute of Prague Published an
analysis in 1871 in which he obtained closed form solutions of these equations
for the special case of sinusoidal volume variations of the working spaces with
respect to the cycle angle . Consider the following diagram showing the volume
variations of the compression and expansion spaces (Vc and Ve) over a single
cycle. Notice the phase advance angle of the expansion space volume
variation with respect to the compression space volume variation:
The sinusoidal volume variations of the compression and expansion spaces are
respectively as follows:
Vc = Vclc + Vswc (1 + cos ) / 2
Ve = Vcle + Vswe (1 + cos( + )) / 2
where Vcl and Vsw represent respectively clearence and swept volumes, and
is the cycle angle. Substituting for Vc and Ve in the pressure equation above and
simplifying we obtain
In order to simplify the pressure equation we now consider a trigonometric
substitution 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 the
extreme 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 working
gas 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 the
compression 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 integrals
and 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 appendix
in this book that deals with the Schmidt analysis has been placed on the web by
Global Cooling, and can be downloaded in Acrobat pdf format. Finally we obtain
Ideal Adiabatic Analysis
In the previous section we considered an ideal Stirling engine model in which the
compression and expansion spaces were maintained at the respective cooler
and heater temperatures. This led to the paradoxical situation that neither the
heater nor the cooler contributed any net heat transfer over the cycle and hence
were redundant. All the required heat transfer occurred across the boundaries of
the isothermal working spaces. Obviously this cannot be correct, since the
cylinder walls are not designed for heat transfer. In real machines the working
spaces 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. We
thus consider an alternative ideal model for Stirling cycle engines, the Ideal
Adiabatic model.
As before the engine is configured as a five component serially connected model
having perfectly effective heat exchangers (including the regenerator) and in this
respect is similar to the Ideal Isothermal model defined previously. However both
the compression and expansion spaces are adiabatic, in which no heat is
transferred to the surroundings. In the following diagram we define the Ideal
Adiabatic 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 the
four interfaces between the cells. Enthalpy is transported across the interfaces in
terms of a mass flow rate m' and an upstream temperature T. The arrows on the
interfaces represent the positive direction of flow, arbitrarily defined from the
compression space to the expansion space.
Notice from the temperature distribution diagram that the temperature in the
compression and expansion spaces (Tc and Te) are not constant, but vary over
the cycle in accordance with the adiabatic compression and expansion occurring
in the working spaces. Thus the enthalpies flowing across the interfaces ck and
he carry the respective adjacent upstream cell temperatures, hence
temperatures Tck and The are conditional on the direction of flow and are
defined algorithmically as follows:
if mck' > 0 then Tck = Tc else Tck = Tk
if 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 system
is constant, and there is no pressure drop, hence p is not suffixed and
represents the instantaneous pressure throughout the system.
Work W is done on the surroundings by virtue of the varying volumes of the
working spaces Vc and Ve, and heat Qk and Qh is transferred from the external
environment to the working gas in the cooler and heater cells, respectively. The
regenerator is externally adiabatic, heat Qr being transferred internally from the
regemerator matrix to the gas flowing through the regenerator void volume Vr.
Development of the equation set
The general approach for deriving the equation set is to apply the equations of
energy and state to each of the cells. The resulting equations are linked by
applying the continuity equation across the entire system. Consider first the
energy equation applied to a generalised cell which may either be reduced to a
working space cell or a heat exchanger cell. Enthalpy is transported into the cell
by means of mass flow mi' and temperature Ti, and out of the cell by means of
mass flow mo' and temperature To. The derivative operator is denoted by d, thus
for 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 the
generalised 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 pressure
and constant volume respectively. This equation is the well known classical form
of the energy equation for non steady flow in which kinetic and potential energy
terms have been neglected.
We assume that the working gas is ideal. This is a reasonable assumption for
Stirling engines since the working gas processes are far removed from the gas
critical point. The equation of state for each cell is presented in both its standard
and 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 is
constant, 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 mean
effective 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 temperatures
are 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 an
explicit 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 to
the mass inflow of gas given by -mck', and the work donw dWc is given by p
dVc, 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 dTc
and 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 is
ideal, 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 Analysis
Once we have done an Ideal Adiabatic analysis on a specific Stirling engine, we
would like to evaluate the heat transfer and flow-friction effects of the three heat
exchangers on the performance of the engine. This will enable us to do a
parametric sensitivity analysis as required for design optimization.
Forced convection heat transfer is fundamental to Stirling engine operation. Heat
is transferred from the external heat source to the working fluid in the heater
section, cyclicly stored and recovered in the regenerator, and rejected by the
working fluid to the external heat sink in the cooler section. All of this is done in
compact heat exchangers (large wetted area to void volume ratio) so as to limit
the "dead space" an acceptable value and thus allow for a reasonable specific
power output of the engine. We find that effective heat exchange comes at a
price of increased flow friction, resulting in the so-called "pumping loss". This loss
refers to the mechanical power required to "pump" the working fluid through the
heat exchangers, and thus reducing the net power output of the engine.
The theory and analysis of these effects is extremely complex, and we find that
we can only rely on the plethora of documented experimental and empirical
studies ( e.g. Kays & London ,"Compact Heat Exchangers"). Almost all of this
vast body of work is based on steady flow conditions and is thus not directly
applicable to the oscillating flow conditions that apply to Stirling engines. In this
section we adopt a "Quasi-Steady Flow" approach, in that we assume that at
each instant of the cycle the fluid behaves as though it is in steady flow. Thus we
have called this analysis a "Simple" analysis because it is a gross simplification
of an extremely complex process. At this stage there is still a major controversy
about this approach, and we need to treat the results of this analysis with a
healthy measure of scepticism. The only alternative for design is the recent
"Similarity and Scaling" approach which has been developed by Allan Organ and
is presented in his book "The Regenerator and the Stirling Engine".
Design Process
To establish a starting point for designing the Idaho Stirling Engine, we
began by modifying an existing Stirling engine that was developed by Ted
Boyl-Davis, a graduate student at the University of Idaho.
Variables
The following independent design variables were manipulated during the
iterative design process of the ISE and empirically evaluated. The variables
were 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 Holder
6: 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 a
silicone gulue.
B: Fix between a piston holder (No.5) and a piston (No.7,8) with a quick drying
glue.
C: Fix a connecting rod (No.13) to a piston holder (No.5) with a bolt (No.24) and
a 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 double
nut 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 the
length of the displacer piston. The displacer piston bore of my engine is 42 mm, and the
stroke is 30mm. The engine is used hot water asthe heat source and have air cooling.
21 May, 1997
T. Ueno
A44: The length of the displacer piston must be decided by the type of the
heatsource and the structure of heat transfer parts. Your engine has a longer
stroke, 30mm, then I think that the length of the displacer piston must be decided
two 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 wall
Twc: Temperature of cold side cylinder wall
A: Section area of displacer cylinder
L: Length of displacer cylinder
In this equation, you see the better size of the length of displacer piston and
cylinder.
"Vintage" - Stirling Cycle Engine
"Vintage" is a 90 degree engine with a horizontal displacer cylinder and a
verticle power cylinder. Connecting rods for both cylinders us a single crank
pin. This makes for easy construction and interesting rod motion. The engine
frame (blue) is made from aluminum plate, most of the rest of the engine is
machined from brass or stainless steel bar stock. Most of the brass parts were
nickel 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 is
water cooled and a belt powered water pump circulates the warm water
through a ring that Miser sits on and back through the engine again. This
powers 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 of
vinyl aquarium air line tubing, etc!
The plans include the entire engine as shown including 2 different water pumps
and piping, a heater ring to operate a Miser engine and an alcohol burner (not
shown). The sole exception is a straight spoke flywheel machined from solid
which is similar to the one on the "Vickie" engine. An optional curved spoke
zinc alloy flywheel casting shown on the above engine photos is available
below.
Vintage runs very easily on a tiny 1/4" diameter by 1/4" high alcohol or propane
gas flame.
The plans set consists of 16 sheets of drawings and 2 sheets of construction
and 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 Kit
Contents 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 - $18.00
post 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 they
run almost ghostly silent. Properly made, they will run flawlessly every time a
source of heat is applied!
"Vickie" is a stirling cycle engine of modified Heinrici type with elegant victorian
styling designed for pleasing looks as was applied to 18th and 19th century
engines and machines. Three fluted columnar legs and two stylish crossheads
of differing style blend perfectly with the curved and angular lines of the engine
frames.
The engine is primarily made of aluminum with accents of polished brass and
stainless steel and trimmed in dark green and maroon paint. A belt driven
brass cooling fan competes with the rod and crosshead action for attention.
Vickie is powered by an attractive horizontal brass alcohol burner which sports
an integral fuel level sight glass.
Vickie is considered by many to be one of the most beautiful stirling engines
ever designed. I hope that you'll agree too! She is a true heirloom engine which
will surely be handed down from generation to generation.
The plans set consists of 16 sheets of drawings and 3 sheets of construction
and assembly notes.
Specifications:
Flywheel Dia.: 4-5/8"
Cylinder Bore: .600
Piston 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 "hot
end" of the engine. As the engine operates due to the temperature difference
between the hot end or hot cap and the cool end or displacer cylinder of the
engine, we want to do what we can to prevent heat from conducting directly to
the cool end without doing any work for us. In electrical terms that would be
called a "short circuit".
The material used to make the hot cap must conduct heat through the wall to
the air inside the engine while at the same time conducting a minimum of heat
to the cooler (displacer cylinder) area of the engine. Historically, mild steel has
been the most widely used material in model engines. It is a fair conductor of
heat as metals go - that is, it is a poorer conductor than brass or aluminum and
most other common metals. Right about here you might be thinking that if we
use a metal that is a poor heat conductor then we won't get much heat to the
inside of the hot cap. The heat will only slightly be hindered going through the
thin wall, but will be greatly hindered traveling the length of the tube.
To further minimize conduction of heat toward the displacer cylinder the wall is
made as thin as practical. This really does help. An example would be that
fewer cars can cross a single lane bridge at a given speed than can cross a
bridge having say 4 lanes at the same given speed. The number of lanes
equals the thickness of the hot cap wall.
Thin wall tubing is usually selected to make the hot cap. A top flange is welded
or brazed to the tube for mounting to the displacer cylinder and a thin plug is
similarly attached to close the bottom end. I machine my hot caps from solid
rod. This eliminates the welding or brazing and I can make the ID any size I
want. I leave the bottom and part of the side wall from .025" to .050" thick. To
minimize the conduction to the displacer I greatly reduce the thickness of the
upper 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 common
metals. Before beginning to machine a hot cap, I make a plug .001" smaller
than 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 is
machined, I insert the plug and bring up the tail stock center as a support. Now
it is possible to reduce the waist to a very thin wall without danger of distorting
or otherwise ruining the work. The plug also prevents the wall from collapsing
from the force of the cutting tool. Use a truly sharp tool bit with a small radius at
the tip (around .010"). Take lighter cuts as the wall gets thinner and use fine
feeds. I routinely produce hot caps with walls at the waist as thin as .006". I
don't try to get thinner than this as I want to leave some metal for strength to
survive bumps etc.!
Titanium is even a poorer heat conductor than stainless steel. Since it is not
much different to machine than stainless steel and the fact that it is beginning
to become readily available, I have been experimenting with it for hot caps and
I like it. It is a better hot cap material. If you can get titanium at a reasonable
price, use it because you will like it too. My "Beamer" and "Vintage" engines
have titanium hot caps and they are the coolest running flame powered
engines I have - the hot cap flange and displacer cylinders runs at LESS than
luke warm.
Power Cylinder - The cylinder must be true and straight, no taper, bell
mouth or barrel shapes allowed! If it was accurately machined with a nice
surface finish then all that is needed is a nice polish. The closer to a mirror
finish the better.
Power Piston - The piston must also be true with no taper, etc. as the
cylinder above. The graphite piston must be within .0005" of the cylinder
diameter. Pistons over .750" can be a little smaller than that and pistons
smaller than .600" should be a little larger than that. The correct fit is when
the piston will fall through the cylinder of it's own weight, but when the piston
is pushed into the cylinder with the bottom closed it feels like there is a spring
under it. Both cylinder and piston must be clean, dry and absolutely oil free.
Mechanical Tightness or Binding - Model stirling engines produce little
power. Because of that if they are to run properly, or at all, the mechanical
aspects must not rob power. If there is any tightness or binding it must be
tracked down and corrected. Even a small amount of tightness or binding will
rob much more power than you would expect. Better a little loose that too
tight.
Displacer Timing - For all practical purposes, the displacer movement
should 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 one
way or the other. Not all engines will be at optimum performance at 90
degrees but it is the best test setting for a new engine.
Engine Balance - If the engine is a "Miser" or other low temperature
difference engine, balance is important. With compression relieved by
loosening or removing the bottom plate, adjust the balance disk so that the
engine will stop at random places when given a spin. If the engine can't be
balanced, gradually enlarge or plug the balance disk holes as needed. If the
engine is unbalanced, it will require more heat and operate at a higher RPM
than it would if balanced.
Air Leakage - Other than minute leakage around the piston and displacer
rod bushing, there should not be any other air leaks. When given a spin, the
engine should exibit some compression by coming to a stop at about the 3:00
o'clock or 9:00 o'clock power piston crankpin positions (vertical engine
example). If it exhibits no compression and all else above is well, there is an
air leak somewhere that must be found and corrected. Don't overlook the
displacer itself. It must be a sealed air tight can. If you put it in the freezer and
get it very cold and then submerge it into near boiling hot water it should not
show any bubbles coming from it. Low temperature "Miser" type engines
should 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 will
be nearly impossible for it NOT to run! One last caveat - be careful not to use
too large a flame to operate your engine as small models can easily be
damaged by overheating. An alcohol or propane flame of 1/4" diameter and
around 1/2" high (or less) will easily operate any of my engine designs. Miser
should NEVER be operated over any flame.