The How and Why of the Stirling Engine

By Lee A. White, P.E.


We begin this discussion with the simplest of reasonably practical heat engine. An engine like this could be built and would function. We will also provide a bit of background regarding heat engines.  Our simple engine will have a quantity of working gas enclosed in a cylinder in the space between a fixed cylinder head and a movable power piston. (see fig 1) The kind of gas chosen for the working gas is unimportant at this point. Any gas (or mixture of gasses) that remains a gas and does not dissociate or condense at the temperatures and pressures encountered within the cylinder will work. From a theoretical point of view this is all we would really need to study engines, but any practical engine capable of doing work would require some additional components.

In most practical engines the pressure exerted by the working gas on the face area of the piston creates a force that is transferred from the piston through a connecting rod and on to the crankshaft. The connecting rod and crankshaft serve to convert the linear motion of the piston into the rotary motion of the crankshaft. While this conversion from linear to rotary motion is not necessary to make an engine, it is typical in modern engine practice. This was not always the case. Early engines were often used to power pumps and reciprocating motion suited this application perfectly. Continuous rotary motion is generally more useful today so we will consider only that configuration. The rotating crankshaft also provides a convenient place to connect auxiliary devices and to synchronize their motions with those of the power piston. We will also add a flywheel to the end of the crankshaft to serve as an energy storage device. The flywheel will adsorb energy during the power stroke and return it during the compression stroke. This simple engine configuration will be added to as required for visualization of the thermodynamics and kinematics involved in practical heat engines.

The purpose of any heat engine is to convert heat into mechanical work. Obviously then, to have a heat engine you need a source of heat. Any source will do, but the higher the temperature the better. The important point here is that we have to get some heat into the working gas inside the cylinder.

The required heat can be produced inside the working gas by burning fuel inside the cylinder. Engines that produce heat inside the cylinder are generally referred to as internal combustion engines (ICEs) This type of engine usually requires a refined and high grade fuel and the engine must provide the means to properly ratio the fuel and the oxidizer, to thoroughly mix them, to induct them into the cylinder, to ignite them at the correct time, and to eject the exhaust products. These requirements bring complication and this type of engine represent a less general case than those which do not produce the required heat internally.

For today’s internal combustion engines the fuel is almost always a liquid hydrocarbon and the oxidizer is almost always the O2 present in normal air. Natural gas is used in some large stationary engines (especially natural gas pumping stations) and LP gas finds use in some mobile applications, but petroleum derivative liquid fuels predominate in the vast majority of cases. This has not always been the case. Very early ICE engines used solid fuels exclusively with gunpowder being the preference. These were single shot engines, manually refueled after each power stroke. Later engine developments switched to coal gas and to gasified liquid fuels like turpentine and alcohol. Solid fuels were abandoned. The original diesel engine design intent (late 1800’s) was to provide an engine to burn coal dust, a solid waste byproduct of mining operations. This concept was abandoned due to the difficulties encountered in metering and feeding solid fuels with precision in the very small volumes required by a cylinder for each single power stroke. That was pretty much the end of the road for solid fuels in ICEs.

Solid fuels continued on as the primary heat source for steam engines. In the steam engine, the heat is released in the firebox and then conducted into the working fluid. In Stirling engines the required heat input is also produced external to the engine and then conducted into the working fluid. In this case the input heat must be transferred to the cylinder and then conducted through the cylinder wall and into the working gas. Engines that get their heat from a source located outside the engine and then conduct it into the working fluid are usually referred to as external combustion engines (ECEs), even though combustion is not strictly required. The external heat can come from essentially anything that is warm; solar or geo-thermally heated water, decomposing compost, a lump of radioactive plutonium, or even the heat from your skin. In this sense, external combustion engines are the most general type conceivable. Stirling and Hot Air engines are of this type and we will restrict this article to External Combustion Engines.

Since the required heat can also come from hot gas combustion products, Stirling engines can be powered from almost anything that can be burned in an external combustor. This includes normal solid fuels like coal and wood, and less common fuels like animal dung, sawdust, straw, dried alga, sewage sludge, spent hops from breweries; the list is pretty endless. Some submarines are even powered by Stirling engines operating off the heat of reaction when Sodium is oxidized by Chlorine (the exhaust is table salt). There is an important concept here. The actual Stirling engine thermodynamic machinery does not care what the source of the heat is. All that really matters is how much heat can be supplied and at what temperature it is available.

Less well understood is the fact that all "heat" engines also need a source of "cool", and the colder the better. Cooling is required for purposes other than just keeping engine parts cool enough to avoid damage. A heat engine can only convert a portion of the input heat into work so there must be a place to reject the remaining unconverted heat. Heat engines can only run when provided with both something that is relatively warmer to provide the input heat and something that is relatively cooler to carry away the unconverted heat. It is the temperature difference between the heater and the cooler that drives the engine. In some senses it would be more correct to call a heat engine a "temperature difference engine". The greater the driving temperature difference is the higher the engine thermal conversion efficiency can be.

In Stirling engines, the cooling considerations are even more critical than they are in internal combustion engines. ICEs reject their waste heat in their exhaust products, which are typically dumped to the atmosphere. ICEs are open cycle machines. The working fluid moves into and out of the engine. They induct their fuel and oxidizer, burn it releasing heat, expand this heated working fluid converting some of the heat into work, and then exhaust the working fluid as spent combustion products. The unconverted heat is carried away by these exhaust products. Typically, about two thirds of the heat released ends up as waste heat carried away by the exhaust. That is why exhaust systems get really hot.

Stirling engines are closed cycle machines. Their working gas is sealed within the engine. Heat is transferred through the engine cylinder and into the working fluid, the working fluid is expanded converting some of the input heat to work, and the unconverted waste heat is then transferred out of the working fluid and into a cooling stream. The working fluid is now cool and ready for the start of the next thermodynamic cycle. All the unconverted waste heat must be conducted away from the engine by a stream of cooling media. This is identical to what happens in a closed cycle coal fired steam power plant. Heat input boils the water producing steam, which is the actual working fluid. This working fluid is expanded converting some of the input heat into work, and the remaining waste heat is rejected to the cooling water stream when the spent steam is condensed and recycled as boiler feed water.

It takes a lot of cooling water from a stream or a lake or a big evaporative cooling tower to carry away all that waste heat. There is a similarly large relative cooling requirement for a Stirling engine. Typically, the required coolant radiator will be 3 or 4 times larger than the one required for an internal combustion engine of comparable power output. If enough cooing is not provided to the engine, the minimum temperature of the working fluid will rise, the temperature difference between the input heat source and the working fluid will decrease, and both the thermal conversion efficiency and the power output will go down. It will now require even more input heat to produce the same power output and even more waste heat must now be rejected. The cooling problem worsens and the engine is considered to be "heat bound". If the radiator is too small, the coolant flow is to small, or the coolant is not very cold, both engine power output and engine efficiency will suffer.

We now return to our simplest form of heat engine as shown in figure 1. Our single vertical cylinder is positioned with the head end up and heating and cooling sources to either side. The crankshaft is located within the crankcase directly below the piston. Since the working gas is always located within the same cylinder, this cylinder must be alternately heated and cooled by the heating and the cooling sources. This application of heating and cooling to the cylinder must occur at the proper time in the cycle and synchronization with piston movement is generally achieved by driving the apparatus from the crankshaft.

THE THERMODYNAMIC CYCLE for this simple engine.

Engine operation begins with the piston at the top of its stroke and located near the cylinder head. This piston position is referred to as Top Dead Center (TDC). The working gas is all contained in the cylinder space between the piston face and the cylinder head and this volume is known as the dead volume or the clearance volume (Vc). The heat source is now applied to the outside of the engine cylinder, heat transfers to the cylinder and into the gas, and the gas temperature and pressure begin to rise. After some period of time, the temperature and pressure in the working gas reach a maximum. This is the beginning point in our thermodynamic cycle. It is position #1 and it is plotted as point #1 on the pressure diagram (figure 2)  and also as point #1 on the temperature diagram (figure 3). The horizontal axes of both of these plots are the volume occupied by the working gas and these plots are knows as P-V and T-V diagrams. Since the volume of the working gas is equal to the cross sectional area of the cylinder times the distance between the cylinder head and the piston face, we see that the horizontal axes of these plots also represent the piston position relative to the cylinder head. Accordingly, TDC and BDC occur at volumes of 1 and 3 cubic inches respectively on these plots.

The piston is now allowed to travel downward until it reaches it most downward position. This is position #2 and this position is known as Bottom Dead Center (BDC). The distance the piston travels from position #1 (TDC) to position #2 (BDC) is know as the Stroke (S). The increase in the gas volume as the piston travels from TDC to BDC is known as the Swept Volume (Vs) and it is equal to the cylinder cross sectional area times the Stroke. The total gas volume (Vt) at BDC is equal to the sum of the clearance volume (Vc) added to the Swept volume (Vs).

As the piston moves downward the gas begins to expand. Normally, an expanding gas would cool but heat is continually transferred into the working gas from the heat source and the hot cylinder wall keeping the gas at its maximum temperature. This expansion (with heating) process is known as an Isothermal (constant temperature) expansion. The total amount of heat transferred into the gas during this expansion is the input energy to the engine and it is given the symbol Q1-2

Even though the gas temperature is maintained at its maximum, the gas pressure falls as the gas volume increases. The pressure falls according to the equation:

P1 * V1 = P2 * V2 Since the pressure force on the piston is acting downward and the piston is traveling downward, work is being done by the gas. In simple terms, the gas is forcing the piston downward, is twisting the crankshaft, and is doing work in the process. The work done is equal to the force exerted times the distance traveled. Since the pressure is continually falling as the piston travels downward, this multiplication is not so simple. The calculation of the work done by the gas requires the use of integral calculus and yields the following solution equation:

W1-2 = P1 * V1 * Ln (V2/V1)

When the piston reaches BDC the next step in the cycle begins. The heating source is removed from the cylinder and the cooling source is applied to the cylinder while the piston remains at BDC. As heat is transferred from the gas to the cooling source the gas temperature and the gas pressure both fall. After some amount of time the lowest gas temperature is attained and we are at point #3 of the engine thermodynamic cycle. The total amount of heat transferred out of the gas is given the symbol Q2-3. Since the piston has not moved, no work has been done during this process.

The next step in this engine thermodynamic cycle is to push the piston back in to the starting position at TDC. As the piston is pushed in the gas is being compressed and the gas temperature would normally begin to rise. However, the cooling source is still applied to the cylinder and prevents this temperature rise. This process is known as an Isothermal Compression. The heat of compression is transferred out of the gas to the cooling source. The amount of this heat transfer is given the symbol Q3-4

As the gas is compressed the gas pressure rises. The pressure rises according to the formula:

P3 * V3 = P4 * V4

Since the pressure force on the piston is acting downward and the piston is traveling upward, work is being done on the gas. In simple terms, the crankshaft is forcing the rod and piston upward, and is doing work on the gas in the process. The work output of the engine is NEGATIVE. Where does this work come from? Typically, it comes from the flywheel and shows up as a reduction in the rotational speed of the flywheel. Once again, the work done is the product of the applied force times the distance traveled. Since the pressure changes with volume, the force is changing with position and calculus is again required. The formula is:

W3-4 = P3 * V3 * Ln (V4/V3)

The piston is now back at the starting position, but we are not yet at the starting thermodynamic point as the working gas is still at the low temperature of the cooling source. It must be warmed up to the starting high temperature.

The next and final step is to remove the cooling source, apply the heating source, and heat the gas back up to the starting high temperature. The total heat transferred during this step is given the symbol Q4-1. Since the piston has not moved, there is no work done in this step.

To review:

  1. We start with a high temperature high pressure gas. We allow the piston to move out expanding the gas while supplying heat to it. When the expansion is complete we remove the heater. The heat supplied is Q1-2 and the work done is W1-2
  2. We then lock the piston in place and cool the gas to a lower temperature by applying the cooling source. The heat removed is Q2-3 The work done is zero.
  3. We then push the piston in compressing the gas while continuing to cool the gas. When the compression is complete the cooling source is removed. The work done is W3-4 and is negative. The heat removed is Q3-4
  4. We then lock the piston in place and apply the heater warming the gas back to the starting temperature. The heat supplied is Q4-1. The work done is zero.
The total heat supplied is Q1-2 + Q4-1

The total heat rejected is Q2-3 + Q3-4

The total heat converted into work is Q1-2 + Q4-1 - Q2-3 - Q3-4

The total work done by the engine is W1-2 - W3-4

The total heat converted is equal to the total work done

The engine thermal efficiency is equal to the total heat converted divided by the total heat supplied.

The engine thermal efficiency is also equal to the work output divided by the work equivalent of the heat input.


It is beyond the scope of this article to derive Carnot theory, but that theory concludes that the ratio of the input heat to the reject heat is the same as the ratio of the heat input temperature to the heat rejection temperature. Beginning with the efficiency equation from above and substituting temperatures for heat quantities results in an expression for engine efficiency based on temperatures only. This equation gives the maximum thermal conversion efficiency that any engine can ever attain. No real engine can do even this well. Real engines will convert less of the input heat to work and will have to reject even more heat than this equation indicates. The equation follows:

Thermal Conversion Efficiency = (T1 - T3) / T1 = 1 - (T3 / T1)

The number this equation calculates is a decimal that represents the theoretical maximum portion of the input heat that could be converted to work by a perfect engine. Multiply this decimal number by 100 to put it in the form of a percent.

From the efficiency equation above we can see that a higher input temperature (T1) or a lower rejection temperature (T3) leads to higher conversion efficiency, as does a higher temperature difference (T1 - T3).

Remember that the temperatures in this equation are not the temperatures of the heater and the cooler. They are the temperatures of the working gas at two points in the thermodynamic cycle. T1 will be lower than the heater temperature and T3 will be higher than the cooler temperature due to the temperature differences required to drive the heat transfers to and from the working gas. Also, this equation is only true when those temperatures are given as absolute temperatures. Absolute temperatures are referenced to absolute zero. To convert degrees Fahrenheit to degrees Rankine (absolute) you must add 460 degrees to the temperature as given in degrees F.

Similarly, we can calculate the theoretical minimum fraction of the input heat that must be rejected by a perfect engine. The equation is as follows:

Heat Rejection Fraction = T3 / T1

Again, this is a decimal number. Multiply it by the input heat to get the reject heat. If you want to see what percent of the input heat must be rejected, multiply the decimal by 100. From the equation we see that the higher the input temperature (T1) the less heat the engine must reject. Similarly, the lower the rejection temperature (T3) the less heat the engine will have to reject. These results seem a bit counterintuitive at first glance but they make perfect sense when we consider the effect these temperatures have on thermal conversion efficiency. Higher conversion efficiency means less unconverted heat to reject.


The first thing one might notice with the design and operation of this simple engine is that the entire cylinder must be alternately heated and then cooled again when the only thing we really need to heat and cool is the working gas. This suggests that a better engine might be made by providing separate heating and cooling areas and shuttling the gas back and forth between these areas. How could we do this?

Lets begin by removing the cylinder head of the simple engine and replacing it with a horizontal cylinder placed on top of and connected to the power cylinder. We now have an engine that has the shape of a "T" as shown in figure 4

We will place a continuously operating heater under one end of this horizontal cylinder and a continuously operating cooler under the opposite end of this cylinder. Inside the horizontal cylinder is a loosely fitted movable piston called a displacer. When the displacer is in the hot end of the cylinder the working gas is displaced to the cool end of the cylinder and is cooled. When the displacer is in the cool end of the cylinder the working gas is in the hot end of the cylinder and is being heated. By pushing the displacer back and forth the gas can be alternately heated and cooled. Since the displacer is loosely fitted inside the cylinder it takes little force to push it back and forth and little work is done in cycling the gas between the hot end and the cold end. By connecting the displacer to the crankshaft, the movement of the displacer can be synchronized to the movement of the piston and the thermodynamic cycle of the simple engine can be duplicated. The process is as follows.
  1. The piston is at TDC, the displacer is at the cool end, the gas is in hot end and is being heated. After a while the gas has reached the maximum temperature and we are at the starting pint. The piston is allowed to move down to BDC. (Step 1-2)
  2. The piston is at BDC, the displacer is moved to the hot end, the gas moves to the cold end, and the gas begins to cool. (Step 2-3)
  3. When the gas reaches minimum temperature, the piston is pushed back in to TDC. (Step 3-4)
  4. The piston is at TDC, the displacer is moved to the cold end, the gas moves to the hot end, and the gas begins to heat. (Step 4-1)
While we have duplicated the thermodynamic cycle of the simple engine, there are some differences. We are now heating and cooling only the gas, not the entire cylinder. Because less material is being alternately heated and cooled, less total heating and cooling are required to achieve the same amount of heating and cooling of the working gas. On the other hand, we have permanently connected to the engine both the heater and the cooler and we have established a thermal conduction path between them. This creates a heat leakage path between the hot end of the cylinder and the cold end of the cylinder reducing engine efficiency.

The displacer cylinder creates a second difficulty. The internal volume of the heater and cooler are dead volumes. Like the cylinder clearance volume, they are not swept by the piston. They add to the power cylinder clearance volume increasing the total dead volume of the engine. This reduces the pressure changes that occur within the engine (given the same bore and stroke in the power cylinder) reducing the engine output.

On the whole this is still a preferred configuration, but we can do even better.


When we review the simple engine thermodynamic cycle we see that there are two steps where heating is involved and two where cooling is involved. We must always supply some heat and we must always remove some heat but perhaps some of the heat could be recycled. This was the genius of Stirling. He realized that some of this heat could be recycled internal to the engine and he invented the device to do so. That device is called a REGENERATOR and it works like this.

We begin with our "T" engine and modify the displacer so that it is tightly fitted in the cylinder. As a result, gas can not pass by it. We provide an external pipe from the hot end to the cold end of the displacer cylinder as shown in figure 5. When the displacer moves, the working gas passes back and for through this pipe on its way to and from the hot end and cold end respectively. We then fill this pipe with a matrix stack of mesh screens, metal beads, wire wool, or some other high heat capacity material. This is the regenerator.

As the displacer moves toward the hot end the hot gas located there passes through the tube and the matrix material of the regenerator on its way to the cold end of the displacer cylinder. The hot gas gives up heat to the matrix material, heats the matrix, and is cooled in the process. As a result, it is now cooler than when it left the hot end. The gas is now cooler than expected when it enters the cold end of the displacer cylinder. It has deposited some of its heat in the matrix and has been pre-cooled by the matrix. This pre-cooling in the regenerator reduces the need for further cooling in the cooler. This reduces the total heat that must be rejected by the engine to produce the same work output.

When the displacer moves back to the cool end, the cool gas there is forced back through the regenerator. The matrix material is hotter than the gas due to the heat that was deposited there earlier. The cool gas is heated by the matrix and enters the hot end hotter than expected. It has been pre-heated by the matrix in the pipe reducing the need to heat it in the hot end. This reduces the total heat that must be supplied to the engine to produce the same work output.

The regenerator is little more than a tube filled with a matrix of high heat capacity material that can temporarily store and the return heat to the working gas. By doing so, it reduces the need to supply input heat and to remove waste heat and it improves the efficiency of the engine considerably. This is the primary reason why Stirling Cycle engines offer the highest possible theoretical thermodynamic efficiency. As always, there is no free lunch. The internal volume of the regenerator adds to the dead volume of the heater and cooler and cylinder and increases the total dead volume of the engine. Pressure losses required to force the gas through the regenerator matrix further decrease engine performance and a greater portion of the working gas never makes it all the way to the heater and back to the cooler. Pressure swings within the engine are further reduced. Even so, it is a better engine with the regenerator installed.


The cooling process in step (2-3) and the heating process in step (4-1) occur in the regenerator as the gas is shuttled from the hot end to the cold in step (2-3) and back again from the cold end to the hot end in step (4-1). Both of these processes occur while the piston is at rest, so there is no net volume change. These processes are referred to as Isochoric (constant volume) and are characteristic of the Stirling cycle. The theoretical Stirling Cycle consists of the following thermodynamic process steps:

  1. An isothermal expansion in the power cylinder
  2. An isochoric cooling in the regenerator
  3. An isothermal compression in the power cylinder
  4. An isochoric heating in the regenerator
It is interesting to note that the Reverend Stirling invented the Regenerator as a device to reduce fuel consumption of ovens, furnaces, and kilns. Later, he designed an engine to use it.


There are a wide variety of engine physical configurations that execute a Stirling like cycle. Some have a single piston and a single displacer, sometimes located in the same cylinder. Some others have two pistons located in two cylinders and no displacer, while others have multiple cylinders cascaded in a ring. There are machines whose pistons or displacers are driven by kinematic mechanisms and some that have electro-magnetically driven moving parts. Some are harmonic resonators that are driven by pressure pulsations or even sound waves. Some use magnetically suspended free pistons and others use liquid pistons in tube cylinders. Examining all these engines is well beyond the scope of a single paper. Each has its own peculiarities from mechanical design as well as a thermodynamic analysis point of view. All these engines are generally referred to as Stirling engines and all real world practical Stirling engines deviate rather substantially from the ideal Stirling cycle. These deviations occur for a variety of reasons.

1) The pistons and displacer in both kinematic and resonant engines are in essentially continuous nearly sinusoidal motion. In kinematic engines they are typically driven by slider cranks operating at reasonably steady rotational speeds. The piston is not at rest while the isochoric processes are occurring so there is volume change occurring during what are supposed to be constant volume processes. The displacer is not at rest while the piston is moving so all the gas is not in the appropriate space during expansion and compression. As a result, the expansion pressure is reduced and the compression pressure is increased, again reducing net engine output.

2) Heat transfer takes time. Modern engines reciprocate so fast that the amount of heat required to maintain a constant gas temperature cannot be transferred in the time available. Gas temperatures change during what are supposed to be isothermal processes.

3) All of the working gas does not move into the heater and then the cooler. Some of the working gas is located in the cylinder clearance volume, the regenerator, the dead volume of the heaters and coolers, and even in the swept volume of the cylinder when the piston is at BDC. As a result, all of the working gas does not undergo the required temperature changes. Failure to fully heat or cool this gas reduces the pressure swing within the engine.

4) The regenerator increases flow losses as the working gas is forced through the matrix material. Make the passages large and the dead volume gets problematic. Make them small and the pressure losses get problematic. The regenerator is a critical component but it is not well understood theoretically. It is difficult to optimize in the design stage.

5) The external combustion nature of the Stirling cycle engine necessitates the need to transfer all the input heat to and through the displacer cylinder wall and then through the gas itself. The reverse path applies to the reject heat at the cooler. Each of these heat transfer steps requires an efficiency reducing temperature difference. Heat transfer through the gas itself is a relatively slow process. Heat transfer considerations are very important in Stirling design.

6) Some of the internal parts of the engine are continuously exposed to very high or sometime very low temperatures increasing component material and lubrication problems.

7) All components of a Stirling engine are interconnected. They are not separated and isolated by valves. They all interact, each affecting the other. The accurate analysis of a real Stirling engine (even a simple one like ours) based on first principles is generally well beyond current technical understanding let alone exact mathematical analysis. Attempts at computer simulation and modeling seem to be in their infancy and generally seem to yield only roughly approximate results; results which have to be calibrated and fudged with experimentally obtained data points. To date, the most promising approach seems to be based on "similarity models" derived from experimental data points.

Even so, several companies have successfully designed Stirling engines and cryocoolers, built limited numbers of prototypes or special purpose devices, and even some production engines. A few companies are currently attempting to produce or are forecasting volume production and amateur hobbyists continue to build a bewildering variety of working model engines.

In the spirit of the hobbyist, we can now modify our simple engine configuration a bit to make a geometry that is somewhat easier to construct. We begin by removing the power cylinder head, detaching the displacer cylinder, rotating it ninety degrees (with the hot end upward), and reattaching it to the top of the power cylinder. We generally locate the hot section on top, the cool section in the middle, and the power section at the bottom to keep the heat as far as possible from the piston seals. We continue by drilling a hole in the piston and installing a seal to accept the displacer rod. The rod actuating the displacer now passes through this hole/seal in the piston and on down into the crankcase, where it attaches to a crank throw. This throw is about ninety degrees out of phase with the main throw for the power piston. Both the piston and the displacer will now move in a continuous nearly sinusoidal fashion, but they move at different times. This is the configuration of many easy to build hobby Stirling engines. It is referred to as the Beta configuration and it is similar to Stirling’s original design. Build one and enjoy.


Copyright 2005 Lee A. White and