Stirling Cycle

Stirling Cycle

Stirling Cycle

Introduction of Stirling Cycle and Engine

One of the most important and useful cycles in thermodynamics is Stirling cycle that establishes and describe the general class of Stirling cycle was invented, developed and patented in 1816 by Robert Stirling with help of his brother. The cycle is reversible, meaning that if supplied with mechanical power, it can function as a heat pump for heating or cooling, and even for cryogenic cooling.


The Stirling cycle is a thermodynamic cycle that describes the general class of Stirling devices. This includes the original Stirling engine that was invented, developed and patented in 1816 by Robert Stirling with help from his brother, an engineer. The ideal Otto and Diesel cycles are not totally reversible because they involve heat transfer through a finite temperature difference during the irreversible isochoric (Constant-Volume)/isobaric heat-addition and heat-rejection processes. The irreversibility renders the thermal efficiency of these cycles less than that of a Carnot engine operating within the same limits of temperature. Another cycle that features isothermal heat-addition and heat-rejection processes is the Stirling cycle, which is an altered version of the Carnot cycle in which the two isentropic processes featured in the Carnot cycle are replaced by two constant-volume regeneration processes.

Robert Stirling

The cycle is reversible, meaning that if supplied with mechanical power, it can function as a heat pump for heating or cooling, and even for cryogenic cooling. The cycle is defined as a closed regenerative cycle with a gaseous working fluid. “Closed cycle” means the working fluid is permanently contained within the thermodynamic system. This also categorizes the engine device as an external heat engine. “Regenerative” refers to the use of an internal heat exchanger called a regenerator which increases the device’s thermal efficiency. The cycle is the same as most other heat cycles in that there are four main processes: compression, heat addition, expansion, and heat removal. However, these processes are not discrete, but rather the transitions overlap. The Stirling cycle is a highly advanced subject that has defied analysis by many experts for over 190 years. Highly advanced thermodynamics is required to describe the cycle. Professor Israel Urieli writes: “…the various ‘ideal’ cycles (such as the Schmidt cycle) are neither physically realizable nor representative of the Stirling cycle”. The analytical problem of the regenerator (the central heat exchanger in the Stirling cycle) is judged by Jakob to rank “among the most difficult and involved that are encountered in engineering”.


A Stirling engine is a specific flavor of heat engine formulated by Robert Stirling in 1816; this means it can transform the flow of heat into mechanical work (such as spinning a crankshaft). The key term is “flow of heat”; there must be two “reservoirs” that are separated, and these reservoirs must be at different temperatures in order for this flow to take place between them. If you put a thermal conductor between the two reservoirs over time, they would both approach the same temperature, indicating that energy is “flowing” from the hot reservoir to the cold reservoir.

The Stirling engine harnesses this flow of energy from hot to cold and siphons some of it off as mechanical work. The Stirling engine needs a hot section and a cold section that are insulated from each other, the clever way a working fluid is routed between the two sections allows the engine to produce mechanical work. Heat is transferred from the hot section to the engine, some of the energy leaves the engine as useful mechanical work, and some of it leaves as heat transfer to the cold section. Remember that energy can never be destroyed so if you add up all the energy leaving the engine (i.e. useful work + heat transfer into the cold section) it must equal the amount of energy entering the engine as heat transfer from the hot section. This energy balance is the first law of thermodynamics and always holds.

Heat engine thermodynamic diagram
Heat engine thermodynamic diagram

First law of thermodynamics for a Stirling engine, the first law is simply an energy balance of the system:

Heat In = Heat Out + Useful Work

$ \displaystyle {{Q}_{{in}}}={{Q}_{{out}}}+{{W}_{{out}}}$

Comments on Thermal Efficiency

The ratio of useful work to the heat transfer into the engine is called thermal efficiency. Think of it as the ratio of what you want (useful mechanical work) divided by what costs (heat transfer into the engine).

Calculating thermal efficiency for a Stirling:

$ \displaystyle \eta =\frac{{Useful\text{ }Mechanical\text{ }Work}}{{Heat\text{ }Transfer\text{ }from\text{ }the\text{ }Hot\text{ }Section}}$

Efficiency can never be higher than 1. An efficiency of 1 would mean that all the heat transfer into the engine becomes useful work and there is no heat transfer to the cold section at all. An efficiency of 0 indicates that no useful work is produced and that all heat transfer from the hot section simply leaves the engine as heat transfer to the cold section. If you put two bricks next to each other, one hot and one cold, inside a perfectly insulated box and left them there for a while you would come back to find two warm bricks. This is technically a heat engine with an efficiency of 0; heat was transferred from the hot brick to the cold brick with a 1:1 ratio, producing no useful work in the process.

It turns out that efficiency can’t ever equal 1 either; sorry about that, the second law of thermodynamics is a real creep. Deriving the relation that bounds physically possible efficiency levels is a whole other topic, but it’s called the Carnot efficiency, named after Nicolas Léonard Sadi Carnot. He was able to postulate the maximum efficiency one could expect without violating the second law of thermodynamics. One can calculate the Carnot efficiency knowing only the temperatures of the hot section and the cold section between which a given heat engine is working. This means you’ll never have a heat engine that doesn’t reject at least some heat to the cold section. When one plots out the possible Carnot efficiency given the hot section temperature it can be seen that the larger the temperature difference between the hot side and the cold side the higher the possible efficiency. Not all engines can even theoretically (not to mention realistically) achieve Carnot efficiency. For example, the perfect diesel engine could never, even in a perfect world, match the efficiency of the theoretical Carnot heat engine. Certain other types of heat engines can match the Carnot engine in theoretical performance. The Stirling engine is one example of this. Therefore, the Carnot efficiency at a given hot section and cold section temperature is equal to the Stirling efficiency between the same hot and cold sections.

Example Carnot or Stirling thermal efficiency vs. hot section temperature
Example Carnot or Stirling thermal efficiency vs. hot section temperature

Ideal Stirling thermal efficiency is equal to the Carnot efficiency

$ \displaystyle {{\eta }_{{Stirling}}}={{\eta }_{{Carnot}}}$

To have a Stirling run continuously you need to have a hot section that gets constantly heated by some source, and a cold section that is cooled in some way. Without constantly heating the hot section and cooling the cold section eventually enough heat would be transferred between the two that you would just end up with two warm sections. Once this happens you no longer have this temperature differential between sections, efficiency would drop to 0, and no heat would be transferred through the engine since no temperature differential exists.

Suppose that the hot section is heated by burning biomass materials, and the cold section is cooled by water that then runs through a radiator. This allows us to keep the hot section at about 900 K and the cold section at about boiling water temperature (373 K). If you do the math to calculate the Carnot efficiency (and therefore the Stirling efficiency) these temperatures mean that one could never expect to get more than an efficiency of 0.58 without the universe imploding. Unfortunately, right away almost half the energy you put into our engine is GUARANTEED to come out as waste heat transfer to the cold section. Right now, we’re just talking about absolutely perfect engines, and that’s all which will be discussed in this article, but in the real world there are all kinds of other factors that make it impossible to reach the Carnot efficiency level.

The Stirling Engine as a Cycle

Heat engines are cyclic, and that’s the case for the Stirling engine. In the case of a reciprocating engine, a process occurs between the hold section and the cold section, which repeats at a certain frequency. Heat is absorbed into the engine in pulses and then rejected to the cold section and as work in pulses. Usually a flywheel is added to the engine to smooth out these pulses and keep the mechanism turning. The heat from the hot section is transferred to the cold section via some type of working fluid (air, helium, hydrogen, nitrogen, or any other type of gas, some are better than others). For a Stirling it’s possible to describe the thermodynamic cycle in four sections.

A loop showing the relation between pressure and volume during a cycle of the Stirling engine
A loop showing the relation between pressure and volume during a cycle of the Stirling engine

State 1 to 2

At state 1 the working fluid is at a maximum volume, minimum temperature, and minimum pressure. From state 1 to state 2 the power piston compresses the working fluid while heat is transferred out of the system which keeps the working fluid at a constant low temperature. When the engine is in state 2 the working fluid is in a compressed state (high pressure and low volume), but remains at the same temperature as state 1. The work that was required to compress the volume is provided by stored energy in the engine’s flywheel.

State 2 to 3

At state 2 the working fluid is at a minimum volume, minimum temperature, and medium pressure. Between states 2 and state 3 the volume is held constant while heat is added by the hot section to increase temperature.

State 3 to 4

At state 3 the working fluid has achieved a maximum temperature, maximum pressure, while also at a minimum volume. From state 3 to 4 the working fluid is allowed to expand, doing useful work while it does so. During the expansion process more heat is added to keep the system at a constant temperature. The energy provided during this expansion outweighs the energy that was required to compress the volume between states 1 and 2, providing a net positive work out.

State 4 to 1

To return the engine to the state 1 where it began heat is removed from the working fluid while volume is kept constant.


– Piston motion variations

Most thermodynamics textbooks describe a highly simplified form of Stirling cycle consisting of four processes. This is known as an “ideal Stirling cycle”, because it is an “idealized” model, and not necessarily an optimized cycle. Theoretically, the “ideal cycle” does have high net-work output, but it is rarely used in practical applications, in part because other cycles are simpler or reduce peak stresses on bearings and other components. For convenience, the designer may elect to use piston motions dictated by system dynamics, such as mechanical linkage mechanisms. At any rate, the efficiency and cycle power are nearly as good as an actual implementation of the idealized case. A typical piston crank or linkage in a so named “kinematic” design often results in a near-sinusoidal piston motion. Some designs will cause the piston to “dwell” at either extreme of travel.

A model of a four-phase Stirling cycle
A model of a four-phase Stirling cycle

Many kinematic linkages, such as the well-known “Ross yoke”, will exhibit near-sinusoidal motion. However, other linkages, such as the “rhombic drive”, will exhibit more non-sinusoidal motion. To a lesser extent, the ideal cycle introduces complications, since it would require somewhat higher piston acceleration and higher viscous pumping losses of the working fluid. The material stresses and pumping losses in an optimized engine, however, would only be intolerable when approaching the “ideal cycle” and/or at high cycle rates. Other issues include the time required for heat transfer, particularly for the isothermal processes. In an engine with a cycle approaching the “ideal cycle”, the cycle rate might have to be reduced to address these issues. In the most basic model of a free piston device, the kinematics will result in simple harmonic motion.

– Volume variations

In beta and gamma engines, generally the phase angle difference between the piston motions is not the same as the phase angle of the volume variations. However, in the alpha Stirling, they are the same. The rest of the article assumes sinusoidal volume variations, as in an alpha Stirling with co-linear pistons, so named an “opposed piston” alpha device.

Alpha type Stirling
Alpha type Stirling

– Pressure-versus-volume graph

This type of plot is used to characterize almost all thermodynamic cycles. The result of sinusoidal volume variations is the quasi-elliptical shaped cycle. Compared to the idealized cycle, this cycle is a more realistic representation of most real Stirling engines. The four points in the graph indicate the crank angle in degrees.

Pressure-versus-volume graph

The adiabatic Stirling cycle is similar to the idealized Stirling cycle; however, the four thermodynamic processes are slightly different:

  • 180° to 270°, pseudo-isothermal expansion. The expansion space is heated externally, and the gas undergoes near-isothermal expansion.
  • 270° to 0°, near-constant-volume (or near-isometric or isochoric) heat removal. The gas is passed through the regenerator, thus cooling the gas, and transferring heat to the regenerator for use in the next cycle.
  • 0° to 90°, pseudo-isothermal compression. The compression space is intercooled, so the gas undergoes near-isothermal compression.
  • 90° to 180°, near-constant-volume (near-isometric or isochoric) heat addition. The compressed air flows back through the regenerator and picks up heat on the way to the heated expansion space.

With the exception of a Stirling thermoacoustic engine, none of the gas particles actually flow through the complete cycle. So, this approach is not amenable to further analysis of the cycle. However, it provides an overview and indicates the cycle work.

Particle/mass motion

– Particle/mass motion

The figure streaklines which indicate how gas flows through a real Stirling engine. The vertical colored lines delineate the volumes of the engine. From left to right, they are: the volume swept by the expansion (power) piston, the clearance volume (which prevents the piston from contacting the hot heat exchanger), the heater, the regenerator, the cooler, the cooler clearance volume, and the compression volume swept by the compression piston.

– Heat-exchanger pressure drop

Also referred to as “pumping losses”, the pressure drops shown in the following figure are caused by viscous flow through the heat exchangers. The red line represents the heater, green is the regenerator, and blue is the cooler. To properly design the heat exchangers, multivariate optimization is required to obtain sufficient heat transfer with acceptable flow losses. The flow losses shown here are relatively low, and they are barely visible in the following image, which will show the overall pressure variations in the cycle.

Pressure versus crank angle

Pressure versus crank angle

The figure shows results from an “adiabatic simulation” with non-ideal heat exchangers. Note that the pressure drop across the regenerator is very low compared to the overall pressure variation in the cycle.

– Temperature versus crank angle

Temperature versus crank angle
Temperature versus crank angle

The figure above, illustrates the adiabatic properties of a real heat exchanger. The straight lines represent the temperatures of the solid portion of the heat exchanger, and the curves are the gas temperatures of the respective spaces. The gas temperature fluctuations are caused by the effects of compression and expansion in the engine, together with non-ideal heat exchangers which have a limited rate of heat transfer. When the gas temperature deviates above and below the heat exchanger temperature, it causes thermodynamic losses known as “heat transfer losses” or “hysteresis losses”. However, the heat exchangers still work well enough to allow the real cycle to be effective, even if the actual thermal efficiency of the overall system is only about half of the theoretical limit.

– Cumulative heat and work energy

The figure shows a graph of the alpha-type Stirling engine data, where ‘Q’ denotes heat energy, and ‘W’ denotes work energy. The blue dotted line shows the work output of the compression space. As the trace dips down, work is done on the gas as it is compressed. During the expansion process of the cycle, some work is actually done on the compression piston, as reflected by the upward movement of the trace. At the end of the cycle, this value is negative, indicating that compression piston requires a net input of work. The blue solid line shows the heat flowing out of the cooler heat exchanger. The heat from the cooler and the work from the compression piston have the same cycle energy. This is consistent with the zero-net heat transfer of the regenerator (solid green line). As would be expected, the heater and the expansion space both have positive energy flow. The black dotted line shows the net-work output of the cycle. On this trace, the cycle ends higher than it started, indicating that the heat engine converts energy from heat into work.

Real Stirling Configurations

In order to control when heat is transferred to or from the working fluid most Stirling engines have what is called a “displacer” piston which simply prevents contact between the working fluid and either the hot section or the cold section depending on its position. To change volume of the system there is usually some type of power piston which reciprocates in a cylinder bore, often this piston is connected to a crankshaft in order to collect the useful work.

There are many ways in which an engineer might choose to mechanically link the power piston, displacer piston, and heat exchangers in order to produce the effects needed during a Stirling cycle. No mechanism perfectly mimics the motions needed, therefore in real Stirling engines one source of loss is the “approximation” of the cycle which is necessary to build a real machine. Two of the most common types of engine configurations are the beta type and the alpha type.

The Stirling Cycle Cooler

One important aspect of Stirling cycle machines that we need to consider is that the cycle can be reversed – if we put net-work into the cycle then it can be used to pump heat from a low temperature source to a high temperature sink. Sunpower, Inc. has been actively involved in the development of Stirling cycle refrigeration systems and produces Stirling cycle cryogenic coolers for liquifying oxygen. In 1984 Sunpower developed a free piston Duplex Stirling Machine having only three moving parts including one piston and two displacers, in which a gas fired Stirling cycle engine powered a Stirling cycle cooler. Global Cooling was established in 1995 as a spinoff of Sunpower, and was formed 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 added advantage of being compact, portable units using helium as the working fluid (and not the HFC refrigerants such as R134a, having a Global Warming Potential of 1,300). More recently Global Cooling decided to concentrate their development efforts on systems in which there are virtually no competitive systems – cooling between -40°C and -80°C, and they established a new company name: Stirling Ultracold. We are fortunate to have obtained two original M100B coolers from Global Cooling. The one is used as a demonstrator unit, and is shown in operation in the following photograph. The second unit is set up as a ME Senior Lab project in which we evaluate the actual performance of the machine under various specified loads and temperatures.

Conceptually the cooler is an extremely simple device, consisting essentially of only two moving parts – a piston and a displacer. The displacer shuttles the working gas (helium) between the compression and expansion spaces. The phasing between the piston and displacer is such that when the most of the gas is in the ambient compression space then the piston compresses the gas while rejecting heat to the ambient. The displacer then displaces the gas through the regenerator to the cold expansion space, and then both displacer and piston allow the gas to expand in this space while absorbing heat at a low temperature.

Applications of the Stirling engine

Applications of the Stirling engine range from mechanical propulsion to heating and cooling to electrical generation systems. A Stirling engine is a heat engine operating by cyclic compression and expansion of air or other gas, the “working fluid”, at different temperature levels such that there is a net conversion of heat to mechanical work. The Stirling cycle heat engine can also be driven in reverse, using a mechanical energy input to drive heat transfer in a reversed direction (i.e. a heat pump, or refrigerator).

A desktop gamma Stirling engine. The working fluid in this engine is air. The hot heat exchange is the glass cylinder on the right, and the cold heat exchanger is the finned cylinder on the top. This engine uses a small alcohol burner (bottom right) as a heat source.
A desktop gamma Stirling engine. The working fluid in this engine is air. The hot heat exchange is the glass cylinder on the right, and the cold heat exchanger is the finned cylinder on the top. This engine uses a small alcohol burner (bottom right) as a heat source.

There are several design configurations for Stirling engines that can be built (many of which require rotary or sliding seals) which can introduce difficult tradeoffs between frictional losses and refrigerant leakage. A free-piston variant of the Stirling engine can be built, which can be completely hermetically sealed, reducing friction losses and completely eliminating refrigerant leakage. For example, a Free Piston Stirling Cooler (FPSC) can convert an electrical energy input into a practical heat pump effect, used for high-efficiency portable refrigerators and freezers. Conversely, a free-piston electrical generator could be built, converting a heat flow into mechanical energy, and then into electricity. In both cases, energy is usually converted from/to electrical energy using magnetic fields in a way that avoids compromising the hermetic seal.

It could be referred to many areas such as: Automotive engines, Electric vehicles, Aircraft engines, Marine engines, Pump engines, Combined heat and power, Solar power generation, Nuclear power, Stirling cryocoolers, Heat pumps, Portable refrigeration, Acoustic Stirling Heat Engine, MicroCHP, Chip cooling, Desalination.

Wikipedia 2w. MIDE Blog 3w. OHIO University 4w. Stirling Cryogenics 5w. Wikipedia
Books & Articles:
Romanelli Alternative thermodynamic cycle for the Stirling machine, American Journal of Physics 85, 926 (2017)
2ba. Jakob, M. (1957) Heat Transfer II John Wiley, New York, USA and Chapman and Hall, London, UK
3ba. Robert Sier (1999). Hot air caloric and stirling engines. Vol.1, A history (1st Edition (Revised) ed.). L.A. Mair.
4ba. Martini, William R. (1983). “Stirling Engine Design Manual” (Second ed.). National Aeronautics and Space Administration. Retrieved 2 November 2014.
5ba. JingwenLubYut, ChunfengSong; kaKitamurab (2015). “Study on the COP of free piston Stirling cooler (FPSC) in the anti-sublimation CO2 capture process”. Renewable Energy. 74: 948–954
6ba. Cairelli, J.E. “NASA Advanced Refrigerator/Freezer Technology Development Project Overview” (PDF). NASA Technical Memorandum 106309: NASA. 8th Intemational Cryocooler Conference sponsored by the ICC Conference Committee Vail, Colorado, June 28-30, 1994.

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