Jiří Škorpík, skorpik.jiri@email.cz
Working pressure equation  
Working volumes at crankshaft engine  
Working temperature equations  
Stirling cycle and Schmidt cycle  
Other comparative cycles for Stirling engine  


Inside a Stirling engine is transformed heat to work through a closed heat cycle. The internal work of the engine, torque and others engine parameters can be predicted through p-V diagram, respectively p-φ diagram, where φ is angle of rotation of crankshaft. An accuracy of these computations is directly proportional to a similarity of shape of the designed cycle to real Stirling engine cycle. The Stirling engine cycle, which is described here is based on several types of heat cycles used to calculation of the Stirling engine cycle, because these cycles have common features (so called assumptions of solving).

Working pressure equation

Working pressure

Working temperature

Temperature inside regenerator

Working pressure inside the engine is changed under a change of temperature and a volume of the working volume. From description of the Stirling engine can be the working volume separated to three volumes, the death volume VD, the cylinder volume on the hot side engine VHV and the cylinder volume on the cold side engine of the engine VCV. The mean working temperature in all volumes is being varied during one cycle between their maximum value and their minimum value, see Figure 1.

Temperatures of working gas
T [K] working temperature; V [m3] volume. Subscript H is term for the hot side engine, subscript C is term for the cold side of the engine, subscript V is term for a cylinder volume, subscript D is term for the death volume, subscript R is term for regenerator, subscript HR is term for the interface between hot volume of the engine and the regenerator, subscript CR is term for the interface between cold volume of the engine and the regenerator. The derivation equation of the mean working gas temperature inside the regenerator is shown in the Appendix 436 or [Martini, 2004, p. 69].

Simplifying assumptions

Temperature ratio

Working pressure

Reduced volume

The calculation of the Stirling engine cycle, which is described here is can use under these simplifying assumptions: (1) Value of mean polytropic index of thermodynamics processes inside working volumes of engine is the same during one cycle. – (2) Temperature ratio at border of regenerator is constant, τ=THR/TCR=const. – (3) There is no pressure loss, pressure of working gas is same in all working volumes. (4) Working gas is ideal gas. – (5) Stirling engine is perfect sealed. – (6) Stirling engine cycle is steady (the same cycle is repeated). Under these simplifying assumptions, Equation 2 can be derived to calculate engine pressure as function of the working volume.

The equation of the working gas pressure inside the Stirling engine as function of the working volume
p [Pa] working pressure; CE [Pa·m3] constant of engine; n [1] mean value of polytropic index (polytropic index is function of change volume); τ [1] temperature ratio between hot and cold side of regenerator; τR [1] temperature ratio between temperature on hot side of regenerator and mean temperature in regenerator; Vred [m3] reduced volume. Derivation of this equation is shown in the Appendix 437. This equations was first published in [Škorpík, 2008], [Škorpík, 2009].

Constant of engine

The integration constant CE can be computed from any point of cycle, in which be knownworking pressure (for expample required mean working pressure) and a value of reduced volume, see Equation 3.

Physical interpretation of the integration constant.

Polytropic index

The polytropic index inside the Stirling engine may be in the interval <1; κ>; (κ [1] heat capacity ratio – adiabatic index) for steady cycle. The polytropic index can not be less than 1. If n is equal κ, then an engine is perfectly thermally isolated and between the hot and cold side of the engine can not occur temperature difference (τ is equal 1 and internal work of the engine is zero). In case n=1 only isothermal processes are performed in the engine, therefore the isothermal processes can be regarded as comparative processes for real processes.

Isothermal ratio

The isothermal ratio determines (see Equation 4) how much polytropic process inside the working volume is similar to isothermal process. Its value can be in interval <0; 1>. If value of isothermal ratio is 1, then thermodynamic process inside the working volume is isothermal process. If value of isothermal ratio is 0, then thermodynamic process in the working volume is adiabatic process. The difference κ-1 is the maximum deviation between polytropic process and isothermal process inside the working volume.

Definition of mean polytropic index.
ν [1] isothermal ratio.

Isothermal ratio values

The isothermal ratio about 0,5 is the usual value for engines with ideal heat transfer between the working gas and heat flow area (e.g. Strojírny Bohdalice Stirling engines, United Stirling V160). The isothermal ratio smaller than 0,5 is the usual value for engines with small heat flow area, higher speed, small death volumes. The isothermal ratio bigger than 0,5 is the usual value for engines with bigger heat flow area, small speed, bigger death volume or engines with control heat flow (difficult to achieve). Polytropic index is function engine speed, engine geometry and working gas.

Working volumes at crankshaft engine

Crankshaft engine

The motion of the piston is often performed through a crankshaft, then the volumes of the engine are function of an angle of rotation φ, length of crank r and length connecting rod c, see Figure 5 and Equation 6.

The kinematic of pistons of an α-configuration of the Stirling engine
φ [rad] angle of rotation; α [rad] phase angle of motion of pistons; r [m] length of crank; c [m] length connecting rod; l [m; %] hot piston position-H and cold piston position-S; S [m2] area cross section of cylinder on hot side-H and area cross section of cylinder on cold side-C.
Cylinder volumes as function φ

Through these equations be can calculating the reduction volume Vred of engine.

Mean working pressure

The mean pressure of cycle be computed according to the mean value theorem which is applied on function p(φ), see Equation 7.

Mean pressure of cycle.

If the enter of calculation contains the mean working pressure of cycle then CE can be determined through iteration from result Equation 7.

Problem 1:

Working pressure profile

An α-configuration of the Stirling engine which is filled by helium, with crankshaft and about this parameters: cylinder diameter 68 mm (on hot side and cold side are the same diameter), lenght of crank 22 mm, length connecting rod 105 mm, death volume on hot side 110 cm3, death volume on cold side 90 cm3, the regenerator volume 68,682 cm3, the mean temperature of the working gas on hot side of the regenerator 900 K, the mean temperature of the working gas on cold side of regenerator 330 K, the mean pressure 15 MPa, the phase angle 105°. Find working pressure profile as function angle of rotation and others significant angles.
pressure profile inside engine stirling engine
Problem 1: results

Working temperature equations

Working temperature

If n≠1 then temperature of the working gas is changed inside individual volumes according equations:

Temperature of the working gas inside individual volumes of the Stirling engine
Subscript mean is term mean temperature of the working gas during one cycle. Derivation of these equations is shown in the Appendix  251. This equations was first published in [Škorpík, 2012].

Mean temperature

From these equations is evident, that temperature change follows pressure change and this change is the bigger the bigger is the mean temperature of the working gas in the individual volume. If mean temperature of the working gas is known only, then for a calculation of the temperature THR,max be must used iteration calculation. It means that during first step of the calculation be must estimate of THR,max and by the Equation 8 computes the temperature THR,mean. This result must be equal with the required mean temperature on hot side the engine.

Problem 2:

Working temperature profile

Find the working temperature profile of the working gas during one cycle on hot and cold side and in the regenerator of the Stirling engine about parameter from Problem 1.
Result of Problem 2.
Results of Problem 2.

Stirling cycle and Schmidt cycle

Stirling cycle

Schmidt cycle

The Stirling cycle and a Schmidt cycle are simplified Stirling engine cycles with assumtion n=1.

Stirling cycle

The Stirling cycle (Figure 9) assumes linear motion of pistons and zero death volumes.

Stirling cycle
(a) p-V diagram; (b) trajectory of pistons. r [J·kg-1·K-1] individual gas constant of working gas; m [kg] mass of working gas inside working volume; t [s] time. A-trajectory of hot piston; B-trajectory of cold piston. Derivation this situation is shown in the Appendix 447.

Schmidt cycle

Gustav Schmidt

The Schmidt cycle assumes sinusoidal move of pistons (c=0, see Equations 10) and non-zero death volumes – Gustav Schmidt (1826-1881), professor at German Polytechnic in Prague. He published his cycle at 1871. Details about these cycles are shown in [Škorpík, 2008], [Walker, 1985]. On

Stirling cycle
φmin [rad] angle of rotation at pmin; φmax [rad] angle of rotation at pmax. Derivation this situation is shown in the Appendix 442.

Other comparative cycles for Stirling engine

Theodor Finkelstein

Israel Urieli

David Berchowitz

Theodor Finkelstein be published cycle where n=κ, this cycle be computed by Finite element methods [Walker, 1985], [Martini, 2004, p. 87]. The most widely is cycle with adiabatic processes in cylinders and isothermal processes in death volumes. Authors this cycle are Israel Urieli and David Berchowitz [Urieli and Berchowitz, 1984] (authors assembled set of differential equations, which are solved by Runge – Kuttak method).


ŠKORPÍK, Jiří, 2008, Příspěvek k návrhu Stirlingova motoru, VUT v Brně, Edice PhD Thesis, ISBN 978-80-214-3763-0.
ŠKORPÍK, Jiří, 2009, A new comparative cycle of a Stirling engine, The 14th International Stirling Engine Conference, Groningen – Netherlands, ISBN: 978-88-8326-022-3.
ŠKORPÍK, Jiří, 2012, Stirling engine cycle-supplement, The 15th International Stirling Engine Conference, Dubrovnik-Croatia, ISBN: 978-88-8326-019-3.
MARTINI, William, 2004, Stirling engine design manual, University press of the Pacific, Honolulu, ISBN: 1-4102-1604-7.
URIELI, Israel, BERCHOWITZ, David, 1984, Stirling Cycle Engine Analysis, Adam Hilger Ltd., Bristol, ISBN 978-0996002196.
WALKER, Graham, 1980, Stirling engine, Oxford University Press, Oxford.

Citation this page

Jiří, Škorpík (first name, surname)
Issue date:
July, 2009; update 2023
Stirling engine cycle
Transformační technologie (on-line journal at transformacni-technolgie.cz; turbomachinery.education; stirling-engine.education)

Copyright©Jiří Škorpík, 2012
All rights reserved.

©Jiří Škorpík, LICENCE