Key wordsSTIRLING ENGINE CYCLE

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 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 *V*_{D}, the cylinder volume on the hot side engine *V*_{HV} and the cylinder volume on the cold side engine of the engine *V*_{CV}. The mean working temperature in all volumes is being varied during one cycle between their maximum value and their minimum value, see Figure 1.

1:

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*=*T*_{HR}/*T*_{CR}=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.

2:

The integration constant *C*_{E} 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.

3:

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.

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.

4:

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.

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.

5:

6:

Through these equations be can calculating the reduction volume *V*_{red} of engine.

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

7:

If the enter of calculation contains the mean working pressure of cycle then *C*_{E} can be determined through iteration from result Equation 7.

Problem 1:

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 cm^{3}, death volume on cold side 90 cm^{3}, the regenerator volume 68,682 cm^{3}, 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.

Problem 1: results

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

8:

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].

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 *T*_{HR,max} be must used iteration calculation. It means that during first step of the calculation be must estimate of *T*_{HR,max} and by the Equation 8 computes the temperature *T*_{HR,mean}. This result must be equal with the required mean temperature on hot side the engine.

Problem 2:

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.

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

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

9:

(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.

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

10:

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.

Author:

Jiří, Škorpík (first name, surname)

Issue date:

July, 2009; update 2023

Title:

Stirling engine cycle

Journal:

Transformační technologie (on-line journal at transformacni-technolgie.cz; turbomachinery.education; stirling-engine.education)

ISSN:

1804-8293

Copyright©Jiří Škorpík, 2012

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©Jiří Škorpík, LICENCE