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Inside a Stirling engine is transformed heat to work through a closed __heat cycle__. From a design of heat cycle in p-V diagram can be approximately calculated an internal work of the engine, torque and others engine parameters. 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).

Pressure of a working gas inside working volume of the Stirling 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, the cylinder volume on the hot side engine and the cylinder volume on the cold side engine of the engine. The mean working gas temperature in all volumes varies during one cycle between their maximum value and their minimum value:

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_{TR}/T_{SR}=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 repeats).

An equation for working gas pressure inside the engine as function its volume can be derived from the assumptions of solvings, which are presented in a previous paragraph:

Last equation shows that only temperatures on boundaries regenerator influences progress of pressure and not mean temperatures in the cylinders. For running engine is necessary difference of temperature between hot and cold side of the engine.

Physical interpretation of the integration constant can be obtainable through a derivation of pressure equation:

4.id438 Physical interpretation of C_{int}.The integration constant can be computed from any point of cycle, in which be known pressure and a value of reduced volume. |

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The polytropic index inside the Stirling engine may be in the interval *<1; κ>*; (**κ** [-] 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 realized in the engine, therefore the isothermal processes can be regarded as comparative processes for real processes:

5.id446 Definition of mean polytropic index.ν [-] isothermal ratio*. |

- *Isothermal ratio
- The isothermal ratio determines 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:

ν [-] ------------ (a) ≐0,5 (b) <0,5 (c) >0,5

Measured of the isothermal ratio can shows constructional deficiencies of the engine.

Equtions for calculation of mass and temperature of the working gas, the internal work of the engine, heat input, heat rejection and regenerated heat during one cycle are shown in article 35. Energy balance of Stirling engine cycle.

The motion of the piston is often realized through a crankshaft, then the volumes of the engine are function of an angle of rotation *φ* (*V _{TV}(φ)*;

In this case for calculation of the pistons position can be used the __equation for piston position conencted with crankshaft__:

Combination of *Equation 3* with *Equation 8* be obtained the equation of pressure as function *φ*. From extremes of function *p(φ)* can be calculate minimum, maximum pressure and pressure ratio during one cycle *φ<0; 2π)*:

The mean pressure of cycle be computed according to the mean value theorem which is applied on function *p(φ):*

10.id443 Mean pressure of cycle. |

If the enter of calculation contains the mean pressure of cycle then *C _{int}* can be determined through iteration from result

Through same procedure can be derived of equations of piston motion for others configurations of the Stirling engine.

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 progress of pressure as function angle of rotation and others significant angles.

**Problem 1.**id444

ST [cm2] 36,3168 τ [-] 2,7273 TR [K] 568,124 τR [-] 1,5842 ν [-] 0,5 κ [-] 1,67 n [-] 1,335 Cint [Pa·m3] 997,548 φ [°] VTV [cm3] VSV [cm3] p [MPa] ------------------------------------------------------- φTVmin 0 0 108,4669 14,554 10 1,4663 95,2595 15,4867 20 5,7988 81,3326 16,5003 30 12,8025 67,1092 17,5696 40 22,1665 53,0694 18,6527 φVmax 36,5639 36,5639 23,955869 19,933 60 46,2786 27,6096 20,6038 70 60,0347 17,2129 21,3098 80 74,2291 8,9836 21,7276 φmin 87,0244 84,1905 4,6979 21,8169 100 101,977 0,3676 21,5106 φSVmin 105 108,4669 0 21,2359 120 126,1756 3,2835 19,9843 130 136,1976 8,9836 18,9003 140 144,5758 17,2129 17,7239 150 151,1881 27,6096 16,5364 160 155,9559 39,7284 15,4005 φTVmax 180 159,7939 67,1092 13,4338 190 158,8326 81,3326 12,6369 200 155,9559 95,2595 11,9684 210 151,1881 108,4669 11,4236 220 144,5758 120,6012 10,995 φVmin 232,5 133,8403 133,8403 10,61 240 126,1756 140,6009 10,4536 250 114,6874 148,1084 10,3262 φmax 259,647 102,4432 153,6366 10,2866 270 88,3611 157,6326 10,3312 280 74,2291 159,5535 10,4573 φSVmax 285 67,1092 159,7939 10,5507 300 46,2786 157,6326 10,9529 310 33,484 153,8061 11,3248 320 22,1665 148,1084 11,7831 340 5,7988 131,3824 12,9747 350 1,4663 120,6012 13,7152 360 0 108,4669 14,554

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

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 _{T,max}* be must used iteration calculation. It means, during first step of the calculation be must estimate of

Find the temperature change 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*.

**Problem 2.**id819

φ [°] TT [K] TS [K] TR [K] φ [°] TT [K] TS [K] TR [K] -------------------------------- -------------------------------- 0 899,07 329,66 567,52 190 867,76 318,18 547,76 10 913,19 334,83 576,44 200 856,01 313,87 540,34 20 927,84 340,20 585,68 210 846,06 310,22 534,06 30 942,57 345,61 594,98 220 837,98 307,26 528,96 40 956,83 350,83 603,98 232,5 830,52 304,52 524,25 52,5 972,90 356,73 614,13 240 827,43 303,39 522,30 60 981,02 359,70 619,25 250 824,88 302,45 520,69 70 989,34 362,76 624,51 259,65 824,09 302,16 520,19 80 994,18 364,53 627,56 270 824,98 302,49 520,76 87,02 995,2 364,90 628,20 280 827,50 303,41 522,35 100 991,68 363,61 625,98 285 829,35 304,09 523,51 105 988,48 362,44 623,96 300 837,17 306,96 528,45 120 973,53 356,96 614,52 310 844,22 309,54 532,90 130 960,00 352,00 605,98 320 852,66 312,64 538,23 140 944,64 346,37 596,29 340 873,52 320,29 551,40 150 928,35 340,39 586,00 350 885,78 324,78 559,13 160 911,91 334,37 575,63 360 899,07 329,66 567,52 180 881,18 323,10 556,23

Results of Problem 2. |

The Stirling cycle and a Schmidt cycle* are simplified Stirling engine cycles with assumtion *n=1*. This methods are very very popular. The Stirling cycle assumes linear motion of pistons and zero death volumes, the Schmidt cycle assumes sinusoidal move of pistons and non-zero death volumes. Details about these cycles are shown in [4], [3].

- Gustav Schmidt; 1826-1881
- Professor at German Polytechnic in Prague. He published his cycle at 1871.

Theodor Finkelstein be published cycle where *n=κ*, this cycle be computed by Finite element methods [3], [2, 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 [1] (authors assembled set of differential equations, which are solved by Runge – Kuttak method).

Thermodynamic cycle described here is based on mean values of temperature ratio and polytropic index, but these quantities are variable during one cycle at real process. Mass of the working gas in the working volume is not constant also at real process (alternate ingress/leakage of the working gas through piston rings, see article 36. Losses in Stirling engines). These factors (especially the last said) significantly influence calculated diagrams.

- URIELI, Israel, BERCHOWITZ, David.
*Stirling Cycle Engine Analysis*, 1984. 1. vydání. Bristol: Adam Hilger Ltd., ISBN 978-0996002196. - MARTINI, William.
*Stirling engine design manual*, 2004. Přetisk vydání z roku 1983. Honolulu: University press of the Pacific, ISBN: 1-4102-1604-7. - WALKER, Graham.
*Dvigateli Stirlinga/Двигатели Стирлинга*, 1985. Doplněný Ruský překlad knihy: WALKER, Graham*Stirling engine*, 1980. Oxford: Oxford University Press. - ŠKORPÍK, Jiří.
*Příspěvek k návrhu Stirlingova motoru*, VUT v Brně, Edice PhD Thesis, 2008, ISBN 978-80-214-3763-0. - ŠKORPÍK, Jiří.
*A new comparative cycle of a Stirling engine*, The 14th International Stirling Engine Conference, in Groningen – Netherlands, 2009, ISBN: 978-88-8326-022-3. - ŠKORPÍK, Jiří.
*Stirling engine cycle-supplement*, The 15th International Stirling Engine Conference, in Dubrovnik–Croatia, 2012, ISBN: 978-88-8326-019-3.

ŠKORPÍK, Jiří. Oběh Stirlingova motoru, *Transformační technologie*, 2009-07, [last updated 2012-01]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z http://www.transformacni-technologie.cz/obeh-stirlingova-motoru.html. English version: Stirling engine cycle. Web: http://www.transformacni-technologie.cz/en_obeh-stirlingova-motoru.html.

©Jiří Škorpík, LICENCE

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