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The power output on shaft of the Stirling engine is influenced by losses. These losses can be separated on thermodynamic losses and mechanical losses. The thermodynamic losses are influencing a shape of p-V diagram, the mechanical losses are caused through friction in mechanisms (in bearings, between piston rings and a cylinder atc.). So, there are other losses, which may not influence on the power output of the engine. There are other losses also , which may not influence on the power output of the engine, but can influence fuel consumption or a heat (efficiency of heat source, conduct of heat inside block engine etc.), this type of losses is described in [5, s. 105].

For base design of the Stirling engine can be used theories of model/dimensionless quantity similar like for a design other machines (see also an article __18. Similarities of turbomachines__). The most popular dimensionless quantity of the Stirling engine is **Beale number*** through it can be predicted the power output of the engine (there are other the dimensionless quantities of the Stirling engine but no use very). For start of calculation can be knew a displacement of the engine on the hot side, a operating speed, a mean pressure of working gas and a heater temperature (outside surface):

1.854 Beale number as function of mean temperature on hot side engine.Bl [-] Beale number; P [W] power output of engine; f [Hz] frequency of engine speed; p [MPa] mean pressure of working gas inside engine; _{st}T [K] mean temperature of working gas on hot side; _{O}V [cm_{TVmax}^{3}] cylinder volume on hot side. a limit for conventional stainless steel of heater of Stirling engine; b border between high-alloy steel and ceramic materials. This function was created for cooler temperature 65 °C and be can use for all types of the Stirling engines and species of working gas. |

- *Beale number
- This method was described by Walker Graham in the year 1979 [2], [3, p. 57]. Name this method was quoted according Williama Beala, which as first man was observed similarities of Stirling engines during his work in Sunpower, Inc. (USA; Ohio; Athens) where he tested of many Stirling engines (more information about historian connections on [5, p. 99]).

Beale number of the engine is near the upper line with smaller death volume and lower temperature of working gas on cold side. Beale number of the engine is near the down line with bigger death volume and higher temperature of working gas on cold side.

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The Stirling engine cycle is influenced by leakage of piston rings, conduct of heat from/to working gas to/from surroundings, conduct of heat in a matrix of regenerator and pressure drop, which are developed during flow of working gas. With increase of these losses is increased different between real cycle and ideal comparing cycle. For calculation of p-V diagram of the Stirling engine cycle are used analytical or numerical methods:

- Analytical method
- In the first step is calculated p-V diagram of Stirling engine cycle for case constant polytropic index inside all working volume through formulas, which are showed in article 34. Stirling engine cycle. This diagram includes some types of losses, which the Schmidt theory; [5, p. 71]; [3, p. 40] does not (the losses of a different between isothermal process and real process). In next step is refined p-V diagram according predicting leakage of piston rings, which has significant influence (the description of this loss is showed in next chapter). The work and efficiency of the engine can be refined with taking on other losses according [5, p. 105], [3].

- Numerical calculation (CFD model)
- This method includes real conditions (heat transfer coefficients; change of thermo-mechanical properties of working gas) and also losses (losses conduct the heat from/to surroundings; losses leakage of the piston rings etc.). This method is difficult on to enter initial and boundaries conditions and there is problem with calculation any losses due to convergence solve yet. Khamid Mahkamov is one of the first man, which made complete CFD model of the Stirling engine cycle including combustion system, this computation be was checked on real engine (γ-configuration,
*0,5 kW*) [1, p. 96].

This paper describes only the calculation of the p-V diagram for case analytical method. From an experience and a measurement of the Stirling engines clearly shows, that greatest influence on change of the shape of p-V diagram has the polytropic exponent and **the leakage of piston rings** (it deforms p-V diagram from all sides). An influence others losses* on the shape of p-V diagram can be negligible.

Great influence can have the pressure loss also. In this case is improper design of the engine (high flow speed). Therefore is calculated with pressure loss on work only, and for calculation p-V diagram the pressure loss is not significant.

The leaks of piston rings has major impact on work of Stirling engine, it was shown the practical experiences with construction of Stirling engines and its operating. It often is the biggest technology problem during developing of new engine. The leakage of pistons is being caused through roughness of cylinder surface, difference between diameter of cylinder and piston ring and vibration of engine during operate.

There are piston rings on hot side and cold side. Piston rings are separated of working volume of engine from the volume under pistons.Pressure of working gas inside working volume is changed during one cycle. If the volume under piston is not working volume then it should be so big for approximate constant pressure here. The working gas is being flowed through the leaks of piston rings from working volume to under the pistons if pressure of working gas is higher than pressure under pistons and on the contrary. It means that the mass of working gas is not constant in working volume of engine.

2.476 The scheme of Stirling engine (α-configuration).a piston ring; b volume under pistons and buffer tank; TS hot side of engine; SS cold side of engine. The figure is shown outflow/inflow of working gas through piston rings during one cycle. |

The mass of working gas is varied between a maximum and a minimum value during an cycle. Therefore the cycle can be divided into two intervals. The mass of working gas is decreased on the one interval of cycle (working gas is flowing from working volume) and is increased on the next interval of cycle (working gas is flowing to working volume). This change of mass of working gas is influenced of p-V diagram shape. Strictly speaking, maximum pressure of cycle is lower, minimum pressure is higher than for case absolute seal of piston rings. The change of pressure is not large usually, nevertheless reducing of the internal work of the Stirling engine is big. This big reduce is caused a shift of maximum pressure left in p-φ diagram. It means that that the maximum pressure of working gas is reached earlier for case a leaks of piston rings than for case absolute seals piston rings:

This diagram is for case α-configuration Stirling engine and for constant pressure under piston. In case double-acting engine is curve

The leak through piston rings is defined as ration between maximum change mass of working gas and mass of working gas in working volume for case absolute seals of piston rings:

There is other leaks in the engine respectively a connection of working volume with volume under pistons. This connection is function construction.

During comparisons the measured p-φ diagram with computed p-φ diagram by method that is shown in article 34. Stirling engine cycle was found a similarity [7, s. 63]. The shape of curve of pressure was similarity to the shape computed of pressure, but it is shifted on difference *Δφ* and it is flattened around pressure extremes about *Δp*. On the base these knowledges was created simplify assumptions of solving Stirling engine cycle with leaks through piston rings:

(1) Pressure inside Stirling engine for case absolute seals of piston rings (ideal cycle) is equaled. (2) Pressure under pistons is constant and equal to the mean pressure of cycle. (3) Leaks through piston rings is brought shift ideal cycle on angle Δφ. (4) Different of pressure is directly proportional to the difference between the pressure and medium pressure (linear model). (5) difference pressure between maximum pressure for case ideal cycle and maximum pressure for case real cycle is equal half of total mass loss of working gas Δm. (6) The working gas outflow/inflow from/in working volume through piston rings only. (7) The working gas outflow/inflow from/in working volume is not brought change temperature of working gas and its change polytropic index. (8) Cycle is steady (the same cycle repeats).

If the pressure is function of rotation of shaft *p(φ)* then the shift of pressure about angle *Δφ* is function of rotation of shaft and the shift *p'(φ+Δφ)*. The shift *Δφ* is computed from minimum of mass of working gas in working volume:

The *V _{red}* equation is function of type of mechanism. The most frequently is used crank shaft and in this case be can derived equation of

The alternates change of working gas in working volume causes a decrease pressure ratio, too. The decrease pressure ratio according to *Assumption 5(4)* is equal and directly proportional to pressure difference and mean pressure:

Find a estimate p-φ and p-V diagram of Stirling engine from Problem 1 [34.] for case *μ''=4*.

**Problem 1.**485

φst [°] 163,72 pI [MPa] 15,63 Δφ [°] 5,76 Vnred(φst) [m3] 6,65E-5 φI [°] 157,97 γ [-] 0,064 μ'' [-] 0,04 Vnred(φI) [m3] 6,38E-5 φ [°] p' [MPa] p'' [MPa] φ [°] p' [MPa] p'' [MPa] --------------------------- --------------------------- 0 15,079932 15,07482 190 12,236765 12,41364 10 16,061489 15,99354 200 11,640273 11,85533 20 17,111215 16,97608 210 11,163332 11,40892 30 18,194737 17,99024 220 10,797822 11,0668 40 19,259821 18,98715 232,5 10,485296 10,77428 52,5 20,456959 20,10766 240 10,369346 10,66575 60 21,041075 20,65439 250 10,293062 10,59435 70 21,589928 21,16811 259,647 10,300495 10,60131 80 21,814108 21,37794 270 10,393907 10,68874 87,0244 21,756338 21,32386 280 10,566627 10,85041 100 21,187625 20,79156 285 10,683327 10,95964 105 20,824310 20,4515 300 11,156593 11,40261 120 19,377154 19,09697 310 11,577794 11,79685 130 18,229126 18,02243 320 12,087572 12,274 140 17,037437 16,90702 340 13,388847 13,49198 150 15,873321 15,81742 350 14,186114 14,23821 160 14,787481 14,80108 360 15,079932 15,07482 180 12,959416 13,09003

It is evident, that wear of piston rings is brought significant decrease of work of cycle, therefore is reasonable create the internal work of the engine as function *μ''* similar as in next problem:

Find an internal work of the engine as function *μ''* for Stirling engine from Problem 1 [34.]. The internal work of engine is calculated in Problem 1 [35.].

**Problem 2.**487

A [J] 876,1982 μ'' [-] A" [J] A''/A [-] μ'' [-] A" [J] A''/A [-] --------------------------- --------------------------- 0 876,1982 1 0,12 314,7611 0,3592 0,04 681,9396 0,7783 0,16 148,0698 0,169 0,08 495,2261 0,5652

Results of Problem 2.A [J] internal work of the engine for case absolute seal of piston rings; A'' [J] internal work of the engine for case the leakage of piston rings. |

From definition of difference

From results of last problem is can deduced that the influence of piston rings on work of cycle is big. Already for a leakage *13%* the internal work of the engine is reduced about *50%*, for a leakage *20 %* engine is not operational really. The leakage is function roughness of cylinder surface, a deformation of piston rings and vibration of engine. The size of leakage can be computed approximately from equation of mass flow rate through the nozzle:

Flow coefficient taken into account of flow losses through flow area. Flow coefficient is function of roughness and shape of flow area. Flow coefficient for ideal jet is

The region around piston rings is being cold and temperature of working gas under piston is approximate constant, therefore it is not necessary doing compute of leakage for each piston ring separately.

From Equation 6. is evident that mass flow is function of specific volume respectively temperature. The working gas on hot side is being flowed through piston rings is more hot than working gas, which is being flowed return. It is being brought a return of more working gas to working volume in part *II* of cycle than was outflow in part *I* of cycle. Therefore is drilled to piston small a nozzle* (about tenth of millimeter-diameter is function *V _{VTmax}* and pressure ratio), which is brought higher leakage and parallel is done balance between outflow/inflow of working gas. On hot side is situation opposite, but difference of temperature is not big and the unbalance mass flow on hot side is more significantly.

The nozzle is made on place where there is not big temperature difference between gas over/under piston-usually on cold side.

Find approximate of leakage (flow area) of Stirling engine from Problem 1 [34.], if the leakage is equal as in *Problem 1*. Speed of rotation is *1530 min*^{-1}, temperature under piston is *85 °C*. Flow area is equivalent to ideal jet *μ=1*.

**Problem 3.**486

π* [-] 0,487 n [min-1] 1530 φst [°] 364,90 π [-] 0,5988 Tp [°C] 85 φII [°] 359,15 Δm [kg] 2,3694E-4 vst [m3·kg-1] 0,0496 A [mm2] 1,1452 μ [-] 1 φ [°] χm [-] φ [°] χm [-] φ [°] χm [-] -------------- -------------- -------------- 157,97 0 232,5 0,6384 300 0,6244 160 0,1148 240 0,6484 310 0,5992 180 0,3629 250 0,6526 320 0,569 190 0,5088 259,647 0,6541 340 0,496 200 0,5615 270 0,6519 350 0,3755 210 0,5971 280 0,646 359,15 0,2228 220 0,6212 285 0,6392

This method of compute of Stirling engine cycle for case a leakage of piston rings is based from measurements of experimental engine from 2002 to 2012 years and these experiments is made company Tedom a.s. [4] respectively company Strojírny Bohdalice, a.s. [6], which in this development is continued. Leader this project is Josef Brož. Some measurements are accessible in [7].

- MAHKAMOV, Khamid, DJUMANOV, D. Three-dimensional CFD modeling of a Stirling engine,
*Proceedings of the 11th International Stirling engine conference*, 19.-21. November 2003. Roma: Department of Mechanical and Aeronautical Engineering University of Rome “La Sapienza”. - WALKER, Graham. Elementary design guidelines for Stirling engines,
*Proceedings of the 14th Intersociety Energy Conversion Engineerng Conference*, 1979. Boston: American Chemical Society. - WALKER, Graham.
*Dvigateli Stirlinga/Двигатели Стирлинга*, 1985. Doplněný Ruský překlad knihy: WALKER, Graham*Stirling engine*, 1980. Oxford: Oxford University Press. *Tedom a.s.*, 2012. Společnost se zabývá výrobou a vývojem kogeneračních jednotek. Adresa: Výčapy 195, Třebíč, 674 01. Web: http://tedom.cz.- MARTINI, William.
*Stirling engine design manual*, 2004. Přetisk vydání z roku 1983. Honolulu: University press of the Pacific, ISBN: 1-4102-1604-7. *Strojírny Bohdalice, a.s.*, 2012. Výroba a vývoj strojních zařízení. Adresa: Bohdalice, 683 41. Web: http://www.strobo.cz.- ŠKORPÍk, Jiří.
*Příspěvek k návrhu Stirlingova motoru*, 2008. Disertační práce obhájená na Vysokém učení technickém v Brně, Fakulta strojního inženýrství, v oboru Konstrukční a procesní inženýrství v roce 2008, ISBN 978-80-214-3763-0 (Ph.D. thesis). - ŠKORPÍk, Jiří.
*A leakage of piston rings and their impact on work Stirling engine*. The 14th International Stirling Engine Conference, in Groningen – Netherlands, 2009, ISBN: 978-88-8326-022-3.

This document is English version of the original in Czech language: ŠKORPÍK, Jiří. Ztráty ve Stirlingových motorech, *Transformační technologie*, 2009-07, [last updated 2012-10]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804–8293. Dostupné z http://www.transformacni-technologie.cz/36.html. English version: Losses in Stirling engines. Web: http://www.transformacni-technologie.cz/en_36.html.

©Jiří Škorpík, LICENCE

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