1912:", 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
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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
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2125:. 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
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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
1895:
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
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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
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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
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This type of plot is used to characterize almost all thermodynamic cycles. The result of sinusoidal volume variations is the quasi-elliptical shaped cycle shown in Figure 1. Compared to the idealized cycle, this cycle is a more realistic representation of most real
Stirling engines. The four
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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
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Figure 5 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
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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
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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.
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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.
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Also referred to as "pumping losses", the pressure drops shown in Figure 3 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,
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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.
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caveat: Among the many inaccuracies in this article, a co-linear alpha configuration is referenced, above. Such a configuration would be beta. Alternatively, it would be an alpha, that has an unacceptably inefficient linkage system.
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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.
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Figure 4 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.
2032:, 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.
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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
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Figure 6 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
1769:(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".
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cycles are not totally reversible because they involve heat transfer through a finite temperature difference during the irreversible
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2024:) heat addition. The compressed air flows back through the regenerator and picks up heat on the way to the heated expansion space.
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heat-addition and heat-rejection processes. The irreversibility renders the thermal efficiency of these cycles less than that of a
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heater, the regenerator, the cooler, the cooler clearance volume, and the compression volume swept by the compression piston.
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Stirling cycle. In real applications of the
Stirling cycles (e.g. Stirling engines) this cycle is quasi-elliptical.
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compression and expansion in the engine, together with non-ideal heat exchangers which have a limited rate of
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Stirling cycle; however, the four thermodynamic processes are slightly different (see graph above):
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Israel Urieli (Dr. Iz), Associate
Professor Mechanical Engineering: Stirling Cycle Machine Analysis
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In beta and gamma engines, generally the phase angle difference between the piston motions is
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The cycle is reversible, meaning that if supplied with mechanical power, it can function as a
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1984:. The expansion space is heated externally, and the gas undergoes near-isothermal expansion.
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working fluid. "Closed cycle" means the working fluid is permanently contained within the
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Organ, "The
Regenerator and the Stirling Engine", p. xxii, Foreword by Urieli
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that describes the general class of Stirling devices. This includes the original
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In the most basic model of a free piston device, the kinematics will result in
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1746:. "Regenerative" refers to the use of an internal heat exchanger called a
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1855: in this section. Unsourced material may be challenged and removed.
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Organ, "The Regenerator and the Stirling Engine", p. 7
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Thermodynamic cycle that includes the basic Stirling engine
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John Wiley, New York, USA and Chapman and Hall, London, UK
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Alternative thermodynamic cycle for the Stirling machine
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Hot air caloric and stirling engines. Vol.1, A history
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Stirling cycle. For the idealized Stirling cycle, see
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that was invented, developed and patented in 1816 by
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acting on the working fluid (See diagram to right):
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2013:, so the gas undergoes near-isothermal compression.
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1904:Many kinematic linkages, such as the well known "
2272:Organ, "The Regenerator and the Stirling Engine"
1949:points in the graph indicate the crank angle in
1742:. This also categorizes the engine device as an
1995:) heat removal. The gas is passed through the
1815:4→1 Isochoric heat addition (constant volume).
2331:
2205:(1st Edition (Revised) ed.). L.A. Mair.
1809:2→3 Isochoric heat removal (constant volume).
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8:
2260:, American Journal of Physics 85, 926 (2017)
1762:nor representative of the Stirling cycle".
1730:cooling. The cycle is defined as a closed
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1987:270° to 0°, near-constant-volume (or near-
1812:3→4 Isothermal heat removal (compression).
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1871:Learn how and when to remove this message
1806:1→2 Isothermal heat addition (expansion).
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2309:Polytropic cycle inside Stirling engine
2016:90° to 180°, near-constant-volume (near-
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1715:constant-volume regeneration processes.
2500:Homogeneous charge compression ignition
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1773:Idealized Stirling cycle thermodynamics
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2063:Alpha type Stirling. Animated version.
1887:A model of a four-phase Stirling cycle
2157:converts energy from heat into work.
7:
1853:adding citations to reliable sources
1667:Applications of the Stirling engine
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355:Intensive and extensive properties
25:
2028:With the exception of a Stirling
1969:Stirling cycle is similar to the
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2085:
2051:
1956:
1829:
1797:Stirling cycle consists of four
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1611:
1610:
930:Table of thermodynamic equations
2305:Stirling Cycle Machine Analysis
2182:Stirling radioisotope generator
2133:Cumulative heat and work energy
2047:
1840:needs additional citations for
1687:with help from his brother, an
1406:Maxwell's thermodynamic surface
2109:Temperature versus crank angle
1765:The analytical problem of the
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1085:
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978:
1:
2177:Solar-powered Stirling engine
1750:which increases the device's
1660:Stirling engine § Theory
1307:Mechanical equivalent of heat
2072:Heat-exchanger pressure drop
2009:. The compression space is
1944:Pressure-versus-volume graph
919:Onsager reciprocal relations
2405:Stirling (pseudo/adiabatic)
2093:Pressure versus crank angle
1411:Entropy as energy dispersal
1222:"Perpetual motion" machines
1161:{\displaystyle G(T,p)=H-TS}
1106:{\displaystyle A(T,V)=U-TS}
1051:{\displaystyle H(S,p)=U+pV}
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2057:
2050:
1664:
1654:This article is about the
1653:
858:{\displaystyle \partial T}
811:{\displaystyle \partial V}
726:{\displaystyle \partial p}
679:{\displaystyle \partial V}
591:{\displaystyle \partial T}
544:{\displaystyle \partial S}
2079:multivariate optimization
1332:An Inquiry Concerning the
1821:Piston motion variations
1345:Heterogeneous Substances
762:{\displaystyle \alpha =}
630:{\displaystyle \beta =-}
1799:thermodynamic processes
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1921:simple harmonic motion
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996:{\displaystyle U(S,V)}
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475:Specific heat capacity
79:Quantum thermodynamics
2311:Stirling engine cycle
2167:Pseudo Stirling cycle
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2030:thermoacoustic engine
1977:180° to 270°, pseudo-
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1783:pressure/volume graph
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1343:On the Equilibrium of
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1061:Helmholtz free energy
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2640:Thermodynamic cycles
2589:Regenerative cooling
2467:combustion / thermal
2366:Without phase change
2357:combustion / thermal
2347:Thermodynamic cycles
2201:Robert Sier (1999).
2036:Particle/mass motion
1914:isothermal processes
1849:improve this article
1744:external heat engine
1740:thermodynamic system
1356:Motive Power of Fire
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924:Bridgman's equations
901:Fundamental relation
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2040:Figure 2 shows the
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1334:Source ... Friction
1266:Loschmidt's paradox
458:Material properties
336:Conjugate variables
2287:2010-06-30 at the
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2002:0° to 90°, pseudo-
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1752:thermal efficiency
1598:Order and disorder
1354:Reflections on the
1261:Heat death paradox
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495:{\displaystyle c=}
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465:Property databases
441:Reduced properties
425:Chemical potential
389:Functions of state
312:Thermal efficiency
48:Carnot heat engine
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2604:Vapor-compression
2530:Staged combustion
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2424:With phase change
2243:Jakob, M. (1957)
2127:theoretical limit
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1927:Volume variations
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834:{\displaystyle V}
787:{\displaystyle 1}
742:Thermal expansion
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702:{\displaystyle V}
655:{\displaystyle 1}
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567:{\displaystyle N}
520:{\displaystyle T}
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364:Process functions
350:Property diagrams
329:System properties
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284:Endoreversibility
176:Equation of state
16:(Redirected from
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2599:Vapor absorption
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1251:Brownian ratchet
1246:Entropy and life
1241:Entropy and time
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1816:
1813:
1810:
1807:
1803:
1774:
1771:
1673:Stirling cycle
1650:
1649:
1647:
1646:
1639:
1632:
1624:
1621:
1620:
1619:
1618:
1605:
1604:
1601:
1600:
1595:
1590:
1585:
1579:
1576:
1575:
1572:
1571:
1567:
1566:
1561:
1556:
1551:
1546:
1541:
1536:
1531:
1526:
1521:
1516:
1511:
1506:
1501:
1496:
1491:
1486:
1481:
1476:
1471:
1466:
1461:
1456:
1451:
1446:
1441:
1436:
1430:
1429:
1426:
1425:
1422:
1421:
1416:
1415:
1414:
1413:
1408:
1400:
1399:
1397:
1396:
1393:
1389:
1386:
1385:
1383:
1382:
1377:
1375:Thermodynamics
1371:
1368:
1367:
1363:
1362:
1361:
1360:
1351:
1349:
1340:
1338:
1329:
1324:
1323:
1317:
1316:
1315:
1314:
1309:
1304:
1292:
1291:
1289:Caloric theory
1285:
1282:
1281:
1277:
1276:
1274:
1273:
1268:
1263:
1258:
1253:
1248:
1243:
1237:
1234:
1233:
1227:
1226:
1225:
1224:
1217:
1216:
1211:
1206:
1200:
1197:
1196:
1190:
1187:
1186:
1183:
1179:
1178:
1177:
1174:
1173:
1169:
1168:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1113:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1058:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1003:
992:
989:
986:
983:
980:
977:
962:
960:
959:
954:
948:
947:
942:
941:
938:
937:
934:
933:
926:
921:
916:
909:
908:
903:
898:
893:
887:
886:
881:
880:
877:
876:
870:
869:
866:
865:
854:
851:
841:
830:
819:
818:
807:
804:
794:
783:
769:
758:
755:
745:
738:
737:
734:
733:
722:
719:
709:
698:
687:
686:
675:
672:
662:
651:
637:
626:
623:
620:
610:
603:
602:
599:
598:
587:
584:
574:
563:
552:
551:
540:
537:
527:
516:
502:
491:
488:
478:
469:
468:
467:
461:
456:
455:
452:
451:
446:
445:
444:
443:
438:
433:
422:
411:
392:
391:
385:
384:
382:
381:
376:
370:
367:
366:
360:
359:
358:
357:
352:
333:
332:
327:
326:
323:
322:
317:
316:
315:
314:
309:
304:
296:
295:
289:
288:
287:
286:
281:
276:
271:
269:Free expansion
266:
261:
256:
251:
246:
241:
236:
231:
223:
222:
216:
215:
214:
213:
208:
206:Control volume
203:
198:
196:Phase (matter)
193:
188:
183:
178:
170:
169:
161:
160:
155:
150:
144:
139:
138:
135:
134:
130:
129:
124:
119:
114:
108:
107:
102:
101:
98:
97:
94:
93:
82:
81:
76:
71:
66:
60:
59:
56:
55:
52:
51:
46:The classical
45:
37:
36:
34:Thermodynamics
26:
24:
18:Stirling-cycle
14:
13:
10:
9:
6:
4:
3:
2:
2657:
2646:
2643:
2641:
2638:
2637:
2635:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2600:
2597:
2595:
2594:Transcritical
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2575:
2574:Hampson–Linde
2572:
2571:
2569:
2567:
2566:Refrigeration
2563:
2557:
2554:
2552:
2549:
2547:
2544:
2543:
2541:
2537:
2531:
2528:
2526:
2523:
2521:
2518:
2516:
2513:
2511:
2508:
2506:
2503:
2501:
2498:
2496:
2495:Gas-generator
2493:
2491:
2488:
2486:
2483:
2481:
2480:Brayton/Joule
2478:
2476:
2473:
2472:
2470:
2468:
2462:
2452:
2449:
2446:
2442:
2439:
2437:
2434:
2432:
2429:
2428:
2426:
2422:
2416:
2413:
2411:
2408:
2406:
2403:
2401:
2398:
2396:
2393:
2391:
2388:
2386:
2385:Brayton/Joule
2383:
2381:
2378:
2377:
2375:
2371:
2363:
2360:
2358:
2352:
2348:
2341:
2336:
2334:
2329:
2327:
2322:
2321:
2318:
2312:
2308:
2306:
2302:
2301:
2297:
2290:
2286:
2283:
2278:
2275:
2269:
2267:
2263:
2259:
2256:A. Romanelli
2253:
2250:
2246:
2240:
2237:
2231:
2228:
2222:
2219:
2214:
2212:0-9526417-0-4
2208:
2204:
2197:
2194:
2187:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2164:
2160:
2158:
2156:
2151:
2147:
2142:
2140:
2132:
2130:
2128:
2124:
2123:heat transfer
2118:
2116:
2108:
2106:
2104:
2099:
2092:
2090:
2088:
2083:
2080:
2071:
2061:
2054:
2049:
2046:
2043:
2035:
2033:
2031:
2023:
2019:
2015:
2012:
2008:
2005:
2001:
1998:
1994:
1990:
1986:
1983:
1980:
1976:
1975:
1974:
1972:
1968:
1967:
1961:
1959:
1954:
1952:
1943:
1941:
1937:
1934:
1926:
1924:
1922:
1917:
1915:
1911:
1910:rhombic drive
1907:
1902:
1899:
1894:
1885:
1875:
1872:
1864:
1854:
1850:
1844:
1843:
1838:This section
1836:
1832:
1827:
1826:
1820:
1814:
1811:
1808:
1805:
1804:
1802:
1800:
1796:
1788:
1784:
1779:
1772:
1770:
1768:
1763:
1759:
1755:
1753:
1749:
1745:
1741:
1737:
1734:cycle with a
1733:
1729:
1725:
1721:
1716:
1713:
1712:Carnot engine
1709:
1705:
1701:
1697:
1692:
1690:
1686:
1682:
1678:
1674:
1668:
1661:
1657:
1645:
1640:
1638:
1633:
1631:
1626:
1625:
1623:
1622:
1617:
1609:
1608:
1607:
1606:
1599:
1596:
1594:
1591:
1589:
1588:Self-assembly
1586:
1584:
1581:
1580:
1574:
1573:
1565:
1562:
1560:
1559:van der Waals
1557:
1555:
1552:
1550:
1547:
1545:
1542:
1540:
1537:
1535:
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1484:von Helmholtz
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1442:
1440:
1437:
1435:
1432:
1431:
1424:
1423:
1412:
1409:
1407:
1404:
1403:
1402:
1401:
1394:
1391:
1390:
1387:
1381:
1378:
1376:
1373:
1372:
1370:
1369:
1364:
1358:
1357:
1350:
1347:
1346:
1339:
1336:
1335:
1328:
1327:
1326:
1325:
1322:
1318:
1313:
1310:
1308:
1305:
1303:
1299:
1295:
1294:
1290:
1287:
1286:
1284:
1283:
1278:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1238:
1236:
1235:
1232:
1228:
1223:
1220:
1219:
1215:
1212:
1210:
1207:
1205:
1202:
1201:
1199:
1198:
1193:
1184:
1181:
1180:
1176:
1175:
1155:
1152:
1149:
1146:
1143:
1137:
1134:
1131:
1125:
1117:
1114:
1100:
1097:
1094:
1091:
1088:
1082:
1079:
1076:
1070:
1062:
1059:
1045:
1042:
1039:
1036:
1033:
1027:
1024:
1021:
1015:
1007:
1004:
987:
984:
981:
975:
967:
964:
963:
958:
955:
953:
950:
949:
945:
940:
939:
932:
931:
927:
925:
922:
920:
917:
915:
912:
911:
907:
906:Ideal gas law
904:
902:
899:
897:
894:
892:
889:
888:
884:
879:
878:
852:
842:
828:
821:
820:
805:
795:
781:
774:
773:
770:
756:
753:
746:
743:
740:
739:
720:
710:
696:
689:
688:
673:
663:
649:
642:
641:
638:
624:
621:
618:
611:
608:
605:
604:
585:
575:
561:
554:
553:
538:
528:
514:
507:
506:
503:
489:
486:
479:
476:
473:
472:
466:
463:
462:
459:
454:
453:
442:
439:
437:
436:Vapor quality
434:
432:
431:
426:
423:
421:
420:
415:
412:
409:
405:
404:
399:
396:
395:
394:
393:
390:
386:
380:
377:
375:
372:
371:
369:
368:
365:
361:
356:
353:
351:
348:
347:
346:
345:
341:
337:
330:
325:
324:
313:
310:
308:
305:
303:
300:
299:
298:
297:
294:
290:
285:
282:
280:
277:
275:
274:Reversibility
272:
270:
267:
265:
262:
260:
257:
255:
252:
250:
247:
245:
242:
240:
237:
235:
232:
230:
227:
226:
225:
224:
221:
217:
212:
209:
207:
204:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
173:
172:
171:
168:
164:
159:
156:
154:
151:
149:
148:Closed system
146:
145:
142:
137:
136:
128:
125:
123:
120:
118:
115:
113:
110:
109:
105:
100:
99:
92:
88:
85:
84:
80:
77:
75:
72:
70:
67:
65:
62:
61:
54:
53:
49:
43:
39:
38:
35:
31:
19:
2451:Regenerative
2399:
2380:Bell Coleman
2277:
2252:
2244:
2239:
2230:
2221:
2202:
2196:
2149:
2143:
2136:
2119:
2112:
2100:
2096:
2084:
2075:
2039:
2027:
1970:
1964:
1962:
1955:
1947:
1938:
1932:
1930:
1918:
1903:
1897:
1890:
1867:
1858:
1847:Please help
1842:verification
1839:
1794:
1792:
1786:
1764:
1760:
1756:
1732:regenerative
1717:
1693:
1672:
1670:
1449:Carathéodory
1380:Heat engines
1352:
1341:
1330:
1312:Motive power
1297:
957:Free entropy
928:
428:
427: /
417:
416: /
408:introduction
401:
400: /
339:
302:Heat engines
89: /
2619:Ionocaloric
2614:Vuilleumier
2436:Hygroscopic
2155:heat engine
2146:work output
2042:streaklines
2011:intercooled
2007:compression
1997:regenerator
1767:regenerator
1748:regenerator
1271:Synergetics
952:Free energy
398:Temperature
259:Quasistatic
254:Isenthalpic
211:Instruments
201:Equilibrium
153:Open system
87:Equilibrium
69:Statistical
2634:Categories
2584:Pulse tube
2556:Mixed/dual
2303:I. Urieli
2188:References
2004:isothermal
1979:isothermal
1694:The ideal
1665:See also:
1583:Nucleation
1427:Scientists
1231:Philosophy
944:Potentials
307:Heat pumps
264:Polytropic
249:Isentropic
239:Isothermal
2579:Kleemenko
2465:Internal
2022:isochoric
2018:isometric
1993:isochoric
1989:isometric
1982:expansion
1971:idealized
1966:adiabatic
1906:Ross yoke
1861:June 2020
1795:idealized
1787:idealized
1728:cryogenic
1720:heat pump
1704:isochoric
1656:adiabatic
1564:Waterston
1514:von Mayer
1469:de Donder
1459:Clapeyron
1439:Boltzmann
1434:Bernoulli
1395:Education
1366:Timelines
1150:−
1095:−
883:Equations
850:∂
803:∂
754:α
718:∂
671:∂
625:−
619:β
583:∂
536:∂
244:Adiabatic
234:Isochoric
220:Processes
181:Ideal gas
64:Classical
2546:Combined
2505:Humphrey
2490:Expander
2475:Atkinson
2410:Stoddard
2400:Stirling
2395:Ericsson
2355:External
2285:Archived
2161:See also
1708:isobaric
1689:engineer
1616:Category
1554:Thompson
1464:Clausius
1444:Bridgman
1298:Vis viva
1280:Theories
1214:Gas laws
1006:Enthalpy
414:Pressure
229:Isobaric
186:Real gas
74:Chemical
57:Branches
2609:Siemens
2525:Scuderi
2441:Rankine
1951:degrees
1785:of the
1736:gaseous
1724:cooling
1539:Smeaton
1534:Rankine
1524:Onsager
1509:Maxwell
1504:Massieu
1209:Entropy
1204:General
1195:History
1185:Culture
1182:History
406: (
403:Entropy
340:italics
141:Systems
2515:Miller
2510:Lenoir
2485:Diesel
2431:Kalina
2415:Manson
2390:Carnot
2209:
1700:Diesel
1529:Planck
1519:Nernst
1494:Kelvin
1454:Carnot
744:
609:
477:
419:Volume
334:Note:
293:Cycles
122:Second
112:Zeroth
2539:Mixed
1898:power
1891:Most
1675:is a
1577:Other
1544:Stahl
1499:Lewis
1489:Joule
1479:Gibbs
1474:Duhem
167:State
127:Third
117:First
2551:HEHC
2520:Otto
2207:ISBN
1963:The
1793:The
1698:and
1696:Otto
1671:The
1549:Tait
379:Heat
374:Work
104:Laws
2020:or
1991:or
1933:not
1851:by
1392:Art
338:in
2636::
2265:^
2150:on
2129:.
1953:.
1923:.
1781:A
1754:.
1691:.
2447:)
2443:(
2372:)
2368:(
2339:e
2332:t
2325:v
2215:.
1874:)
1868:(
1863:)
1859:(
1845:.
1706:/
1662:.
1643:e
1636:t
1629:v
1156:S
1153:T
1147:H
1144:=
1141:)
1138:p
1135:,
1132:T
1129:(
1126:G
1101:S
1098:T
1092:U
1089:=
1086:)
1083:V
1080:,
1077:T
1074:(
1071:A
1046:V
1043:p
1040:+
1037:U
1034:=
1031:)
1028:p
1025:,
1022:S
1019:(
1016:H
991:)
988:V
985:,
982:S
979:(
976:U
853:T
829:V
806:V
782:1
757:=
721:p
697:V
674:V
650:1
622:=
586:T
562:N
539:S
515:T
490:=
487:c
410:)
20:)
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