2043:), the gas is in thermal contact with the hot temperature reservoir, and is thermally isolated from the cold temperature reservoir. The gas is allowed to expand, doing work on the surroundings by pushing up the piston (Stage One figure, right). Although the pressure drops from points 1 to 2 (figure 1) the temperature of the gas does not change during the process because the heat transferred from the hot temperature reservoir to the gas is exactly used to do work on the surroundings by the gas. There is no change in the gas internal energy, and no change in the gas temperature if it is an ideal gas. Heat
2348:
2191:
2134:
1998:
2682:
2383:
1967:
31:
1601:
2715:
2766:, is the amount of work energy exchanged by the system with its surroundings. The amount of heat exchanged with the hot reservoir is the sum of the two. If the system is behaving as an engine, the process moves clockwise around the loop, and moves counter-clockwise if it is behaving as a refrigerator. The efficiency to the cycle is the ratio of the white area (work) divided by the sum of the white and red areas (heat absorbed from the hot reservoir).
2369:) Once again the gas in the engine is thermally insulated from the hot and cold reservoirs, and the engine is assumed to be frictionless and the process is slow enough, hence reversible. During this step, the surroundings do work on the gas, pushing the piston down further (Stage Four figure, right), increasing its internal energy, compressing it, and causing its temperature to rise back to the temperature infinitesimally less than
4306:
3093:
3123:
2662:), the isothermal stages follow the isotherm lines for the working fluid, the adiabatic stages move between isotherms, and the area bounded by the complete cycle path represents the total work that can be done during one cycle. From point 1 to 2 and point 3 to 4 the temperature is constant (isothermal process). Heat transfer from point 4 to 1 and point 2 to 3 are equal to zero (adiabatic process).
2166:. The gas continues to expand with reduction of its pressure, doing work on the surroundings (raising the piston; Stage Two figure, right), and losing an amount of internal energy equal to the work done. The loss of internal energy causes the gas to cool. In this step it is cooled to a temperature that is infinitesimally higher than the cold reservoir temperature
4349:
Looking at this formula an interesting fact becomes apparent: Lowering the temperature of the cold reservoir will have more effect on the ceiling efficiency of a heat engine than raising the temperature of the hot reservoir by the same amount. In the real world, this may be difficult to achieve since the cold reservoir is often an existing ambient temperature.
4348:
Rearranging the right side of the equation gives what may be a more easily understood form of the equation, namely that the theoretical maximum efficiency of a heat engine equals the difference in temperature between the hot and cold reservoir divided by the absolute temperature of the hot reservoir.
4384:
In mesoscopic heat engines, work per cycle of operation in general fluctuates due to thermal noise. If the cycle is performed quasi-statically, the fluctuations vanish even on the mesoscale. However, if the cycle is performed faster than the relaxation time of the working medium, the fluctuations of
2899:
which is the amount of heat transferred in the process. If the process moves the system to greater entropy, the area under the curve is the amount of heat absorbed by the system in that process; otherwise, it is the amount of heat removed from or leaving the system. For any cyclic process, there is
2908:
diagrams for a clockwise cycle, the area under the upper portion will be the energy absorbed by the system during the cycle, while the area under the lower portion will be the energy removed from the system during the cycle. The area inside the cycle is then the difference between the two (the
4226:. This time, the cycle remains exactly the same except that the directions of any heat and work interactions are reversed. Heat is absorbed from the low-temperature reservoir, heat is rejected to a high-temperature reservoir, and a work input is required to accomplish all this. The
3088:
diagram is (a) equal to the total work performed by the system on the surroundings if the loop is traversed in a clockwise direction, and (b) is equal to the total work done on the system by the surroundings as the loop is traversed in a counterclockwise direction.
4686:
can improve the thermal efficiency of steam power plants and why the thermal efficiency of combined-cycle power plants (which incorporate gas turbines operating at even higher temperatures) exceeds that of conventional steam plants. The first prototype of the
5171:
2806:). For a simple closed system (control mass analysis), any point on the graph represents a particular state of the system. A thermodynamic process is represented by a curve connecting an initial state (A) and a final state (B). The area under the curve is:
4976:...since the Carnot heat engine, setting an upper bound on the efficiency of a heat engine is an ideal, reversible engine of which a single cycle must be performed in infinite time which is impractical and so the Carnot engine has zero power.
3658:
4213:
This is the Carnot heat engine working efficiency definition as the fraction of the work done by the system to the thermal energy received by the system from the hot reservoir per cycle. This thermal energy is the cycle initiator.
3054:
4583:
4507:
2710:. The vertical axis is the system temperature, the horizontal axis is the system entropy. A-to-B (isothermal expansion), B-to-C (isentropic expansion), C-to-D (isothermal compression), D-to-A (isentropic compression).
1925:
4385:
work are inevitable. Nevertheless, when work and heat fluctuations are counted, an exact equality relates the exponential average of work performed by any heat engine to the heat transfer from the hotter heat bath.
5168:
2470:
3244:
4112:
3442:
2544:
4011:
3341:
4409:. In addition, real engines that operate along the Carnot cycle style (isothermal expansion / isentropic expansion / isothermal compression / isentropic compression) are rare. Nevertheless, Equation
2909:
absorbed net heat energy), but since the internal energy of the system must have returned to its initial value, this difference must be the amount of work done by the system per cycle. Referring to
4187:
1320:
2881:
3916:
1790:
done by the system or engine to the environment per Carnot cycle depends on the temperatures of the thermal reservoirs and the entropy transferred from the hot reservoir to the system
2304:
2126:
4585:
at which the first integral is over a part of a cycle where heat goes into the system and the second integral is over a cycle part where heat goes out from the system. Then, replace
2237:. The surroundings do work on the gas, pushing the piston down (Stage Three figure, right). An amount of energy earned by the gas from this work exactly transfers as a heat energy
2340:
3516:
4511:
2642:
1155:
1100:
1045:
4435:
852:
805:
720:
673:
585:
538:
1762:, merely transferred between the thermal reservoirs and the system without gain or loss. When work is applied to the system, heat moves from the cold to hot reservoir (
1811:
756:
624:
990:
4300:
4273:
4222:
A Carnot heat-engine cycle described is a totally reversible cycle. That is, all the processes that compose it can be reversed, in which case it becomes the Carnot
4207:
3859:
3830:
3801:
3768:
3739:
3710:
3548:
3538:
2600:
2573:
1952:
1748:
1721:
4352:
In other words, the maximum efficiency is achieved if and only if entropy does not change per cycle. An entropy change per cycle is made, for example, if there is
4312:: A real engine (left) compared to the Carnot cycle (right). The entropy of a real material changes with temperature. This change is indicated by the curve on a
2772:(energy lost to the cold reservoir) can be seen as a direct subtraction, or expressed as the sum of a negative quantity, which can lead to different conventions.
489:
3464:
2068:
1788:
828:
781:
696:
649:
561:
514:
5120:
Kostic, M (2011). "Revisiting The Second Law of Energy
Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization".
2393:
to illustrate the work done. 1-to-2 (isothermal expansion), 2-to-3 (isentropic expansion), 3-to-4 (isothermal compression), 4-to-1 (isentropic compression).
1750:(referred to as the hot and cold reservoirs, respectively), and a part of this transferred energy is converted to the work done by the system. The cycle is
2925:
2404:
4728:, the Carnot engine may be thought as the theoretical limit of macroscopic scale heat engines rather than any practical device that could ever be built.
2027:. (The infinitesimal temperature difference allows the heat to transfer into the gas without a significant change in the gas temperature. This is called
3138:
2376:
due solely to the work added to the system, but the entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.
3346:
2475:
4327:). Irreversible systems and losses of energy (for example, work due to friction and heat losses) prevent the ideal from taking place at every step.
1630:
1816:
3251:
5223:
4720:. However, on a macroscopic scale limitations placed by the model's assumptions prove it impractical, and, ultimately, incapable of doing any
4234:
diagram of the reversed Carnot cycle is the same as for the Carnot heat-engine cycle except that the directions of the processes are reversed.
1331:
1219:
4368:, the required dumping of heat into the environment to dispose of excess entropy leads to a (minimal) reduction in efficiency. So Equation
453:
4344:
gives the maximum efficiency possible for any engine using the corresponding temperatures. A corollary to Carnot's theorem states that:
4016:
2162:) the gas in the engine is thermally insulated from both the hot and cold reservoirs, thus they neither gain nor lose heat. It is an
1309:
4336:
No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between those same reservoirs.
5385:
5021:
1342:
3921:
4415:
is extremely useful for determining the maximum efficiency that could ever be expected for a given set of thermal reservoirs.
5161:
5110:
5084:
5057:
912:
4117:
4725:
4243:
1658:
1623:
1210:
879:
446:
324:
2814:
4742:
4716:. On a practical human-scale level the Carnot cycle has proven a valuable model, as in advancing the development of the
4375:
2398:
1751:
262:
3133:
Evaluation of the above integral is particularly simple for a Carnot cycle. The amount of energy transferred as work is
5216:
3343:
and the total amount of heat transferred from the system to the cold reservoir (in the isothermal compression) will be
2696:), illustrated on a TS (temperature T–entropy S) diagram. The cycle takes place between a hot reservoir at temperature
1394:
1368:
889:
343:
5451:
4223:
3469:
295:
3868:
2008:
expansion. Heat (as an energy) is transferred reversibly from the hot temperature reservoir at constant temperature
1447:
918:
317:
4700:
2198:
Isothermal compression. Heat is transferred reversibly to the low temperature reservoir at a constant temperature
5489:
2777:
2676:
1616:
3248:
The total amount of heat transferred from the hot reservoir to the system (in the isothermal expansion) will be
1547:
79:
2251:
2073:
1654:
1442:
5525:
5209:
1522:
1295:
272:
1770:). When heat moves from the hot to the cold reservoir, the system applies work to the environment. The work
5436:
2917:, mathematically, for a reversible process, we may write the amount of work done over a cyclic process as:
2736:
2145:
907:
110:
100:
4683:
4400:
2653:
115:
105:
2762:|, is the amount of energy exchanged between the system and the cold reservoir. The area in white,
5350:
5240:
2309:
2233:
to allow heat transfer from the gas to the cold reservoir. There is no change in temperature, it is an
2226:, and is thermally isolated from the hot reservoir. The gas temperature is infinitesimally higher than
1399:
1363:
141:
75:
1437:
5504:
1192:
940:
386:
199:
189:
5142:
4822:
Holubec Viktor and Ryabov Artem (2018). "Cycling Tames Power
Fluctuations near Optimum Efficiency".
5265:
5484:
5469:
5459:
5415:
2605:
1604:
1432:
1229:
1110:
1055:
1000:
932:
871:
407:
396:
62:
5185:
834:
787:
702:
655:
567:
520:
5137:
2050:> 0 is absorbed from the hot temperature reservoir, resulting in an increase in the entropy
1537:
1254:
338:
92:
67:
5076:
5069:
3653:{\displaystyle \eta ={\frac {W}{Q_{H}}}={\frac {Q_{H}-Q_{C}}{Q_{H}}}=1-{\frac {T_{C}}{T_{H}}}}
3080:, its integral over any closed loop is zero and it follows that the area inside the loop on a
1793:
1457:
738:
603:
5330:
5290:
1674:
1472:
1049:
362:
208:
57:
960:
5474:
5192:
5129:
4902:
4841:
4713:
4278:
4251:
4192:
3837:
3808:
3779:
3771:
3746:
3717:
3688:
3523:
2578:
2551:
1930:
1726:
1699:
1685:
1681:
system in creating a temperature difference through the application of work to the system.
1552:
1477:
1467:
267:
129:
8:
5479:
5380:
5232:
5196:
4914:
2689:
1984:
1650:
1497:
1259:
281:
247:
242:
155:
5133:
4906:
4845:
4346:
All reversible engines operating between the same heat reservoirs are equally efficient.
4248:
It can be seen from the above diagram that for any cycle operating between temperatures
2721:: A generalized thermodynamic cycle taking place between a hot reservoir at temperature
1492:
471:
5336:
5026:
4949:
4918:
4892:
4865:
4831:
4737:
4706:
3449:
3077:
2693:
2234:
2182:
2141:
2053:
1988:
1773:
1759:
1662:
1586:
1249:
1244:
1197:
813:
766:
681:
634:
546:
499:
429:
413:
300:
252:
237:
227:
36:
30:
2219:), the gas in the engine is in thermal contact with the cold reservoir at temperature
1970:
5499:
5321:
5157:
5106:
5080:
5053:
4967:
4963:
4938:"Effects of dark energy on the efficiency of charged AdS black holes as heat engines"
4922:
4857:
3049:{\displaystyle W=\oint PdV=\oint (dQ-dU)=\oint (TdS-dU)=\oint TdS-\oint dU=\oint TdS}
2163:
1693:
1581:
1542:
1532:
1104:
902:
730:
232:
222:
164:
5165:
5039:
4869:
4578:{\displaystyle \langle T_{C}\rangle ={\frac {1}{\Delta S}}\int _{Q_{\text{out}}}TdS}
4394:
5441:
5147:
5034:
4959:
4910:
4853:
4849:
4502:{\displaystyle \langle T_{H}\rangle ={\frac {1}{\Delta S}}\int _{Q_{\text{in}}}TdS}
4361:
2788:
diagram), in which the thermodynamic state is specified by a point on a graph with
1502:
1487:
1427:
1422:
1239:
1234:
884:
352:
217:
4403:
engine. So, real heat engines are even less efficient than indicated by
Equation
5464:
5098:
4937:
4790:
2401:
thermodynamic cycle (no net change in the system and its surroundings per cycle)
1452:
1300:
954:
595:
418:
179:
146:
5049:
5431:
5390:
5375:
5360:
5295:
5285:
5280:
5255:
5094:
5030:
4721:
4365:
2347:
2190:
2133:
1997:
1507:
1277:
377:
257:
194:
184:
52:
22:
2776:
The behavior of a Carnot engine or refrigerator is best understood by using a
5519:
5494:
5410:
5365:
5326:
5300:
5270:
4971:
4717:
4688:
4364:
becomes an inequality rather than an equality. Otherwise, since entropy is a
4323:
2248:) to the cold reservoir so the entropy of the system decreases by the amount
1767:
1678:
1576:
894:
463:
424:
136:
5400:
5395:
5370:
5316:
4861:
2681:
2382:
1920:{\displaystyle W=(T_{H}-T_{C})\Delta S=(T_{H}-T_{C}){\frac {Q_{H}}{T_{H}}}}
1527:
1512:
1462:
945:
4320:
diagram. For this figure, the curve indicates a vapor-liquid equilibrium (
3712:< 0 is the heat taken from the system (heat energy leaving the system),
2342:
because the isothermal compression decreases the multiplicity of the gas.
4773:
4772:
Planck, M. (1945). "equations 39, 40 and 65 in sections §90 & §137".
4428:
as the expression of the Carnot efficiency is still useful. Consider the
4378:
4357:
3741:> 0 is the heat put into the system (heat energy entering the system),
1666:
1482:
290:
5405:
5201:
4989:
4679:
2714:
2390:
2005:
1571:
1517:
5151:
5042:
4360:
of work into heat. In that case, the cycle is not reversible and the
3099:: A Carnot cycle taking place between a hot reservoir at temperature
1763:
169:
2465:{\displaystyle \Delta S_{H}+\Delta S_{C}=\Delta S_{\text{cycle}}=0,}
1954:
is heat transferred from the hot reservoir to the system per cycle.
4954:
4836:
4622:
For the Carnot cycle, or its equivalent, the average value ⟨
4353:
3683:
is the work done by the system (energy exiting the system as work),
2602:
are both smaller in magnitude and in fact are in the same ratio as
1285:
1202:
994:
402:
174:
4990:"Power, efficiency, and fluctuations in steady-state heat engines"
4897:
4619:⟩, respectively, to estimate the efficiency a heat engine.
5040:
Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1963).
4429:
4305:
2789:
2181:= 0) between the system (the gas) and its surroundings. It is an
1755:
391:
4883:
N. A. Sinitsyn (2011). "Fluctuation
Relation for Heat Engines".
4399:
Carnot realized that, in reality, it is not possible to build a
3239:{\displaystyle W=\oint PdV=\oint TdS=(T_{H}-T_{C})(S_{B}-S_{A})}
3092:
4821:
4107:{\displaystyle Q_{C}=T_{C}(S_{A}-S_{B})=T_{C}\Delta S_{C}<0}
3437:{\displaystyle Q_{C}=T_{C}(S_{A}-S_{B})=T_{C}\Delta S_{C}<0}
2539:{\displaystyle {\frac {Q_{H}}{T_{H}}}=-{\frac {Q_{C}}{T_{C}}}.}
1689:
1657:
in 1824 and expanded upon by others in the 1830s and 1840s. By
4629:⟩ will equal the highest temperature available, namely
3122:
2244:< 0 (negative as leaving from the system, according to the
3918:
can be derived from the expressions above with the entropy:
2739:, the cycle cannot extend outside the temperature band from
2245:
1670:
367:
4006:{\displaystyle Q_{H}=T_{H}(S_{B}-S_{A})=T_{H}\Delta S_{H}}
3336:{\displaystyle Q_{H}=T_{H}(S_{B}-S_{A})=T_{H}\Delta S_{H}}
4650:. For other less efficient thermodynamic cycles, ⟨
5066:
5093:
4988:
Benenti, Giuliano; Casati, Giulio; Wang, Jiao (2020).
4182:{\displaystyle \Delta S_{C}=S_{A}-S_{B}=-\Delta S_{H}}
3446:
Due to energy conservation, the net heat transferred,
2900:
an upper portion of the cycle and a lower portion. In
2017:
to the gas at a temperature infinitesimally less than
5005:
However, fluctuations make impractical such engines.
4812:. 7th ed. New York: McGraw-Hill, 2011. p. 299. Print.
4514:
4438:
4281:
4254:
4195:
4120:
4019:
3924:
3871:
3840:
3811:
3782:
3749:
3720:
3691:
3551:
3526:
3472:
3452:
3349:
3254:
3141:
2928:
2817:
2608:
2581:
2554:
2478:
2407:
2312:
2254:
2076:
2056:
1933:
1819:
1796:
1776:
1729:
1702:
1113:
1058:
1003:
963:
837:
816:
790:
769:
741:
705:
684:
658:
637:
606:
570:
549:
523:
502:
474:
4709:
is, ultimately, a theoretical construct based on an
4701:
Carnot heat engine § As a macroscopic construct
4302:, none can exceed the efficiency of a Carnot cycle.
2876:{\displaystyle Q=\int _{A}^{B}dQ=\int _{A}^{B}T\,dS}
4189:, a minus sign appears in the final expression for
5068:
5041:
4577:
4501:
4388:
4294:
4267:
4201:
4181:
4106:
4005:
3910:
3853:
3824:
3795:
3762:
3733:
3704:
3652:
3532:
3510:
3458:
3436:
3335:
3238:
3048:
2875:
2670:
2636:
2594:
2567:
2538:
2464:
2334:
2298:
2120:
2062:
1946:
1919:
1805:
1782:
1742:
1715:
1149:
1094:
1039:
984:
846:
822:
799:
775:
750:
714:
690:
667:
643:
618:
579:
555:
532:
508:
483:
4987:
4691:was based on the principles of the Carnot cycle.
3803:is the absolute temperature of the hot reservoir.
5517:
5075:(3rd ed.). John Wiley & Sons. pp.
2148:) expansion of the gas (isentropic work output).
4678:. This can help illustrate, for example, why a
5048:. Addison-Wesley Publishing Company. pp.
4882:
3911:{\displaystyle \eta =1-{\frac {T_{C}}{T_{H}}}}
2665:
5217:
4694:
1624:
4767:
4765:
4763:
4528:
4515:
4452:
4439:
2647:
2346:
2189:
2132:
1996:
2173:. The entropy remains unchanged as no heat
5224:
5210:
4936:Liu, Hang; Meng, Xin-He (18 August 2017).
2796:) as the horizontal axis and temperature (
1971:Carnot cycle from The Mechanical Universe
1631:
1617:
29:
5141:
5067:Halliday, David; Resnick, Robert (1978).
4953:
4896:
4835:
4760:
2866:
5231:
4808:Çengel, Yunus A., and Michael A. Boles.
4782:
4304:
4217:
3121:
3091:
2713:
2680:
2381:
2299:{\displaystyle \Delta S_{C}=Q_{C}/T_{C}}
2121:{\displaystyle \Delta S_{H}=Q_{H}/T_{H}}
5386:Homogeneous charge compression ignition
5105:(2nd ed.). W. H. Freeman Company.
5022:Reflections on the Motive Power of Fire
4935:
4810:Thermodynamics: An Engineering Approach
4778:. Dover Publications. pp. 75, 135.
5518:
5119:
4771:
2246:universal convention in thermodynamics
2029:isothermal heat addition or absorption
5205:
5156:American Institute of Physics, 2011.
4788:
1991:, consisting of the following steps:
1677:, or conversely, the efficiency of a
4334:is a formal statement of this fact:
3865:The expression with the temperature
3542:
3106:and a cold reservoir at temperature
2919:
2808:
2728:and a cold reservoir at temperature
2703:and a cold reservoir at temperature
2652:When a Carnot cycle is plotted on a
1661:, it provides an upper limit on the
4929:
4743:Reversible process (thermodynamics)
4237:
3129:: A visualization of a Carnot cycle
3117:
13:
5031:The Steam-Engine and Other Engines
4540:
4464:
4309:
4166:
4121:
4085:
3990:
3415:
3320:
3126:
3096:
2912:
2802:
2718:
2685:
2658:
2440:
2424:
2408:
2389:: A Carnot cycle illustrated on a
2386:
2365:
2359:
2313:
2255:
2215:
2209:
2158:
2152:
2077:
2039:
2033:
1855:
1797:
838:
791:
706:
659:
571:
524:
344:Intensive and extensive properties
14:
5537:
5179:
4798:. Dover Publications. p. 48.
4789:Fermi, E. (1956). "equation 64".
4244:Carnot's theorem (thermodynamics)
4224:heat pump and refrigeration cycle
3466:, is equal to the work performed
2688:: A Carnot cycle as an idealized
2335:{\displaystyle \Delta S_{C}<0}
1965:
1692:in the form of heat between two
1600:
1599:
919:Table of thermodynamic equations
5044:The Feynman Lectures on Physics
4942:The European Physical Journal C
4389:Efficiency of real heat engines
3511:{\displaystyle W=Q=Q_{H}-Q_{C}}
2671:The temperature–entropy diagram
2031:.) During this step (1 to 2 on
1983:A Carnot cycle as an idealized
1395:Maxwell's thermodynamic surface
5124:. AIP Conference Proceedings.
4981:
4964:10.1140/epjc/s10052-017-5134-9
4915:10.1088/1751-8113/44/40/405001
4876:
4854:10.1103/PhysRevLett.121.120601
4815:
4802:
4069:
4043:
3974:
3948:
3399:
3373:
3304:
3278:
3233:
3207:
3204:
3178:
3001:
2980:
2971:
2953:
1890:
1864:
1852:
1826:
1129:
1117:
1074:
1062:
1019:
1007:
979:
967:
1:
4748:
4671:⟩ will be higher than
4422:is an idealization, Equation
4395:Heat Engine § Efficiency
3861:is the minimum system entropy
3832:is the maximum system entropy
1653:proposed by French physicist
1296:Mechanical equivalent of heat
16:Idealized thermodynamic cycle
5188:article on the Carnot cycle.
4657:⟩ will be lower than
4643:⟩ the lowest, namely
4401:thermodynamically reversible
4374:gives the efficiency of any
2737:second law of thermodynamics
2397:In this case, since it is a
2205:(isothermal heat rejection).
908:Onsager reciprocal relations
7:
5291:Stirling (pseudo/adiabatic)
4731:
4601:
4424:
4411:
4405:
4370:
4340:
3666:
3062:
2889:
2778:temperature–entropy diagram
2677:Temperature–entropy diagram
2666:Properties and significance
2637:{\displaystyle Q_{H}/T_{H}}
1400:Entropy as energy dispersal
1211:"Perpetual motion" machines
1150:{\displaystyle G(T,p)=H-TS}
1095:{\displaystyle A(T,V)=U-TS}
1040:{\displaystyle H(S,p)=U+pV}
10:
5542:
4775:Treatise on Thermodynamics
4698:
4695:As a macroscopic construct
4392:
4241:
3774:of the cold reservoir, and
2674:
847:{\displaystyle \partial T}
800:{\displaystyle \partial V}
715:{\displaystyle \partial p}
668:{\displaystyle \partial V}
580:{\displaystyle \partial T}
533:{\displaystyle \partial S}
5450:
5424:
5349:
5309:
5250:
5239:
5193:Carnot Cycle on Ideal Gas
5167:. Full article (24 pages
2753:. The area in red, |
2648:The pressure–volume graph
2150:For this step (2 to 3 on
2070:of the gas by the amount
1978:
1964:
1959:
1669:during the conversion of
1321:An Inquiry Concerning the
5033:edition 3, page 62, via
2800:) as the vertical axis (
2207:In this step (3 to 4 on
1806:{\displaystyle \Delta S}
1334:Heterogeneous Substances
751:{\displaystyle \alpha =}
619:{\displaystyle \beta =-}
4885:J. Phys. A: Math. Theor
2654:pressure–volume diagram
2355:Isentropic compression.
4579:
4503:
4328:
4296:
4269:
4203:
4183:
4108:
4007:
3912:
3855:
3826:
3797:
3764:
3735:
3706:
3654:
3534:
3512:
3460:
3438:
3337:
3240:
3130:
3114:
3050:
2877:
2773:
2711:
2638:
2596:
2569:
2540:
2466:
2394:
2351:
2336:
2300:
2194:
2137:
2122:
2064:
2001:
1948:
1921:
1807:
1784:
1744:
1717:
1151:
1096:
1041:
986:
985:{\displaystyle U(S,V)}
848:
824:
801:
777:
752:
716:
692:
669:
645:
620:
581:
557:
534:
510:
485:
464:Specific heat capacity
68:Quantum thermodynamics
4612:⟩ and ⟨
4580:
4504:
4308:
4297:
4295:{\displaystyle T_{C}}
4270:
4268:{\displaystyle T_{H}}
4218:Reversed Carnot cycle
4204:
4202:{\displaystyle \eta }
4184:
4109:
4008:
3913:
3856:
3854:{\displaystyle S_{A}}
3827:
3825:{\displaystyle S_{B}}
3798:
3796:{\displaystyle T_{H}}
3765:
3763:{\displaystyle T_{C}}
3736:
3734:{\displaystyle Q_{H}}
3707:
3705:{\displaystyle Q_{C}}
3655:
3535:
3533:{\displaystyle \eta }
3513:
3461:
3439:
3338:
3241:
3125:
3095:
3051:
2878:
2717:
2684:
2639:
2597:
2595:{\displaystyle T_{C}}
2570:
2568:{\displaystyle Q_{C}}
2541:
2467:
2385:
2350:
2337:
2301:
2193:
2136:
2123:
2065:
2000:
1949:
1947:{\displaystyle Q_{H}}
1922:
1808:
1785:
1745:
1743:{\displaystyle T_{C}}
1718:
1716:{\displaystyle T_{H}}
1684:In a Carnot cycle, a
1332:On the Equilibrium of
1152:
1097:
1050:Helmholtz free energy
1042:
987:
849:
825:
802:
778:
753:
717:
693:
670:
646:
621:
582:
558:
535:
511:
486:
5526:Thermodynamic cycles
5475:Regenerative cooling
5353:combustion / thermal
5252:Without phase change
5243:combustion / thermal
5233:Thermodynamic cycles
4714:thermodynamic system
4512:
4436:
4279:
4252:
4193:
4118:
4017:
3922:
3869:
3838:
3809:
3780:
3772:absolute temperature
3747:
3718:
3689:
3549:
3524:
3470:
3450:
3347:
3252:
3139:
2926:
2815:
2606:
2579:
2552:
2476:
2405:
2310:
2252:
2146:reversible adiabatic
2074:
2054:
1931:
1817:
1794:
1774:
1727:
1700:
1688:or engine transfers
1667:thermodynamic engine
1345:Motive Power of Fire
1111:
1056:
1001:
961:
913:Bridgman's equations
890:Fundamental relation
835:
814:
788:
767:
739:
703:
682:
656:
635:
604:
568:
547:
521:
500:
472:
5197:Wolfram Mathematica
5134:2011AIPC.1411..327K
4907:2011JPhA...44N5001S
4846:2018PhRvL.121l0601H
2862:
2838:
2690:thermodynamic cycle
1985:thermodynamic cycle
1651:thermodynamic cycle
1323:Source ... Friction
1255:Loschmidt's paradox
447:Material properties
325:Conjugate variables
4738:Carnot heat engine
4707:Carnot heat engine
4575:
4499:
4329:
4292:
4265:
4199:
4179:
4104:
4003:
3908:
3851:
3822:
3793:
3760:
3731:
3702:
3650:
3540:is defined to be:
3530:
3508:
3456:
3434:
3333:
3236:
3131:
3115:
3078:exact differential
3046:
2873:
2848:
2824:
2774:
2712:
2694:Carnot heat engine
2634:
2592:
2565:
2536:
2462:
2395:
2352:
2332:
2296:
2235:isothermal process
2195:
2183:isentropic process
2138:
2118:
2060:
2002:
1989:Carnot heat engine
1944:
1917:
1813:per cycle such as
1803:
1780:
1740:
1713:
1694:thermal reservoirs
1587:Order and disorder
1343:Reflections on the
1250:Heat death paradox
1147:
1092:
1037:
982:
844:
820:
797:
773:
748:
712:
688:
665:
641:
616:
577:
553:
530:
506:
484:{\displaystyle c=}
481:
454:Property databases
430:Reduced properties
414:Chemical potential
378:Functions of state
301:Thermal efficiency
37:Carnot heat engine
5513:
5512:
5490:Vapor-compression
5416:Staged combustion
5345:
5344:
5310:With phase change
5162:978-0-7354-0985-9
5152:10.1063/1.3665247
5112:978-0-7167-1088-2
5086:978-0-471-02456-9
5059:978-0-201-02116-5
4997:Physical Review E
4561:
4547:
4485:
4471:
3906:
3674:
3673:
3648:
3615:
3573:
3459:{\displaystyle Q}
3070:
3069:
2897:
2896:
2531:
2501:
2450:
2164:adiabatic process
2063:{\displaystyle S}
1976:
1975:
1915:
1783:{\displaystyle W}
1665:of any classical
1641:
1640:
1582:Self-organization
1407:
1406:
1105:Gibbs free energy
903:Maxwell relations
861:
860:
857:
856:
823:{\displaystyle V}
776:{\displaystyle 1}
731:Thermal expansion
725:
724:
691:{\displaystyle V}
644:{\displaystyle 1}
590:
589:
556:{\displaystyle N}
509:{\displaystyle T}
437:
436:
353:Process functions
339:Property diagrams
318:System properties
308:
307:
273:Endoreversibility
165:Equation of state
5533:
5485:Vapor absorption
5248:
5247:
5226:
5219:
5212:
5203:
5202:
5155:
5145:
5116:
5099:Kroemer, Herbert
5090:
5074:
5063:
5047:
5035:Internet Archive
5008:
5007:
4994:
4985:
4979:
4978:
4957:
4933:
4927:
4926:
4900:
4880:
4874:
4873:
4839:
4819:
4813:
4806:
4800:
4799:
4797:
4786:
4780:
4779:
4769:
4726:Carnot's theorem
4584:
4582:
4581:
4576:
4565:
4564:
4563:
4562:
4559:
4548:
4546:
4535:
4527:
4526:
4508:
4506:
4505:
4500:
4489:
4488:
4487:
4486:
4483:
4472:
4470:
4459:
4451:
4450:
4362:Clausius theorem
4332:Carnot's theorem
4311:
4301:
4299:
4298:
4293:
4291:
4290:
4274:
4272:
4271:
4266:
4264:
4263:
4238:Carnot's theorem
4208:
4206:
4205:
4200:
4188:
4186:
4185:
4180:
4178:
4177:
4159:
4158:
4146:
4145:
4133:
4132:
4113:
4111:
4110:
4105:
4097:
4096:
4084:
4083:
4068:
4067:
4055:
4054:
4042:
4041:
4029:
4028:
4012:
4010:
4009:
4004:
4002:
4001:
3989:
3988:
3973:
3972:
3960:
3959:
3947:
3946:
3934:
3933:
3917:
3915:
3914:
3909:
3907:
3905:
3904:
3895:
3894:
3885:
3860:
3858:
3857:
3852:
3850:
3849:
3831:
3829:
3828:
3823:
3821:
3820:
3802:
3800:
3799:
3794:
3792:
3791:
3769:
3767:
3766:
3761:
3759:
3758:
3740:
3738:
3737:
3732:
3730:
3729:
3711:
3709:
3708:
3703:
3701:
3700:
3682:
3668:
3659:
3657:
3656:
3651:
3649:
3647:
3646:
3637:
3636:
3627:
3616:
3614:
3613:
3604:
3603:
3602:
3590:
3589:
3579:
3574:
3572:
3571:
3559:
3543:
3539:
3537:
3536:
3531:
3517:
3515:
3514:
3509:
3507:
3506:
3494:
3493:
3465:
3463:
3462:
3457:
3443:
3441:
3440:
3435:
3427:
3426:
3414:
3413:
3398:
3397:
3385:
3384:
3372:
3371:
3359:
3358:
3342:
3340:
3339:
3334:
3332:
3331:
3319:
3318:
3303:
3302:
3290:
3289:
3277:
3276:
3264:
3263:
3245:
3243:
3242:
3237:
3232:
3231:
3219:
3218:
3203:
3202:
3190:
3189:
3128:
3118:The Carnot cycle
3098:
3064:
3055:
3053:
3052:
3047:
2920:
2916:
2891:
2882:
2880:
2879:
2874:
2861:
2856:
2837:
2832:
2809:
2761:
2720:
2687:
2643:
2641:
2640:
2635:
2633:
2632:
2623:
2618:
2617:
2601:
2599:
2598:
2593:
2591:
2590:
2574:
2572:
2571:
2566:
2564:
2563:
2548:This is true as
2545:
2543:
2542:
2537:
2532:
2530:
2529:
2520:
2519:
2510:
2502:
2500:
2499:
2490:
2489:
2480:
2471:
2469:
2468:
2463:
2452:
2451:
2448:
2436:
2435:
2420:
2419:
2388:
2341:
2339:
2338:
2333:
2325:
2324:
2305:
2303:
2302:
2297:
2295:
2294:
2285:
2280:
2279:
2267:
2266:
2127:
2125:
2124:
2119:
2117:
2116:
2107:
2102:
2101:
2089:
2088:
2069:
2067:
2066:
2061:
1969:
1968:
1957:
1956:
1953:
1951:
1950:
1945:
1943:
1942:
1926:
1924:
1923:
1918:
1916:
1914:
1913:
1904:
1903:
1894:
1889:
1888:
1876:
1875:
1851:
1850:
1838:
1837:
1812:
1810:
1809:
1804:
1789:
1787:
1786:
1781:
1749:
1747:
1746:
1741:
1739:
1738:
1722:
1720:
1719:
1714:
1712:
1711:
1696:at temperatures
1659:Carnot's theorem
1633:
1626:
1619:
1603:
1602:
1310:Key publications
1291:
1290:("living force")
1240:Brownian ratchet
1235:Entropy and life
1230:Entropy and time
1181:
1180:
1156:
1154:
1153:
1148:
1101:
1099:
1098:
1093:
1046:
1044:
1043:
1038:
991:
989:
988:
983:
885:Clausius theorem
880:Carnot's theorem
853:
851:
850:
845:
829:
827:
826:
821:
806:
804:
803:
798:
782:
780:
779:
774:
761:
760:
757:
755:
754:
749:
721:
719:
718:
713:
697:
695:
694:
689:
674:
672:
671:
666:
650:
648:
647:
642:
629:
628:
625:
623:
622:
617:
586:
584:
583:
578:
562:
560:
559:
554:
539:
537:
536:
531:
515:
513:
512:
507:
494:
493:
490:
488:
487:
482:
460:
459:
333:
332:
152:
151:
33:
19:
18:
5541:
5540:
5536:
5535:
5534:
5532:
5531:
5530:
5516:
5515:
5514:
5509:
5446:
5420:
5352:
5341:
5331:Organic Rankine
5305:
5259:
5256:hot air engines
5253:
5242:
5235:
5230:
5182:
5164:. Abstract at:
5143:10.1.1.405.1945
5113:
5103:Thermal Physics
5095:Kittel, Charles
5087:
5060:
5012:
5011:
4992:
4986:
4982:
4934:
4930:
4881:
4877:
4824:Phys. Rev. Lett
4820:
4816:
4807:
4803:
4795:
4787:
4783:
4770:
4761:
4751:
4734:
4724:. As such, per
4703:
4697:
4676:
4669:
4662:
4655:
4648:
4641:
4634:
4627:
4617:
4610:
4597:
4590:
4558:
4554:
4553:
4549:
4539:
4534:
4522:
4518:
4513:
4510:
4509:
4482:
4478:
4477:
4473:
4463:
4458:
4446:
4442:
4437:
4434:
4433:
4397:
4391:
4338:Thus, Equation
4286:
4282:
4280:
4277:
4276:
4259:
4255:
4253:
4250:
4249:
4246:
4240:
4220:
4212:
4194:
4191:
4190:
4173:
4169:
4154:
4150:
4141:
4137:
4128:
4124:
4119:
4116:
4115:
4092:
4088:
4079:
4075:
4063:
4059:
4050:
4046:
4037:
4033:
4024:
4020:
4018:
4015:
4014:
3997:
3993:
3984:
3980:
3968:
3964:
3955:
3951:
3942:
3938:
3929:
3925:
3923:
3920:
3919:
3900:
3896:
3890:
3886:
3884:
3870:
3867:
3866:
3845:
3841:
3839:
3836:
3835:
3816:
3812:
3810:
3807:
3806:
3787:
3783:
3781:
3778:
3777:
3754:
3750:
3748:
3745:
3744:
3725:
3721:
3719:
3716:
3715:
3696:
3692:
3690:
3687:
3686:
3680:
3642:
3638:
3632:
3628:
3626:
3609:
3605:
3598:
3594:
3585:
3581:
3580:
3578:
3567:
3563:
3558:
3550:
3547:
3546:
3525:
3522:
3521:
3520:The efficiency
3502:
3498:
3489:
3485:
3471:
3468:
3467:
3451:
3448:
3447:
3422:
3418:
3409:
3405:
3393:
3389:
3380:
3376:
3367:
3363:
3354:
3350:
3348:
3345:
3344:
3327:
3323:
3314:
3310:
3298:
3294:
3285:
3281:
3272:
3268:
3259:
3255:
3253:
3250:
3249:
3227:
3223:
3214:
3210:
3198:
3194:
3185:
3181:
3140:
3137:
3136:
3120:
3112:
3105:
2927:
2924:
2923:
2910:
2857:
2852:
2833:
2828:
2816:
2813:
2812:
2771:
2767:
2760:
2754:
2752:
2745:
2734:
2727:
2709:
2702:
2692:performed by a
2679:
2673:
2668:
2650:
2628:
2624:
2619:
2613:
2609:
2607:
2604:
2603:
2586:
2582:
2580:
2577:
2576:
2559:
2555:
2553:
2550:
2549:
2525:
2521:
2515:
2511:
2509:
2495:
2491:
2485:
2481:
2479:
2477:
2474:
2473:
2447:
2443:
2431:
2427:
2415:
2411:
2406:
2403:
2402:
2380:
2374:
2320:
2316:
2311:
2308:
2307:
2290:
2286:
2281:
2275:
2271:
2262:
2258:
2253:
2250:
2249:
2243:
2232:
2225:
2204:
2172:
2112:
2108:
2103:
2097:
2093:
2084:
2080:
2075:
2072:
2071:
2055:
2052:
2051:
2049:
2025:
2016:
1987:performed by a
1981:
1966:
1960:External videos
1938:
1934:
1932:
1929:
1928:
1909:
1905:
1899:
1895:
1893:
1884:
1880:
1871:
1867:
1846:
1842:
1833:
1829:
1818:
1815:
1814:
1795:
1792:
1791:
1775:
1772:
1771:
1734:
1730:
1728:
1725:
1724:
1707:
1703:
1701:
1698:
1697:
1637:
1592:
1591:
1567:
1559:
1558:
1557:
1417:
1409:
1408:
1387:
1373:
1348:
1344:
1337:
1333:
1326:
1322:
1289:
1282:
1264:
1245:Maxwell's demon
1207:
1178:
1177:
1161:
1160:
1159:
1112:
1109:
1108:
1107:
1057:
1054:
1053:
1052:
1002:
999:
998:
997:
962:
959:
958:
957:
955:Internal energy
950:
935:
925:
924:
899:
874:
864:
863:
862:
836:
833:
832:
815:
812:
811:
789:
786:
785:
768:
765:
764:
740:
737:
736:
704:
701:
700:
683:
680:
679:
657:
654:
653:
636:
633:
632:
605:
602:
601:
596:Compressibility
569:
566:
565:
548:
545:
544:
522:
519:
518:
501:
498:
497:
473:
470:
469:
449:
439:
438:
419:Particle number
372:
331:
320:
310:
309:
268:Irreversibility
180:State of matter
147:Isolated system
132:
122:
121:
120:
95:
85:
84:
80:Non-equilibrium
72:
47:
39:
17:
12:
11:
5:
5539:
5529:
5528:
5511:
5510:
5508:
5507:
5502:
5497:
5492:
5487:
5482:
5477:
5472:
5467:
5462:
5456:
5454:
5448:
5447:
5445:
5444:
5439:
5434:
5428:
5426:
5422:
5421:
5419:
5418:
5413:
5408:
5403:
5398:
5393:
5388:
5383:
5378:
5373:
5368:
5363:
5357:
5355:
5347:
5346:
5343:
5342:
5340:
5339:
5334:
5324:
5319:
5313:
5311:
5307:
5306:
5304:
5303:
5298:
5293:
5288:
5283:
5278:
5273:
5268:
5262:
5260:
5251:
5245:
5237:
5236:
5229:
5228:
5221:
5214:
5206:
5200:
5199:
5191:S. M. Blinder
5189:
5181:
5180:External links
5178:
5177:
5176:
5175:
5174:
5128:(1): 327–350.
5122:AIP Conf. Proc
5117:
5111:
5091:
5085:
5064:
5058:
5037:
5024:
5020:Carnot, Sadi,
5016:
5010:
5009:
4980:
4928:
4891:(40): 405001.
4875:
4830:(12): 120601.
4814:
4801:
4792:Thermodynamics
4781:
4758:
4757:
4756:
4755:
4750:
4747:
4746:
4745:
4740:
4733:
4730:
4699:Main article:
4696:
4693:
4674:
4667:
4664:, and ⟨
4660:
4653:
4646:
4639:
4636:, and ⟨
4632:
4625:
4615:
4608:
4595:
4588:
4574:
4571:
4568:
4557:
4552:
4545:
4542:
4538:
4533:
4530:
4525:
4521:
4517:
4498:
4495:
4492:
4481:
4476:
4469:
4466:
4462:
4457:
4454:
4449:
4445:
4441:
4432:temperatures,
4420:Carnot's cycle
4390:
4387:
4366:state function
4289:
4285:
4262:
4258:
4242:Main article:
4239:
4236:
4219:
4216:
4198:
4176:
4172:
4168:
4165:
4162:
4157:
4153:
4149:
4144:
4140:
4136:
4131:
4127:
4123:
4103:
4100:
4095:
4091:
4087:
4082:
4078:
4074:
4071:
4066:
4062:
4058:
4053:
4049:
4045:
4040:
4036:
4032:
4027:
4023:
4000:
3996:
3992:
3987:
3983:
3979:
3976:
3971:
3967:
3963:
3958:
3954:
3950:
3945:
3941:
3937:
3932:
3928:
3903:
3899:
3893:
3889:
3883:
3880:
3877:
3874:
3863:
3862:
3848:
3844:
3833:
3819:
3815:
3804:
3790:
3786:
3775:
3757:
3753:
3742:
3728:
3724:
3713:
3699:
3695:
3684:
3672:
3671:
3662:
3660:
3645:
3641:
3635:
3631:
3625:
3622:
3619:
3612:
3608:
3601:
3597:
3593:
3588:
3584:
3577:
3570:
3566:
3562:
3557:
3554:
3529:
3505:
3501:
3497:
3492:
3488:
3484:
3481:
3478:
3475:
3455:
3433:
3430:
3425:
3421:
3417:
3412:
3408:
3404:
3401:
3396:
3392:
3388:
3383:
3379:
3375:
3370:
3366:
3362:
3357:
3353:
3330:
3326:
3322:
3317:
3313:
3309:
3306:
3301:
3297:
3293:
3288:
3284:
3280:
3275:
3271:
3267:
3262:
3258:
3235:
3230:
3226:
3222:
3217:
3213:
3209:
3206:
3201:
3197:
3193:
3188:
3184:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3119:
3116:
3110:
3103:
3068:
3067:
3058:
3056:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2895:
2894:
2885:
2883:
2872:
2869:
2865:
2860:
2855:
2851:
2847:
2844:
2841:
2836:
2831:
2827:
2823:
2820:
2769:
2758:
2750:
2743:
2732:
2725:
2707:
2700:
2675:Main article:
2672:
2669:
2667:
2664:
2649:
2646:
2631:
2627:
2622:
2616:
2612:
2589:
2585:
2562:
2558:
2535:
2528:
2524:
2518:
2514:
2508:
2505:
2498:
2494:
2488:
2484:
2461:
2458:
2455:
2446:
2442:
2439:
2434:
2430:
2426:
2423:
2418:
2414:
2410:
2379:
2378:
2372:
2344:
2331:
2328:
2323:
2319:
2315:
2293:
2289:
2284:
2278:
2274:
2270:
2265:
2261:
2257:
2241:
2230:
2223:
2202:
2187:
2170:
2130:
2115:
2111:
2106:
2100:
2096:
2092:
2087:
2083:
2079:
2059:
2047:
2021:
2012:
1993:
1980:
1977:
1974:
1973:
1962:
1961:
1941:
1937:
1912:
1908:
1902:
1898:
1892:
1887:
1883:
1879:
1874:
1870:
1866:
1863:
1860:
1857:
1854:
1849:
1845:
1841:
1836:
1832:
1828:
1825:
1822:
1802:
1799:
1779:
1737:
1733:
1710:
1706:
1639:
1638:
1636:
1635:
1628:
1621:
1613:
1610:
1609:
1608:
1607:
1594:
1593:
1590:
1589:
1584:
1579:
1574:
1568:
1565:
1564:
1561:
1560:
1556:
1555:
1550:
1545:
1540:
1535:
1530:
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1485:
1480:
1475:
1470:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1430:
1425:
1419:
1418:
1415:
1414:
1411:
1410:
1405:
1404:
1403:
1402:
1397:
1389:
1388:
1386:
1385:
1382:
1378:
1375:
1374:
1372:
1371:
1366:
1364:Thermodynamics
1360:
1357:
1356:
1352:
1351:
1350:
1349:
1340:
1338:
1329:
1327:
1318:
1313:
1312:
1306:
1305:
1304:
1303:
1298:
1293:
1281:
1280:
1278:Caloric theory
1274:
1271:
1270:
1266:
1265:
1263:
1262:
1257:
1252:
1247:
1242:
1237:
1232:
1226:
1223:
1222:
1216:
1215:
1214:
1213:
1206:
1205:
1200:
1195:
1189:
1186:
1185:
1179:
1176:
1175:
1172:
1168:
1167:
1166:
1163:
1162:
1158:
1157:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1102:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1047:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
992:
981:
978:
975:
972:
969:
966:
951:
949:
948:
943:
937:
936:
931:
930:
927:
926:
923:
922:
915:
910:
905:
898:
897:
892:
887:
882:
876:
875:
870:
869:
866:
865:
859:
858:
855:
854:
843:
840:
830:
819:
808:
807:
796:
793:
783:
772:
758:
747:
744:
734:
727:
726:
723:
722:
711:
708:
698:
687:
676:
675:
664:
661:
651:
640:
626:
615:
612:
609:
599:
592:
591:
588:
587:
576:
573:
563:
552:
541:
540:
529:
526:
516:
505:
491:
480:
477:
467:
458:
457:
456:
450:
445:
444:
441:
440:
435:
434:
433:
432:
427:
422:
411:
400:
381:
380:
374:
373:
371:
370:
365:
359:
356:
355:
349:
348:
347:
346:
341:
322:
321:
316:
315:
312:
311:
306:
305:
304:
303:
298:
293:
285:
284:
278:
277:
276:
275:
270:
265:
260:
258:Free expansion
255:
250:
245:
240:
235:
230:
225:
220:
212:
211:
205:
204:
203:
202:
197:
195:Control volume
192:
187:
185:Phase (matter)
182:
177:
172:
167:
159:
158:
150:
149:
144:
139:
133:
128:
127:
124:
123:
119:
118:
113:
108:
103:
97:
96:
91:
90:
87:
86:
83:
82:
71:
70:
65:
60:
55:
49:
48:
45:
44:
41:
40:
35:The classical
34:
26:
25:
23:Thermodynamics
15:
9:
6:
4:
3:
2:
5538:
5527:
5524:
5523:
5521:
5506:
5503:
5501:
5498:
5496:
5493:
5491:
5488:
5486:
5483:
5481:
5480:Transcritical
5478:
5476:
5473:
5471:
5468:
5466:
5463:
5461:
5460:Hampson–Linde
5458:
5457:
5455:
5453:
5452:Refrigeration
5449:
5443:
5440:
5438:
5435:
5433:
5430:
5429:
5427:
5423:
5417:
5414:
5412:
5409:
5407:
5404:
5402:
5399:
5397:
5394:
5392:
5389:
5387:
5384:
5382:
5381:Gas-generator
5379:
5377:
5374:
5372:
5369:
5367:
5366:Brayton/Joule
5364:
5362:
5359:
5358:
5356:
5354:
5348:
5338:
5335:
5332:
5328:
5325:
5323:
5320:
5318:
5315:
5314:
5312:
5308:
5302:
5299:
5297:
5294:
5292:
5289:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5271:Brayton/Joule
5269:
5267:
5264:
5263:
5261:
5257:
5249:
5246:
5244:
5238:
5234:
5227:
5222:
5220:
5215:
5213:
5208:
5207:
5204:
5198:
5194:
5190:
5187:
5184:
5183:
5172:
5169:
5166:
5163:
5159:
5153:
5149:
5144:
5139:
5135:
5131:
5127:
5123:
5118:
5114:
5108:
5104:
5100:
5096:
5092:
5088:
5082:
5078:
5073:
5072:
5065:
5061:
5055:
5051:
5046:
5045:
5038:
5036:
5032:
5028:
5025:
5023:
5019:
5018:
5017:
5014:
5013:
5006:
5002:
4998:
4991:
4984:
4977:
4973:
4969:
4965:
4961:
4956:
4951:
4947:
4943:
4939:
4932:
4924:
4920:
4916:
4912:
4908:
4904:
4899:
4894:
4890:
4886:
4879:
4871:
4867:
4863:
4859:
4855:
4851:
4847:
4843:
4838:
4833:
4829:
4825:
4818:
4811:
4805:
4794:
4793:
4785:
4777:
4776:
4768:
4766:
4764:
4759:
4753:
4752:
4744:
4741:
4739:
4736:
4735:
4729:
4727:
4723:
4719:
4718:diesel engine
4715:
4712:
4708:
4702:
4692:
4690:
4689:diesel engine
4685:
4681:
4677:
4670:
4663:
4656:
4649:
4642:
4635:
4628:
4620:
4618:
4611:
4604:
4603:
4598:
4591:
4572:
4569:
4566:
4555:
4550:
4543:
4536:
4531:
4523:
4519:
4496:
4493:
4490:
4479:
4474:
4467:
4460:
4455:
4447:
4443:
4431:
4427:
4426:
4421:
4416:
4414:
4413:
4408:
4407:
4402:
4396:
4386:
4382:
4380:
4377:
4373:
4372:
4367:
4363:
4359:
4355:
4350:
4347:
4343:
4342:
4337:
4333:
4326:
4325:
4324:Rankine cycle
4319:
4315:
4307:
4303:
4287:
4283:
4260:
4256:
4245:
4235:
4233:
4229:
4225:
4215:
4210:
4196:
4174:
4170:
4163:
4160:
4155:
4151:
4147:
4142:
4138:
4134:
4129:
4125:
4101:
4098:
4093:
4089:
4080:
4076:
4072:
4064:
4060:
4056:
4051:
4047:
4038:
4034:
4030:
4025:
4021:
3998:
3994:
3985:
3981:
3977:
3969:
3965:
3961:
3956:
3952:
3943:
3939:
3935:
3930:
3926:
3901:
3897:
3891:
3887:
3881:
3878:
3875:
3872:
3846:
3842:
3834:
3817:
3813:
3805:
3788:
3784:
3776:
3773:
3755:
3751:
3743:
3726:
3722:
3714:
3697:
3693:
3685:
3679:
3678:
3677:
3670:
3663:
3661:
3643:
3639:
3633:
3629:
3623:
3620:
3617:
3610:
3606:
3599:
3595:
3591:
3586:
3582:
3575:
3568:
3564:
3560:
3555:
3552:
3545:
3544:
3541:
3527:
3518:
3503:
3499:
3495:
3490:
3486:
3482:
3479:
3476:
3473:
3453:
3444:
3431:
3428:
3423:
3419:
3410:
3406:
3402:
3394:
3390:
3386:
3381:
3377:
3368:
3364:
3360:
3355:
3351:
3328:
3324:
3315:
3311:
3307:
3299:
3295:
3291:
3286:
3282:
3273:
3269:
3265:
3260:
3256:
3246:
3228:
3224:
3220:
3215:
3211:
3199:
3195:
3191:
3186:
3182:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3134:
3124:
3109:
3102:
3094:
3090:
3087:
3083:
3079:
3075:
3066:
3059:
3057:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
2998:
2995:
2992:
2989:
2986:
2983:
2977:
2974:
2968:
2965:
2962:
2959:
2956:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2922:
2921:
2918:
2915:
2914:
2907:
2903:
2893:
2886:
2884:
2870:
2867:
2863:
2858:
2853:
2849:
2845:
2842:
2839:
2834:
2829:
2825:
2821:
2818:
2811:
2810:
2807:
2805:
2804:
2799:
2795:
2791:
2787:
2783:
2779:
2765:
2757:
2749:
2742:
2738:
2731:
2724:
2716:
2706:
2699:
2695:
2691:
2683:
2678:
2663:
2661:
2660:
2655:
2645:
2629:
2625:
2620:
2614:
2610:
2587:
2583:
2560:
2556:
2546:
2533:
2526:
2522:
2516:
2512:
2506:
2503:
2496:
2492:
2486:
2482:
2459:
2456:
2453:
2444:
2437:
2432:
2428:
2421:
2416:
2412:
2400:
2392:
2384:
2377:
2375:
2368:
2367:
2362:
2361:
2356:
2349:
2345:
2343:
2329:
2326:
2321:
2317:
2291:
2287:
2282:
2276:
2272:
2268:
2263:
2259:
2247:
2240:
2236:
2229:
2222:
2218:
2217:
2212:
2211:
2206:
2201:
2192:
2188:
2186:
2184:
2180:
2176:
2169:
2165:
2161:
2160:
2155:
2154:
2149:
2147:
2143:
2135:
2131:
2129:
2113:
2109:
2104:
2098:
2094:
2090:
2085:
2081:
2057:
2046:
2042:
2041:
2036:
2035:
2030:
2026:
2024:
2020:
2015:
2011:
2007:
1999:
1995:
1994:
1992:
1990:
1986:
1972:
1963:
1958:
1955:
1939:
1935:
1910:
1906:
1900:
1896:
1885:
1881:
1877:
1872:
1868:
1861:
1858:
1847:
1843:
1839:
1834:
1830:
1823:
1820:
1800:
1777:
1769:
1768:refrigeration
1765:
1761:
1757:
1753:
1735:
1731:
1708:
1704:
1695:
1691:
1687:
1682:
1680:
1679:refrigeration
1676:
1672:
1668:
1664:
1660:
1656:
1652:
1648:
1643:
1634:
1629:
1627:
1622:
1620:
1615:
1614:
1612:
1611:
1606:
1598:
1597:
1596:
1595:
1588:
1585:
1583:
1580:
1578:
1577:Self-assembly
1575:
1573:
1570:
1569:
1563:
1562:
1554:
1551:
1549:
1548:van der Waals
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1524:
1521:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1499:
1496:
1494:
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1474:
1473:von Helmholtz
1471:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1420:
1413:
1412:
1401:
1398:
1396:
1393:
1392:
1391:
1390:
1383:
1380:
1379:
1377:
1376:
1370:
1367:
1365:
1362:
1361:
1359:
1358:
1354:
1353:
1347:
1346:
1339:
1336:
1335:
1328:
1325:
1324:
1317:
1316:
1315:
1314:
1311:
1308:
1307:
1302:
1299:
1297:
1294:
1292:
1288:
1284:
1283:
1279:
1276:
1275:
1273:
1272:
1268:
1267:
1261:
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1227:
1225:
1224:
1221:
1218:
1217:
1212:
1209:
1208:
1204:
1201:
1199:
1196:
1194:
1191:
1190:
1188:
1187:
1183:
1182:
1173:
1170:
1169:
1165:
1164:
1144:
1141:
1138:
1135:
1132:
1126:
1123:
1120:
1114:
1106:
1103:
1089:
1086:
1083:
1080:
1077:
1071:
1068:
1065:
1059:
1051:
1048:
1034:
1031:
1028:
1025:
1022:
1016:
1013:
1010:
1004:
996:
993:
976:
973:
970:
964:
956:
953:
952:
947:
944:
942:
939:
938:
934:
929:
928:
921:
920:
916:
914:
911:
909:
906:
904:
901:
900:
896:
895:Ideal gas law
893:
891:
888:
886:
883:
881:
878:
877:
873:
868:
867:
841:
831:
817:
810:
809:
794:
784:
770:
763:
762:
759:
745:
742:
735:
732:
729:
728:
709:
699:
685:
678:
677:
662:
652:
638:
631:
630:
627:
613:
610:
607:
600:
597:
594:
593:
574:
564:
550:
543:
542:
527:
517:
503:
496:
495:
492:
478:
475:
468:
465:
462:
461:
455:
452:
451:
448:
443:
442:
431:
428:
426:
425:Vapor quality
423:
421:
420:
415:
412:
410:
409:
404:
401:
398:
394:
393:
388:
385:
384:
383:
382:
379:
376:
375:
369:
366:
364:
361:
360:
358:
357:
354:
351:
350:
345:
342:
340:
337:
336:
335:
334:
330:
326:
319:
314:
313:
302:
299:
297:
294:
292:
289:
288:
287:
286:
283:
280:
279:
274:
271:
269:
266:
264:
263:Reversibility
261:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
224:
221:
219:
216:
215:
214:
213:
210:
207:
206:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
162:
161:
160:
157:
154:
153:
148:
145:
143:
140:
138:
137:Closed system
135:
134:
131:
126:
125:
117:
114:
112:
109:
107:
104:
102:
99:
98:
94:
89:
88:
81:
77:
74:
73:
69:
66:
64:
61:
59:
56:
54:
51:
50:
43:
42:
38:
32:
28:
27:
24:
21:
20:
5337:Regenerative
5275:
5266:Bell Coleman
5186:Hyperphysics
5125:
5121:
5102:
5070:
5043:
5027:Ewing, J. A.
5004:
5000:
4996:
4983:
4975:
4945:
4941:
4931:
4888:
4884:
4878:
4827:
4823:
4817:
4809:
4804:
4791:
4784:
4774:
4710:
4704:
4672:
4665:
4658:
4651:
4644:
4637:
4630:
4623:
4621:
4613:
4606:
4600:
4599:in Equation
4593:
4586:
4423:
4419:
4417:
4410:
4404:
4398:
4383:
4369:
4351:
4345:
4339:
4335:
4331:
4330:
4321:
4317:
4313:
4247:
4231:
4227:
4221:
4211:
3864:
3675:
3664:
3519:
3445:
3247:
3135:
3132:
3107:
3100:
3085:
3081:
3073:
3071:
3060:
2911:
2905:
2901:
2898:
2887:
2801:
2797:
2793:
2785:
2781:
2775:
2763:
2755:
2747:
2740:
2729:
2722:
2704:
2697:
2657:
2651:
2547:
2396:
2370:
2364:
2363:, D to A on
2358:
2354:
2353:
2238:
2227:
2220:
2214:
2213:, C to D on
2208:
2199:
2197:
2196:
2178:
2174:
2167:
2157:
2156:, B to C in
2151:
2140:
2139:
2044:
2038:
2037:, A to B in
2032:
2028:
2022:
2018:
2013:
2009:
2004:
2003:
1982:
1683:
1649:is an ideal
1647:Carnot cycle
1646:
1644:
1642:
1438:Carathéodory
1369:Heat engines
1341:
1330:
1319:
1301:Motive power
1286:
946:Free entropy
917:
417:
416: /
406:
405: /
397:introduction
390:
389: /
328:
291:Heat engines
78: /
5505:Ionocaloric
5500:Vuilleumier
5322:Hygroscopic
5195:powered by
5170:), also at
4684:regenerator
4605:by ⟨
4379:heat engine
4358:dissipation
4356:leading to
2357:(4 to 1 on
2177:transfers (
1655:Sadi Carnot
1260:Synergetics
941:Free energy
387:Temperature
248:Quasistatic
243:Isenthalpic
200:Instruments
190:Equilibrium
142:Open system
76:Equilibrium
58:Statistical
5470:Pulse tube
5442:Mixed/dual
5050:Chapter 44
4955:1704.04363
4948:(8): 556.
4837:1805.00848
4749:References
4393:See also:
4376:reversible
2399:reversible
2391:PV diagram
2142:Isentropic
2006:Isothermal
1752:reversible
1663:efficiency
1572:Nucleation
1416:Scientists
1220:Philosophy
933:Potentials
296:Heat pumps
253:Polytropic
238:Isentropic
228:Isothermal
5465:Kleemenko
5351:Internal
5138:CiteSeerX
4972:1434-6052
4923:119261929
4898:1111.7014
4711:idealized
4551:∫
4541:Δ
4529:⟩
4516:⟨
4475:∫
4465:Δ
4453:⟩
4440:⟨
4418:Although
4197:η
4167:Δ
4164:−
4148:−
4122:Δ
4086:Δ
4057:−
3991:Δ
3962:−
3882:−
3873:η
3624:−
3592:−
3553:η
3528:η
3496:−
3416:Δ
3387:−
3321:Δ
3292:−
3221:−
3192:−
3164:∮
3149:∮
3035:∮
3023:∮
3020:−
3008:∮
2993:−
2978:∮
2963:−
2951:∮
2936:∮
2850:∫
2826:∫
2735:. By the
2507:−
2441:Δ
2425:Δ
2409:Δ
2314:Δ
2256:Δ
2078:Δ
1878:−
1856:Δ
1840:−
1798:Δ
1764:heat pump
1760:conserved
1553:Waterston
1503:von Mayer
1458:de Donder
1448:Clapeyron
1428:Boltzmann
1423:Bernoulli
1384:Education
1355:Timelines
1139:−
1084:−
872:Equations
839:∂
792:∂
743:α
707:∂
660:∂
614:−
608:β
572:∂
525:∂
233:Adiabatic
223:Isochoric
209:Processes
170:Ideal gas
53:Classical
5520:Category
5432:Combined
5391:Humphrey
5376:Expander
5361:Atkinson
5296:Stoddard
5286:Stirling
5281:Ericsson
5241:External
5101:(1980).
4870:52943273
4862:30296120
4732:See also
4680:reheater
4354:friction
4310:Figure 6
4114:. Since
3127:Figure 5
3097:Figure 4
2913:Figure 1
2803:Figure 2
2719:Figure 3
2686:Figure 2
2659:Figure 1
2387:Figure 1
2366:Figure 2
2360:Figure 1
2216:Figure 2
2210:Figure 1
2159:Figure 2
2153:Figure 1
2040:Figure 2
2034:Figure 1
1927:, where
1605:Category
1543:Thompson
1453:Clausius
1433:Bridgman
1287:Vis viva
1269:Theories
1203:Gas laws
995:Enthalpy
403:Pressure
218:Isobaric
175:Real gas
63:Chemical
46:Branches
5495:Siemens
5411:Scuderi
5327:Rankine
5130:Bibcode
5077:541–548
5071:Physics
5029:(1910)
5015:Sources
4903:Bibcode
4842:Bibcode
4430:average
3770:is the
2790:entropy
1756:entropy
1528:Smeaton
1523:Rankine
1513:Onsager
1498:Maxwell
1493:Massieu
1198:Entropy
1193:General
1184:History
1174:Culture
1171:History
395: (
392:Entropy
329:italics
130:Systems
5401:Miller
5396:Lenoir
5371:Diesel
5317:Kalina
5301:Manson
5276:Carnot
5160:
5140:
5109:
5083:
5056:
4970:
4921:
4868:
4860:
3676:where
3076:is an
3072:Since
1979:Stages
1754:, and
1690:energy
1686:system
1518:Planck
1508:Nernst
1483:Kelvin
1443:Carnot
733:
598:
466:
408:Volume
323:Note:
282:Cycles
111:Second
101:Zeroth
5425:Mixed
5003:(4).
4993:(PDF)
4950:arXiv
4919:S2CID
4893:arXiv
4866:S2CID
4832:arXiv
4796:(PDF)
4754:Notes
4682:or a
2449:cycle
1673:into
1566:Other
1533:Stahl
1488:Lewis
1478:Joule
1468:Gibbs
1463:Duhem
156:State
116:Third
106:First
5437:HEHC
5406:Otto
5158:ISBN
5126:1411
5107:ISBN
5081:ISBN
5054:ISBN
4968:ISSN
4858:PMID
4722:work
4705:The
4592:and
4322:See
4275:and
4099:<
4013:and
3429:<
2575:and
2472:or,
2327:<
1723:and
1675:work
1671:heat
1538:Tait
368:Heat
363:Work
93:Laws
5148:doi
5001:102
4960:doi
4911:doi
4850:doi
4828:121
4560:out
2746:to
1766:or
1758:is
1381:Art
327:in
5522::
5146:.
5136:.
5097:;
5079:.
5052:.
4999:.
4995:.
4974:.
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4958:.
4946:77
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4940:.
4917:.
4909:.
4901:.
4889:44
4887:.
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4848:.
4840:.
4826:.
4762:^
4484:in
4381:.
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3074:dU
2768:Q
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2306:.
2185:.
2128:.
1645:A
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5329:(
5258:)
5254:(
5225:e
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5150::
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5115:.
5089:.
5062:.
4962::
4952::
4925:.
4913::
4905::
4895::
4872:.
4852::
4844::
4834::
4675:C
4673:T
4668:C
4666:T
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4645:T
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4607:T
4602:3
4596:C
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4589:H
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4570:d
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4537:1
4532:=
4524:C
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4494:d
4491:T
4480:Q
4468:S
4461:1
4456:=
4448:H
4444:T
4425:3
4412:3
4406:3
4371:3
4341:3
4318:S
4316:–
4314:T
4288:C
4284:T
4261:H
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4232:V
4230:–
4228:P
4175:H
4171:S
4161:=
4156:B
4152:S
4143:A
4139:S
4135:=
4130:C
4126:S
4102:0
4094:C
4090:S
4081:C
4077:T
4073:=
4070:)
4065:B
4061:S
4052:A
4048:S
4044:(
4039:C
4035:T
4031:=
4026:C
4022:Q
3999:H
3995:S
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3982:T
3978:=
3975:)
3970:A
3966:S
3957:B
3953:S
3949:(
3944:H
3940:T
3936:=
3931:H
3927:Q
3902:H
3898:T
3892:C
3888:T
3879:1
3876:=
3847:A
3843:S
3818:B
3814:S
3789:H
3785:T
3756:C
3752:T
3727:H
3723:Q
3698:C
3694:Q
3681:W
3669:)
3667:3
3665:(
3644:H
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3634:C
3630:T
3621:1
3618:=
3611:H
3607:Q
3600:C
3596:Q
3587:H
3583:Q
3576:=
3569:H
3565:Q
3561:W
3556:=
3504:C
3500:Q
3491:H
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3483:=
3480:Q
3477:=
3474:W
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3432:0
3424:C
3420:S
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3407:T
3403:=
3400:)
3395:B
3391:S
3382:A
3378:S
3374:(
3369:C
3365:T
3361:=
3356:C
3352:Q
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3312:T
3308:=
3305:)
3300:A
3296:S
3287:B
3283:S
3279:(
3274:H
3270:T
3266:=
3261:H
3257:Q
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3229:A
3225:S
3216:B
3212:S
3208:(
3205:)
3200:C
3196:T
3187:H
3183:T
3179:(
3176:=
3173:S
3170:d
3167:T
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3158:V
3155:d
3152:P
3146:=
3143:W
3113:.
3111:C
3108:T
3104:H
3101:T
3086:S
3084:–
3082:T
3065:)
3063:2
3061:(
3044:S
3041:d
3038:T
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3029:U
3026:d
3017:S
3014:d
3011:T
3005:=
3002:)
2999:U
2996:d
2990:S
2987:d
2984:T
2981:(
2975:=
2972:)
2969:U
2966:d
2960:Q
2957:d
2954:(
2948:=
2945:V
2942:d
2939:P
2933:=
2930:W
2906:S
2904:-
2902:T
2892:)
2890:1
2888:(
2871:S
2868:d
2864:T
2859:B
2854:A
2846:=
2843:Q
2840:d
2835:B
2830:A
2822:=
2819:Q
2798:T
2794:S
2792:(
2786:S
2784:–
2782:T
2780:(
2770:C
2764:W
2759:C
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2698:T
2656:(
2630:H
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2621:/
2615:H
2611:Q
2588:C
2584:T
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2557:Q
2534:.
2527:C
2523:T
2517:C
2513:Q
2504:=
2497:H
2493:T
2487:H
2483:Q
2460:,
2457:0
2454:=
2445:S
2438:=
2433:C
2429:S
2422:+
2417:H
2413:S
2373:H
2371:T
2330:0
2322:C
2318:S
2292:C
2288:T
2283:/
2277:C
2273:Q
2269:=
2264:C
2260:S
2242:C
2239:Q
2231:C
2228:T
2224:C
2221:T
2203:C
2200:T
2179:Q
2175:Q
2171:C
2168:T
2144:(
2114:H
2110:T
2105:/
2099:H
2095:Q
2091:=
2086:H
2082:S
2058:S
2048:H
2045:Q
2023:H
2019:T
2014:H
2010:T
1940:H
1936:Q
1911:H
1907:T
1901:H
1897:Q
1891:)
1886:C
1882:T
1873:H
1869:T
1865:(
1862:=
1859:S
1853:)
1848:C
1844:T
1835:H
1831:T
1827:(
1824:=
1821:W
1801:S
1778:W
1736:C
1732:T
1709:H
1705:T
1632:e
1625:t
1618:v
1145:S
1142:T
1136:H
1133:=
1130:)
1127:p
1124:,
1121:T
1118:(
1115:G
1090:S
1087:T
1081:U
1078:=
1075:)
1072:V
1069:,
1066:T
1063:(
1060:A
1035:V
1032:p
1029:+
1026:U
1023:=
1020:)
1017:p
1014:,
1011:S
1008:(
1005:H
980:)
977:V
974:,
971:S
968:(
965:U
842:T
818:V
795:V
771:1
746:=
710:p
686:V
663:V
639:1
611:=
575:T
551:N
528:S
504:T
479:=
476:c
399:)
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