CARNOT CYCLE I am teaching Engineering Thermodynamics using the textbook by Cengel and Boles. This set of slides overlap somewhat with Chapter 6. But here I assume that we have established the concept of entropy, and use the concept to analyze the Carnot cycle in the same way as we analyze any other thermodynamic process. An isolated system conserves energy and generates entropy. I did add a few slides to show how Carnot motivated his idea of entropy using the analogy of waterfall. I used the Dover edition of his book. I went through these slides in one 90-minute lecture. Zhigang Suo, Harvard University

Thermodynamics relates heat and motion thermo = heat dynamics = motion

Stirling engine Please watch this video

Carnot’s question How much work can be produced from a given quantity of heat? 4 “…whether the motive power of heat is unbounded, whether the possible improvements in steam-engines have an assignable limit, a limit which the nature of things will not allow to be passed by any means whatever...” Carnot, Reflections on the Motive Power of Fire (1824) Modern translations Motive power: work Motive power of heat: work produced by heat Limit: Carnot limit, Carnot efficiency

Device runs in cycle So they can run steadily over many, many cycles 5 Heat, Q Device Work, W

Isolated system When confused, isolate. 6 Isolated system conserves mass over time: Isolated system conserves energy over time: Isolated system generates entropy over time: Define more words: Isolated system IS

7 Reservoir of energy. Reservoir of entropy A (purely) thermal system with a fixed temperature The reservoir has a fixed temperature: The reservoir receives energy by heat Conservation of energy: The reservoir increases entropy (Reversible process. Clausius-Gibbs equation): reservoir of energy and entropy Fixed T R Changing U R, S R QRQR

8 Heat, Q Weight goes down. Thermodynamics permits heater A device runs in cycle to convert work to heat Device Work, W Heat, Q Reservoir of energy, T R Isolated system Device runs in cycle: Isolated system conserves energy: Isolated system generates entropy:

Thermodynamics forbids perpetual motion of the second kind a device runs in cycle to produce work by receiving heat from a single reservoir 9 Device Work, W Heat, Q Reservoir of energy, T R Isolated system Heat, Q Weight goes up Device runs in cycle: Isolated system conserves energy: Isolated system generates entropy:

Carnot’s remarks 10 1.“Wherever there exists a difference of temperature, motive power can be produced.” 1.To maximize motive power, “contact (between bodies of different temperatures) should be avoided as much as possible” Low-temperature sink, T L High-temperature source, T H Q Thermal contact of reservoirs of different temperatures generates entropy, and does no work.

Two reservoirs 11 For an engine running in cycle to convert heat to work, a single reservoir will not do; we need reservoirs of different temperatures. Engine High-temperature source, T H Low-temperature sink, T L Isolated system conserves energy isolated system generates entropy Isolated system of 4 parts W out

12 Carnot cycle Carnot (1824) Clapeyron (1834) Gibbs (1873)

13 Steam power plant

14 Thermal efficiency

15 Carnot efficiency Isolated system conserves energy: Isolated system generates entropy: All reversible engines running in cycle between reservoirs of two fixed temperatures T H and T L have the same thermal efficiency (Carnot efficiency): All real engines are irreversible. For an irreversible (i.e. real) engine running in cycle between reservoirs of two fixed temperatures T H and T L, the thermal efficiency is below the Carnot efficiency: Isolated system

16 All real processes are irreversible So many ways to generate entropy (i.e., to be irreversible) FrictionHeat transfer through a temperature difference

Carnot (1824): Two reservoirs Reflections on the Motive Power of Fire. 17 …the re-establishing of equilibrium in the caloric; that is, its passage from a body in which the temperature is more or less elevated, to another in which it is lower. What happens in fact in a steam-engine actually in motion? The caloric developed in the furnace by the effect of the combustion traverses the walls of the boiler, produces steam, and in some way incorporates itself with it. The latter carrying it away, takes it first into the cylinder, where it performs some function, and from thence into the condenser, where it is liquefied by contact with the cold water which it encounters there. Then, as a final result, the cold water of the condenser takes possession of the caloric developed by the combustion... The steam is here only a means of transporting the caloric. These two bodies, to which we can give or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of caloric. Carnot ( ) Modern translation Caloric: entropy Reservoir of caloric: Thermal reservoir

Carnot: “The steam is here only a means of transporting the caloric.” 18 Engine High-temperature source, T H Low-temperature sink, T L Thermal contact generates entropy Reversible engine transports entropy Low-temperature sink, T L High-temperature source, T H Q

Carnot’s analogy in his own words 19 The motive power of a waterfall depends on its height and on the quantity of the liquid; the motive power of heat depends also on the quantity of caloric used, and on what may be termed, on what in fact we will call, the height of its fall, that is to say, the difference of temperature of the bodies between which the exchange of caloric is made. In the waterfall the motive power is exactly proportional to the difference of level between the higher and lower reservoirs. In the fall of caloric the motive power undoubtedly increases with the difference of temperature between the warm and the cold bodies; but we do not know whether it is proportional to this difference. We do not know, for example, whether the fall of caloric from 100 to 50 degrees furnishes more or less motive power than the fall of this same caloric from 50 to zero. It is a question which we propose to examine hereafter. Modern translations Motive power: work Caloric: entropy Carnot, Reflections on the Motive Power of Fire (1824)

Carnot’s analogy in pictures 20 Low-height sink high-height source W out

Carnot’s analogy in modern terms 21 Fall of waterFall of caloric (entropy) ReservoirsTwo reservoirs of waterTwo reservoirs of caloric (entropy) Height of fallz H - z L T H - T L What is falling?Quantity of water, (m 2 g – m 1 g)Quantity of entropy, (S 2 – S 1 ) Work produced by the fall(z H – z L )(m 2 g – m 1 g)(T H – T L )(S 2 – S 1 ) 1212Gain water from sourceGain entropy from source 2323Drop elevation at constant quantity of water m 2 gDrop temperature at constant entropy S 2 3434Lose water to sinkLose entropy to sink 4141Raise elevation at constant quantity of water m 1 gRaise temperature at constant entropy S 1 THTH TLTL zHzH zLzL Fall of water Fall of caloric (entropy) S2S2 m2gm2g S1S1 m1gm1g

Carnot efficiency reversible engine running between two reservoirs of fixed temperatures T H and T L 22 Carnot efficiency: Low-temperature reservoir is the atmosphere: High-temperature reservoir is limited by materials (Melting point of iron is 1811 K. Metals creep at temperatures much below the melting point.) Carnot efficiency in numbers

24 What you need to know about energy, The National Academies.

25 Yang, Stabler, Journal of Electronic Materials. 38, 1245 (2009) Wasted energy

26 Refrigerator Isolated system conserves energy: Isolated system generates entropy: Carnot limit: Isolated system

27 Heat pump Isolated system conserves energy: Isolated system generates entropy: Carnot limit: Isolated system

Summary Thermodynamics permits heater (a device running in cycle to convert work to heat). Thermodynamics forbids perpetual motion of the second kind (a device running in cycle to produce work by receiving heat from a single reservoir of a fixed temperature). Carnot cycle: A reversible cycle consisting of isothermal processes at two temperatures T H and T L, and two isentropic processes. All reversible engines running in cycle between reservoirs of two fixed temperatures T H and T L have the same thermal efficiency (Carnot efficiency): All real engines are irreversible. For an irreversible (i.e. real) engine running in cycle between reservoirs of two fixed temperatures T H and T L, the thermal efficiency is below the Carnot efficiency (Carnot limit): Carnot cycle also limits the coefficients of performance of refrigerators and heat pumps. 28