02. Equilibrium Thermodynamics II: Engines

Transcription

1 University of Rhode Island Equilibrium Statistical Physics Physics Course Materials Equilibrium Thermodynamics II: Engines Gerhard Müller University of Rhode Island, Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Follow this and additional works at: htt://digitalcommons.uri.edu/equilibrium_statistical_hysics Abstract Part two of course materials for Statistical Physics I: PHY525, taught by Gerhard Müller at the University of Rhode Island. Documents will be udated eriodically as more entries become resentable. Recommended Citation Müller, Gerhard, "02. Equilibrium Thermodynamics II: Engines" (205). Equilibrium Statistical Physics. Paer 3. htt://digitalcommons.uri.edu/equilibrium_statistical_hysics/3 This Course Material is brought to you for free and oen access by the Physics Course Materials at DigitalCommons@URI. It has been acceted for inclusion in Equilibrium Statistical Physics by an authorized administrator of DigitalCommons@URI. For more information, lease contact digitalcommons@etal.uri.edu.

2 Contents of this Document [ttc2] 2. Equilibrium Thermodynamics II: Engines Carnot engine. [tln] Maximum efficiency. [tln2] Absolute temerature. [tln3] Entroy change caused by exanding ideal gas. [tex] Carnot cycle of the classical ideal gas. [tex3] Carnot cycle of an ideal aramagnet. [tex4] Reversible rocesses in fluid systems. [tln5] Adiabates of the classical ideal gas. [tex7] Roads from to 2: isothermal, isentroic, isochoric, isobaric. [tex25] Room heater: electric radiator versus heat um. [tex3] Mayer s relation for the heat caacities of the classical ideal gas. [tex2] Positive and negative heat caacities. [tex26] Work extracted from finite heat reservoir in infinite environment. [tex9] Work extracted from finite heat reservoir in finite environment. [tex0] Heating the air in a room. [tex2] Gasoline engine. [tln65] Idealized gasoline engine(otto cycle). [tex8] Diesel engine. [tln66] Idealized Diesel engine. [tex6] Escher-Wyss gas turbine. [tln75] Joule cycle. [tex08] Stirling engine. [tln76] Idealized Stirling cycle. [tex3] Ideal-gas engine with two-ste cycle I. [tex06] Ideal-gas engine with two-ste cycle II. [tex07]

3 Circular heat engine I. [tex47] Circular heat engine II. [tex48] Square heat engine. [tex49]

4 Carnot engine [tln] Second law: Heat flows sontaneously from high to low temeratures. Thermal contact: Temerature differences disaear without roducing work. Heat engine: Part of the heat flowing from high to low temeratures is converted into work via a cyclic rocess. Carnot engine: All wasteful heat flows are eliminated (reversible rocesses). The four stes of a Carnot rocess: 2: Isothermal absortion of heat: Q 2 > 0 at Θ H. 2 3: Adiabatic cooling: Θ H Θ L with Q 23 = 0 and W 23 < : Isothermal exulsion of heat: Q 34 < 0 at Θ L. 4 : Adiabatic heating: Θ L Θ H with Q 4 = 0 and W 4 > 0. Total heat inut: Q in = Q 2. Use first law: U = Q 2 + W 2 + W 23 + Q 34 + W 34 + W 4 = 0. Net work outut: W out W 2 W 23 W 34 W 4 = Q 2 Q 34 Efficiency: η W out Q in = Q 34 Q 2. Θ H Y Θ H Q 2 C W out 2 4 Θ L Q 34 3 Θ L X

5 Maximum efficiency [tln2] Is it ossible to construct a heat engine A which is more efficient than the Carnot engine C? Use engine A to drive engine C in the reverse i.e. as a refrigerator. Θ H Q A A W C Q 2 W Q A Q 34 Θ L Heat transfers: Q A > 0, Q 2 < 0, Q 34 > 0. Work erformance: W = W (A) out Efficiencies: η A = W Q A, η C = W Q 2 = W (C) in > 0. Since engine C oerates reversibly, η C is the same in the forward and reverse directions. Note: η C is not an efficiency in the reverse mode. η A > η C would imly Q A < Q 2. The two engines combined would then cause heat to flow from low to high temerature without work inut, which is a violation of the second law. Conclusions: Engine A cannot be more efficient than engine C. All Carnot engines oerating between Θ H and Θ L must have the same efficiency.

6 Absolute temerature [tln3] Reservoir temeratures: Θ H, Θ M, Θ L. Efficiency: η = Q L Q H = f(θ L, Θ H ). Likewise: Q M Q H = f(θ M, Θ H ), Q L Q M = f(θ L, Θ M ). Θ H Q H C Q H Q M Θ M C Q M C Q L Q L Θ L Second law imlies: If Q L = Q L then Q H = Q H. Q L Q M Q M Q H = Q L Q H f(θ L, Θ M )f(θ M, Θ H ) = f(θ L, Θ H ) Functional form: f(θ L, Θ H ) = g(θ L) g(θ H ) T L T H η = T L T H. Definition of absolute temerature: T L = Q L. T H Q H Kelvin scale is fixed by trile oint of water: T tr = 273.6K. Note: η = imlies T L = 0. However, the third law states δq = T ds = 0 at T = 0. Hence, all reversible rocesses at T = 0 are adiabatic. Heat cannot be absorbed reversibly at T = 0.

7 [tex] Entroy change caused by exanding ideal gas Consider the amount n = mol of a classical ideal gas in a box of volume with heat-conducting walls. The gas is described by the equation of state = nrt and the internal energy U = C T with C = const. Now we let the gas exand to the volume 2 =2 via two different rocesses: (a) by quasi-static isothermal exansion; (b) by leakage through a hole in one wall. Calculate the change in entroy S G of the gas and S E of the environment during each rocess. Exress the results in SI units. (a) (b)

8 [tex3] Carnot cycle of the classical ideal gas Consider the four stes of a Carnot engine with the oerating material in the form of a classical ideal gas [ = nrt, U = C T with C = const]. (a) Determine the heat transfer, Q, the work erformance, W, and the change in internal energy, U, for each of the four stes: 2 isothermal exansion: T = T H = const, 2 >. 2 3 adiabatic exansion: S = const, 3 > isothermal comression: T = T L = const, 4 < 3. 4 adiabatic comression: S = const, < 4. (b) Sketch the Carnot cycle in the (, )-lane and in the (U, S)-lane. (c) Show that the efficiency is η C = T L /T H.

9 [tex4] Carnot cycle of an ideal aramagnet Consider the four stes of a Carnot engine with the oerating material in the form of an ideal aramagnet. The equation of state is Curie s law, M = DH/T, where H is the magnetic field, T the absolute temerature, and D a constant. The internal energy is a monotonically increasing function, U(T ), of temerature. (a) Determine the heat transfer, Q, the work erformance, W, and the change in internal energy, U, for each of the four stes: 2 isothermal demagnetization: T = T H = const, M 2 < M. 2 3 adiabatic demagnetization: S = const, M 3 < M isothermal magnetization: T = T L = const, M 4 > M 3. 4 adiabatic magnetization: S = const, M > M 4. (b) Sketch the Carnot cycle in the (M, H)-lane and in the (U, S)-lane. (c) Show that the efficiency is η C = T L /T H.

10 Reversible rocesses in fluid system [tln5] Isothermal rocess: T = const. δq 0 in general. Isochoric rocess: = const. δq = C dt, du = C dt. Isobaric rocess: = const. δq = C dt. Isentroic (adiabatic) rocess: S = const. δq = 0. Internal energy: du = δq + δw = T ds d. = const. δw = 0 du = δq (no work erformed). S = const. δq = 0 du = δw (no heat transferred). Classical ideal gas: Equation of state: = nrt. Internal energy: U = C T, C = αnr = const. Isotherm: T = const. = const. Adiabate: S = const. γ = const., γ = + /α monatomic gas: α = 3, γ = diatomic gas: α = 5, γ = olyatomic gas: α = 3, γ = 4. 3 = const = const T = const S = const

11 [tex7] Adiabates of the classical ideal gas The classical ideal gas is secified by the thermodynamic equation of state = nrt and by the internal energy (caloric equation of state) U = C T with C = αnr = const [α = 3 2 (monatomic), α = 5 2 (diatomic), α = 3 (olyatomic)]. A reversible rocess with S = const is called isentroic or adiabatic and is characterized by the curve γ = const. No heat is exchanged in an adiabatic rocess: du = δw, δq = 0. Find γ as a function of α.

12 [tex25] Roads from to 2: isothermal, isentroic, isochoric, isobaric The amount n = mol of an ideal gas undergoes three different quasistatic rocesses (see Figure) from the initial state (,,T ) to the final state ( 2, 2,T 2 ): (i) A 2; (ii) B 2; (iii) C 2. Find the work W done on the system and the heat Q added to the system in each rocess. Exress all results in terms of (T,,T 2, 2 ). isochoric isothermal isentroic B C A isobaric 2 2

13 [tex3] Room heater: Electric radiator versus heat um A room is to be ket at temerature T H = 294K, (2 C). The outdoor temerature is T L. Heat, which leaks through the windows and doors at the rate Q leak = γ T, must be relaced by a room heater at the same rate. The electric radiator converts electric ower W el into heat with 00% efficiency. The electric heat um uses the amount W su of electric ower to drive a Carnot cycle in the reverse, which extracts heat Q L at temerature T L from the exterior and converts it (reversibly) into heat Q H = Q L + W h at temerature T H. In the relation W h = ( λ) W su, λ reresents the energy loss in the gears of the heat um. Quantitites with overbars denote energy transfers er time unit. (a) Find W el as a function of γ, T H, T L, and W su as a function of γ, λ, T H, T L. (b) Plot W el /γ and W su /γ versus t L T L 273K (measured in C) for fixed T H = 294K and λ = 0.8 (20% efficiency). (c) Determine the range of T L where the heat um is more economical than the radiator.

14 [tex2] Mayer s relation for the heat caacities of the classical ideal gas The amount n = mol of a classical ideal gas [ = nrt, U = C T with C = const] is initially confined to a volume at ressure. In ste 2 of Mayer s cycle, the gas undergoes free exansion to volume 2 while it is thermally isolated (δq =0, δw = 0) The ressure decreases from to 2 during this ste. In ste 2 3 the gas is quasi-statically comressed back to volume, while the ressure is maintained at 2. With the temerature decreasing during this ste, heat is exelled. In ste 3 the gas is heated u quasi-statically at constant volume until the ressure returns to. Use the first law to derive Mayer s relation, C C = R, between the heat caacities of the classical ideal gas

15 [tex26] Positive and negative heat caacities The diagram shows an isotherm and an adiabate for the classical ideal gas. Show that a quasistatic rocess of the tye 2 is characterized by a ositive heat caacity and a rocess of the tye 3 4 by a negative heat caacity. 3 adiabate 2 isotherm 4

16 [tex9] Work extracted from finite heat reservoir in infinite environment A (finite) heat reservoir with heat caacity C = const is initially at temerature T H and the (infinite) environment at the lower temerature T 0. Now the reservoir is connected to the environment by a heat engine, which absorbs an infinitesimal amount of heat δq er cycle, converts art of it into work δw, and dums the rest into the environment. During each cycle the temerature of the reservoir decreases infinitesimally: δq = CdT. Determine the maximum amount of work W that can be extracted from the reservoir before its temerature has droed to that of the environment. The fraction of the excess internal energy U ex = C(T H T 0 ) that can be converted into work is characterized by the quantity W/U ex. Plot this quantity versus the reduced temerature (T H T 0 )/T 0 for T 0 < T H < 3T 0. Set T H /T 0 = + ɛ with ɛ and find the deendence of W/U ex on ɛ to leading order.

17 [tex0] Work extracted from finite heat reservoir in finite environment A (finite) heat reservoir with heat caacity C H = const is initially at temerature T H and the (finite) environment with heat caacity C L at the lower temerature T L. (a) When heat is allowed to flow from the reservoir to the environment, both will end u at the temerature T f = (C H T H + C L T L )/(C H + C L ) (arithmetic mean). erify this and determine the total amount of heat Q that has been transferred. (b) When the reservoir is connected to the environment by a Carnot engine which absorbs an infinitesimal amount of heat δq er cycle, converts art of it into work δw, and dums the rest into the environment, the final common temerature of the reservoir and the environment will be T f = T C H/(C H +C L ) H T C L/(C H +C L ) L (geometric mean). erify this and determine the total amout of work W that has been extracted from the system. The fraction of the excess internal energy U ex = C H (T H T L ) that can be converted into work is characterized by the quantity W/U ex. Plot this quantity versus the reduced temerature (T H T L )/T L for C H = C L and T L < T H < 3T L. Discuss the roerties of this function in the limit T H T L.

18 [tex2] Heating the air in a room Calculate the amount of energy Q that must be sulied to heat the air in a room from 0 C to 20 C under three different circumstances. For each case, calculate also the change in internal energy U of the air in the room. Mass density of air at STP (0 C and atm): ρ = g/cm 3. Secific heats of air: c = 0.69cal/gK, c /c γ =.4. Exress all results in SI units. (a) The room has rigid, insulating walls. The volume is 27m 3. The initial ressure is atm. (b) The room has insulating walls. One wall is mobile. The rocess takes lace at constant ressure (atm). The initial volume is 27m 3. (c) The room has rigid, insulating walls. The volume is 27m 3. One wall has a small hole through which air leaks out slowly. The rocess takes lace at constant ressure (atm).

19 Gasoline engine (Otto cycle) [tln65] Four-stroke Otto cycle (left) -2: comression stroke 2-3-4: ower stroke (sark lug ignites at 2) 4- -5: exhaust stroke (exhaust valve oens at 4) 5-: intake stroke (intake valve oens at 5) Idealized Otto cycle (right) -2: adiabatic comression of air-fuel mixture (S = const) 2-3: exlosion of air-fuel mixture ( = const) 3-4: adiabatic exansion of exhaust gas (S = const) 4-: isochoric release of exhaust gas ( = const). -5-: intake stroke (thermodynamically ignored) Parameter: K. = / 2 (comression ratio). The comression ratio K must not be chosen too large to revent the air-fuel mixture from igniting sontaneously, i.e. rematurely.

20 [tex8] Idealized Otto cycle Consider the four stes of the idealized Otto cycle for a classical ideal gas [ = nrt, U = C T with C = αnr]. (a) Determine the heat transfer, Q, the work erformance, W, and the change in internal energy, U, for each of the four stes: 2 adiabatic comression of air-fuel mixture: S = const. 2 3 exlosion of air-fuel mixture: = const. 3 4 adiabatic exansion of exhaust gas: S = const. 4 isochoric release of exhaust gas: = const. (c) Calculate the efficiency η and exress it as a function of the comression ratio K / 2. 3 S = const 2 4 S = const 2

21 Diesel engine [tln66] Four-stroke Diesel cycle (left) -2: comression stroke (fuel injected and sontaneously ignited at 2) 2-3-4: ower stroke (Diesel fuel burns more slowly than gasoline) 4- -5: exhaust stroke (exhaust valve oens at 4) 5-: intake stroke (intake valve oens at 5) Idealized Diesel cycle (right) -2: adiabatic comression of air (S = const) 2-3: isobaric exansion as fuel exlodes ( = const) 3-4: adiabatic exansion of exhaust gas (S = const) 4-: isochoric release of exhaust gas ( = const). -5-: intake stroke (thermodynamically ignored) Parameter: K. = / 2 (comression ratio), L. = 3 / 2

22 [tex6] Idealized Diesel cycle Consider the four stes of the Diesel cycle for a classical ideal gas [ = nrt, U = C T with C = αnr]. (a) Determine the heat transfer, Q, the work erformance, W, and the change in internal energy, U, for each of the four stes: 2 adiabatic comression of air: S = const. 2 3 isobaric exansion during exlosion: = const. 3 4 adiabatic exansion after exlosion: S = const. 4 isochoric release of exhaust gas: = const. Calculate the efficiency η and exress it as a function of the two arameters K / 2 and L 3 / S = const 4 S = const

23 Escher-Wyss gas turbine [tln75] A gas flows in a closed system from the boiler via the turbine to the radiator and then via the comressor back into the boiler. As the beam of hot gas hits the blades of the turbine during the ower stroke it exands with little heat transfer. The comression of the cooled gas is also roughly adiabatic. The gas is heated u inside the boiler and cooled down inside the radiator at different but roughly constant ressures Idealized rocess (Joule cycle) -2: Adiabatic exansion of the hot gas after ejection from the boiler as it drives the turbine (S = const). 2-3: Isobaric contraction as the gas flows through the radiator and cools down further in the rocess ( = const). 3-4: Adiabatic comression of the cooled gas for injection into the boiler (S = const). 4-: Isobaric exansion of the gas as it heats u inside the boiler ( = const). Notes: The ressure inside the boiler is regulated by the rates of gas injection and ejection and the rate of heat transfer from the energy source to the gas. The injection and ejection rates are the same in mass units but the ejection rate is larger than the injection rate in volume units. This accounts for the exansion of the gas inside the boiler as described in ste 4-.

24 [tex08] Joule cycle Consider the four stes of the Joule cycle for the classical ideal gas [ = Nk B T, C = αnk B, γ. = C /C = (α + )/α]. It reresents an idealized version of the Escher-Wyss gas turbine. (a) Calculate the work erformance, W, the heat transfer, Q, and the change in internal energy, U, for each ste. 2 adiabatic exansion: S = const. 2 3 isobaric contraction: = const. 3 4 adiabatic comression: S = const. 4 isobaric exansion: = const. (b) Calculate the efficiency η and exress it as a function of the ressure ratio 2 /. (c) Sketch the Joule cycle in the (U, S)-lane

25 Stirling engine [tln76] The Stirling engine is an external combustion engine. It isolates the working fluid from the heat source. Combustion is better controlled than in internal combustion engines. T H R T L 3 T H D P 4 2 T L 2 Piston P exands gas at high temerature T H and comresses gas at low temerature T L. Dislacer D moves gas between regions of high temerature T H and low temerature T L through the regenerator. Regenerator R acts as a heat exchanger. It stores heat when hot gas flows from left to right and releases heat when colder gas flows from right to left. Idealized Stirling cycle -2: Isothermal comression at temerature T L. Dislacer stationary at left. Piston moving left. 2-3: Isochoric heating u at volume 2. Piston stationary at left. Dislacer moving right. 3-4: Isothermal exansion at temerature T H. Dislacer and iston moving right. 4-: Isochoric cooling down at volume. Piston stationary at right. Dislacer moving left. Note: Some of the heat is recycled in the regenerator. This amount should not be counted in the exression η = W out / Q in of the efficiency.

26 [tex3] Idealized Stirling cycle Consider the four stes of the idealized Stirling cycle for the classical ideal gas [ = Nk B T, C = αnk B, γ. = C /C = (α + )/α]. (a) Calculate the work erformance, W, the heat transfer, Q, and the change in internal energy, U, for each ste. 2 isothermal comression: T = T L, 2 3 isochoric heating u: = 2, 3 4 isothermal exansion: T = T H, 4 isochoric cooling down: =. (b) Calculate the efficiency η and exress it as a function of T H and T L. 3 T H 2 4 T L 2

27 [tex06] Ideal-gas engine with two-ste cycle I Consider the two-ste cycle for a classical ideal gas [ = Nk B T, C = αnk B, γ =. C /C = (α + )/α] as shown. The first ste (A) is an adiabatic comression and the second ste (B) an exansion along a straight line segment in the (, )-lane. (a) Show that the difference in internal energy U =. U U 2 is determined by the exression [ ( ) ] γ U = α. (b) Show that the heat transfer δq between system and environment during a volume increase from to + δ along the straight line segment is given by the exression [ δq = ( + α)( + σ) ( + 2α)σ ] d, σ = ( / 2 ) γ 2 /. (c) Show that along the straight-line segment the system absorbs heat if < < c and exells heat if c < < 2, where c / = [( + α)( + σ)]/[( + 2α)σ]. 2 A B 2 2

28 [tex07] Ideal-gas engine with two-ste cycle II Consider the two-ste cycle for a classical ideal gas [ = Nk B T, C = αnk B, γ. = C /C = (α + )/α] as reviously discussed in [tex06]. For the following we assume that the comression ratio is 2 / = 2 and that the gas is monatomic (α = 3 2 ). (a) Show that the net work outut along the adiabate and along the straight line segment are W A / and W B / (b) Show that the total heat absorbed during the cycle is Q in / (c) Determine the efficiency η 2 of the two-ste cycle and comare it with the efficiency η C of the Carnot engine oerating between the same two temeratures. A B 2 2

29 [tex47] Circular heat engine I Consider mol of a classical ideal gas [ = RT ] confined to a cylinder by a iston. The cylinder is in thermal contact with a heat bath of adjustable temerature. As the iston moves back and forth between volume = 0 ( r) and = 0 ( + r) quasistatically, the temerature of the gas is being adjusted via thermal contact such that the cycle becomes circular in the (, )-lane and roceeds in clockwise direction (φ from 0 to 2π). (a) Calculate the net work outut W out during one cycle. (b) Set r = 0.5 and identify the segments along the circle where the temerature of the gas rises and the segments where it falls. (c) Reeat the revious art for r = 0.9. Note that there now are more segments. / 0 r φ 0 0 / 0

30 [tex48] Circular heat engine II Consider mol of a monatomic classical ideal gas [ = RT, U = 3 2RT ] confined to a cylinder by a iston. The cylinder is in thermal contact with a heat bath of adjustable temerature. As the iston moves back and forth between volume = 0 ( r) and = 0 ( + r) quasistatically, the temerature of the gas is being adjusted via thermal contact such that the cycle becomes circular in the (, )-lane and roceeds in clockwise direction (φ from 0 to 2π). (a) Calculate the rate dw/dφ at which work is being erformed, the rate du/dφ at which the internal energy changes, and the rate dq/dφ at which heat is being transferred. (b) Set r = 0.5 and identify the segments along the circle where each rate is ositive or negative. (c) Reeat the revious art for r = 0.9. (d) Plot all three rates as functions of φ/π for r = 0.5 in one grah and then for r = 0.9 in a second grah. / 0 r φ 0 0 / 0

31 [tex49] Square heat engine Consider mol of a monatomic classical ideal gas [ = RT, U = 3 2RT ] confined to a cylinder by a iston. The cylinder is in thermal contact with a heat bath of adjustable temerature. The gas undergoes a quasistatic, cyclic rocess as shown. Use,, T as units for ressure, volume, and temerature, resectively. (a) Find 2, 2, T 2, 3, 3, T 3, and 4, 4, T 4 in these units. (b) Find the work erformance, W 2, W 23, W 34, W 4, the change in internal energy, U 2, U 23, U 34, U 4, and the heat transfer, Q 2, Q 23, Q 34, Q 4, along the legs of the cycle. Exress these quantities in units of RT. (c) Find the net work W net erformed during the cycle. Find also the heat Q in absorbed and the heat Q out exelled by the gas during the cycle. (d) Find the efficiency η S of this cycle in the role of heat engine. / /