Ch.8

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Ch.8

  1. 1. 8-1 Chapter 8 EXERGY – A MEASURE OF WORK POTENTIALExergy, Irreversibility, Reversible Work, and Second-Law Efficiency8-1C Reversible work differs from the useful work by irreversibilities. For reversible processes both areidentical. Wu = Wrev − I.8-2C Reversible work and irreversibility are identical for processes that involve no actual useful work.8-3C The dead state.8-4C Yes; exergy is a function of the state of the surroundings as well as the state of the system.8-5C Useful work differs from the actual work by the surroundings work. They are identical for systemsthat involve no surroundings work such as steady-flow systems.8-6C Yes.8-7C No, not necessarily. The well with the higher temperature will have a higher exergy.8-8C The system that is at the temperature of the surroundings has zero exergy. But the system that is at alower temperature than the surroundings has some exergy since we can run a heat engine between thesetwo temperature levels.8-9C They would be identical.8-10C The second-law efficiency is a measure of the performance of a device relative to its performanceunder reversible conditions. It differs from the first law efficiency in that it is not a conversion efficiency.8-11C No. The power plant that has a lower thermal efficiency may have a higher second-law efficiency.8-12C No. The refrigerator that has a lower COP may have a higher second-law efficiency.8-13C A processes with Wrev = 0 is reversible if it involves no actual useful work. Otherwise it isirreversible.PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  2. 2. 8-28-14C Yes.8-15 Windmills are to be installed at a location with steady winds to generate power. The minimumnumber of windmills that need to be installed is to be determined.Assumptions Air is at standard conditions of 1 atm and 25°CProperties The gas constant of air is 0.287 kPa.m3/kg.K (Table A-1).Analysis The exergy or work potential of the blowing air is the kinetic energy it possesses, V 2 (8 m/s) 2 ⎛ 1 kJ/kg ⎞ Exergy = ke = = ⎜ ⎟ = 0.032 kJ/kg 2 2 ⎝ 1000 m 2 / s 2 ⎠At standard atmospheric conditions (25°C, 101 kPa), the density andthe mass flow rate of air are P 101 kPa ρ= = = 118 m 3 / kg . RT (0.287 kPa ⋅ m 3 / kg ⋅ K)(298 K)and π D2 m = ρAV1 = ρ & V1 = (1.18 kg/m 3 )(π / 4)(10 m) 2 (8 m/s) = 742 kg/s 4Thus, Available Power = mke = (742 kg/s)(0.032 kJ/kg) = 23.74 kW &The minimum number of windmills that needs to be installed is & W total 600 kW N= = = 25.3 ≅ 26 windmills W& 23.74 kWPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  3. 3. 8-38-16E Saturated steam is generated in a boiler by transferring heat from the combustion gases. The wastedwork potential associated with this heat transfer process is to be determined. Also, the effect of increasingthe temperature of combustion gases on the irreversibility is to be discussed.Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible.Analysis The properties of water at the inlet and outlet of the boiler and at the dead state are (Tables A-4Ethrough A-6E) P1 = 200 psia ⎫ h1 = h f = 355.46 Btu/lbm ⎬ q x1 = 0 (sat. liq.) ⎭ s1 = s f = 0.54379 Btu/lbm ⋅ R Water P2 = 200 psia ⎫ h2 = h g = 1198.8 Btu/lbm 200 psia ⎬ 200 psia x 2 = 1 (sat. vap.) ⎭ s 2 = s g = 1.5460 Btu/lbm ⋅ R sat. liq. sat. vap. T0 = 80°F ⎫ h0 ≅ h f @ 80°F = 48.07 Btu/lbm ⎬ P0 = 14.7 psia ⎭ s 0 ≅ s f @ 80°F = 0.09328 Btu/lbm ⋅ RThe heat transfer during the process is qin = h2 − h1 = 1198.8 − 355.46 = 843.3 Btu/lbmThe entropy generation associated with this process is qin sgen = Δsw + ΔsR = ( s2 − s1 ) − TR 843.3 Btu/lbm = (1.5460 − 0.54379)Btu/lbm ⋅ R − (500 + 460)R = 0.12377 Btu/lbm ⋅ RThe wasted work potential (exergy destruction is) xdest = T0 sgen = (80 + 460 R)(0.12377 Btu/lbm ⋅ R) = 66.8 Btu/lbmThe work potential (exergy) of the steam stream is Δψ w = h2 − h1 − T0 ( s2 − s1 ) = (1198.8 − 355.46)Btu/lbm − (540 R )(1.5460 − 0.54379)Btu/lbm ⋅ R = 302.1 Btu/lbmIncreasing the temperature of combustion gases does not effect the work potential of steam stream since itis determined by the states at which water enters and leaves the boiler.Discussion This problem may also be solved as follows:Exergy transfer by heat transfer: ⎛ T ⎞ ⎛ 540 ⎞ xheat = q⎜1 − 0 ⎟ = (843.3)⎜1 − ⎜ T ⎟ ⎟ = 368.9 Btu/lbm ⎝ R ⎠ ⎝ 960 ⎠Exergy increase of steam: Δψ w = 302.1 Btu/lbmThe net exergy destruction: xdest = xheat − Δψ w = 368.9 − 302.1 = 66.8 Btu/lbmPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  4. 4. 8-48-17 Water is to be pumped to a high elevation lake at times of low electric demand for use in ahydroelectric turbine at times of high demand. For a specified energy storage capacity, the minimumamount of water that needs to be stored in the lake is to be determined.Assumptions The evaporation of water from the lake is negligible.Analysis The exergy or work potential of the water is the potentialenergy it possesses, 75 m Exergy = PE = mghThus, PE 5 × 10 6 kWh ⎛ 3600 s ⎞⎛ 1000 m 2 / s 2 ⎞ m= = ⎜ ⎟⎜ ⎟ = 2.45 × 10 10 kg gh (9.8 m/s 2 )(75 m) ⎝ 1 h ⎠⎜ 1 kW ⋅ s/kg ⎝ ⎟ ⎠8-18 A heat reservoir at a specified temperature can supply heat at a specified rate. The exergy of this heatsupplied is to be determined.Analysis The exergy of the supplied heat, in the rate form, is theamount of power that would be produced by a reversible heat engine, 1500 K T0 298 K η th,max = η th, rev = 1 − = 1− = 0.8013 TH 1500 K HE & Wrev & Exergy = W max,out & & = W rev,out = η th, rev Qin = (0.8013)(150,000 / 3600 kJ/s) 298 K = 33.4 kW8-19 [Also solved by EES on enclosed CD] A heat engine receives heat from a source at a specifiedtemperature at a specified rate, and rejects the waste heat to a sink. For a given power output, the reversiblepower, the rate of irreversibility, and the 2nd law efficiency are to be determined.Analysis (a) The reversible power is the power produced by a reversible heat engine operating between thespecified temperature limits, TL 320 K η th, max = η th,rev = 1 − = 1− = 0.787 TH 1500 K 1500 K & & W rev,out = η th, rev Qin = (0.787)(700 kJ/s) = 550.7 kW 700 kJ/s(b) The irreversibility rate is the difference between the reversible HE 320 kWpower and the actual power output: & & & I = W rev,out − W u,out = 550.7 − 320 = 230.7 kW(c) The second law efficiency is determined from its definition, 320 K W u,out 320 kW η II = = = 58.1% W rev,out 550.7 kWPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  5. 5. 8-58-20 EES Problem 8-19 is reconsidered. The effect of reducing the temperature at which the waste heat isrejected on the reversible power, the rate of irreversibility, and the second law efficiency is to be studiedand the results are to be plotted.Analysis The problem is solved using EES, and the solution is given below."Input Data"T_H= 1500 [K]Q_dot_H= 700 [kJ/s]{T_L=320 [K]}W_dot_out = 320 [kW]T_Lsurr =25 [C]"The reversible work is the maximum work done by the Carnot Enginebetween T_H and T_L:"Eta_Carnot=1 - T_L/T_HW_dot_rev=Q_dot_H*Eta_Carnot"The irreversibility is given as:"I_dot = W_dot_rev-W_dot_out"The thermal efficiency is, in percent:"Eta_th = Eta_Carnot*Convert(, %)"The second law efficiency is, in percent:"Eta_II = W_dot_out/W_dot_rev*Convert(, %) ηII [%] I [kJ/s] Wrev [kJ/s] TL [K] 68.57 146.7 466.7 500 67.07 157.1 477.1 477.6 65.63 167.6 487.6 455.1 64.25 178.1 498.1 432.7 62.92 188.6 508.6 410.2 61.65 199 519 387.8 60.43 209.5 529.5 365.3 59.26 220 540 342.9 58.13 230.5 550.5 320.4 57.05 240.9 560.9 298PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  6. 6. 8-6 570 548 Wrev [kJ/s] 526 504 482 460 275 320 365 410 455 500 TL [K] 70 68 66ηII [%] 64 62 60 58 56 275 320 365 410 455 500 TL [K] 250 228 I [kJ/s] 206 184 162 140 275 320 365 410 455 500 TL [K]PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  7. 7. 8-78-21E The thermal efficiency and the second-law efficiency of a heat engine are given. The sourcetemperature is to be determined.Analysis From the definition of the second law efficiency, TH η th η 0.36 η II = ⎯→ η th, rev = th = ⎯ = 0.60 η th, rev η II 0.60 HEThus, η th = 36% TL η II = 60% η th,rev = 1 − ⎯→ T H = T L /(1 − η th, rev ) = (530 R)/0.40 = 1325 R ⎯ TH 530 R8-22 A body contains a specified amount of thermal energy at a specified temperature. The amount that canbe converted to work is to be determined.Analysis The amount of heat that can be converted to work is simplythe amount that a reversible heat engine can convert to work, 800 K T0 298 K 100 kJ η th,rev = 1 − = 1− = 0.6275 TH 800 K HE W max,out = W rev,out = η th, rev Qin = (0.6275)(100 kJ) 298 K = 62.75 kJ8-23 The thermal efficiency of a heat engine operating between specified temperature limits is given. Thesecond-law efficiency of a engine is to be determined.Analysis The thermal efficiency of a reversible heat engineoperating between the same temperature reservoirs is 1200°C T0 293 K η th,rev = 1 − = 1− = 0.801 η th = 0.40 TH 1200 + 273 K HEThus, η th 0.40 20°C η II = = = 49.9% η th, rev 0.801PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  8. 8. 8-88-24 A house is maintained at a specified temperature by electric resistance heaters. The reversible workfor this heating process and irreversibility are to be determined.Analysis The reversible work is the minimum work required to accomplish this process, and theirreversibility is the difference between the reversible work and the actual electrical work consumed. Theactual power input is & & & Win = Qout = Q H = 80,000 kJ/h = 22.22 kWThe COP of a reversible heat pump operating between the 80,000 kJ/hspecified temperature limits is 1 1 House COPHP, rev = = = 42.14 1 − T L / T H 1 − 288 / 295 22 °C 15 °CThus, & QH 22.22 kW & W rev,in = = = 0.53 kW COPHP, rev 42.14 ·and W & & & I = W u,in − W rev,in = 22.22 − 0.53 = 21.69 kW8-25E A freezer is maintained at a specified temperature by removing heat from it at a specified rate. Thepower consumption of the freezer is given. The reversible power, irreversibility, and the second-lawefficiency are to be determined.Analysis (a) The reversible work is the minimum work required to accomplish this task, which is thework that a reversible refrigerator operating between the specified temperature limits would consume, 1 1 COPR, rev = = = 8.73 T H / T L − 1 535 / 480 − 1 & 75°F & QL 75 Btu/min ⎛ 1 hp ⎞ W rev,in = = ⎜ ⎟ = 0.20 hp COPR, rev 8.73 ⎝ 42.41 Btu/min ⎠(b) The irreversibility is the difference between the reversible R 0.70 hpwork and the actual electrical work consumed, 75 Btu/min & & & I = W u,in − W rev,in = 0.70 − 0.20 = 0.50 hp Freezer(c) The second law efficiency is determined from its definition, 20°F & W rev 0.20 hp η II = = = 28.9% Wu& 0.7 hpPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  9. 9. 8-98-26 It is to be shown that the power produced by a wind turbine is proportional to the cube of the windvelocity and the square of the blade span diameter.Analysis The power produced by a wind turbine is proportional to the kinetic energy of the wind, which isequal to the product of the kinetic energy of air per unit mass and the mass flow rate of air through theblade span area. Therefore, Wind power = (Efficiency)(Kinetic energy)(Mass flow rate of air) V2 V 2 π D2 = η wind ( ρAV ) = η wind ρ V 2 2 4 πV 3 D 2 = η wind ρ = (Constant )V 3 D 2 8which completes the proof that wind power is proportional to the cube of the wind velocity and to thesquare of the blade span diameter.8-27 A geothermal power produces 14 MW power while the exergy destruction in the plant is 18.5 MW.The exergy of the geothermal water entering to the plant, the second-law efficiency of the plant, and theexergy of the heat rejected from the plant are to be determined.Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. 3Water properties are used for geothermal water.Analysis (a) The properties of geothermal water at the inlet of the plant and at the dead state are (Tables A-4 through A-6) T1 = 160°C ⎫ h1 = 675.47 kJ/kg ⎬ x1 = 0 ⎭ s1 = 1.9426 kJ/kg.K T0 = 25°C⎫ h0 = 104.83 kJ/kg ⎬ x0 = 0 ⎭ s 0 = 0.36723 kJ/kg.KThe exergy of geothermal water entering the plant is X in = m[h1 − h0 − T0 ( s1 − s 0 ] & & = (440 kg/s)[(675.47 − 104.83) kJ/kg + 0 − (25 + 273 K )(1.9426 − 0.36723)kJ/kg.K ] = 44,525 kW = 44.53 MW(b) The second-law efficiency of the plant is the ratio of power produced to the exergy input to the plant & Wout 14,000 kW η II = = = 0.314 & X in 44,525 kW(c) The exergy of the heat rejected from the plant may be determined from an exergy balance on the plant & & & & X heat,out = X in − Wout − X dest = 44,525 − 14,000 − 18,500 = 12,025 kW = 12.03 MWPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  10. 10. 8-10Second-Law Analysis of Closed Systems8-28C Yes.8-29C Yes, it can. For example, the 1st law efficiency of a reversible heat engine operating between thetemperature limits of 300 K and 1000 K is 70%. However, the second law efficiency of this engine, likeall reversible devices, is 100%.PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  11. 11. 8-118-30E Air is expanded in an adiabatic closed system with an isentropic efficiency of 95%. The second lawefficiency of the process is to be determined.Assumptions 1 Kinetic and potential energy changes are negligible. 2 The process is adiabatic, and thusthere is no heat transfer. 3 Air is an ideal gas with constant specific heats.Properties The properties of air at room temperature are cp = 0.240 Btu/lbm·R, cv = 0.171 Btu/lbm·R, k =1.4, and R = 0.06855 Btu/lbm·R (Table A-2Ea).Analysis We take the air as the system. This is a closed system since no mass crosses the boundaries of thesystem. The energy balance for this system can be expressed as E in − E out = ΔE system 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, by heat, work, and mass potential, etc. energies − Wb,out = ΔU = mcv (T2 − T1 ) Air 150 psiaThe final temperature for the isentropic case is 100°F ( k −1) / k 0.4 / 1.4 ⎛P ⎞ ⎛ 15 psia ⎞ T2 s = T1 ⎜ 2 ⎟ ⎜P ⎟ = (560 R)⎜ ⎜ 150 psia ⎟ ⎟ = 290.1 R ⎝ 1⎠ ⎝ ⎠ T 150 psiaThe actual exit temperature from the isentropic relation is 1 T −T 15 psia η= 1 2 T1 − T2s 2 T2 = T1 − η (T1 − T2 s ) = 560 − (0.95)(560 − 290.1) = 303.6 R 2sThe boundary work output is s wb,out = cv (T1 − T2 ) = (0.171 Btu/lbm ⋅ R)(560 − 303.6)R = 43.84 Btu/lbmThe entropy change of air is T P Δs air = c p ln 2 − R ln 2 T1 P1 303.6 R 15 psia = (0.240 Btu/lbm ⋅ R)ln − (0.06855 Btu/lbm ⋅ R)ln 560 R 150 psia = 0.01091 Btu/lbm ⋅ RThe exergy difference between states 1 and 2 is φ1 − φ2 = u1 − u2 + P0 (v1 − v 2 ) − T0 (s1 − s2 ) ⎛T T ⎞ = cv (T1 − T2 ) + P0 R⎜ 1 − 2 ⎟ − T0 ( s1 − s2 ) ⎜ ⎟ ⎝ P P2 ⎠ 1 ⎛ 560 R 303.6 R ⎞ = 43.84 Btu/lbm + (14.7 psia)(0.06855 Btu/lbm ⋅ R)⎜ ⎜ 150 psia − 15 psia ⎟ − (537 R)(−0.01091 Btu/lbm ⋅ R) ⎟ ⎝ ⎠ = 33.07 Btu/lbmThe useful work is determined from ⎛T T ⎞ wu = wb,out − wsurr = cv (T1 − T2 ) − P0 (v 2 − v1 ) = cv (T1 − T2 ) − P0 R⎜ 2 − 1 ⎟ ⎜P P ⎟ ⎝ 2 1⎠ ⎛ 303.6 R 560 R ⎞ = 43.84 Btu/lbm − (14.7 psia)(0.06855 Btu/lbm ⋅ R)⎜ ⎜ 15 psia − 150 psia ⎟ ⎟ ⎝ ⎠ = 27.21 Btu/lbmThe second law efficiency is then w 27.21 Btu/lbm η II = u = = 0.823 Δφ 33.07 Btu/lbmPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  12. 12. 8-128-31E Air and helium at specified states are considered. The gas with the higher exergy content is to beidentified.Assumptions 1 Kinetic and potential energy changes are negligible. 2 Air and helium are ideal gases withconstant specific heats.Properties The properties of air at room temperature are cp = 0.240 Btu/lbm·R, cv = 0.171 Btu/lbm·R, k =1.4, and R = 0.06855 Btu/lbm·R = 0.3704 psia⋅ft3/lbm·R. For helium, cp = 1.25 Btu/lbm·R, cv = 0.753Btu/lbm·R, k = 1.667, and R = 0.4961 Btu/lbm·R= 2.6809 psia⋅ft3/lbm·R. (Table A-2E).Analysis The mass of air in the system is PV (100 psia)(10 ft 3 ) m= = = 3.552 lbm RT (0.3704 psia ⋅ ft 3 /lbm ⋅ R)(760 R) Air 10 ft3The entropy change of air between the given state and the dead state is 100 psia 300°F T P s − s 0 = c p ln − R ln T0 P0 760 R 100 psia = (0.240 Btu/lbm ⋅ R)ln − (0.06855 Btu/lbm ⋅ R)ln 537 R 14.7 psia = −0.04808 Btu/lbm ⋅ RThe air’s specific volumes at the given state and dead state are RT (0.3704 psia ⋅ ft 3 /lbm ⋅ R)(760 R) v= = = 2.815 ft 3 /lbm P 100 psia RT0 (0.3704 psia ⋅ ft 3 /lbm ⋅ R)(537 R) v0 = = = 13.53 ft 3 /lbm P0 14.7 psiaThe specific closed system exergy of the air is then φ = u − u 0 + P0 (v − v 0 ) − T0 ( s − s 0 ) = cv (T − T0 ) + P0 (v − v 0 ) − T0 ( s − s 0 ) ⎛ 1 Btu ⎞ = (0.171 Btu/lbm ⋅ R )(300 − 77)R + (14.7 psia)(2.815 − 13.53)ft 3 /lbm⎜ ⎟ ⎜ 5.404 psia ⋅ ft 3 ⎟ ⎝ ⎠ − (537 R)( −0.04808) Btu/lbm ⋅ R = 34.80 Btu/lbmThe total exergy available in the air for the production of work is then Φ = mφ = (3.552 lbm)(34.80 Btu/lbm) = 123.6 BtuWe now repeat the calculations for helium: PV (80 psia)(20 ft 3 ) m= = = 0.9043 lbm RT (2.6809 psia ⋅ ft 3 /lbm ⋅ R)(660 R) Helium 20 ft3 T P 80 psia s − s 0 = c p ln − R ln 200°F T0 P0 660 R 80 psia = (1.25 Btu/lbm ⋅ R)ln − (0.4961 Btu/lbm ⋅ R)ln 537 R 14.7 psia = −0.5827 Btu/lbm ⋅ RPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  13. 13. 8-13 RT (2.6809 psia ⋅ ft 3 /lbm ⋅ R)(660 R) v= = = 22.12 ft 3 /lbm P 80 psia RT0 (2.6809 psia ⋅ ft 3 /lbm ⋅ R)(537 R) v0 = = = 97.93 ft 3 /lbm P0 14.7 psia φ = u − u 0 + P0 (v − v 0 ) − T0 ( s − s 0 ) = cv (T − T0 ) + P0 (v − v 0 ) − T0 ( s − s 0 ) ⎛ 1 Btu ⎞ = (0.753 Btu/lbm ⋅ R )(200 − 77)R + (14.7 psia)(22.12 − 97.93)ft 3 /lbm⎜ ⎟ ⎜ 5.404 psia ⋅ ft 3 ⎟ ⎝ ⎠ − (537 R)(−0.5827) Btu/lbm ⋅ R = 199.3 Btu/lbm Φ = mφ = (0.9043 lbm)(199.3 Btu/lbm) = 180.2 BtuComparison of two results shows that the helium system has a greater potential for the production of work.PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  14. 14. 8-148-32 Steam and R-134a at the same states are considered. The fluid with the higher exergy content is to beidentified.Assumptions Kinetic and potential energy changes are negligible.Analysis The properties of water at the given state and at the dead state are u = 2594.7 kJ/kg P = 800 kPa ⎫ 3 Steam ⎬ v = 0.24720 m /kg (Table A - 6) 1 kg T = 180°C ⎭ s = 6.7155 kJ/kg ⋅ K 800 kPa u 0 ≅ u f @ 25°C = 104.83 kJ/kg 180°C T0 = 25°C ⎫ 3 ⎬ v 0 ≅ v f @ 25°C = 0.001003 m /kg (Table A - 4) P0 = 100 kPa ⎭ s 0 ≅ s f @ 25°C = 0.3672 kJ/kg ⋅ KThe exergy of steam is Φ = m[u − u 0 + P0 (v − v 0 ) − T0 ( s − s 0 )] ⎡ 3 ⎛ 1 kJ ⎞⎤ ⎢(2594.7 − 104.83)kJ/kg + (100 kPa)(0.24720 − 0.001003)m /kg⎜ ⎟⎥ = (1 kg) ⎢ ⎝ 1 kPa ⋅ m 3 ⎠⎥ ⎢− (298 K)(6.7155 − 0.3672)kJ/kg ⋅ K ⎣ ⎥ ⎦ = 622.7 kJFor R-134a; u = 386.99 kJ/kg P = 800 kPa ⎫ 3 ⎬ v = 0.044554 m /kg (Table A - 13) T = 180°C ⎭ R-134a s = 1.3327 kJ/kg ⋅ K 1 kg u 0 ≅ u f @ 25°C = 85.85 kJ/kg 800 kPa T0 = 25°C ⎫ ⎬ v 0 ≅ v f @ 25°C = 0.0008286 m 3 /kg (Table A - 11) 180°C P0 = 100 kPa ⎭ s 0 ≅ s f @ 25°C = 0.32432 kJ/kg ⋅ K Φ = m[u − u 0 + P0 (v − v 0 ) − T0 ( s − s 0 )] ⎡ 3 ⎛ 1 kJ ⎞⎤ ⎢(386.99 − 85.85)kJ/kg + (100 kPa)(0.044554 − 0.0008286)m /kg⎜ ⎟⎥ = (1 kg) ⎢ ⎝ 1 kPa ⋅ m 3 ⎠⎥ ⎢− (298 K)(1.3327 − 0.32432)kJ/kg ⋅ K ⎣ ⎥ ⎦ = 5.02 kJThe steam can therefore has more work potential than the R-134a.PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  15. 15. 8-158-33 A cylinder initially contains air at atmospheric conditions. Air is compressed to a specified state andthe useful work input is measured. The exergy of the air at the initial and final states, and the minimumwork input to accomplish this compression process, and the second-law efficiency are to be determinedAssumptions 1 Air is an ideal gas with constant specific heats. 2 The kinetic and potential energies arenegligible.Properties The gas constant of air is R = 0.287 kPa.m3/kg.K (Table A-1). The specific heats of air at theaverage temperature of (298+423)/2=360 K are cp = 1.009 kJ/kg·K and cv = 0.722 kJ/kg·K (Table A-2).Analysis (a) We realize that X1 = Φ1 = 0 since air initially is at the dead state. The mass of air is P1V1 (100 kPa)(0.002 m 3 ) m= = = 0.00234 kg RT1 (0.287 kPa ⋅ m 3 / kg ⋅ K)(298 K)Also, P2V 2 P1V1 PT (100 kPa)(423 K) = ⎯→ V 2 = 1 2 V 1 = ⎯ (2 L) = 0.473 L T2 T1 P2 T1 (600 kPa)(298 K) AIRand V1 = 2 L P1 = 100 kPa T P T1 = 25°C s 2 − s 0 = c p ,avg ln 2 − R ln 2 T0 P0 423 K 600 kPa = (1.009 kJ/kg ⋅ K) ln − (0.287 kJ/kg ⋅ K) ln 298 K 100 kPa = −0.1608 kJ/kg ⋅ KThus, the exergy of air at the final state is [ ] X 2 = Φ 2 = m cv ,avg (T2 − T0 ) − T0 ( s 2 − s 0 ) + P0 (V 2 −V 0 ) = (0.00234 kg)[(0.722 kJ/kg ⋅ K)(423 - 298)K - (298 K)(-0.1608 kJ/kg ⋅ K)] + (100 kPa)(0.000473 - 0.002)m 3 [kJ/m 3 ⋅ kPa] = 0.171 kJ(b) The minimum work input is the reversible work input, which can be determined from the exergybalance by setting the exergy destruction equal to zero, X in − X out − X destroyed 0 (reversible) = ΔX system 1 24 4 3 144 24444 3 123 4 4 Net exergy transfer Exergy Change by heat, work,and mass destruction in exergy Wrev,in = X 2 − X1 = 0.171− 0 = 0.171kJ(c) The second-law efficiency of this process is Wrev,in 0.171 kJ ηII = = = 14.3% Wu,in 1.2 kJPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  16. 16. 8-168-34 A cylinder is initially filled with R-134a at a specified state. The refrigerant is cooled and condensedat constant pressure. The exergy of the refrigerant at the initial and final states, and the exergy destroyedduring this process are to be determined.Assumptions The kinetic and potential energies are negligible.Properties From the refrigerant tables (Tables A-11 through A-13), v = 0.034875 m 3 / kg P1 = 0.7 MPa ⎫ 1 ⎬ u1 = 274.01 kJ/kg T1 = 60°C ⎭ s = 1.0256 kJ/kg ⋅ K 1 v ≅ v f @ 24°C = 0.0008261 m 3 / kg P2 = 0.7 MPa ⎫ 2 R-134a ⎬ u 2 ≅ u f @ 24°C = 84.44 kJ/kg 0.7 MPa T2 = 24°C ⎭s ≅s 2 f @ 24°C = 0.31958 kJ/kg ⋅ K P = const. Q v = 0.23718 m 3 / kg P0 = 0.1 MPa ⎫ 0 ⎬ u 0 = 251.84 kJ/kg T0 = 24°C ⎭ s 0 = 1.1033 kJ/kg ⋅ KAnalysis (a) From the closed system exergy relation, X 1 = Φ 1 = m{(u1 − u 0 ) − T0 ( s1 − s 0 ) + P0 (v 1 − v 0 )} = (5 kg){(274.01 − 251.84) kJ/kg − (297 K)(1.0256 − 1.1033) kJ/kg ⋅ K ⎛ 1 kJ ⎞ + (100 kPa)(0.034875 − 0.23718)m 3 /kg⎜ ⎟} ⎝ 1 kPa ⋅ m 3 ⎠ = 125.1 kJand, X 2 = Φ 2 = m{(u 2 − u 0 ) − T0 ( s 2 − s 0 ) + P0 (v 2 − v 0 )} = (5 kg){(84.44 − 251.84) kJ/kg - (297 K)(0.31958 − 1.1033) kJ/kg ⋅ K ⎛ 1 kJ ⎞ + (100 kPa)(0.0008261 − 0.23718)m 3 /kg⎜ ⎟} ⎝ 1 kPa ⋅ m 3 ⎠ = 208.6 kJ(b) The reversible work input, which represents the minimum work input Wrev,in in this case can bedetermined from the exergy balance by setting the exergy destruction equal to zero, X in − X out − X destroyed 0 (reversible) = ΔX system 1 24 4 3 144 24444 3 123 4 4 Net exergy transfer Exergy Change by heat, work,and mass destruction in exergy Wrev,in = X 2 − X1 = 208.6 − 125.1 = 83.5 kJNoting that the process involves only boundary work, the useful work input during this process is simplythe boundary work in excess of the work done by the surrounding air, W u,in = Win − Wsurr,in = Win − P0 (V1 −V 2 ) = P(V1 −V 2 ) − P0 m(v 1 − v 2 ) = m( P − P0 )(v 1 − v 2 ) ⎛ 1 kJ ⎞ = (5 kg)(700 - 100 kPa)(0.034875 − 0.0008261 m 3 / kg)⎜ ⎟ = 102.1 kJ ⎝ 1 kPa ⋅ m 3 ⎠Knowing both the actual useful and reversible work inputs, the exergy destruction or irreversibility that isthe difference between the two is determined from its definition to be X destroyed = I = W u,in − W rev,in = 102.1 − 83.5 = 18.6 kJPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  17. 17. 8-178-35E An insulated rigid tank contains saturated liquid-vapor mixture of water at a specified pressure. Anelectric heater inside is turned on and kept on until all the liquid is vaporized. The exergy destruction andthe second-law efficiency are to be determined.Assumptions Kinetic and potential energies are negligible.Properties From the steam tables (Tables A-4 through A-6) v = v f + x1v fg = 0.01708 + 0.25 × (11.901 − 0.01708) = 2.9880 ft 3 / lbm P1 = 35 psia ⎫ 1 ⎬ u1 = u f + x1u fg = 227.92 + 0.25 × 862.19 = 443.47 Btu / lbm x1 = 0.25 ⎭ s1 = s f + x1 s fg = 0.38093 + 0.25 × 1.30632 = 0.70751 Btu / lbm ⋅ R v 2 = v 1 ⎫ u 2 = u g @ v g = 2.9880 ft 3 /lbm = 1110.9 Btu/lbm ⎬ sat. vapor ⎭ s 2 = s g @ v g = 2.9880 ft 3 /lbm = 1.5692 Btu/lbm ⋅ RAnalysis (a) The irreversibility can be determined from its definition H2OXdestroyed = T0Sgen where the entropy generation is determined from an 35 psiaentropy balance on the tank, which is an insulated closed system, We S in − S out + S gen = ΔS system 1 24 4 3 { 123 4 4 Net entropy transfer Entropy Change by heat and mass generation in entropy S gen = ΔS system = m( s 2 − s1 )Substituting, X destroyed = T0 S gen = mT0 ( s 2 − s1 ) = (6 lbm)(535 R)(1.5692 - 0.70751)Btu/lbm ⋅ R = 2766 Btu(b) Noting that V = constant during this process, the W and Wu are identical and are determined from theenergy balance on the closed system energy equation, E in − E out = ΔE system 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, by heat, work, and mass potential, etc. energies We,in = ΔU = m(u 2 − u1 )or, We,in = (6 lbm)(1110.9 - 443.47)Btu/lbm = 4005 BtuThen the reversible work during this process and the second-law efficiency become W rev,in = W u,in − X destroyed = 4005 − 2766 = 1239 BtuThus, Wrev 1239 Btu ηII = = = 30.9% Wu 4005 BtuPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  18. 18. 8-188-36 A rigid tank is divided into two equal parts by a partition. One part is filled with compressed liquidwhile the other side is evacuated. The partition is removed and water expands into the entire tank. Theexergy destroyed during this process is to be determined.Assumptions Kinetic and potential energies are negligible.Analysis The properties of the water are (Tables A-4 through A-6) 1.5 kg v 1 ≅ v f @ 60°C = 0.001017 m 3 / kg 300 kPa P1 = 300 kPa ⎫ Vacuum ⎬ u1 ≅ u f @ 60°C = 251.16 kJ/kg 60°C T1 = 60°C ⎭ s1 ≅ s f @ 60°C = 0.8313 kJ/kg ⋅ K WATERNoting that v 2 = 2v 1 = 2 × 0.001017 = 0.002034 m 3 / kg , v 2 −v f 0.002034 − 0.001014 x2 = = = 0.0001017 v fg 10.02 − 0.001014 P2 = 15 kPa ⎫ ⎬ u 2 = u f + x 2 u fg = 225.93 + 0.0001017 × 2222.1 = 226.15 kJ/kg v 2 = 0.002034 ⎭ s 2 = s f + x 2 s fg = 0.7549 + 0.0001017 × 7.2522 = 0.7556 kJ/kg ⋅ KTaking the direction of heat transfer to be to the tank, the energy balance on this closed system becomes E in − E out = ΔE system 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, by heat, work, and mass potential, etc. energies Qin = ΔU = m(u 2 − u1 )or, Qin = (1.5 kg)(226.15 - 251.16)kJ/kg = -37.51 kJ → Qout = 37.51 kJThe irreversibility can be determined from its definition Xdestroyed = T0Sgen where the entropy generation isdetermined from an entropy balance on an extended system that includes the tank and its immediatesurroundings so that the boundary temperature of the extended system is the temperature of thesurroundings at all times, S in − S out + S gen = ΔS system 1 24 4 3 { 123 4 4 Net entropy transfer Entropy Change by heat and mass generation in entropy Qout − + S gen = ΔS system = m( s 2 − s1 ) Tb,out Qout S gen = m( s 2 − s1 ) + TsurrSubstituting, ⎛ Q ⎞ X destroyed = T0 S gen = T0 ⎜ m( s 2 − s1 ) + out ⎟ ⎜ ⎝ Tsurr ⎟ ⎠ ⎡ 37.51 kJ ⎤ = (298 K) ⎢(1.5 kg)(0.7556 - 0.8313)kJ/kg ⋅ K + ⎣ 298 K ⎥ ⎦ = 3.67 kJPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  19. 19. 8-198-37 EES Problem 8-36 is reconsidered. The effect of final pressure in the tank on the exergy destroyedduring the process is to be investigated.Analysis The problem is solved using EES, and the solution is given below.T_1=60 [C]P_1=300 [kPa]m=1.5 [kg]P_2=15 [kPa]T_o=25 [C]P_o=100 [kPa]T_surr = T_o"Conservation of energy for closed system is:"E_in - E_out = DELTAEDELTAE = m*(u_2 - u_1)E_in=0E_out= Q_outu_1 =intenergy(steam_iapws,P=P_1,T=T_1)v_1 =volume(steam_iapws,P=P_1,T=T_1)s_1 =entropy(steam_iapws,P=P_1,T=T_1)v_2 = 2*v_1u_2 = intenergy(steam_iapws, v=v_2,P=P_2)s_2 = entropy(steam_iapws, v=v_2,P=P_2)S_in -S_out+S_gen=DELTAS_sysS_in=0 [kJ/K]S_out=Q_out/(T_surr+273)DELTAS_sys=m*(s_2 - s_1)X_destroyed = (T_o+273)*S_gen P2 Xdestroyed Qout 4 [kPa] [kJ] [kJ] 3 15 3.666 37.44 16.11 2.813 28.07 2 Xdestroyed [kJ] 17.22 1.974 19.25 1 18.33 1.148 10.89 19.44 0.336 2.95 0 20.56 -0.4629 -4.612 -1 21.67 -1.249 -11.84 22.78 -2.022 -18.75 -2 23.89 -2.782 -25.39 -3 25 -3.531 -31.77 -4 15 17.2 19.4 21.6 23.8 26 P2 [kPa]PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  20. 20. 8-208-38 An insulated cylinder is initially filled with saturated liquid water at a specified pressure. The water isheated electrically at constant pressure. The minimum work by which this process can be accomplished andthe exergy destroyed are to be determined.Assumptions 1 The kinetic and potential energy changes are negligible. 2 The cylinder is well-insulatedand thus heat transfer is negligible. 3 The thermal energy stored in the cylinder itself is negligible. 4 Thecompression or expansion process is quasi-equilibrium.Analysis (a) From the steam tables (Tables A-4 through A-6), u1 = u f @ 150 kPa = 466.97 kJ / kg 3 P1 = 150 kPa ⎫ v 1 = v f @ 150 kPa = 0.001053 m /kg ⎬ sat. liquid ⎭ h1 = h f @ 150 kPa = 467.13 kJ/kg Saturated s1 = s f @ 150 kPa = 1.4337 kJ/kg ⋅ K Liquid H2O WeThe mass of the steam is P = 150 kPa V 0.002 m 3 m= = = 1.899 kg v 1 0.001053 m 3 / kgWe take the contents of the cylinder as the system. This is a closed system since no mass enters or leaves.The energy balance for this stationary closed system can be expressed as E in − E out = ΔE system 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, by heat, work, and mass potential, etc. energies We,in − W b,out = ΔU We,in = m(h2 − h1 )since ΔU + Wb = ΔH during a constant pressure quasi-equilibrium process. Solving for h2, We,in 2200 kJ h2 = h1 + = 467.13 + = 1625.1 kJ/kg m 1.899 kgThus, h2 − h f 1625.1 − 467.13 x2 = = = 0.5202 h fg 2226.0P2 = 150 kPa ⎫ ⎬ s 2 = s f + x 2 s fg = 1.4337 + 0.5202 × 5.7894 = 4.4454 kJ/kg ⋅ Kh2 = 1625.1 kJ/kg ⎭ u 2 = u f + x 2 u fg = 466.97 + 0.5202 × 2052.3 = 1534.6 kJ/kg v 2 = v f + x 2v fg = 0.001053 + 0.5202 × (1.1594 − 0.001053) = 0.6037 m 3 /kgThe reversible work input, which represents the minimum work input Wrev,in in this case can be determinedfrom the exergy balance by setting the exergy destruction equal to zero, X in − X out − X destroyed 0 (reversible) = ΔX system → Wrev,in = X 2 − X1 1 24 4 3 144 24444 3 123 4 4 Net exergy transfer Exergy Change by heat, work,and mass destruction in exergySubstituting the closed system exergy relation, the reversible work input during this process is determinedto bePROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  21. 21. 8-21 W rev,in = −m[(u1 − u 2 ) − T0 ( s1 − s 2 ) + P0 (v 1 − v 2 )] = −(1.899 kg){(466.97 − 1534.6) kJ/kg − (298 K)(1.4337 − 4.4454) kJ/kg ⋅ K + (100 kPa)(0.001053 − 0.6037)m 3 / kg[1 kJ/1 kPa ⋅ m 3 ]} = 437.7 kJ(b) The exergy destruction (or irreversibility) associated with this process can be determined from itsdefinition Xdestroyed = T0Sgen where the entropy generation is determined from an entropy balance on thecylinder, which is an insulated closed system, S in − S out + S gen = ΔS system 1 24 4 3 { 123 4 4 Net entropy transfer Entropy Change by heat and mass generation in entropy S gen = ΔS system = m( s 2 − s1 )Substituting, X destroyed = T0 S gen = mT0 ( s 2 − s1 ) = (298 K)(1.899 kg)(4.4454 − 1.4337)kJ/kg ⋅ K = 1705 kJPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  22. 22. 8-228-39 EES Problem 8-38 is reconsidered. The effect of the amount of electrical work on the minimum workand the exergy destroyed is to be investigated.Analysis The problem is solved using EES, and the solution is given below.x_1=0P_1=150 [kPa]V=2 [L]P_2=P_1{W_Ele = 2200 [kJ]}T_o=25 [C]P_o=100 [kPa]"Conservation of energy for closed system is:"E_in - E_out = DELTAEDELTAE = m*(u_2 - u_1)E_in=W_EleE_out= W_bW_b = m*P_1*(v_2-v_1)u_1 =intenergy(steam_iapws,P=P_1,x=x_1)v_1 =volume(steam_iapws,P=P_1,x=x_1)s_1 =entropy(steam_iapws,P=P_1,x=x_1)u_2 = intenergy(steam_iapws, v=v_2,P=P_2)s_2 = entropy(steam_iapws, v=v_2,P=P_2)m=V*convert(L,m^3)/v_1 450W_rev_in=m*(u_2 - u_1 -(T_o+273.15) 400 *(s_2-s_1)+P_o*(v_2-v_1)) 350 300 Wrev,in [kJ]"Entropy Balance:"S_in - S_out+S_gen = DELTAS_sys 250DELTAS_sys = m*(s_2 - s_1) 200S_in=0 [kJ/K] 150S_out= 0 [kJ/K] 100"The exergy destruction or irreversibility is:" 50X_destroyed = (T_o+273.15)*S_gen 0 0 450 900 1350 1800 2250 WEle [kJ] WEle Wrev,in Xdestroyed 1800 [kJ] [kJ] [kJ] 0 0 0 1600 244.4 48.54 189.5 1400 Xdestroyed [kJ/K] 488.9 97.07 379.1 1200 733.3 145.6 568.6 977.8 194.1 758.2 1000 1222 242.7 947.7 800 1467 291.2 1137 600 1711 339.8 1327 400 1956 388.3 1516 2200 436.8 1706 200 0 0 450 900 1350 1800 2250 WEle [kJ]PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  23. 23. 8-238-40 An insulated cylinder is initially filled with saturated R-134a vapor at a specified pressure. Therefrigerant expands in a reversible manner until the pressure drops to a specified value. The change in theexergy of the refrigerant during this process and the reversible work are to be determined.Assumptions 1 The kinetic and potential energy changes are negligible. 2 The cylinder is well-insulatedand thus heat transfer is negligible. 3 The thermal energy stored in the cylinder itself is negligible. 4 Theprocess is stated to be reversible.Analysis This is a reversible adiabatic (i.e., isentropic) process, and thus s2 = s1. From the refrigerant tables(Tables A-11 through A-13), v 1 = v g @ 0.8 MPa = 0.02562 m 3 / kg P1 = 0.8 MPa ⎫ ⎬ u1 = u g @ 0.8 MPa = 246.79 kJ/kg sat. vapor ⎭s =s 1 g @ 0.8 MPa = 0.9183 kJ/kg ⋅ KThe mass of the refrigerant is R-134a 3 0.8 MPa V 0.05 m m= = = 1.952 kg Reversible v 1 0.02562 m 3 / kg s2 − s f 0.9183 − 0.15457 x2 = = = 0.9753 s fg 0.78316P2 = 0.2 MPa ⎫ 3 ⎬ v 2 = v f + x 2v fg = 0.0007533 + 0.099867 × (0.099867 − 0.0007533) = 0.09741 m /kgs 2 = s1 ⎭ u = u + x u = 38.28 + 0.9753 × 186.21 = 219.88 kJ/kg 2 f 2 fgThe reversible work output, which represents the maximum work output Wrev,out can be determined fromthe exergy balance by setting the exergy destruction equal to zero, X in − X out − X destroyed 0 (reversible) = ΔX system 1 24 4 3 144 24444 3 123 4 4 Net exergy transfer Exergy Change by heat, work,and mass destruction in exergy - Wrev,out = X 2 − X1 Wrev,out = X1 − X 2 = Φ1 − Φ2Therefore, the change in exergy and the reversible work are identical in this case. Using the definition ofthe closed system exergy and substituting, the reversible work is determined to be [ W rev,out = Φ 1 − Φ 2 = m (u1 − u 2 ) − T0 ( s1 − s 2 ) 0 ] + P0 ( v 1 − v 2 ) = m[(u1 − u 2 ) + P0 (v 1 − v 2 )] = (1.952 kg)[(246.79 − 219.88) kJ/kg + (100 kPa)(0.02562 − 0.09741)m 3 / kg[kJ/kPa ⋅ m 3 ] = 38.5 kJPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  24. 24. 8-248-41E Oxygen gas is compressed from a specified initial state to a final specified state. The reversiblework and the increase in the exergy of the oxygen during this process are to be determined.Assumptions At specified conditions, oxygen can be treated as an ideal gas with constant specific heats.Properties The gas constant of oxygen is R = 0.06206 Btu/lbm.R (Table A-1E). The constant-volumespecific heat of oxygen at the average temperature is Tavg = (T1 + T2 ) / 2 = (75 + 525) / 2 = 300°F ⎯ ⎯→ cv ,avg = 0.164 Btu/lbm ⋅ RAnalysis The entropy change of oxygen is ⎛T ⎞ ⎛v ⎞ s 2 − s1 = cv, avg ln⎜ 2 ⎟ + R ln⎜ 2 ⎟ ⎜T ⎟ ⎜v ⎟ O2 ⎝ 1⎠ ⎝ 1⎠ 12 ft3/lbm ⎛ 985 R ⎞ ⎛ 1.5 ft 3 /lbm ⎞ = (0.164 Btu/lbm ⋅ R) ln⎜ ⎟ + (0.06206 Btu/lbm ⋅ R) ln⎜ ⎜ 12 ft 3 /lbm ⎟ ⎟ 75°F ⎝ 535 R ⎠ ⎝ ⎠ = −0.02894 Btu/lbm ⋅ RThe reversible work input, which represents the minimum work input Wrev,in in this case can be determinedfrom the exergy balance by setting the exergy destruction equal to zero, X in − X out − X destroyed 0 (reversible) = ΔX system → Wrev,in = X 2 − X1 1 24 4 3 144 24444 3 123 4 4 Net exergy transfer Exergy Change by heat, work,and mass destruction in exergyTherefore, the change in exergy and the reversible work are identical in this case. Substituting the closedsystem exergy relation, the reversible work input during this process is determined to be w rev,in = φ 2 − φ1 = −[(u1 − u 2 ) − T0 ( s1 − s 2 ) + P0 (v 1 − v 2 )] = −{(0.164 Btu/lbm ⋅ R)(535 - 985)R − (535 R)(0.02894 Btu/lbm ⋅ R) + (14.7 psia)(12 − 1.5)ft 3 /lbm[Btu/5.4039 psia ⋅ ft 3 ]} = 60.7 Btu/lbmAlso, the increase in the exergy of oxygen is φ 2 − φ1 = wrev,in = 60.7 Btu/lbmPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  25. 25. 8-258-42 An insulated tank contains CO2 gas at a specified pressure and volume. A paddle-wheel in the tankstirs the gas, and the pressure and temperature of CO2 rises. The actual paddle-wheel work and theminimum paddle-wheel work by which this process can be accomplished are to be determined.Assumptions 1 At specified conditions, CO2 can be treated asan ideal gas with constant specific heats at the averagetemperature. 2 The surroundings temperature is 298 K.Analysis (a) The initial and final temperature of CO2 are 1.2 m3 2.13 kg P1V1 (100 kPa)(1.2 m 3 ) CO2 T1 = = = 298.2 K Wpw mR (2.13 kg)(0.1889 kPa ⋅ m 3 / kg ⋅ K ) 100 kPa P2V 2 (120 kPa)(1.2 m 3 ) T2 = = = 357.9 K mR (2.13 kg)(0.1889 kPa ⋅ m 3 / kg ⋅ K ) Tavg = (T1 + T2 ) / 2 = (298.2 + 357.9) / 2 = 328 K ⎯ ⎯→ cv ,avg = 0.684 kJ/kg ⋅ KThe actual paddle-wheel work done is determined from the energy balance on the CO gas in the tank, We take the contents of the cylinder as the system. This is a closed system since no mass enters orleaves. The energy balance for this stationary closed system can be expressed as E in − E out = ΔE system 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, by heat, work, and mass potential, etc. energies W pw,in = ΔU = mcv (T2 − T1 )or, W pw,in = (2.13 kg)(0.684 kJ/kg ⋅ K)(357.9 − 298.2)K = 87.0 kJ(b) The minimum paddle-wheel work with which this process can be accomplished is the reversible work,which can be determined from the exergy balance by setting the exergy destruction equal to zero, X in − X out − X destroyed 0 (reversible) = ΔX system → Wrev,in = X 2 − X1 1 24 4 3 144 24444 3 123 4 4 Net exergy transfer Exergy Change by heat, work,and mass destruction in exergySubstituting the closed system exergy relation, the reversible work input for this process is determined tobe [ W rev,in = m (u 2 − u1 ) − T0 ( s 2 − s1 ) + P0 (v 2 − v 1 ) 0 ] [ = m cv ,avg (T2 − T1 ) − T0 ( s 2 − s1 ) ] = (2.13 kg)[(0.684 kJ/kg ⋅ K)(357.9 − 298.2)K − (298.2)(0.1253 kJ/kg ⋅ K)] = 7.74 kJsince T2 v 0 ⎛ 357.9 K ⎞ s 2 − s1 = cv ,avg ln + R ln 2 = (0.684 kJ/kg ⋅ K) ln⎜ ⎟ = 0.1253 kJ/kg ⋅ K T1 v1 ⎝ 298.2 K ⎠PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  26. 26. 8-268-43 An insulated cylinder initially contains air at a specified state. A resistance heater inside the cylinderis turned on, and air is heated for 15 min at constant pressure. The exergy destruction during this process isto be determined.Assumptions Air is an ideal gas with variable specific heats.Properties The gas constant of air is R = 0.287 kJ/kg.K (Table A-1).Analysis The mass of the air and the electrical workdone during this process are AIR PV (120 kPa)(0.03 m 3 ) m= 1 1 = = 0.0418 kg 120 kPa We RT1 (0.287 kPa ⋅ m 3 /kg ⋅ K)(300 K) P = const & We = We Δt = (−0.05 kJ / s)(5 × 60 s) = −15 kJAlso, o T1 = 300 K ⎯→ ⎯ h1 = 30019 kJ / kg . and s1 = 1.70202 kJ / kg ⋅ KThe energy balance for this stationary closed system can be expressed as E in − E out = ΔE system 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, by heat, work, and mass potential, etc. energies We,in − W b,out = ΔU We,in = m(h2 − h1 )since ΔU + Wb = ΔH during a constant pressure quasi-equilibrium process. Thus, We,in 15 kJ T2 = 650 K h2 = h1 + = 300.19 + = 659.04 kJ/kg ⎯ ⎯→ o m 0.0418 kg s 2 = 2.49364 kJ/kg ⋅ KAlso, 0 ⎛P ⎞ s 2 − s1 = o s2 − o s1 ⎜ − R ln⎜ 2 ⎟ ⎟ o o = s 2 − s1 = 2.49364 − 1.70202 = 0.79162 kJ/kg ⋅ K ⎝ P1 ⎠The exergy destruction (or irreversibility) associated with this process can be determined from itsdefinition Xdestroyed = T0Sgen where the entropy generation is determined from an entropy balance on thecylinder, which is an insulated closed system, S in − S out + S gen = ΔS system 1 24 4 3 { 123 4 4 Net entropy transfer Entropy Change by heat and mass generation in entropy S gen = ΔS system = m( s 2 − s1 )Substituting, X destroyed = T0 S gen = mT0 ( s 2 − s1 ) == (300 K)(0.0418 kg)(0.79162 kJ/kg ⋅ K) = 9.9 kJPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  27. 27. 8-278-44 A fixed mass of helium undergoes a process from a specified state to another specified state. Theincrease in the useful energy potential of helium is to be determined.Assumptions 1 At specified conditions, helium can be treated as an idealgas. 2 Helium has constant specific heats at room temperature.Properties The gas constant of helium is R = 2.0769 kJ/kg.K (Table A-1). Theconstant volume specific heat of helium is cv = 3.1156 kJ/kg.K (Table A-2). HeAnalysis From the ideal-gas entropy change relation, 8 kg 288 K T2 v s 2 − s1 = cv ,avg ln + R ln 2 T1 v1 353 K 0.5 m 3 /kg = (3.1156 kJ/kg ⋅ K) ln + (2.0769 kJ/kg ⋅ K) ln = −3.087 kJ/kg ⋅ K 288 K 3 m 3 /kgThe increase in the useful potential of helium during this process is simply the increase in exergy, Φ 2 − Φ 1 = −m[(u1 − u 2 ) − T0 ( s1 − s 2 ) + P0 (v 1 − v 2 )] = −(8 kg){(3.1156 kJ/kg ⋅ K)(288 − 353) K − (298 K)(3.087 kJ/kg ⋅ K) + (100 kPa)(3 − 0.5)m 3 / kg[kJ/kPa ⋅ m 3 ]} = 6980 kJPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  28. 28. 8-288-45 One side of a partitioned insulated rigid tank contains argon gas at a specified temperature andpressure while the other side is evacuated. The partition is removed, and the gas fills the entire tank. Theexergy destroyed during this process is to be determined.Assumptions Argon is an ideal gas with constant specific heats, and thus ideal gas relations apply.Properties The gas constant of argon is R = 0.208 kJ/kg.K (Table A-1).Analysis Taking the entire rigid tank as the system, the energy balance can be expressed as Ein − Eout = ΔEsystem 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, by heat, work, and mass potential, etc. energies Argon 0 = ΔU = m(u2 − u1 ) 300 kPa Vacuum u2 = u1 → T2 = T1 70°Csince u = u(T) for an ideal gas. The exergy destruction (or irreversibility) associated with this process can be determined from itsdefinition Xdestroyed = T0Sgen where the entropy generation is determined from an entropy balance on theentire tank, which is an insulated closed system, S in − S out + S gen = ΔS system 1 24 4 3 { 123 4 4 Net entropy transfer Entropy Change by heat and mass generation in entropy S gen = ΔS system = m( s 2 − s1 )where ⎛ T 0 V ⎞ V ΔS system = m( s 2 − s1 ) = m⎜ cv ,avg ln 2 + R ln 2 ⎟ = mR ln 2 ⎜ T1 V1 ⎟ V1 ⎝ ⎠ = (3 kg)(0.208 kJ/kg ⋅ K) ln (2) = 0.433 kJ/KSubstituting, X destroyed = T0 S gen = mT0 ( s 2 − s1 ) = (298 K)(0.433 kJ/K) = 129 kJPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  29. 29. 8-298-46E A hot copper block is dropped into water in an insulated tank. The final equilibrium temperature ofthe tank and the work potential wasted during this process are to be determined.Assumptions 1 Both the water and the copper block are incompressible substances with constant specificheats at room temperature. 2 The system is stationary and thus the kinetic and potential energies arenegligible. 3 The tank is well-insulated and thus there is no heat transfer.Properties The density and specific heat of water at the anticipated average temperature of 90°F are ρ =62.1 lbm/ft3 and cp = 1.00 Btu/lbm.°F. The specific heat of copper at the anticipated average temperature of100°F is cp = 0.0925 Btu/lbm.°F (Table A-3E).Analysis We take the entire contents of the tank, water + copper block, as the system, which is a closedsystem. The energy balance for this system can be expressed as E in − E out = ΔE system 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, by heat, work, and mass potential, etc. energies Water 0 = ΔU 75°For, Copper ΔU Cu + ΔU water = 0 250°F [mc(T2 − T1 )] Cu + [mc(T2 − T1 )] water = 0where m w = ρV = (62.1 lbm/ft 3 )(1.5 ft 3 ) = 93.15 lbmSubstituting, 0 = (70 lbm)(0.0925 Btu/lbm ⋅ °F)(T2 − 250°F) + (93.15 lbm)(1.0 Btu/lbm ⋅ °F)(T2 − 75°F) T2 = 86.4°F = 546.4 RThe wasted work potential is equivalent to the exergy destruction (or irreversibility), and it can bedetermined from its definition Xdestroyed = T0Sgen where the entropy generation is determined from an entropybalance on the system, which is an insulated closed system, S in − S out + S gen = ΔS system 1 24 4 3 { 123 4 4 Net entropy transfer Entropy Change by heat and mass generation in entropy S gen = ΔS system = ΔS water + ΔS copperwhere ⎛T ⎞ ⎛ 546.4 R ⎞ ΔS copper = mc avg ln⎜ 2 ⎟ = (70 lbm)(0.092 Btu/lbm ⋅ R) ln⎜ ⎜T ⎟ ⎟ = −1.696 Btu/R ⎝ 1⎠ ⎝ 710 R ⎠ ⎛T ⎞ ⎛ 546.4 R ⎞ ΔS water = mc avg ln⎜ 2 ⎟ = (93.15 lbm)(1.0 Btu/lbm ⋅ R) ln⎜ ⎜T ⎟ ⎟ = 1.960 Btu/R ⎝ 1⎠ ⎝ 535 R ⎠Substituting, X destroyed = (535 R)(−1.696 + 1.960)Btu/R = 140.9 BtuPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  30. 30. 8-308-47 A hot iron block is dropped into water in an insulated tank that is stirred by a paddle-wheel. The massof the iron block and the exergy destroyed during this process are to be determined. √Assumptions 1 Both the water and the iron block are incompressible substances with constant specificheats at room temperature. 2 The system is stationary and thus the kinetic and potential energies arenegligible. 3 The tank is well-insulated and thus there is no heat transfer.Properties The density and specific heat of water at 25°C are ρ = 997 kg/m3 and cp = 4.18 kJ/kg.°F. Thespecific heat of iron at room temperature (the only value available in the tables) is cp = 0.45 kJ/kg.°C(Table A-3).Analysis We take the entire contents of the tank, water + iron block, as the system, which is a closedsystem. The energy balance for this system can be expressed as E in − E out = ΔE system 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, by heat, work, and mass potential, etc. energies 100 L W pw,in = ΔU = ΔU iron + ΔU water 20°C W pw,in = [mc(T2 − T1 )] iron + [mc(T2 − T1 )] water Iron 85°C Wpwwhere Water m water = ρV = (997 kg/m 3 )(0.1 m 3 ) = 99.7 kg & W pw = W pw,in Δt = (0.2 kJ/s)(20 × 60 s) = 240 kJSubstituting, 240 kJ = miron (0.45 kJ/kg ⋅ °C)(24 − 85)°C + (99.7 kg)(4.18 kJ/kg ⋅ °C)(24 − 20)°C miron = 52.0 kg(b) The exergy destruction (or irreversibility) can be determined from its definition Xdestroyed = T0Sgen wherethe entropy generation is determined from an entropy balance on the system, which is an insulated closedsystem, S in − S out + S gen = ΔS system 1 24 4 3 { 123 4 4 Net entropy transfer Entropy Change by heat and mass generation in entropy S gen = ΔS system = ΔS iron + ΔS waterwhere ⎛T ⎞ ⎛ 297 K ⎞ ΔS iron = mc avg ln⎜ 2 ⎟ = (52.0 kg)(0.45 kJ/kg ⋅ K) ln⎜ ⎜T ⎟ ⎟ = −4.371 kJ/K ⎝ 1⎠ ⎝ 358 K ⎠ ⎛T ⎞ ⎛ 297 K ⎞ ΔS water = mc avg ln⎜ 2 ⎟ = (99.7 kg)(4.18 kJ/kg ⋅ K) ln⎜ ⎜T ⎟ ⎟ = 5.651 kJ/K ⎝ 1⎠ ⎝ 293 K ⎠Substituting, X destroyed = T0 S gen = (293 K)(−4.371 + 5.651) kJ/K = 375.0 kJPROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  31. 31. 8-318-48 An iron block and a copper block are dropped into a large lake where they cool to lake temperature.The amount of work that could have been produced is to be determined.Assumptions 1 The iron and copper blocks and water are incompressible substances with constant specificheats at room temperature. 2 Kinetic and potential energies are negligible.Properties The specific heats of iron and copper at room temperature are cp, iron = 0.45 kJ/kg.°C and cp,copper= 0.386 kJ/kg.°C (Table A-3).Analysis The thermal-energy capacity of the lake is very large, and thus the temperatures of both the ironand the copper blocks will drop to the lake temperature (15°C) when the thermal equilibrium is established. We take both the iron and the copper blocks as the system, which is a closed system. The energybalance for this system can be expressed as E in − E out = ΔE system 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, Lake by heat, work, and mass potential, etc. energies 15°C − Qout = ΔU = ΔU iron + ΔU copper Iron Copper 85°C Ironor, Qout = [mc(T1 − T2 )] iron + [mc(T1 − T2 )] copperSubstituting, Qout = (50 kg )(0.45 kJ/kg ⋅ K )(353 − 288)K + (20 kg )(0.386 kJ/kg ⋅ K )(353 − 288)K = 1964 kJThe work that could have been produced is equal to the wasted work potential. It is equivalent to theexergy destruction (or irreversibility), and it can be determined from its definition Xdestroyed = T0Sgen . Theentropy generation is determined from an entropy balance on an extended system that includes the blocksand the water in their immediate surroundings so that the boundary temperature of the extended system isthe temperature of the lake water at all times, S in − S out + S gen = ΔS system 1 24 4 3 { 123 4 4 Net entropy transfer Entropy Change by heat and mass generation in entropy Qout − + S gen = ΔS system = ΔS iron + ΔS copper Tb,out Qout S gen = ΔS iron + ΔS copper + Tlakewhere ⎛T ⎞ ⎛ 288 K ⎞ ΔS iron = mc avg ln⎜ 2 ⎟ = (50 kg )(0.45 kJ/kg ⋅ K ) ln⎜ ⎜T ⎟ ⎜ 353 K ⎟ = −4.579 kJ/K ⎟ ⎝ 1⎠ ⎝ ⎠ ⎛T ⎞ ⎛ 288 K ⎞ ΔS copper = mc avg ln⎜ 2 ⎜T ⎟ = (20 kg )(0.386 kJ/kg ⋅ K ) ln⎜ ⎟ ⎜ 353 K ⎟ = −1.571 kJ/K ⎟ ⎝ 1 ⎠ ⎝ ⎠Substituting, ⎛ 1964 kJ ⎞ X destroyed = T0 S gen = (293 K)⎜ − 4.579 − 1.571 + ⎜ ⎟kJ/K = 196 kJ ⎝ 288 K ⎟⎠PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.
  32. 32. 8-328-49E A rigid tank is initially filled with saturated mixture of R-134a. Heat is transferred to the tank from asource until the pressure inside rises to a specified value. The amount of heat transfer to the tank from thesource and the exergy destroyed are to be determined.Assumptions 1 The tank is stationary and thus the kinetic and potential energy changes are zero. 2 There isno heat transfer with the environment.Properties From the refrigerant tables (Tables A-11E through A-13E), u1 = u f + x1u fg = 21.246 + 0.55 × 77.307 = 63.76 Btu / lbm P1 = 40 psia ⎫ ⎬ s1 = s f + x1 s fg = 0.04688 + 0.55 × 0.17580 = 0.1436 Btu / lbm ⋅ R x1 = 0.55 ⎭ v 1 = v f + x1 v fg = 0.01232 + 0.55 × 1.16368 = 0.65234 ft 3 / lbm v 2 −v f 0.65234 − 0.01270 x2 = = = 0.8191 0.79361 − 0.01270 v fg P2 = 60 psia ⎫ ⎬ s 2 = s f + x 2 s fg = 0.06029 + 0.8191× 0.16098 = 0.1922 Btu/lbm ⋅ R (v 2 = v 1 ) ⎭ u 2 = u f + x 2 u fg = 27.939 + 0.8191× 73.360 = 88.03 Btu/lbmAnalysis (a) The mass of the refrigerant is V 12 ft 3 Source m= = = 18.40 lbm v 1 0.65234 ft 3 / lbm 120°C R-134aWe take the tank as the system, which is a closed system. The 40 psiaenergy balance for this stationary closed system can be expressed as Q E in − E out = ΔE system 1 24 4 3 1 24 4 3 Net energy transfer Change in internal, kinetic, by heat, work, and mass potential, etc. energies Qin = ΔU = m(u 2 − u1 )Substituting, Qin = m(u 2 − u1 ) = (18.40 lbm)(88.03 - 63.76) Btu/lbm = 446.3 Btu(b) The exergy destruction (or irreversibility) can be determined from its definition Xdestroyed = T0Sgen . Theentropy generation is determined from an entropy balance on an extended system that includes the tank andthe region in its immediate surroundings so that the boundary temperature of the extended system whereheat transfer occurs is the source temperature, S in − S out + S gen = ΔS system 1 24 4 3 { 123 4 4 Net entropy transfer Entropy Change by heat and mass generation in entropy Qin + S gen = ΔS system = m( s 2 − s1 ) , Tb,in Qin S gen = m( s 2 − s1 ) − TsourceSubstituting, ⎡ 446.3 Btu ⎤ X destroyed = T0 S gen = (535 R) ⎢(18.40 lbm)(0.1922 − 0.1436)Btu/lbm ⋅ R − = 66.5 Btu ⎣ 580 R ⎥ ⎦PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andeducators for course preparation. If you are a student using this Manual, you are using it without permission.

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