Chapter 8 EXERGY – A MEASURE OF WORK POTENTIAL PROPRIETARY AND CONFIDENTIAL

8-1

Solutions Manual for

Thermodynamics: An Engineering Approach Seventh Edition Yunus A. Cengel, Michael A. Boles McGraw-Hill, 2011

Chapter 8 EXERGY – A MEASURE OF WORK POTENTIAL

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8-2

Exergy, Irreversibility, Reversible Work, and Second-Law Efficiency 8-1C Reversible work and irreversibility are identical for processes that involve no actual useful work.

8-2C The dead state.

8-3C Yes; exergy is a function of the state of the surroundings as well as the state of the system.

8-4C Useful work differs from the actual work by the surroundings work. They are identical for systems that involve no surroundings work such as steady-flow systems.

8-5C Yes.

8-6C No, not necessarily. The well with the higher temperature will have a higher exergy.

8-7C The system that is at the temperature of the surroundings has zero exergy. But the system that is at a lower temperature than the surroundings has some exergy since we can run a heat engine between these two temperature levels.

8-8C They would be identical.

8-9C The second-law efficiency is a measure of the performance of a device relative to its performance under reversible conditions. It differs from the first law efficiency in that it is not a conversion efficiency.

8-10C No. The power plant that has a lower thermal efficiency may have a higher second-law efficiency.

8-11C No. The refrigerator that has a lower COP may have a higher second-law efficiency.


8-3

8-12C A processes with Wrev = 0 is reversible if it involves no actual useful work. Otherwise it is irreversible.

8-13C Yes.

8-14 Windmills are to be installed at a location with steady winds to generate power. The minimum number of windmills that need to be installed is to be determined. Assumptions Air is at standard conditions of 1 atm and 25°C Properties 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,

Exergy = ke =

V 2 (6 m/s) 2 ⎛ 1 kJ/kg ⎞ = ⎜ ⎟ = 0.0180 kJ/kg 2 2 ⎝ 1000 m 2 / s 2 ⎠

At standard atmospheric conditions (25°C, 101 kPa), the density and the mass flow rate of air are

ρ=

P 101 kPa = = 1.18 m 3 / kg RT (0.287 kPa ⋅ m 3 / kg ⋅ K)(298 K)

and

m& = ρAV1 = ρ

π D2 4

V1 = (1.18 kg/m 3 )(π / 4)(20 m) 2 (6 m/s) = 2225 kg/s

Thus,

Available Power = m& ke = (2225 kg/s)(0.0180 kJ/kg) = 40.05 kW The minimum number of windmills that needs to be installed is

N=

W& total 900 kW = = 22.5 ≅ 23 windmills & 40.05 kW W


8-4

8-15E Saturated steam is generated in a boiler by transferring heat from the combustion gases. The wasted work potential associated with this heat transfer process is to be determined. Also, the effect of increasing the 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-4E through A-6E) ⎫ h1 = h f = 355.46 Btu/lbm ⎬ x1 = 0 (sat. liq.) ⎭ s1 = s f = 0.54379 Btu/lbm ⋅ R P2 = 200 psia ⎫ h2 = h g = 1198.8 Btu/lbm ⎬ x 2 = 1 (sat. vap.) ⎭ s 2 = s g = 1.5460 Btu/lbm ⋅ R

P1 = 200 psia

⎫ h0 ≅ h f @ 80°F = 48.07 Btu/lbm ⎬ P0 = 14.7 psia ⎭ s 0 ≅ s f @ 80°F = 0.09328 Btu/lbm ⋅ R

q

Water 200 psia sat. liq.

200 psia sat. vap.

T0 = 80°F

The heat transfer during the process is qin = h2 − h1 = 1198.8 − 355.46 = 843.3 Btu/lbm

The entropy generation associated with this process is sgen = ∆sw + ∆sR = ( s2 − s1 ) −

qin TR

= (1.5460 − 0.54379)Btu/lbm ⋅ R −

843.3 Btu/lbm (500 + 460)R

= 0.12377 Btu/lbm ⋅ R

The wasted work potential (exergy destruction is) xdest = T0 sgen = (80 + 460 R)(0.12377 Btu/lbm ⋅ R) = 66.8 Btu/lbm

The work potential (exergy) of the steam stream is ∆ψ w = h2 − h1 − T0 ( s2 − s1 ) = (1198.8 − 355.46)Btu/lbm − (540 R )(1.5460 − 0.54379)Btu/lbm ⋅ R = 302.1 Btu/lbm

Increasing the temperature of combustion gases does not effect the work potential of steam stream since it is 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 − ⎟ = 368.9 Btu/lbm ⎝ 960 ⎠ ⎝ TR ⎠

Exergy increase of steam: ∆ψ w = 302.1 Btu/lbm

The net exergy destruction: xdest = xheat − ∆ψ w = 368.9 − 302.1 = 66.8 Btu/lbm


8-5

8-16 Water is to be pumped to a high elevation lake at times of low electric demand for use in a hydroelectric turbine at times of high demand. For a specified energy storage capacity, the minimum amount 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 potential energy it possesses,

75 m

Exergy = PE = mgh

Thus, m=

PE 5 × 10 6 kWh ⎛ 3600 s ⎞⎛⎜ 1000 m 2 / s 2 = ⎜ ⎟ gh (9.8 m/s 2 )(75 m) ⎝ 1 h ⎠⎜⎝ 1 kW ⋅ s/kg

⎞ ⎟ = 2.45 × 10 10 kg ⎟ ⎠

8-17 A body contains a specified amount of thermal energy at a specified temperature. The amount that can be converted to work is to be determined. Analysis The amount of heat that can be converted to work is simply the amount that a reversible heat engine can convert to work,

η th, rev = 1 −

T0 298 K = 1− = 0.6275 TH 800 K

800 K 100 kJ HE

W max,out = W rev,out = η th, rev Qin = (0.6275)(100 kJ)

298 K

= 62.75 kJ

8-18 The thermal efficiency of a heat engine operating between specified temperature limits is given. The second-law efficiency of a engine is to be determined. Analysis The thermal efficiency of a reversible heat engine operating between the same temperature reservoirs is

η th, rev = 1 −

T0 293 K = 1− = 0.801 1200 + 273 K TH

1200°C HE

η th = 0.40

Thus,

η II =

η th 0.40 = = 49.9% η th, rev 0.801

20°C


8-6

8-19 A heat reservoir at a specified temperature can supply heat at a specified rate. The exergy of this heat supplied is to be determined. Analysis The exergy of the supplied heat, in the rate form, is the amount of power that would be produced by a reversible heat engine, T0 298 K = 1− = 0.8013 1500 K TH = W& rev,out = η th, rev Q& in

η th, max = η th, rev = 1 − Exergy = W& max,out

= (0.8013)(150,000 / 3600 kJ/s)

1500 K & W rev

HE 298 K

= 33.4 kW

A heat engine receives heat from a source at a specified temperature at a specified rate, and rejects the waste 8-20 heat to a sink. For a given power output, the reversible power, 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 the specified temperature limits,

η th,max = η th,rev = 1 −

TL 320 K =1− = 0.7091 TH 1100 K

1100 K

W& rev,out = η th,rev Q& in = (0.7091)(400 kJ/s) = 283.6 kW (b) The irreversibility rate is the difference between the reversible power and the actual power output:

400 kJ/s HE

120 kW

I& = W& rev,out − W& u,out = 283.6 − 120 = 163.6 kW

(c) The second law efficiency is determined from its definition,

η II =

Wu,out Wrev,out

=

320 K

120 kW = 0.423 = 42.3% 283.6 kW


8-7

8-21 Problem 8-20 is reconsidered. The effect of reducing the temperature at which the waste heat is rejected on the reversible power, the rate of irreversibility, and the second law efficiency is to be studied and the results are to be plotted. Analysis The problem is solved using EES, and the solution is given below. "Input Data" T_H= 1100 [K] Q_dot_H= 400 [kJ/s] {T_L=320 [K]} W_dot_out = 120 [kW] T_Lsurr =25 [C] "The reversible work is the maximum work done by the Carnot Engine between T_H and T_L:" Eta_Carnot=1 - T_L/T_H W_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(, %)

TL [K] 500 477.6 455.1 432.7 410.2 387.8 365.3 342.9 320.4 298

Wrev [kJ/s] 218.2 226.3 234.5 242.7 250.8 259 267.2 275.3 283.5 291.6

I [kJ/s] 98.18 106.3 114.5 122.7 130.8 139 147.2 155.3 163.5 171.6

ηII [%] 55 53.02 51.17 49.45 47.84 46.33 44.92 43.59 42.33 41.15


8-8

300 290

Wrev [kJ/s]

280 270 260 250 240 230 220 210 275

320

365

410

455

500

410

455

500

455

500

TL [K] 180

I [kJ/s]

160

140

120

100 275

320

365

TL [K] 56 54

η II [%]

52 50 48 46 44 42 40 275

320

365

410

TL [K]


8-9

8-22E The thermal efficiency and the second-law efficiency of a heat engine are given. The source temperature is to be determined. TH Analysis From the definition of the second law efficiency,

η II =

η th η 0.36 ⎯ ⎯→ η th, rev = th = = 0.60 η th, rev η II 0.60

HE

Thus,

η th, rev = 1 −

TL ⎯ ⎯→ T H = T L /(1 − η th, rev ) = (530 R)/0.40 = 1325 R TH

η th = 36% η II = 60%

530 R

8-23 A house is maintained at a specified temperature by electric resistance heaters. The reversible work for this heating process and irreversibility are to be determined. Analysis The reversible work is the minimum work required to accomplish this process, and the irreversibility is the difference between the reversible work and the actual electrical work consumed. The actual power input is

W& in = Q& out = Q& H = 50,000 kJ/h = 13.89 kW 50,000 kJ/h

The COP of a reversible heat pump operating between the specified temperature limits is

COPHP,rev =

1 1 = = 14.20 1 − TL / TH 1 − 277.15 / 298.15

House 25 °C

4 °C

Thus, W& rev,in =

Q& H 13.89 kW = = 0.978 kW COPHP, rev 14.20

and

· W

I& = W& u,in − W& rev,in = 13.89 − 0.978 = 12.91 kW


8-10

8-24E A freezer is maintained at a specified temperature by removing heat from it at a specified rate. The power consumption of the freezer is given. The reversible power, irreversibility, and the second-law efficiency are to be determined. Analysis (a) The reversible work is the minimum work required to accomplish this task, which is the work that a reversible refrigerator operating between the specified temperature limits would consume, COPR, rev = W& rev,in =

1 1 = = 8.73 T H / T L − 1 535 / 480 − 1

75°F

Q& L 1 hp 75 Btu/min ⎛ ⎞ = ⎟ = 0.20 hp ⎜ COPR, rev 8.73 42.41 Btu/min ⎠ ⎝

(b) The irreversibility is the difference between the reversible work and the actual electrical work consumed, I& = W& u,in − W& rev,in = 0.70 − 0.20 = 0.50 hp

0.70 hp

R

75 Btu/min Freezer 20°F

(c) The second law efficiency is determined from its definition,

η II =

W& rev 0.20 hp = = 28.9% 0.7 hp W& u

8-25 A geothermal power produces 5.1 MW power while the exergy destruction in the plant is 7.5 MW. The exergy of the geothermal water entering to the plant, the second-law efficiency of the plant, and the exergy 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. 3 Water 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 = 150°C ⎫ h1 = 632.18 kJ/kg ⎬ x1 = 0 ⎭ s1 = 1.8418 kJ/kg.K T0 = 25°C⎫ h0 = 104.83 kJ/kg ⎬ x0 = 0 ⎭ s 0 = 0.36723 kJ/kg.K The exergy of geothermal water entering the plant is X& in = m& [h1 − h0 − T0 ( s1 − s 0 ]

= (210 kg/s)[(632.18 − 104.83) kJ/kg − (25 + 273 K )(1.8418 − 0.36723)kJ/kg.K ] = 18,460 kW = 18.46 MW

(b) The second-law efficiency of the plant is the ratio of power produced to the exergy input to the plant

η II =

W& out 5100 kW = = 0.276 = 27.6% & X in 18,460 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 − W& out − X& dest = 18,460 − 5100 − 7500 = 5864 kW = 5.86 MW


8-11

8-26 It is to be shown that the power produced by a wind turbine is proportional to the cube of the wind velocity 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 is equal to the product of the kinetic energy of air per unit mass and the mass flow rate of air through the blade span area. Therefore, Wind power = (Efficiency)(Kinetic energy)(Mass flow rate of air) V2 V 2 π D2 V ( ρAV ) = η wind ρ 2 2 4 πV 3 D 2 = η wind ρ = (Constant )V 3 D 2 8

= η wind

which completes the proof that wind power is proportional to the cube of the wind velocity and to the square of the blade span diameter.


8-12

Exergy Analysis of Closed Systems

8-27C Yes, it can. For example, the 1st law efficiency of a reversible heat engine operating between the temperature limits of 300 K and 1000 K is 70%. However, the second law efficiency of this engine, like all reversible devices, is 100%.

8-28 A fixed mass of helium undergoes a process from a specified state to another specified state. The increase in the useful energy potential of helium is to be determined. Assumptions 1 At specified conditions, helium can be treated as an ideal gas. 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). The constant volume specific heat of helium is cv = 3.1156 kJ/kg.K (Table A-2). Analysis From the ideal-gas entropy change relation, s 2 − s1 = cv ,avg ln

v T2 + R ln 2 v1 T1

= (3.1156 kJ/kg ⋅ K) ln

He 8 kg 288 K

0.5 m 3 /kg 353 K + (2.0769 kJ/kg ⋅ K) ln = −3.087 kJ/kg ⋅ K 288 K 3 m 3 /kg

The 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 kJ


8-13

8-29E Air is expanded in an adiabatic closed system with an isentropic efficiency of 95%. The second law efficiency of the process is to be determined. Assumptions 1 Kinetic and potential energy changes are negligible. 2 The process is adiabatic, and thus there 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 the system. The energy balance for this system can be expressed as

E −E 1in424out 3

=

Net energy transfer by heat, work, and mass

∆E system 1 424 3

Change in internal, kinetic, potential, etc. energies

Air 150 psia 100°F

− Wb,out = ∆U = mcv (T2 − T1 ) The final temperature for the isentropic case is

T2 s

⎛P = T1 ⎜⎜ 2 ⎝ P1

⎞ ⎟⎟ ⎠

( k −1) / k

⎛ 15 psia ⎞ ⎟⎟ = (560 R)⎜⎜ ⎝ 150 psia ⎠

0.4 / 1.4

= 290.1 R

The actual exit temperature from the isentropic relation is

T

T −T η= 1 2 T1 − T2s T2 = T1 − η (T1 − T2 s ) = 560 − (0.95)(560 − 290.1) = 303.6 R

The boundary work output is

150 psia

1

15 psia 2

2s s

wb ,out = cv (T1 − T2 ) = (0.171 Btu/lbm ⋅ R)(560 − 303.6 ) R = 43.84 Btu/lbm

The entropy change of air is ∆s air = c p ln

T2 P − R ln 2 T1 P1

= (0.240 Btu/lbm ⋅ R)ln

15 psia 303.6 R − (0.06855 Btu/lbm ⋅ R)ln 560 R 150 psia

= 0.01091 Btu/lbm ⋅ R

The 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 ) ⎝ P1 P2 ⎠ ⎛ 560 R 303.6 R ⎞ ⎟⎟ − (537 R)(−0.01091 Btu/lbm ⋅ R) = 43.84 Btu/lbm + (14.7 psia)(0.06855 Btu/lbm ⋅ R)⎜⎜ − ⎝ 150 psia 15 psia ⎠ = 33.07 Btu/lbm

The 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 ⎝ 2 P1 ⎠ ⎛ 303.6 R 560 R ⎞ ⎟⎟ = 43.84 Btu/lbm − (14.7 psia)(0.06855 Btu/lbm ⋅ R)⎜⎜ − ⎝ 15 psia 150 psia ⎠ = 27.21 Btu/lbm

The second law efficiency is then

η II =

wu 27.21 Btu/lbm = = 0.823 ∆φ 33.07 Btu/lbm


8-14

8-30E Air and helium at specified states are considered. The gas with the higher exergy content is to be identified. Assumptions 1 Kinetic and potential energy changes are negligible. 2 Air and helium are ideal gases 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 = 0.3704 psia⋅ft3/lbm·R. For helium, cp = 1.25 Btu/lbm·R, cv = 0.753 Btu/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 (100 psia)(15 ft 3 ) PV = = 5.704 lbm RT (0.3704 psia ⋅ ft 3 /lbm ⋅ R)(710 R)

m=

The entropy change of air between the given state and the dead state is s − s 0 = c p ln

Air 15 ft3 100 psia 250°F

T P − R ln T0 P0

= (0.240 Btu/lbm ⋅ R)ln

100 psia 710 R − (0.06855 Btu/lbm ⋅ R)ln 537 R 14.7 psia

= −0.06441 Btu/lbm ⋅ R The air’s specific volumes at the given state and dead state are

v=

RT (0.3704 psia ⋅ ft 3 /lbm ⋅ R)(710 R) = = 2.630 ft 3 /lbm P 100 psia

v0 =

RT0 (0.3704 psia ⋅ ft 3 /lbm ⋅ R)(537 R) = = 13.53 ft 3 /lbm P0 14.7 psia

The 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)(2.630 − 13.53)ft 3 /lbm⎜ ⎜ 5.404 psia ⋅ ft 3 ⎝ − (537 R)( −0.06441) Btu/lbm ⋅ R

⎞ ⎟ ⎟ ⎠

= 34.52 Btu/lbm

The total exergy available in the air for the production of work is then Φ = mφ = (5.704 lbm)(34.52 Btu/lbm) = 197 Btu

We now repeat the calculations for helium: m=

(60 psia)(20 ft 3 ) PV = = 0.6782 lbm RT (2.6809 psia ⋅ ft 3 /lbm ⋅ R)(660 R)

s − s 0 = c p ln

T P − R ln T0 P0

= (1.25 Btu/lbm ⋅ R)ln

Helium 20 ft3 60 psia 200°F

60 psia 660 R − (0.4961 Btu/lbm ⋅ R)ln 537 R 14.7 psia

= −0.4400 Btu/lbm ⋅ R

v=

RT (2.6809 psia ⋅ ft 3 /lbm ⋅ R)(660 R) = = 29.49 ft 3 /lbm P 60 psia


8-15

v0 =

3

RT0 (2.6809 psia ⋅ ft /lbm ⋅ R)(537 R) = = 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)(29.49 − 97.93)ft 3 /lbm⎜ ⎜ 5.404 psia ⋅ ft 3 ⎝ − (537 R)( −0.4400) Btu/lbm ⋅ R

⎞ ⎟ ⎟ ⎠

= 142.7 Btu/lbm Φ = mφ = (0.6782 lbm)(142.7 Btu/lbm) = 96.8 Btu

Comparison of two results shows that the air system has a greater potential for the production of work.


8-16

8-31 Steam and R-134a at the same states are considered. The fluid with the higher exergy content is to be identified. 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 ⎬ v = 0.24720 m /kg T = 180°C ⎭ s = 6.7155 kJ/kg ⋅ K

Steam 1 kg 800 kPa 180°C

(Table A - 6)

u 0 ≅ u f @ 25°C = 104.83 kJ/kg ⎫ 3 ⎬ v 0 ≅ v f @ 25°C = 0.001003 m /kg P0 = 100 kPa ⎭ s 0 ≅ s f @ 25°C = 0.3672 kJ/kg ⋅ K

T0 = 25°C

(Table A - 4)

The exergy of steam is Φ = m[u − u 0 + P0 (v − v 0 ) − T0 ( s − s 0 )] ⎡ ⎛ 1 kJ 3 ⎢(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 kJ

For R-134a; u = 386.99 kJ/kg P = 800 kPa ⎫ 3 ⎬ v = 0.044554 m /kg T = 180°C ⎭ s = 1.3327 kJ/kg ⋅ K

(Table A - 13)

u 0 ≅ u f @ 25°C = 85.85 kJ/kg ⎫ 3 ⎬ v 0 ≅ v f @ 25°C = 0.0008286 m /kg P0 = 100 kPa ⎭ s 0 ≅ s f @ 25°C = 0.32432 kJ/kg ⋅ K

T0 = 25°C

(Table A - 11)

R-134a 1 kg 800 kPa 180°C

Φ = m[u − u 0 + P0 (v − v 0 ) − T0 ( s − s 0 )] ⎡ ⎛ 1 kJ 3 ⎢(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 kJ

The steam can therefore has more work potential than the R-134a.


8-17

8-32 A cylinder is initially filled with R-134a at a specified state. The refrigerant is cooled and condensed at constant pressure. The exergy of the refrigerant at the initial and final states, and the exergy destroyed during 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 1 = 0.034875 m 3 / kg P1 = 0.7 MPa ⎫ ⎬ u1 = 274.01 kJ/kg T1 = 60°C ⎭ s = 1.0256 kJ/kg ⋅ K 1 v 2 ≅ v f @ 24°C = 0.0008261 m 3 / kg P2 = 0.7 MPa ⎫ ⎬ u 2 ≅ u f @ 24°C = 84.44 kJ/kg T2 = 24°C ⎭s ≅s 2 f @ 24° C = 0.31958 kJ/kg ⋅ K

R-134a 0.7 MPa 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 ⋅ K Analysis (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 kJ and

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 be determined from the exergy balance by setting the exergy destruction equal to zero, X − X out 1in 4243

Net exergy transfer by heat, work, and mass

− X destroyed 0 (reversible) = ∆X system 144424443 1 424 3 Exergy destruction

Change in exergy

Wrev,in = X 2 − X1 = 208.6 − 125.1 = 83.5 kJ

Noting that the process involves only boundary work, the useful work input during this process is simply the boundary work in excess of the work done by the surrounding air, W u,in = Win − Wsurr,in = Win − P0 (V 1 −V 2 ) = P(V 1 −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 is the 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 kJ


8-18

8-33E An insulated rigid tank contains saturated liquid-vapor mixture of water at a specified pressure. An electric heater inside is turned on and kept on until all the liquid is vaporized. The exergy destruction and the 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 1 = v f + x1v fg = 0.01708 + 0.25 × (11.901 − 0.01708) = 2.9880 ft 3 / lbm P1 = 35 psia ⎫ ⎬ 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 ⋅ R

Analysis (a) The irreversibility can be determined from its definition Xdestroyed = T0Sgen where the entropy generation is determined from an entropy balance on the tank, which is an insulated closed system,

S − S out 1in424 3

Net entropy transfer by heat and mass

H 2O 35 psia

We

+ S gen = ∆S system { 1 424 3 Entropy generation

Change in entropy

S gen = ∆S 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 the energy balance on the closed system energy equation, E −E 1in424out 3


=

∆E system 1 424 3


We,in = ∆U = m(u 2 − u1 ) or, W e,in = (6 lbm)(1110.9 - 443.47)Btu /lbm = 4005 Btu

Then the reversible work during this process and the second-law efficiency become

W rev,in = W u,in − X destroyed = 4005 − 2766 = 1239 Btu Thus,

ηII =

Wrev 1239 Btu = = 30.9% 4005 Btu Wu


8-19

8-34 A rigid tank is divided into two equal parts by a partition. One part is filled with compressed liquid while the other side is evacuated. The partition is removed and water expands into the entire tank. The exergy 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)

v 1 ≅ v f @ 80°C = 0.001029 m 3 / kg

P1 = 200 kPa ⎫ ⎬ u1 ≅ u f @ 80°C = 334.97 kJ/kg T1 = 80°C ⎭ s1 ≅ s f @ 80°C = 1.0756 kJ/kg ⋅ K

4 kg 200 kPa 80°C WATER

Vacuum

Noting that v 2 = 2v 1 = 2 × 0.001029 = 0.002058 m 3 / kg ,

v2 −v f

0.002058 − 0.001026 = 0.0002584 v fg 3.9933 − 0.001026 P2 = 40 kPa ⎫⎪ ⎬ u 2 = u f + x 2 u fg = 317.58 + 0.0002584 × 2158.8 = 318.14 kJ/kg v 2 = 0.002058 m 3 / kg ⎪⎭ s 2 = s f + x 2 s fg = 1.0261 + 0.0002584 × 6.6430 = 1.0278 kJ/kg ⋅ K x2 =

=

Taking the direction of heat transfer to be to the tank, the energy balance on this closed system becomes E −E 1in424out 3

=


∆E system 1 424 3


Qin = ∆U = m(u 2 − u1 )

or Qin = (4 kg)(318.14 − 334.97)kJ/kg = −67.30 kJ → Qout = 67.30 kJ

The irreversibility can be determined from its definition Xdestroyed = T0Sgen where the entropy generation is determined from an entropy balance on an extended system that includes the tank and its immediate surroundings so that the boundary temperature of the extended system is the temperature of the surroundings at all times, S in − S out 1424 3


−


Change in entropy

Qout + S gen = ∆S system = m( s 2 − s1 ) Tb,out S gen = m( s 2 − s1 ) +

Qout Tsurr

Substituting, ⎛ Q X destroyed = T0 S gen = T0 ⎜⎜ m( s 2 − s1 ) + out Tsurr ⎝

⎞ ⎟ ⎟ ⎠

67.30 kJ ⎤ ⎡ = (298 K) ⎢(4 kg)(1.0278 − 1.0756)kJ/kg ⋅ K + 298 K ⎥⎦ ⎣ = 10.3 kJ


8-20

8-35 Problem 8-34 is reconsidered. The effect of final pressure in the tank on the exergy destroyed during the process is to be investigated. Analysis The problem is solved using EES, and the solution is given below. T_1=80 [C] P_1=200 [kPa] m=4 [kg] P_2=40 [kPa] T_o=25 [C] P_o=100 [kPa] T_surr = T_o

800 700 600

Qout [kJ]

"Conservation of energy for closed system is:" E_in - E_out = DELTAE DELTAE = m*(u_2 - u_1) E_in=0 E_out= Q_out u_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_1 u_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_sys S_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

500 400 300 200 100 0 5

10

15

20

25

30

35

40

45

P2 [kPa]

Xdestroyed [kJ] 74.41 64.4 53.8 43.85 34.61 25.99 17.91 10.3 3.091

Qout [kJ] 788.4 571.9 435.1 332.9 250.5 181 120.7 67.15 18.95

80 70 60

Xdestroyed [kJ]

P2 [kPa] 5 10 15 20 25 30 35 40 45

50 40 30 20 10 0 5

10

15

20

25

30

35

40

45

P2 [kPa]


8-21

8-36 An insulated cylinder is initially filled with saturated liquid water at a specified pressure. The water is heated electrically at constant pressure. The minimum work by which this process can be accomplished and the exergy destroyed are to be determined. Assumptions 1 The kinetic and potential energy changes are negligible. 2 The cylinder is well-insulated and thus heat transfer is negligible. 3 The thermal energy stored in the cylinder itself is negligible. 4 The compression or expansion process is quasi-equilibrium. Analysis (a) From the steam tables (Tables A-4 through A-6), u1 = u f @ 120 kPa = 439.27 kJ / kg 3 P1 = 120 kPa ⎫ v 1 = v f @ 120 kPa = 0.001047 m /kg ⎬ sat. liquid ⎭ h1 = h f @ 120 kPa = 439.36 kJ/kg s1 = s f @ 120 kPa = 1.3609 kJ/kg ⋅ K

Saturated Liquid H2O P = 120 kPa

The mass of the steam is We 3 0.008 m V m= = = 7.639 kg v 1 0.001047 m 3 / kg We 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 −E = ∆E system 1in424out 3 1 424 3 Net energy transfer by heat, work, and mass


We,in − W b,out = ∆U We,in = m(h2 − h1 )

since ∆U + Wb = ∆H during a constant pressure quasi-equilibrium process. Solving for h2, We,in 1400 kJ = 439.36 + = 622.63 kJ/kg h2 = h1 + m 7.639 kg Thus, h2 − h f 622.63 − 439.36 x2 = = = 0.08168 h fg 2243.7 P2 = 120 kPa ⎫ ⎬ s 2 = s f + x 2 s fg = 1.3609 + 0.08168 × 5.93687 = 1.8459 kJ/kg ⋅ K h2 = 622.63 kJ/kg ⎭ u = u + x u = 439.24 + 0.08168 × 2072.4 = 608.52 kJ/kg 2

f

2

fg

v 2 = v f + x 2v fg = 0.001047 + 0.08168 × (1.4285 − 0.001047) = 0.1176 m 3 /kg The reversible work input, which represents the minimum work input Wrev,in in this case can be determined from the exergy balance by setting the exergy destruction equal to zero, X − X out 1in 4243


− X destroyed 0 (reversible) = ∆X system → Wrev,in = X 2 − X1 144424443 1 424 3 Exergy destruction

Change in exergy

Substituting the closed system exergy relation, the reversible work input during this process is determined to be Wrev,in = − m[(u1 − u 2 ) − T0 ( s1 − s 2 ) + P0 (v 1 − v 2 )] = −(7.639 kg){(439.27 − 608.52) kJ/kg − (298 K)(1.3609 − 1.8459) kJ/kg ⋅ K + (100 kPa)(0.001047 − 0.1176)m 3 / kg[1 kJ/1 kPa ⋅ m 3 ]} = 278 kJ (b) The exergy destruction (or irreversibility) associated with this process can be determined from its definition Xdestroyed = T0Sgen where the entropy generation is determined from an entropy balance on the cylinder, which is an insulated closed system, S − S out + S gen = ∆S system 1in424 3 { 1 424 3 Net entropy transfer by heat and mass

Entropy generation

Change in entropy

S gen = ∆S system = m( s 2 − s1 ) Substituting, X destroyed = T0 S gen = mT0 ( s 2 − s1 ) = ( 298 K)(7.639 kg)(1.8459 − 1.3609)kJ/ kg ⋅ K = 1104 kJ


8-22

8-37 Problem 8-36 is reconsidered. The effect of the amount of electrical work on the minimum work and the exergy destroyed is to be investigated. Analysis The problem is solved using EES, and the solution is given below. x_1=0 P_1=120 [kPa] V=8 [L] P_2=P_1 {W_Ele = 1400 [kJ]} T_o=25 [C] P_o=100 [kPa] "Conservation of energy for closed system is:" E_in - E_out = DELTAE DELTAE = m*(u_2 - u_1) E_in=W_Ele E_out= W_b W_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 W_rev_in=m*(u_2 - u_1 -(T_o+273.15) *(s_2-s_1)+P_o*(v_2-v_1))

1750

Xdestroyed [kJ]

1350

"Entropy Balance:" S_in - S_out+S_gen = DELTAS_sys DELTAS_sys = m*(s_2 - s_1) S_in=0 [kJ/K] S_out= 0 [kJ/K]

950

550

150

-250 0

400

"The exergy destruction or irreversibility is:" X_destroyed = (T_o+273.15)*S_gen Wrev,in [kJ] 0 39.68 79.35 119 158.7 198.4 238.1 277.7 317.4 357.1 396.8

Xdestroyed [kJ] 0 157.8 315.6 473.3 631.1 788.9 946.7 1104 1262 1420 1578

1200

W Ele [kJ]

1600

2000

400 350 300

Wrev,in [kJ]

WEle [kJ] 0 200 400 600 800 1000 1200 1400 1600 1800 2000

800

250 200 150 100 50 0 0

400

800

1200

1600

2000

W Ele [kJ]


8-23

8-38 An insulated cylinder is initially filled with saturated R-134a vapor at a specified pressure. The refrigerant expands in a reversible manner until the pressure drops to a specified value. The change in the exergy 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-insulated and thus heat transfer is negligible. 3 The thermal energy stored in the cylinder itself is negligible. 4 The process 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 ⋅ K The mass of the refrigerant is m=

R-134a 0.8 MPa Reversible

0.05 m 3 V = = 1.952 kg v 1 0.02562 m 3 / kg

x2 =

s2 − s f s fg

=

0.9183 − 0.15457 = 0.9753 0.78316

P2 = 0.2 MPa ⎫ 3 ⎬ v 2 = v f + x 2v fg = 0.0007533 + 0.099867 × (0.099867 − 0.0007533) = 0.09741 m /kg s 2 = s1 ⎭ u = u + x u = 38.28 + 0.9753 × 186.21 = 219.88 kJ/kg 2 f 2 fg

The reversible work output, which represents the maximum work output Wrev,out can be determined from the exergy balance by setting the exergy destruction equal to zero, X − X out 1in 4243



Change in exergy

- Wrev,out = X 2 − X1 Wrev,out = X1 − X 2 = Φ1 − Φ2

Therefore, the change in exergy and the reversible work are identical in this case. Using the definition of the 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 kJ


8-24

8-39E Oxygen gas is compressed from a specified initial state to a final specified state. The reversible work 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-volume specific heat of oxygen at the average temperature is

Tavg = (T1 + T2 ) / 2 = (75 + 525) / 2 = 300°F ⎯ ⎯→ cv ,avg = 0.164 Btu/lbm ⋅ R Analysis The entropy change of oxygen is ⎛T s 2 − s1 = cv, avg ln⎜⎜ 2 ⎝ T1

⎞ ⎛v ⎞ ⎟⎟ + R ln⎜⎜ 2 ⎟⎟ ⎠ ⎝ v1 ⎠ ⎛ 1.5 ft 3 /lbm ⎞ ⎛ 985 R ⎞ ⎟ = (0.164 Btu/lbm ⋅ R) ln⎜ ⎟ + (0.06206 Btu/lbm ⋅ R) ln⎜⎜ 3 ⎟ ⎝ 535 R ⎠ ⎝ 12 ft /lbm ⎠ = −0.02894 Btu/lbm ⋅ R

O2 12 ft3/lbm 75°F

The reversible work input, which represents the minimum work input Wrev,in in this case can be determined from the exergy balance by setting the exergy destruction equal to zero, X − X out 1in 4243



Change in exergy

Therefore, the change in exergy and the reversible work are identical in this case. Substituting the closed system 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/lbm

Also, the increase in the exergy of oxygen is

φ 2 − φ1 = w rev,in = 60.7 Btu/lbm


8-25

8-40 A cylinder initially contains air at atmospheric conditions. Air is compressed to a specified state and the useful work input is measured. The exergy of the air at the initial and final states, and the minimum work input to accomplish this compression process, and the second-law efficiency are to be determined Assumptions 1 Air is an ideal gas with constant specific heats. 2 The kinetic and potential energies are negligible. Properties The gas constant of air is R = 0.287 kPa.m3/kg.K (Table A-1). The specific heats of air at the average 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 ) = = 0.00234 kg RT1 (0.287 kPa ⋅ m 3 / kg ⋅ K)(298 K)

m=

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)

AIR

V1 = 2 L

and s 2 − s 0 = c p ,avg

P1 = 100 kPa T1 = 25°C

T P ln 2 − R ln 2 T0 P0

= (1.009 kJ/kg ⋅ K) ln

600 kPa 423 K − (0.287 kJ/kg ⋅ K) ln 100 kPa 298 K

= −0.1608 kJ/kg ⋅ K

Thus, 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 exergy balance by setting the exergy destruction equal to zero,

X − X out 1in 4243

Net exergy transfer by heat, work,and mass

− X destroyed 0 (reversible) = ∆X system 144424443 1 424 3 Change in exergy

Exergy destruction

Wrev,in = X 2 − X1 = 0.171− 0 = 0.171kJ (c) The second-law efficiency of this process is

η II =

Wrev,in Wu,in

=

0.171 kJ = 14.3% 1.2 kJ


8-26

8-41 An insulated tank contains CO2 gas at a specified pressure and volume. A paddle-wheel in the tank stirs the gas, and the pressure and temperature of CO2 rises. The actual paddle-wheel work and the minimum paddle-wheel work by which this process can be accomplished are to be determined. Assumptions 1 At specified conditions, CO2 can be treated as an ideal gas with constant specific heats at the average temperature. 2 The surroundings temperature is 298 K. Properties The gas constant of CO2 is 0.1889 kJ/kg·K (Table A-1) Analysis (a) The initial and final temperature of CO2 are

T1 =

P1V1 (100 kPa)(1.2 m 3 ) = = 298.2 K mR (2.13 kg)(0.1889 kPa ⋅ m 3 / kg ⋅ K )

T2 =

P2V 2 (120 kPa)(1.2 m 3 ) = = 357.9 K mR (2.13 kg)(0.1889 kPa ⋅ m 3 / kg ⋅ K )

1.2 m3 2.13 kg CO2 100 kPa

Wpw

Tavg = (T1 + T2 ) / 2 = (298.2 + 357.9) / 2 = 328 K ⎯ ⎯→ cv ,avg = 0.684 kJ/kg ⋅ K (Table A-2b) The 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 or leaves. The energy balance for this stationary closed system can be expressed as E −E 1in424out 3

=


∆E system 1 424 3


W pw,in = ∆U = mcv (T2 − T1 ) or

Wpw,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 − X out 1in 4243



Change in exergy

Substituting the closed system exergy relation, the reversible work input for this process is determined to be

[

0

W rev,in = m (u 2 − u1 ) − T0 ( s 2 − s1 ) + P0 (v 2 − v 1 )

[

= 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 kJ since s 2 − s1 = cv ,avg ln

T2 v + R ln 2 T1 v1

0

⎛ 357.9 K ⎞ = (0.684 kJ/kg ⋅ K) ln⎜ ⎟ = 0.1253 kJ/kg ⋅ K ⎝ 298.2 K ⎠


8-27

8-42 An insulated cylinder initially contains air at a specified state. A resistance heater inside the cylinder is turned on, and air is heated for 10 min at constant pressure. The exergy destruction during this process is to 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 work done during this process are m=

P1V1 (140 kPa)(0.020 m 3 ) = = 0.03250 kg RT1 (0.287kPa ⋅ m 3 /kg ⋅ K)(300 K)

AIR 140 kPa P = const

We = W& e ∆t = (0.100 kJ/s)(10 × 60 s) = 60 kJ Also, T1 = 300 K ⎯ ⎯→ h1 = 300.19 kJ/kg

and

We

s1o = 1.70202 kJ/kg ⋅ K

The energy balance for this stationary closed system can be expressed as E −E 1in424out 3

=


∆E system 1 424 3


We,in − W b,out = ∆U We,in = m(h2 − h1 )

since ∆U + Wb = ∆H during a constant pressure quasi-equilibrium process. Thus, h2 = h1 +

We,in m

= 300.19 +

T = 1915 K 60 kJ −17 = 2146.3 kJ/kg ⎯Table ⎯ ⎯A⎯ ⎯→ o2 s 2 = 3.7452 kJ/kg ⋅ K 0.03250 kg

Also, ⎛P s 2 − s1 = s 2o − s1o − R ln⎜⎜ 2 ⎝ P1

⎞ ⎟⎟ ⎠

0

= s 2o − s1o = 3.7452 − 1.70202 = 2.0432 kJ/kg ⋅ K

The exergy destruction (or irreversibility) associated with this process can be determined from its definition Xdestroyed = T0Sgen where the entropy generation is determined from an entropy balance on the cylinder, which is an insulated closed system, S − S out 1in424 3



Change in entropy

S gen = ∆S system = m( s 2 − s1 ) Substituting, X destroyed = T0 S gen = mT0 ( s 2 − s1 ) = (0.03250 kg)(300 K)(2.0432 kJ/kg ⋅ K) = 19.9 kJ


8-28

8-43 One side of a partitioned insulated rigid tank contains argon gas at a specified temperature and pressure while the other side is evacuated. The partition is removed, and the gas fills the entire tank. The exergy 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 = 0.2081 kJ/kg.K (Table A-1). Analysis Taking the entire rigid tank as the system, the energy balance can be expressed as E − Eout 1in 424 3

=


∆Esystem 1 424 3


0 = ∆U = m(u2 − u1 ) u2 = u1

→ T2 = T1

Argon 300 kPa 70°C

Vacuum

since u = u(T) for an ideal gas. The exergy destruction (or irreversibility) associated with this process can be determined from its definition Xdestroyed = T0Sgen where the entropy generation is determined from an entropy balance on the entire tank, which is an insulated closed system, S − S out 1in424 3



Change in entropy

S gen = ∆S system = m( s 2 − s1 ) where ⎛ V ⎞ V T 0 ∆S system = m( s2 − s1 ) = m⎜ cv , avg ln 2 + R ln 2 ⎟ = mR ln 2 ⎜ ⎟ V V1 T 1 1 ⎠ ⎝ = (3 kg)(0.2081 kJ/kg ⋅ K) ln (2) = 0.433 kJ/K

Substituting,

X destroyed = T0 S gen = mT0 ( s 2 − s1 ) = (298 K)(0.433 kJ/K) = 129 kJ


8-29

8-44E A hot copper block is dropped into water in an insulated tank. The final equilibrium temperature of the 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 specific heats at room temperature. 2 The system is stationary and thus the kinetic and potential energies are negligible. 3 The tank is wellinsulated 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 of 100°F is cp = 0.0925 Btu/lbm.°F (Table A-3E). Analysis (a) We take the entire contents of the tank, water + copper block, as the system, which is a closed system. The energy balance for this system can be expressed as E −E 1in424out 3

=


∆E system 1 424 3


0 = ∆U

or ∆U Cu + ∆U water = 0

Water 75°F

Copper 180°F

[mc(T2 − T1 )] Cu + [mc(T2 − T1 )] water = 0

where

m w = ρV = (62.1 lbm/ft 3 )(1.2 ft 3 ) = 74.52 lbm Substituting, 0 = (55 lbm)(0.0925 Btu/lbm ⋅ °F)(T2 − 180°F) + (74.52 lbm)(1.0 Btu/lbm ⋅ °F)(T2 − 75°F) T2 = 81.7°F = 541.7 R

(b) The wasted work potential is equivalent to the exergy destruction (or irreversibility), and it can be determined from its definition Xdestroyed = T0Sgen where the entropy generation is determined from an entropy balance on the system, which is an insulated closed system, S in − S out 1424 3



Change in entropy

S gen = ∆S system = ∆S water + ∆S copper where

⎛T ⎞ ⎛ 541.7 R ⎞ ∆S copper = mcavg ln⎜⎜ 2 ⎟⎟ = (55 lbm)(0.092 Btu/lbm ⋅ R) ln⎜ ⎟ = −0.8483 Btu/R T ⎝ 640 R ⎠ ⎝ 1⎠ ⎛T ⎞ ⎛ 541.7 R ⎞ ∆S water = mcavg ln⎜⎜ 2 ⎟⎟ = (74.52 lbm)(1.0 Btu/lbm ⋅ R) ln⎜ ⎟ = 0.9250 Btu/R T ⎝ 535 R ⎠ ⎝ 1⎠ Substituting,

X destroyed = (535 R)(−0.8483 + 0.9250)Btu/R = 43.1 Btu


8-30

8-45 A hot iron block is dropped into water in an insulated tank that is stirred by a paddle-wheel. The mass of 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 specific heats at room temperature. 2 The system is stationary and thus the kinetic and potential energies are negligible. 3 The tank is wellinsulated 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. The specific 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 closed system. The energy balance for this system can be expressed as

E −E 1in424out 3

=


∆E system 1 424 3


W pw,in = ∆U = ∆U iron + ∆U water

100 L 20°C

Iron 85°C

Wpw,in = [mc(T2 − T1 )]iron + [mc(T2 − T1 )] water where

m water = ρV = (997 kg/m 3 )(0.1 m 3 ) = 99.7 kg Wpw = W& pw,in ∆t = (0.2 kJ/s)(20 × 60 s) = 240 kJ

Wpw

Water

Substituting, 240 kJ = m iron (0.45 kJ/kg ⋅ °C)(24 − 85)°C + (99.7 kg)(4.18 kJ/kg ⋅ °C)(24 − 20)°C m iron = 52.0 kg

(b) The exergy destruction (or irreversibility) can be determined from its definition Xdestroyed = T0Sgen where the entropy generation is determined from an entropy balance on the system, which is an insulated closed system, S − S out 1in424 3



Change in entropy

S gen = ∆S system = ∆S iron + ∆S water where ⎛T ⎞ ⎛ 297 K ⎞ ∆S iron = mc avg ln⎜⎜ 2 ⎟⎟ = (52.0 kg)(0.45 kJ/kg ⋅ K) ln⎜ ⎟ = −4.371 kJ/K ⎝ 358 K ⎠ ⎝ T1 ⎠ ⎛T ⎞ ⎛ 297 K ⎞ ∆S water = mc avg ln⎜⎜ 2 ⎟⎟ = (99.7 kg)(4.18 kJ/kg ⋅ K) ln⎜ ⎟ = 5.651 kJ/K T ⎝ 293 K ⎠ ⎝ 1⎠ Substituting,

X destroyed = T0 S gen = (293 K)(−4.371 + 5.651) kJ/K = 375.0 kJ


8-31

8-46 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 specific heats 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 iron and 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 energy balance for this system can be expressed as E −E 1in424out 3

=


∆E system 1 424 3

Lake 15°C


− Qout = ∆U = ∆U iron + ∆U copper or,

Iron 85°C

Copper Iron

Qout = [mc(T1 − T2 )]iron + [mc(T1 − T2 )]copper Substituting, Qout = (50 kg )(0.45 kJ/kg ⋅ K )(353 − 288)K + (20 kg )(0.386 kJ/kg ⋅ K )(353 − 288)K = 1964 kJ

The work that could have been produced is equal to the wasted work potential. It is equivalent to the exergy destruction (or irreversibility), and it can be determined from its definition Xdestroyed = T0Sgen . The entropy generation is determined from an entropy balance on an extended system that includes the blocks and the water in their immediate surroundings so that the boundary temperature of the extended system is the temperature of the lake water at all times, S in − S out 1424 3


−


Change in entropy

Qout + S gen = ∆S system = ∆S iron + ∆S copper Tb,out S gen = ∆S iron + ∆S copper +

Qout Tlake

where ⎛T ⎞ ⎛ 288 K ⎞ ⎟⎟ = −4.579 kJ/K ∆S iron = mc avg ln⎜⎜ 2 ⎟⎟ = (50 kg )(0.45 kJ/kg ⋅ K ) ln⎜⎜ T ⎝ 353 K ⎠ ⎝ 1⎠ ⎛T ⎞ ⎛ 288 K ⎞ ⎟⎟ = −1.571 kJ/K ∆S copper = mc avg ln⎜⎜ 2 ⎟⎟ = (20 kg )(0.386 kJ/kg ⋅ K ) ln⎜⎜ T ⎝ 353 K ⎠ ⎝ 1⎠ Substituting, ⎛ 1964 kJ ⎞ ⎟kJ/K = 196 kJ X destroyed = T0 S gen = (293 K)⎜⎜ − 4.579 − 1.571 + 288 K ⎟⎠ ⎝


8-32

8-47E A rigid tank is initially filled with saturated mixture of R-134a. Heat is transferred to the tank from a source until the pressure inside rises to a specified value. The amount of heat transfer to the tank from the source 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 is no heat transfer with the environment. Properties From the refrigerant tables (Tables A-11E through A-13E),

u = u f + x1u fg = 21.246 + 0.55 × 77.307 = 63.76 Btu / lbm P1 = 40 psia ⎫ 1 ⎬ 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 x2 =

v 2 −v f v

=

0.65234 − 0.01270 = 0.8191 0.79361 − 0.01270 = 0.06029 + 0.8191× 0.16098 = 0.1922 Btu/lbm ⋅ R

fg P2 = 60 psia ⎫ ⎬ s 2 = s f + x 2 s fg (v 2 = v 1 ) ⎭ u 2 = u f + x 2 u fg = 27.939 + 0.8191× 73.360 = 88.03 Btu/lbm

Analysis (a) The mass of the refrigerant is

m=

12 ft 3 V = = 18.40 lbm v 1 0.65234 ft 3 / lbm

We take the tank as the system, which is a closed system. The energy balance for this stationary closed system can be expressed as E −E 1in424out 3

=


Source 120°C

R-134a 40 psia Q

∆E system 1 424 3


Qin = ∆U = 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 . The entropy generation is determined from an entropy balance on an extended system that includes the tank and the region in its immediate surroundings so that the boundary temperature of the extended system where heat transfer occurs is the source temperature, S in − S out 1424 3



Change in entropy

Qin + S gen = ∆S system = m( s 2 − s1 ) , Tb,in S gen = m( s 2 − s1 ) −

Qin Tsource

Substituting, 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 ⎥⎦ ⎣


8-33

8-48 Chickens are to be cooled by chilled water in an immersion chiller that is also gaining heat from the surroundings. The rate of heat removal from the chicken and the rate of exergy destruction during this process are to be determined. Assumptions 1 Steady operating conditions exist. 2 Thermal properties of chickens and water are constant. 3 The temperature of the surrounding medium is 25°C. Properties The specific heat of chicken is given to be 3.54 kJ/kg.°C. The specific heat of water at room temperature is 4.18 kJ/kg.°C (Table A-3). Analysis (a) Chickens are dropped into the chiller at a rate of 700 per hour. Therefore, chickens can be considered to flow steadily through the chiller at a mass flow rate of m& chicken = (700 chicken/h)(1.6 kg/chicken) = 1120 kg/h = 0.3111 kg/s

Taking the chicken flow stream in the chiller as the system, the energy balance for steadily flowing chickens can be expressed in the rate form as E& − E& = ∆E& system 0 (steady) = 0 → E& in = E& out 1in424out 3 1442444 3 Rate of net energy transfer by heat, work, and mass

Rate of change in internal, kinetic, potential, etc. energies

m& h1 = Q& out + m& h2 (since ∆ke ≅ ∆pe ≅ 0) Q& out = Q& chicken = m& chicken c p (T1 − T2 ) Then the rate of heat removal from the chickens as they are cooled from 15°C to 3ºC becomes Q& =(m& c ∆T ) = (0.3111 kg/s)(3.54 kJ/kg.º C)(15 − 3)º C = 13.22 kW chicken

p

chicken

The chiller gains heat from the surroundings as a rate of 200 kJ/h = 0.0556 kJ/s. Then the total rate of heat gain by the water is Q& = Q& + Q& = 13.22 kW + (400 / 3600) kW = 13.33 kW water

chicken

heat gain

Noting that the temperature rise of water is not to exceed 2ºC as it flows through the chiller, the mass flow rate of water must be at least Q& water 13.33 kW = = 1.594 kg/s m& water = (c p ∆T ) water (4.18 kJ/kg.º C)(2º C) (b) The exergy destruction can be determined from its definition Xdestroyed = T0Sgen. The rate of entropy generation during this chilling process is determined by applying the rate form of the entropy balance on an extended system that includes the chiller and the immediate surroundings so that the boundary temperature is the surroundings temperature: S& in − S& out + S& gen = ∆S& system 0 (steady) 1424 3 { 1442443 Rate of net entropy transfer by heat and mass

Rate of entropy generation

m& 1 s1 + m& 3 s 3 − m& 2 s 2 − m& 3 s 4 +

Qin + S& gen = 0 Tsurr

m& chicken s1 + m& water s 3 − m& chicken s 2 − m& water s 4 +

Qin + S& gen = 0 Tsurr

Rate of change of entropy

Q S& gen = m& chicken ( s 2 − s1 ) + m& water ( s 4 − s 3 ) − in Tsurr

Noting that both streams are incompressible substances, the rate of entropy generation is determined to be Q& T T S& gen = m& chicken c p ln 2 + m& water c p ln 4 − in T1 T3 Tsurr = (0.3111 kg/s)(3.54 kJ/kg.K) ln

276 275.5 (400 / 3600) kW + (1.594 kg/s)(4.18 kJ/kg.K) ln − 288 273.5 298 K

= 0.001306 kW/K Finally, X& destroyed = T0 S& gen = (298 K)(0.001306 kW/K) = 0.389 kW


8-34

8-49 Carbon steel balls are to be annealed at a rate of 2500/h by heating them first and then allowing them to cool slowly in ambient air at a specified rate. The total rate of heat transfer from the balls to the ambient air and the rate of exergy destruction due to this heat transfer are to be determined. Assumptions 1 The thermal properties of the balls are constant. 2 There are no changes in kinetic and potential energies. 3 The balls are at a uniform temperature at the end of the process. Properties The density and specific heat of the balls are given to be ρ = 7833 kg/m3 and cp = 0.465 kJ/kg.°C. Analysis (a) We take a single ball as the system. The energy balance for this closed system can be expressed as

Ein − Eout 1424 3

=


∆Esystem 1 424 3


− Qout = ∆U ball = m(u2 − u1 ) Qout = mc p (T1 − T2 ) The amount of heat transfer from a single ball is m = ρV = ρ Qout

πD 3

= (7833 kg/m 3 )

π (0.008 m) 3

= 0.00210 kg 6 6 = mc p (T1 − T2 ) = (0.0021 kg)(0.465 kJ/kg.°C)(900 − 100)°C = 781 J = 0.781 kJ (per ball)

Then the total rate of heat transfer from the balls to the ambient air becomes

Q& out = n& ball Qout = (1200 balls/h) × (0.781 kJ/ball) = 936 kJ/h = 260 W (b) The exergy destruction (or irreversibility) can be determined from its definition Xdestroyed = T0Sgen. The entropy generated during this process can be determined by applying an entropy balance on an extended system that includes the ball and its immediate surroundings so that the boundary temperature of the extended system is at 35°C at all times: S − S out 1in424 3


−


Change in entropy

Q Qout + S gen = ∆S system → S gen = out + ∆S system Tb Tb

where

∆Ssystem = m( s2 − s1 ) = mc p ln

T2 100 + 273 = (0.00210 kg)(0.465 kJ/kg.K)ln = −0.00112 kJ/K T1 900 + 273

Substituting, S gen =

Qout 0.781 kJ + ∆S system = − 0.00112 kJ/K = 0.00142 kJ/K (per ball) Tb 308 K

Then the rate of entropy generation becomes S& gen = S gen n& ball = (0.00142 kJ/K ⋅ ball)(1200 balls/h) = 1.704 kJ/h.K = 0.000473 kW/K

Finally, X& destroyed = T0 S& gen = (308 K)(0.00047 3 kW/K) = 0.146 kW = 146 W


8-35

8-50 Heat is transferred to a piston-cylinder device with a set of stops. The work done, the heat transfer, the exergy destroyed, and the second-law efficiency are to be determined. Assumptions 1 The device is stationary and kinetic and potential energy changes are zero. 2 There is no friction between the piston and the cylinder. 3 Heat is transferred to the refrigerant from a source at 150˚C. Analysis (a) The properties of the refrigerant at the initial and final states are (Tables A-11 through A-13) 3 P1 = 120 kPa ⎫ v 1 = 0.19390 m /kg ⎬ T1 = 20°C ⎭ u1 = 248.51 kJ/kg s1 = 1.0760 kJ/kg.K 3 P2 = 140 kPa ⎫ v 2 = 0.20847 m /kg ⎬ T2 = 90°C ⎭ u 2 = 305.38 kJ/kg s 2 = 1.2553 kJ/kg.K

Noting that pressure remains constant at 140 kPa as the piston moves, the boundary work is determined to be Wb,out = mP2 (v 2 − v 1 ) = (0.75 kg)(140 kPa)(0.20847 − 0.19390)m 3 /kg = 1.53 kJ

R-134a 0.75 kg 120 kPa 20°C

Q

(b) The heat transfer can be determined from an energy balance on the system Qin = m(u2 − u1 ) + Wb, out = (0.75 kg)(305.38 − 248.51)kJ/ kg + 1.53 kJ = 44.2 kJ

(c) The exergy destruction associated with this process can be determined from its definition Xdestroyed = T0Sgen . The entropy generation is determined from an entropy balance on an extended system that includes the piston-cylinder device and the region in its immediate surroundings so that the boundary temperature of the extended system where heat transfer occurs is the source temperature, S − S out + S gen = ∆S system 1in424 3 { 1 424 3 Net entropy transfer by heat and mass

Entropy generation

Change in entropy

Qin + S gen = ∆S system = m( s 2 − s1 ) , Tb,in S gen = m( s 2 − s1 ) −

Qin Tsource

Substituting, 44.2 kJ ⎤ ⎡ X destroyed = T0 S gen = (298 K) ⎢(0.75 kg)(1.2553 − 1.0760)kJ/kg ⋅ K − = 8.935 kJ 150 + 273 K ⎥⎦ ⎣ (d) Exergy expended is the work potential of the heat extracted from the source at 150˚C,

⎛ T ⎞ 25 + 273 K ⎞ ⎛ X expended = X Q = η th, rev Q = ⎜⎜1 − L ⎟⎟Q = ⎜1 − ⎟(44.2 kJ ) = 13.06 kJ T + 273 K ⎠ 150 ⎝ H ⎠ ⎝ Then the 2nd law efficiency becomes X destroyed X 8.935 kJ η II = recovered = 1 − = 1− = 0.316 or 31.6% X expended X expended 13.06 kJ

Discussion The second-law efficiency can also be determine as follows: The exergy increase of the refrigerant is the exergy difference between the initial and final states, ∆X = m[u 2 − u1 − T0 ( s 2 − s1 ) + P0 (v 2 − v 1 )]

[

= (0.75 kg) (305.38 − 248.51)kJ/kg − (298 K)(1.2553 − 1.0760)kg.K + (100 kPa)(0.20847 − 0.19390)m 3 /kg

]

= 3.666 kJ

The useful work output for the process is W u,out = W b,out − mP0 (v 2 − v 1 ) = 1.53 kJ − (0.75 kg)(100 kPa)(0.20847 − 0.19390)m 3 /kg = 0.437 kJ The exergy recovered is the sum of the exergy increase of the refrigerant and the useful work output, X recovered = ∆X + W u,out = 3.666 + 0.437 = 4.103 kJ Then the second-law efficiency becomes X 4.103 kJ η II = recovered = = 0.314 or 31.4% X expended 13.06 kJ PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

8-36

8-51 A tank containing hot water is placed in a larger tank. The amount of heat lost to the surroundings and the exergy destruction during the process are to be determined. Assumptions 1 Kinetic and potential energy changes are negligible. 2 Air is an ideal gas with constant specific heats. 3 The larger tank is well-sealed. Properties The properties of air at room temperature are R = 0.287 kPa.m3/kg.K, cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg.K (Table A-2). The properties of water at room temperature are ρ = 997 kg/m3, cw = 4.18 kJ/kg.K (Table A-3). Analysis (a) The final volume of the air in the tank is

V a 2 = V a1 −V w = 0.04 − 0.015 = 0.025 m 3

Air, 22°C

The mass of the air in the large tank is ma =

Water 85°C 15 L

P1V a1 (100 kPa)(0.04 m 3 ) = = 0.04724 kg RTa1 (0.287 kPa ⋅ m 3 /kg ⋅ K)(22 + 273 K)

Q

The pressure of air at the final state is

Pa 2 =

m a RTa 2

V a2

=

(0.04724 kg)(0.287 kPa ⋅ m 3 /kg ⋅ K)(44 + 273 K) 0.025 m 3

= 171.9 kPa

The mass of water is

mw = ρ wV w = (997 kg/m3 )(0.015 m 3 ) = 14.96 kg An energy balance on the system consisting of water and air is used to determine heat lost to the surroundings Qout = −[mw c w (T2 − Tw1 ) + ma cv (T2 − Ta1 )] = −(14.96 kg)(4.18 kJ/kg.K)(44 − 85) − (0.04724 kg)(0.718 kJ/kg.K)(44 − 22) = 2563 kJ

(b) An exergy balance written on the (system + immediate surroundings) can be used to determine exergy destruction. But we first determine entropy and internal energy changes

∆S w = mw cw ln

Tw1 (85 + 273) K = (14.96 kg)(4.18 kJ/kg.K)ln = 7.6059 kJ/K T2 (44 + 273) K

⎡ P ⎤ T ∆S a = m a ⎢c p ln a1 − R ln a1 ⎥ P2 ⎦ T2 ⎣ ⎡ (22 + 273) K 100 kPa ⎤ = (0.04724 kg) ⎢(1.005 kJ/kg.K)ln − (0.287 kJ/kg.K)ln ⎥ (44 + 273) K 171.9 kPa ⎦ ⎣ = 0.003931 kJ/K ∆U w = mw cw (T1w − T2 ) = (14.96 kg)(4.18 kJ/kg.K)(85 - 44)K = 2564 kJ ∆U a = ma cv (T1a − T2 ) = (0.04724 kg)(0.718 kJ/kg.K)(22 - 44)K = −0.7462 kJ

X dest = ∆X w + ∆X a = ∆U w − T0 ∆S w + ∆U a − T0 ∆S a = 2564 kJ − (295 K)(7.6059 kJ/K) + ( −0.7462 kJ) − (295 K)(0.003931 kJ/K) = 318.4 kJ


8-37

Exergy Analysis of Control Volumes 8-52 R-134a is is throttled from a specified state to a specified pressure. The temperature of R-134a at the outlet of the expansion valve, the entropy generation, and the exergy destruction are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 Heat transfer is negligible.

(a) The properties of refrigerant at the inlet and exit states of the throttling valve are (from R134a tables) P1 = 1200 kPa ⎫ h1 = 117.77 kJ/kg ⎬ x1 = 0 ⎭ s1 = 0.4244 kJ/kg ⋅ K P2 = 200 kPa ⎫ T2 = −10.1°C ⎬ h2 = h1 = 117.77 kJ/kg ⎭ s 2 = 0.4562 kJ/kg ⋅ K

(b) Noting that the throttling valve is adiabatic, the entropy generation is determined from s gen = s 2 − s1 = (0.4562 − 0.4244 )kJ/kg ⋅ K = 0.03176 kJ/kg ⋅ K

Then the irreversibility (i.e., exergy destruction) of the process becomes

exdest = T0 s gen = (298 K)(0.03176 kJ/kg ⋅ K) = 9.464 kJ/kg


8-38

8-53 Heium expands in an adiabatic turbine from a specified inlet state to a specified exit state. The maximum work output is to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 The device is adiabatic and thus heat transfer is negligible. 3 Helium is an ideal gas. 4 Kinetic and potential energy changes are negligible. Properties The properties of helium are cp = 5.1926 kJ/kg.K and R = 2.0769 kJ/kg.K (Table A-1). Analysis The entropy change of helium is

s 2 − s1 = c p ln

T2 P − R ln 2 T1 P1

= (5.1926 kJ/kg ⋅ K) ln = 2.2295 kJ/kg ⋅ K

1500 kPa 300°C 298 K 100 kPa − (2.0769 kJ/kg ⋅ K) ln 573 K 1500 kPa Helium

The maximum (reversible) work is the exergy difference between the inlet and exit states wrev,out = h1 − h2 − T0 ( s1 − s 2 )

100 kPa 25°C

= c p (T1 − T2 ) − T0 ( s1 − s 2 ) = (5.1926 kJ/kg ⋅ K)(300 − 25)K − (298 K)(−2.2295 kJ/kg ⋅ K) = 2092 kJ/kg

There is only one inlet and one exit, and thus m& 1 = m& 2 = m& . We take the turbine as the system, which is a control volume since mass crosses the boundary. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

=

Rate of net energy transfer by heat, work, and mass

∆E& system 0 (steady) 1442443

=0


E& in = E& out m& h1 = W& out + Q& out + m& h2 W& = m& (h − h ) − Q& out

1

2

out

wout = (h1 − h2 ) − q out

Inspection of this result reveals that any rejection of heat will decrease the work that will be produced by the turbine since inlet and exit states (i.e., enthalpies) are fixed. If there is heat loss from the turbine, the maximum work output is determined from the rate form of the exergy balance applied on the turbine and setting the exergy destruction term equal to zero, X& − X& 1in424out 3

Rate of net exergy transfer by heat, work,and mass

− X& destroyed 0 (reversible) = ∆X& system 0 (steady) = 0 144424443 144 42444 3 Rate of exergy destruction

Rate of change of exergy

X& in = X& out ⎛ T ⎞ m& ψ 1 = W& rev,out + Q& out ⎜⎜1 − 0 ⎟⎟ + m& ψ 2 ⎝ T ⎠ ⎛ T ⎞ wrev,out = (ψ 1 −ψ 2 ) − qout ⎜⎜1 − 0 ⎟⎟ ⎝ T ⎠ ⎛ T = (h1 − h2 ) − T0 (s1 − s 2 ) − qout ⎜⎜1 − 0 ⎝ T

⎞ ⎟⎟ ⎠

Inspection of this result reveals that any rejection of heat will decrease the maximum work that could be produced by the turbine. Therefore, for the maximum work, the turbine must be adiabatic.


8-39

8-54 Air is compressed steadily by an 8-kW compressor from a specified state to another specified state. The increase in the exergy of air and the rate of exergy destruction are to be determined. Assumptions 1 Air is an ideal gas with variable specific heats. 2 Kinetic and potential energy changes are negligible. Properties The gas constant of air is R = 0.287 kJ/kg.K (Table A-1). From the air table (Table A-17)

T1 = 290 K

⎯ ⎯→

h1 = 29016 . kJ / kg s1o = 1.66802 kJ / kg ⋅ K

T2 = 440 K

⎯ ⎯→

600 kPa 167°C

h2 = 441.61 kJ / kg s2o = 2.0887 kJ / kg ⋅ K

Analysis The increase in exergy is the difference between the exit and inlet flow exergies,

AIR 8 kW

Increase in exergy = ψ 2 −ψ 1 = [(h2 − h1 ) + ∆ke

0

+ ∆pe

0

− T0 ( s 2 − s1 )]

= (h2 − h1 ) − T0 ( s 2 − s1 )

100 kPa 17°C

where s 2 − s1 = ( s 2o − s1o ) − R ln

P2 P1

= (2.0887 − 1.66802)kJ/kg ⋅ K - (0.287 kJ/kg ⋅ K) ln = −0.09356 kJ/kg ⋅ K

600 kPa 100 kPa

Substituting, Increase in exergy = ψ 2 −ψ 1

= [(441.61 − 290.16)kJ/kg - (290 K)(−0.09356 kJ/kg ⋅ K)]

= 178.6 kJ/kg

Then the reversible power input is W& rev,in = m& (ψ 2 − ψ 1 ) = ( 2.1 / 60 kg/s)(178.6 kJ/kg) = 6.25 kW

(b) The rate of exergy destruction (or irreversibility) is determined from its definition, X& destroyed = W& in − W& rev,in = 8 − 6.25 = 1.75 kW

Discussion Note that 1.75 kW of power input is wasted during this compression process.


8-40

8-55 Problem 8-54 is reconsidered. The problem is to be solved and the actual heat transfer, its direction, the minimum power input, and the compressor second-law efficiency are to be determined. Analysis The problem is solved using EES, and the solution is given below. Function Direction$(Q) If Q<0 then Direction$='out' else Direction$='in' end Function Violation$(eta) If eta>1 then Violation$='You have violated the 2nd Law!!!!!' else Violation$='' end {"Input Data from the Diagram Window" T_1=17 [C] P_1=100 [kPa] W_dot_c = 8 [kW] P_2=600 [kPa] S_dot_gen=0 Q_dot_net=0} {"Special cases" T_2=167 [C] m_dot=2.1 [kg/min]} T_o=T_1 P_o=P_1 m_dot_in=m_dot*Convert(kg/min, kg/s) "Steady-flow conservation of mass" m_dot_in = m_dot_out "Conservation of energy for steady-flow is:" E_dot_in - E_dot_out = DELTAE_dot DELTAE_dot = 0 E_dot_in=Q_dot_net + m_dot_in*h_1 +W_dot_c "If Q_dot_net < 0, heat is transferred from the compressor" E_dot_out= m_dot_out*h_2 h_1 =enthalpy(air,T=T_1) h_2 = enthalpy(air, T=T_2) W_dot_net=-W_dot_c W_dot_rev=-m_dot_in*(h_2 - h_1 -(T_1+273.15)*(s_2-s_1)) "Irreversibility, entropy generated, second law efficiency, and exergy destroyed:" s_1=entropy(air, T=T_1,P=P_1) s_2=entropy(air,T=T_2,P=P_2) s_2s=entropy(air,T=T_2s,P=P_2) s_2s=s_1"This yields the isentropic T_2s for an isentropic process bewteen T_1, P_1 and P_2"I_dot=(T_o+273.15)*S_dot_gen"Irreversiblility for the Process, KW" S_dot_gen=(-Q_dot_net/(T_o+273.15) +m_dot_in*(s_2-s_1)) "Entropy generated, kW" Eta_II=W_dot_rev/W_dot_net"Definition of compressor second law efficiency, Eq. 7_6" h_o=enthalpy(air,T=T_o) s_o=entropy(air,T=T_o,P=P_o) Psi_in=h_1-h_o-(T_o+273.15)*(s_1-s_o) "availability function at state 1" Psi_out=h_2-h_o-(T_o+273.15)*(s_2-s_o) "availability function at state 2" X_dot_in=Psi_in*m_dot_in X_dot_out=Psi_out*m_dot_out DELTAX_dot=X_dot_in-X_dot_out "General Exergy balance for a steady-flow system, Eq. 7-47" (1-(T_o+273.15)/(T_o+273.15))*Q_dot_net-W_dot_net+m_dot_in*Psi_in - m_dot_out*Psi_out =X_dot_dest "For the Diagram Window" Text$=Direction$(Q_dot_net) Text2$=Violation$(Eta_II)


8-41

ηII 0.7815 0.8361 0.8908 0.9454 1

I [kW] 1.748 1.311 0.874 0.437 1.425E-13

Xdest [kW] 1.748 1.311 0.874 0.437 5.407E-15

T2s [C] 209.308 209.308 209.308 209.308 209.308

T2 [C] 167 200.6 230.5 258.1 283.9

Qnet [kW] -2.7 -1.501 -0.4252 0.5698 1.506

How can entropy decrease? 250

2s

200

2 ideal

T [C]

150

100 kPa 100

600 kPa

actual

50

1 0 5.0

5.5

6.0

6.5

s [kJ/kg-K]

300

2.0

280

T2

240 1.0

Xdest

1.5

260

220 200

0.5

180 160 0.75

0.80

0.85

0.90

0.95

0.0 1.00

ηII

2

2.0

1

0 1.0

X dest

Q net

1.5

-1 0.5

-2

-3 0.75

0.80

0.85

0.90

0.95

0.0 1.00

ηII


8-42

8-56 Steam is decelerated in a diffuser. The second law efficiency of the diffuser is to be determined. Assumptions 1 The diffuser operates steadily. 2 The changes in potential energies are negligible. Properties The properties of steam at the inlet and the exit of the diffuser are (Tables A-4 through A-6) P1 = 500 kPa ⎫ h1 = 2855.8 kJ/kg ⎬ T1 = 200°C ⎭ s1 = 7.0610 kJ/kg ⋅ K P2 = 200 kPa

⎫ h2 = 2706.3 kJ/kg ⎬ x 2 = 1 (sat. vapor) ⎭ s 2 = 7.1270 kJ/kg ⋅ K

Analysis We take the diffuser to be the system, which is a control volume. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

∆E& system 0 (steady) 144 42444 3

=


=0


E& in = E& out

500 kPa 200°C 30 m/s

m& (h1 + V12 / 2) = m& (h2 + V 22 /2) V 22 − V12 = h1 − h2 = ∆ke actual 2

H 2O

200 kPa sat. vapor

Substituting, ∆ke actual = h1 − h2 = 2855.8 − 2706.3 = 149.5 kJ/kg

An exergy balance on the diffuser gives X& − X& out 1in 4243


− X& destroyed 0 (reversible) = ∆X& system 0 (steady) = 0 144424443 1442443 Rate of exergy destruction


X& in = X& out m& ψ1 = m& ψ 2 h1 − h0 +

V12 V2 − T0 (s1 − s0 ) = h2 − h0 + 2 − T0 (s2 − s0 ) 2 2 2 2 V2 − V1 = h1 − h2 − T0 (s1 − s2 ) 2 ∆kerev = h1 − h2 − T0 (s1 − s2 )

Substituting, ∆ke rev = h1 − h2 − T0 ( s1 − s 2 ) = (2855.8 − 2706.3)kJ/kg − (298 K)(7.0610 − 7.1270) kJ/kg ⋅ K = 169.2 kJ/kg


η II =

∆ke actual 149.5 kJ/kg = = 0.884 169.2 kJ/kg ∆ke rev


8-43

8-57 Air is accelerated in a nozzle while losing some heat to the surroundings. The exit temperature of air and the exergy destroyed during the process are to be determined. Assumptions 1 Air is an ideal gas with variable specific heats. 2 The nozzle operates steadily. Properties The gas constant of air is R = 0.287 kJ/kg.K (Table A-1). The properties of air at the nozzle inlet are (Table A17) T1 = 338 K ⎯ ⎯→ h1 = 338.40 kJ/kg s1o = 1.8219 kJ/kg ⋅ K

Analysis (a) We take the nozzle as the system, which is a control volume. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& out 1in 424 3

=



=0

3 kJ/kg


E& in = E& out m& (h1 + V12 / 2) = m& ( h2 + V22 /2) + Q& out

35 m/s

AIR

240 m/s

or 0 = q out + h2 − h1 +

V 22 − V12 2

Therefore,

h2 = h1 − q out −

V22 − V12 (240 m/s) 2 − (35 m/s) 2 ⎛ 1 kJ/kg ⎞ = 338.40 − 3 − ⎜ ⎟ = 307.21 kJ/kg 2 2 ⎝ 1000 m 2 / s 2 ⎠ T2 = 307.0 K = 34.0°C and s 2o = 1.7251 kJ/kg ⋅ K

At this h2 value we read, from Table A-17,

(b) The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0S gen where the entropy generation Sgen is determined from an entropy balance on an extended system that includes the device and its immediate surroundings so that the boundary temperature of the extended system is Tsurr at all times. It gives

S& − S& out 1in424 3

+

Rate of net entropy transfer by heat and mass

ms1 − m& s 2 −

S& gen {


= ∆S& system 0 = 0 14243 Rate of change of entropy

Q& out + S& gen = 0 Tb,surr Q& S& gen = m& (s 2 − s1 ) + out Tsurr

where ∆s air = s 2o − s1o − R ln

P2 95 kPa = (1.7251 − 1.8219)kJ/kg ⋅ K − (0.287 kJ/kg ⋅ K) ln = 0.1169 kJ/kg ⋅ K P1 200 kPa

Substituting, the entropy generation and exergy destruction per unit mass of air are determined to be x destroyed = T0 s gen = Tsurr s gen ⎛ q = T0 ⎜⎜ s 2 − s1 + surr Tsurr ⎝

⎞ 3 kJ/kg ⎞ ⎛ ⎟ = (290 K)⎜ 0.1169 kJ/kg ⋅ K + ⎟ = 36.9 kJ/kg ⎟ 290 K ⎠ ⎝ ⎠


8-44

Alternative solution The exergy destroyed during a process can be determined from an exergy balance applied on the extended system that includes the device and its immediate surroundings so that the boundary temperature of the extended system is environment temperature T0 (or Tsurr) at all times. Noting that exergy transfer with heat is zero when the temperature at the point of transfer is the environment temperature, the exergy balance for this steady-flow system can be expressed as X& − X& out 1in 4243


− X& destroyed = ∆X& system 0 (steady) = 0 → X& destroyed = X& in − X& out = m& ψ1 − m& ψ 2 = m& (ψ1 −ψ 2 ) 1 424 3 1442443 Rate of exergy destruction


= m& [(h1 − h2 ) − T0 (s1 − s2 ) − ∆ke − ∆pe 0 ] = m& [T0 (s2 − s1) − (h2 − h1 + ∆ke)] = m& [T0 (s2 − s1) + qout ] since, from energy balance, − qout = h2 − h1 + ∆ke ⎛ Q& ⎞ = T0 ⎜⎜ m& (s2 − s1) + out ⎟⎟ = T0S&gen T0 ⎠ ⎝

Therefore, the two approaches for the determination of exergy destruction are identical.


8-45

8-58 Problem 8-57 is reconsidered. The effect of varying the nozzle exit velocity on the exit temperature and exergy destroyed is to be investigated. Analysis The problem is solved using EES, and the solution is given below. "Knowns:" WorkFluid$ = 'Air' P[1] = 200 [kPa] T[1] =65 [C] P[2] = 95 [kPa] Vel[1] = 35 [m/s] {Vel[2] = 240 [m/s]} T_o = 17 [C] T_surr = T_o q_loss = 3 [kJ/kg] "Conservation of Energy - SSSF energy balance for nozzle -- neglecting the change in potential energy:" h[1]=enthalpy(WorkFluid$,T=T[1]) s[1]=entropy(WorkFluid$,P=P[1],T=T[1]) ke[1] = Vel[1]^2/2 ke[2]=Vel[2]^2/2 h[1]+ke[1]*convert(m^2/s^2,kJ/kg) = h[2] + ke[2]*convert(m^2/s^2,kJ/kg)+q_loss T[2]=temperature(WorkFluid$,h=h[2]) s[2]=entropy(WorkFluid$,P=P[2],h=h[2]) "The entropy generated is detemined from the entropy balance:" s[1] - s[2] - q_loss/(T_surr+273) + s_gen = 0 x_destroyed = (T_o+273)*s_gen T2 [C] 57.66 52.89 46.53 38.58 29.02 17.87

xdestroyed [kJ/kg] 58.56 54.32 48.56 41.2 32.12 21.16

60 55 50 45

T[2] [C]

Vel2 [m/s] 100 140 180 220 260 300

40 35 30 25 20 15 100

140

180

220

260

300

Vel[2] [m/s]


8-46 60 55

xdestroyed [kJ/kg]

50 45 40 35 30 25 20 100

140

180

220

260

300

Vel[2] [m/s]


8-47

8-59 Steam is decelerated in a diffuser. The mass flow rate of steam and the wasted work potential during the process are to be determined. Assumptions 1 The diffuser operates steadily. 2 The changes in potential energies are negligible. Properties The properties of steam at the inlet and the exit of the diffuser are (Tables A-4 through A-6) P1 = 10 kPa ⎫ h1 = 2592.0 kJ/kg ⎬ T1 = 50°C ⎭ s1 = 8.1741 kJ/kg ⋅ K h2 = 2591.3 kJ/kg T2 = 50°C ⎫ ⎬ s 2 = 8.0748 kJ/kg ⋅ K sat.vapor ⎭ v 2 = 12.026 m 3 /kg

300 m/s

H 2O

70 m/s

Analysis (a) The mass flow rate of the steam is m& =

1

v2

A2V 2 =

1 3

12.026 m / kg

(3 m 2 )(70 m/s) = 17.46 kg/s

(b) We take the diffuser to be the system, which is a control volume. Assuming the direction of heat transfer to be from the stem, the energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

=


∆E& system 0 (steady) 144 42444 3

=0


E& in = E& out m& (h1 + V12 / 2) = m& (h2 + V22 /2) + Q& out ⎛ V 2 − V12 Q& out = −m& ⎜ h2 − h1 + 2 ⎜ 2 ⎝

⎞ ⎟ ⎟ ⎠

Substituting, ⎡ (70 m/s) 2 − (300 m/s) 2 ⎛ 1 kJ/kg Q& out = −(17.46 kg/s) ⎢2591.3 − 2592.0 + ⎜ 2 ⎝ 1000 m 2 / s 2 ⎢⎣

⎞⎤ ⎟⎥ = 754.8 kJ/s ⎠⎥⎦

The wasted work potential is equivalent to exergy destruction. The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0S gen where the entropy generation Sgen is determined from an entropy balance on an extended system that includes the device and its immediate surroundings so that the boundary temperature of the extended system is Tsurr at all times. It gives S& in − S& out 1424 3


m& s1 − m& s 2 −

+

S& gen {



Q& out Q& + S& gen = 0 → S& gen = m& (s 2 − s1 ) + out Tb,surr Tsurr

Substituting, the exergy destruction is determined to be ⎛ Q& X& destroyed = T0 S& gen = T0 ⎜ m& ( s 2 − s1 ) + out ⎜ T0 ⎝

⎞ ⎟ ⎟ ⎠

754.8 kW ⎞ ⎛ = (298 K)⎜ (17.46 kg/s)(8.0748 - 8.1741)kJ/kg ⋅ K + ⎟ 298 K ⎠ ⎝ = 238.3 kW


8-48

8-60E Air is compressed steadily by a compressor from a specified state to another specified state. The minimum power input required for the compressor is to be determined. Assumptions 1 Air is an ideal gas with variable specific heats. 2 Kinetic and potential energy changes are negligible. Properties The gas constant of air is R = 0.06855 Btu/lbm.R (Table A-1E). From the air table (Table A-17E) T1 = 520 R ⎯ ⎯→ h1 = 124.27 Btu/lbm s1o = 0.59173 Btu/lbm ⋅ R

100 psia 480°F

T2 = 940 R ⎯ ⎯→ h2 = 226.11 Btu/lbm s 2o = 0.73509 Btu/lbm ⋅ R

AIR 22 lbm/min

Analysis The reversible (or minimum) power input is determined from the rate form of the exergy balance applied on the compressor and setting the exergy destruction term equal to zero, X& − X& out 1in 4243



14.7 psia 60°F


X& in = X& out m& ψ1 + W&rev,in = m& ψ 2 W&rev,in = m& (ψ 2 −ψ1) = m& [(h2 − h1) − T0 (s2 − s1) + ∆ke

0

+ ∆pe 0 ]

where ∆s air = s 2o − s1o − R ln

P2 P1

= (0.73509 − 0.59173)Btu/lbm ⋅ R − (0.06855 Btu/lbm ⋅ R) ln

100 psia 14.7 psia

= 0.01193 Btu/lbm ⋅ R

Substituting, W& rev,in = (22/60 lbm/s)[(226.11 − 124.27) Btu/lbm − (520 R)(0.01193 Btu/lbm ⋅ R)] = 35.1 Btu/s = 49.6 hp

Discussion Note that this is the minimum power input needed for this compressor.


8-49

8-61 Steam expands in a turbine from a specified state to another specified state. The actual power output of the turbine is given. The reversible power output and the second-law efficiency are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Potential energy change is negligible. 3 The temperature of the surroundings is given to be 25°C. Properties From the steam tables (Tables A-4 through A-6) P1 = 6 MPa ⎫ h1 = 3658.8 kJ/kg ⎬ T1 = 600°C ⎭ s1 = 7.1693 kJ/kg ⋅ K P2 = 50 kPa ⎫ h2 = 2682.4 kJ/kg ⎬ T2 = 100°C ⎭ s 2 = 7.6953 kJ/kg ⋅ K &1 = m &2 = m & . We take the turbine as the system, which is a Analysis (b) There is only one inlet and one exit, and thus m control volume since mass crosses the boundary. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

=


∆E& system 0 (steady) 1442444 3

=0

80 m/s 6 MPa 600°C


E& in = E& out

m& (h1 + V12 / 2) = W& out + m& (h2 + V 22 / 2) W& out

STEAM

⎡ V 2 − V 22 ⎤ = m& ⎢h1 − h2 + 1 ⎥ 2 ⎥⎦ ⎢⎣

5 MW

Substituting, ⎛ (80 m/s) 2 − (140 m/s) 2 5000 kJ/s = m& ⎜ 3658.8 − 2682.4 + ⎜ 2 ⎝ m& = 5.156 kg/s

⎛ 1 kJ/kg ⎜ ⎝ 1000 m 2 / s 2

⎞ ⎞⎟ ⎟ ⎠ ⎟⎠

50 kPa 100°C 140 m/s

The reversible (or maximum) power output is determined from the rate form of the exergy balance applied on the turbine and setting the exergy destruction term equal to zero, X& − X& 1in424out 3


− X& destroyed 0 (reversible) = ∆X& system 0 (steady) = 0 144424443 144 42444 3 Rate of exergy destruction


X& in = X& out & ψ = W& m 1

rev,out

&ψ2 +m

& (ψ 1 −ψ 2 ) = m & [(h1 − h2 ) − T0 ( s1 − s2 ) − ∆ke − ∆pe 0 ] W& rev,out = m

Substituting, W& rev, out = W& out − m& T0 ( s1 − s2 ) = 5000 kW − (5.156 kg/s)(298 K)(7.1693 − 7.6953) kJ/kg ⋅ K = 5808 kW

(b) The second-law efficiency of a turbine is the ratio of the actual work output to the reversible work,

W& out

η II = & W

rev, out

=

5 MW = 86.1% 5.808 MW

Discussion Note that 13.9% percent of the work potential of the steam is wasted as it flows through the turbine during this process.


8-50

8-62 Steam is throttled from a specified state to a specified pressure. The decrease in the exergy of the steam during this throttling process is to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 The temperature of the surroundings is given to be 25°C. 4 Heat transfer is negligible. Properties The properties of steam before and after throttling are (Tables A-4 through A-6) P1 = 6 MPa ⎫ h1 = 3178.3 kJ/kg ⎬ T1 = 400°C ⎭ s1 = 6.5432 kJ/kg ⋅ K

Steam 1

P2 = 2 MPa ⎫ ⎬ s 2 = 7.0225 kJ/kg ⋅ K h2 = h1 ⎭

Analysis The decrease in exergy is of the steam is the difference between the inlet and exit flow exergies, Decrease in exergy = ψ 1 − ψ 2 = −[∆h

0

− ∆ke

0

− ∆pe

0

2

− T0 ( s1 − s 2 )] = T0 ( s 2 − s1 )

= (298 K)(7.0225 − 6.5432)kJ/kg ⋅ K = 143 kJ/kg

Discussion Note that 143 kJ/kg of work potential is wasted during this throttling process.


8-51

8-63 CO2 gas is compressed steadily by a compressor from a specified state to another specified state. The power input to the compressor if the process involved no irreversibilities is to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 The device is adiabatic and thus heat transfer is negligible. 4 CO2 is an ideal gas with constant specific heats. Properties At the average temperature of (300 + 450)/2 = 375 K, the constant pressure specific heat and the specific heat ratio of CO2 are cp = 0.917 kJ/kg.K and k = 1.261 (Table A-2b). Also, cp = 0.1889 kJ/kg.K (Table A-2a).

600 kPa 450 K

Analysis The reversible (or minimum) power input is determined from the exergy balance applied on the compressor, and setting the exergy destruction term equal to zero, X& − X& out 1in 4243


CO2 0.2 kg/s



X& in = X& out

100 kPa 300 K

m& ψ1 + W&rev,in = m& ψ 2 W&rev,in = m& (ψ 2 −ψ1) = m& [(h2 − h1) − T0 (s2 − s1) + ∆ke

0

+ ∆pe 0 ]

where s 2 − s1 = c p ln

T2 P − R ln 2 T1 P1

= (0.917 kJ/kg ⋅ K) ln = 0.03335 kJ/kg ⋅ K

450 K 600 kPa − (0.1889 kJ/kg ⋅ K) ln 300 K 100 kPa

Substituting, W& rev,in = (0.2 kg/s)[(0.917 kJ/kg ⋅ K)(450 − 300)K − (298 K)(0.03335 kJ/kg ⋅ K)] = 25.5 kW

Discussion Note that a minimum of 25.5 kW of power input is needed for this compressor.


8-52

8-64 Combustion gases expand in a turbine from a specified state to another specified state. The exergy of the gases at the inlet and the reversible work output of the turbine are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Potential energy changes are negligible. 3 The temperature of the surroundings is given to be 25°C. 4 The combustion gases are ideal gases with constant specific heats. Properties The constant pressure specific heat and the specific heat ratio are given to be cp = 1.15 kJ/kg.K and k = 1.3. The gas constant R is determined from

R = c p − cv = c p − c p / k = c p (1 − 1 / k ) = (1.15 kJ/kg ⋅ K)(1 − 1/1.3) = 0.265 kJ/kg ⋅ K Analysis (a) The exergy of the gases at the turbine inlet is simply the flow exergy,

ψ 1 = h1 − h0 − T0 ( s1 − s 0 ) +

V12 + gz1 2

800 kPa 900°C

0

where T1 P − R ln 1 T0 P0 1173 K 800 kPa = (1.15 kJ/kg ⋅ K)ln − (0.265 kJ/kg ⋅ K)ln 298 K 100 kPa = 1.025 kJ/kg ⋅ K

s1 − s 0 = c p ln

Thus,

ψ 1 = (1.15 kJ/kg.K)(900 − 25)°C − (298 K)(1.025 kJ/kg ⋅ K) +

GAS TURBINE

400 kPa 650°C

(100 m/s) 2 ⎛ 1 kJ/kg ⎞ ⎜ ⎟ = 705.8 kJ/kg 2 ⎝ 1000 m 2 / s 2 ⎠

(b) The reversible (or maximum) work output is determined from an exergy balance applied on the turbine and setting the exergy destruction term equal to zero, X& − X& out 1in 4243

Rate of net exergy transfer by heat, work, and mass



X& in = X& out m& ψ = W& 1

rev,out

+ m& ψ 2

W&rev,out = m& (ψ1 −ψ 2 ) = m& [(h1 − h2 ) − T0 (s1 − s2 ) − ∆ke − ∆pe 0 ]

where

∆ke =

V22 − V12 (220 m/s) 2 − (100 m/s) 2 = 2 2

⎛ 1 kJ/kg ⎞ ⎜ ⎟ = 19.2 kJ/kg ⎝ 1000 m 2 / s 2 ⎠

and s 2 − s1 = c p ln

T2 P − R ln 2 T1 P1

400 kPa 923 K − (0.265 kJ/kg ⋅ K)ln 800 kPa 1173 K = −0.09196 kJ/kg ⋅ K = (1.15 kJ/kg ⋅ K)ln

Then the reversible work output on a unit mass basis becomes wrev,out = h1 − h2 + T0 ( s 2 − s1 ) − ∆ke = c p (T1 − T2 ) + T0 ( s 2 − s1 ) − ∆ke = (1.15 kJ/kg ⋅ K)(900 − 650)°C + (298 K)(−0.09196 kJ/kg ⋅ K) − 19.2 kJ/kg = 240.9 kJ/kg


8-53

8-65E Refrigerant-134a enters an adiabatic compressor with an isentropic efficiency of 0.80 at a specified state with a specified volume flow rate, and leaves at a specified pressure. The actual power input and the second-law efficiency to the compressor are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 The device is adiabatic and thus heat transfer is negligible. Properties From the refrigerant tables (Tables A-11E through A-13E) h1 = h g @ 30 psia = 105.32 Btu / lbm P1 = 30 psia ⎫ ⎬ s1 = s g @ 30 psia = 0.2238 Btu/lbm ⋅ R sat.vapor ⎭ v 1 = v g @ 30 psia = 1.5492 ft 3 /lbm

P2 = 70 psia ⎫ ⎬ h2 s = 112.80 Btu/lbm s 2 s = s1 ⎭

Analysis From the isentropic efficiency relation,

ηc =

70 psia s2 = s1

h2 s − h1 ⎯ ⎯→ h2 a = h1 + (h2 s − h1 ) / η c h2 a − h1 = 105.32 + (112.80 − 105.32) / 0.80 = 114.67 Btu/lbm

R-134a 20 ft3/min

Then, P2 = 70 psia ⎫ ⎬ s 2 = 0.2274 Btu/lbm h2a = 114.67 ⎭

Also,

30 psia sat. vapor

V& 20 / 60 ft 3 / s m& = 1 = = 0.2152 lbm/s v 1 1.5492 ft 3 / lbm

&1 = m &2 = m & . We take the actual compressor as the system, which is a control There is only one inlet and one exit, and thus m volume. The energy balance for this steady-flow system can be expressed as E& − E& = ∆E& system 0 (steady) =0 1in424out 3 1442444 3 Rate of net energy transfer by heat, work, and mass


E& in = E& out

& 1 = mh & 2 (since Q& ≅ ∆ke ≅ ∆pe ≅ 0) W& a,in + mh W& a,in = m& (h2 − h1 ) Substituting, the actual power input to the compressor becomes 1 hp ⎛ ⎞ W& a,in = (0.2152 lbm/s)(114.67 − 105.32) Btu/lbm⎜ ⎟ = 2.85 hp ⎝ 0.7068 Btu/s ⎠

(b) The reversible (or minimum) power input is determined from the exergy balance applied on the compressor and setting the exergy destruction term equal to zero, X& − X& out 1in 4243




X& in = X& out W&rev,in + m& ψ1 = m& ψ 2 W&rev,in = m& (ψ 2 −ψ1) = m& [(h2 − h1) − T0 (s2 − s1) + ∆ke

Substituting, W&

rev,in

η II =

+ ∆pe 0 ]

= (0.2152 lbm/s)[(114.67 − 105.32) Btu/lbm − (535 R)(0.2274 − 0.2238) Btu/lbm ⋅ R ] = 1.606 Btu/s = 2.27 hp

Thus,

0

(since 1 hp = 0.7068 Btu/s)

W& rev,in 2.27 hp = = 79.8% W& act,in 2.85 hp


8-54

8-66 Refrigerant-134a enters an adiabatic compressor at a specified state with a specified volume flow rate, and leaves at a specified state. The power input, the isentropic efficiency, the rate of exergy destruction, and the second-law efficiency are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 The device is adiabatic and thus heat transfer is negligible. Analysis (a) The properties of refrigerant at the inlet and exit states of the compressor are obtained from R-134a tables: h = 234.68 kJ/kg T1 = −26°C⎫ s1 = 0.9514 kJ/kg ⋅ K ⎬ 1 x1 = 1 ⎭ v = 0.18946 m 3 /kg 1 P2 = 800 kPa ⎫ h2 = 286.69 kJ/kg ⎬ ⎭ s 2 = 0.9802 kJ/kg ⋅ K

T2 = 50°C

P2 = 800 kPa

⎫ ⎬h2 s = 277.53 kJ/kg s 2 = s1 = 0.9514 kJ/kg ⋅ K ⎭

The mass flow rate of the refrigerant and the actual power input are m& =

V&1 (0.45 / 60) m 3 /s = = 0.03959 kg/s v 1 0.18946 m 3 /kg

W& act = m& (h2 − h1 ) = (0.03959 kg/s)(286.69 − 234.68)kJ/kg = 2.059 kW (b) The power input for the isentropic case and the isentropic efficiency are

W& isen = m& (h2 s − h1 ) = (0.03959 kg/s)(277.53 − 234.68)kJ/kg = 1.696 kW

η Comp,isen =

W& isen 1.696 kW = = 0.8238 = 82.4% 2.059 kW W& act

(c) The exergy destruction is

X& dest = m& T0 ( s 2 − s1 ) = (0.03959 kg/s)(300 K)(0.9802 − 0.9514)kJ/kg ⋅ K = 0.3417 kW The reversible power and the second-law efficiency are

W& rev = W& act − X& dest = 2.059 − 0.3417 = 1.717 kW

η Comp,II =

W& rev 1.717 kW = = 0.8341 = 83.4% W& act 2.059 kW


8-55

8-67 Refrigerant-134a is condensed in a refrigeration system by rejecting heat to ambient air. The rate of heat rejected, the COP of the refrigeration cycle, and the rate of exergy destruction are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. Analysis (a) The properties of refrigerant at the inlet and exit states of the condenser are (from R134a tables) P1 = 700 kPa ⎫ h1 = 288.53 kJ/kg ⎬ T1 = 50°C ⎭ s1 = 0.9954 kJ/kg ⋅ K P2 = 700 kPa ⎫ h2 = 88.82 kJ/kg ⎬ x2 = 0 ⎭ s 2 = 0.3323 kJ/kg ⋅ K

The rate of heat rejected in the condenser is

Q& H = m& R (h1 − h2 ) = (0.05 kg/s)(288.53 − 88.82)kJ/kg = 9.985 kW (b) From the definition of COP for a refrigerator, COP =

Q& L Q& L 6 kW = = = 1.506 & & & Win Q H − Q L (9.985 − 6) kW

(c) The entropy generation and the exergy destruction in the condenser are Q& S& gen = m& R ( s 2 − s1 ) + H TH 9.985 kW = 0.0003516 kW/K 298 K = (298 K)(0.00035 16 kJ/kg ⋅ K) = 0.1048 kW

= (0.05 kg/s)(0.3323 − 0.9954) kJ/kg ⋅ K + X& dest = T0 S& gen


8-56

8-68E Refrigerant-134a is evaporated in the evaporator of a refrigeration system. the rate of cooling provided, the rate of exergy destruction, and the second-law efficiency of the evaporator are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. Analysis (a) The rate of cooling provided is

Q& L = m& (h2 − h1 ) = (0.08 lbm/s)(172.1 − 107.5)Btu/lbm = 5.162 Btu/s = 18,580 Btu/h (b) The entropy generation and the exergy destruction are Q& S& gen = m& ( s 2 − s1 ) − L TL = (0.08 lbm/s)(0.4225 − 0.2851) Btu/lbm ⋅ R −

5.162 Btu/s (50 + 460) R

= 0.0008691 Btu/s ⋅ R X& dest = T0 S& gen = (537 R)(0.00086 91 Btu/s ⋅ R) = 0.4667 Btu/s

(c) The exergy supplied (or expended) during this cooling process is the exergy decrease of the refrigerant as it evaporates in the evaporator: X& 1 − X& 2 = m& (h1 − h2 ) − m& T0 ( s1 − s 2 ) = −5.162 − (0.08 lbm/s)(537 R)(0.2851 − 0.4225) Btu/lbm ⋅ R = 0.7400 Btu/s

The exergy efficiency is then

η II,Evap = 1 −

X& dest 0.4667 =1− = 0.3693 = 36.9% 0.7400 X& 1 − X& 2


8-57

8-69 Air is compressed steadily by a compressor from a specified state to another specified state. The reversible power is to be determined. Assumptions 1 Air is an ideal gas with variable specific heats. 2 Kinetic and potential energy changes are negligible. Properties The gas constant of air is R = 0.287 kJ/kg.K (Table A-1). From the air table (Table A-17) ⎯→ h1 = 300.19 kJ/kg T1 = 300 K ⎯

400 kPa 220°C

s1o = 1.702 kJ/kg ⋅ K ⎯→ h2 = 495.82 kJ/kg T2 = 493 K ⎯

AIR 0.15 kg/s

s 2o = 2.20499 kJ/kg ⋅ K

Analysis The reversible (or minimum) power input is determined from the rate form of the exergy balance applied on the compressor and setting the exergy destruction term equal to zero, X& − X& out 1in 4243



101 kPa 27°C


X& in = X& out m& ψ1 + W&rev,in = m& ψ 2 W&rev,in = m& (ψ 2 −ψ1) = m& [(h2 − h1) − T0 (s2 − s1) + ∆ke

0

+ ∆pe 0 ]

where s 2 − s1 = s 2o − s1o − R ln

P2 P1

= (2.205 − 1.702)kJ/kg ⋅ K − (0.287 kJ/kg ⋅ K)ln = 0.1080 kJ/kg ⋅ K

400 kPa 101 kPa

Substituting, W& rev,in = (0.15 kg/s)[(495.82 − 300.19)kJ/kg − (298 K)(0.1080 kJ/kg ⋅ K)] = 24.5 kW

Discussion Note that a minimum of 24.5 kW of power input is needed for this compression process.


8-58

8-70

Problem 8-69 is reconsidered. The effect of compressor exit pressure on reversible power is to be investigated.

Analysis The problem is solved using EES, and the solution is given below. T_1=27 [C] P_1=101 [kPa] m_dot = 0.15 [kg/s] {P_2=400 [kPa]} T_2=220 [C] T_o=25 [C] P_o=100 [kPa] m_dot_in=m_dot "Steady-flow conservation of mass" m_dot_in = m_dot_out h_1 =enthalpy(air,T=T_1) h_2 = enthalpy(air, T=T_2) W_dot_rev=m_dot_in*(h_2 - h_1 -(T_1+273.15)*(s_2-s_1)) s_1=entropy(air, T=T_1,P=P_1) s_2=entropy(air,T=T_2,P=P_2) Wrev [kW] 15.55 18.44 20.79 22.79 24.51 26.03 27.4 28.63 29.75

30 28 26

Wrev [kW]

P2 [kPa] 200 250 300 350 400 450 500 550 600

24 22 20 18 16 200

250

300

350

400

450

500

550

600

P2 [kPa]


8-59

8-71 A rigid tank initially contains saturated liquid of refrigerant-134a. R-134a is released from the vessel until no liquid is left in the vessel. The exergy destruction associated with this process is to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process. It can be analyzed as a uniform-flow process since the state of fluid leaving the device remains constant. 2 Kinetic and potential energies are negligible. 3 There are no work interactions involved. Properties The properties of R-134a are (Tables A-11 through A-13)

v 1 = v f @ 20°C = 0.0008161 m 3 / kg T1 = 20°C ⎫ ⎬ u1 = u f @ 20°C = 78.86 kJ/kg sat. liquid ⎭ s1 = s f @ 20°C = 0.30063 kJ/kg ⋅ K 3

v 2 = v g @ 20°C = 0.035969 m / kg T2 = 20°C ⎫ u 2 = u g @ 20°C = 241.02 kJ/kg ⎬ sat. vapor ⎭ s 2 = s e = s g @ 20°C = 0.92234 kJ/kg ⋅ K

R-134a 1 kg 20°C sat. liq.

he = h g @ 20°C = 261.59 kJ/kg Analysis The volume of the container is

me 3

V = m1v 1 = (1 kg)(0.0008161 m /kg) = 0.0008161 m

3

The mass in the container at the final state is m2 =

0.0008161 m 3 V = = 0.02269 kg v 2 0.035969 m 3 /kg

The amount of mass leaving the container is m e = m1 − m 2 = 1 − 0.02269 = 0.9773 kg

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen . The entropy generation Sgen in this case is determined from an entropy balance on the system: S − S out 1in424 3



Change in entropy

− m e s e + S gen = ∆S tank = (m 2 s 2 − m1 s1 ) tank S gen = m 2 s 2 − m1 s1 + m e s e

Substituting, X destroyed = T0 S gen = T0 (m 2 s 2 − m1 s1 + m e s e ) = (293 K)(0.02269 × 0.92234 − 1× 0.30063 + 0.9773 × 0.92234) = 182.2 kJ


8-60

8-72E An adiabatic rigid tank that is initially evacuated is filled by air from a supply line. The work potential associated with this process is to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process. It can be analyzed as a uniform-flow process since the state of fluid entering the device remains constant. 2 Kinetic and potential energies are negligible. 3 There are no work interactions involved. 4 Air is an ideal gas with constant specific heats. Properties The properties of air at room temperature are cp = 0.240 Btu/lbm⋅R, k = 1.4, and R = 0.06855 Btu/lbm⋅R = 0.3704 kPa⋅m3/lbm⋅R (Table A-2Ea). Analysis We take the tank as the system, which is a control volume since mass crosses the boundary. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances for this uniform-flow system can be expressed as

Mass balance:

min − mout = ∆msystem → mi = m 2

Air

150 psia, 90°F

Energy balance: E −E 1in424out 3


=

∆E system 1 424 3


m i hi = m 2 u 2

40 ft3

Combining the two balances:

hi = u 2 ⎯ ⎯→ c p Ti = cv T2 ⎯ ⎯→ T2 =

cp cv

Ti = kTi

Substituting, T2 = kTi = (1.4)(550 R ) = 770 R

The final mass in the tank is m 2 = mi =

(150 psia )(40 ft 3 ) PV = = 21.04 lbm RT2 (0.3704 psia ⋅ ft 3 /lbm ⋅ R )(770 R )

The work potential associated with this process is equal to the exergy destroyed during the process. The exergy destruction during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen . The entropy generation Sgen in this case is determined from an entropy balance on the system: S in − S out 1424 3



Change in entropy

m i s i + S gen = ∆S tank = m 2 s 2 S gen = m 2 s 2 − m i s i S gen = m 2 ( s 2 − s i )

Substituting, ⎛ T ⎞ Wrev = X destyroyed = m 2T0 ( s 2 − s i ) = m 2T0 ⎜⎜ c p ln 2 ⎟⎟ Ti ⎠ ⎝ 770 R ⎤ ⎡ = (21.04 lbm)(540 R) ⎢(0.240 Btu/lbm ⋅ R)ln 550 R ⎥⎦ ⎣ = 917 Btu


8-61

8-73E An rigid tank that is initially evacuated is filled by air from a supply line. The work potential associated with this process is to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process. It can be analyzed as a uniform-flow process since the state of fluid entering the device remains constant. 2 Kinetic and potential energies are negligible. 3 There are no work interactions involved. 4 Air is an ideal gas with constant specific heats. Properties The properties of air at room temperature are cp = 0.240 Btu/lbm⋅R and R = 0.06855 Btu/lbm⋅R = 0.3704 kPa⋅m3/lbm⋅R (Table A-2Ea). Analysis We take the tank as the system, which is a control volume since mass crosses the boundary. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances for this uniform-flow system can be expressed as

Mass balance: min − mout = ∆msystem → mi = m 2 Energy balance: E −E 1in424out 3


=

∆E system 1 424 3

Air

200 psia, 100°F


m i hi − Qout = m 2 u 2 Qout = mi hi − m 2 u 2

10 ft3

Combining the two balances: Qout = m 2 (hi − u 2 )

The final mass in the tank is m 2 = mi =

(150 psia )(40 ft 3 ) PV = = 29.45 lbm RT2 (0.3704 psia ⋅ ft 3 /lbm ⋅ R )(550 R )

Substituting, Qout = m 2 (hi − u 2 ) = m 2 (c p Ti − c v Ti ) = m 2Ti (c p − c v ) = m 2Ti R = (29.45 lbm)(550 R)(0.06855 Btu/lbm ⋅ R) = 1110 Btu

The work potential associated with this process is equal to the exergy destroyed during the process. The exergy destruction during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen . The entropy generation Sgen in this case is determined from an entropy balance on the system: S − S out 1in424 3


mi s i −


Change in entropy

Qout + S gen = ∆S tank = m 2 s 2 T0 S gen = m 2 s 2 − mi s i + S gen = m 2 ( s 2 − s i ) +

Qout T0

Qout T0

Noting that both the temperature and pressure in the tank is same as those in the supply line at the final state, substituting gives, ⎡ Q ⎤ Wrev = X destroyed = T0 ⎢m 2 ( s 2 − s i ) + out ⎥ T0 ⎦ ⎣ ⎛ Q = T0 ⎜⎜ 0 + out T0 ⎝

⎞ ⎛Q ⎟ = T0 ⎜ out ⎟ ⎜ T ⎠ ⎝ 0

⎞ ⎟ = Qout = 1110 Btu ⎟ ⎠


8-62

8-74 Steam expands in a turbine steadily at a specified rate from a specified state to another specified state. The power potential of the steam at the inlet conditions and the reversible power output are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 The temperature of the surroundings is given to be 25°C. Properties From the steam tables (Tables A-4 through 6)

8 MPa 450°C

P1 = 8 MPa ⎫ h1 = 3273.3 kJ/kg ⎬ T1 = 450°C ⎭ s1 = 6.5579 kJ/kg ⋅ K P2 = 50 kPa ⎫ h2 = 2645.2 kJ/kg ⎬ sat. vapor ⎭ s 2 = 7.5931 kJ/kg ⋅ K

STEAM 15,000 kg/h

P0 = 100 kPa ⎫ h0 ≅ h f @ 25°C = 104.83 kJ/kg ⎬ T0 = 25°C ⎭ s 0 ≅ s f @ 25°C = 0.36723 kJ/kg ⋅ K

Analysis (a) The power potential of the steam at the inlet conditions is equivalent to its exergy at the inlet state,

50 kPa sat. vapor

0 ⎞ ⎛ 0 V12 ⎜ & & & Ψ = mψ 1 = m⎜ h1 − h0 − T0 ( s1 − s 0 ) + + gz1 ⎟⎟ = m& (h1 − h0 − T0 ( s1 − s 0 ) ) 2 ⎟ ⎜ ⎠ ⎝ = (15,000 / 3600 kg/s)[(3273.3 − 104.83)kJ/kg − (298 K)(6.5579 - 0.36723) kJ/kg ⋅ K ]

= 5515 kW

(b) The power output of the turbine if there were no irreversibilities is the reversible power, is determined from the rate form of the exergy balance applied on the turbine and setting the exergy destruction term equal to zero, X& − X& out 1in 4243




X& in = X& out m& ψ = W& 1

rev,out

+ m& ψ 2

W&rev,out = m& (ψ1 −ψ 2 ) = m& [(h1 − h2 ) − T0 (s1 − s2 ) − ∆ke 0 − ∆pe 0 ]

Substituting, W& rev,out = m& [(h1 − h2 ) − T0 ( s1 − s 2 )]

= (15,000/3600 kg/s)[(3273.3 − 2645.2) kJ/kg − (298 K)(6.5579 − 7.5931) kJ/kg ⋅ K ] = 3902 kW


8-63

8-75E Air is compressed steadily by a 400-hp compressor from a specified state to another specified state while being cooled by the ambient air. The mass flow rate of air and the part of input power that is used to just overcome the irreversibilities are to be determined. Assumptions 1 Air is an ideal gas with variable specific heats. 2 Potential energy changes are negligible. 3 The temperature of the surroundings is given to be 60°F. Properties The gas constant of air is R = 0.06855 Btu/lbm.R (Table A-1E). From the air table (Table A-17E) T1 = 520 R T2 = 1080 R

h1 = 124.27 Btu/lbm s1o = 0.59173 Btu/lbm ⋅ R

} }

350 ft/s 150 psia 620°F

h2 = 260.97 Btu/lbm s 2o = 0.76964 Btu/lbm ⋅ R

Analysis (a) There is only one inlet and one exit, and thus &1 = m &2 = m & . We take the actual compressor as the system, which is a m

1500 Btu/min AIR 400 hp

control volume. The energy balance for this steady-flow system can be expressed as E& − E& 1in424out 3

=



=0


15 psia 60°F

E& in = E& out

⎛ V 2 − V12 W& a ,in + m& (h1 + V12 / 2) = m& (h2 + V 22 / 2) + Q& out → W& a ,in − Q& out = m& ⎜ h2 − h1 + 2 ⎜ 2 ⎝

⎞ ⎟ ⎟ ⎠

Substituting, the mass flow rate of the refrigerant becomes ⎛ ⎛ 0.7068 Btu/s ⎞ (350 ft/s) 2 − 0 1 Btu/lbm ⎟⎟ − (1500 / 60 Btu/s) = m& ⎜ 260.97 − 124.27 + (400 hp)⎜⎜ ⎜ 1 hp 2 25,037 ft 2 / s 2 ⎝ ⎠ ⎝

⎞ ⎟ ⎟ ⎠

It yields m& = 1.852 lbm / s (b) The portion of the power output that is used just to overcome the irreversibilities is equivalent to exergy destruction, which can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen where the entropy generation Sgen is determined from an entropy balance on an extended system that includes the device and its immediate surroundings. It gives S& − S& out 1in424 3


m& s1 − m& s 2 −

+

S& gen {



Q& out Q& + S& gen = 0 → S& gen = m& (s 2 − s1 ) + out Tb,surr T0

where s 2 − s1 = s 20 − s10 − R ln

P2 150 psia = (0.76964 − 0.59173) Btu/lbm − (0.06855 Btu/lbm.R) ln = 0.02007 Btu/lbm.R P1 15 psia

Substituting, the exergy destruction is determined to be ⎛ Q& X& destroyed = T0 S& gen = T0 ⎜ m& ( s 2 − s1 ) + out ⎜ T0 ⎝

⎞ ⎟ ⎟ ⎠

1 hp 1500 / 60 Btu/s ⎞⎛ ⎞ ⎛ = (520 R)⎜ (1.852 lbm/s)(0.02007 Btu/lbm ⋅ R) + ⎟ = 62.72 hp ⎟⎜ 520 R ⎝ ⎠⎝ 0.7068 Btu/s ⎠


8-64

8-76 Hot combustion gases are accelerated in an adiabatic nozzle. The exit velocity and the decrease in the exergy of the gases are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Potential energy changes are negligible. 3 The device is adiabatic and thus heat transfer is negligible. 4 The combustion gases are ideal gases with constant specific heats. Properties The constant pressure specific heat and the specific heat ratio are given to be cp = 1.15 kJ/kg.K and k = 1.3. The gas constant R is determined from

R = c p − cv = c p − c p / k = c p (1 − 1 / k ) = (1.15 kJ/kg ⋅ K)(1 − 1/1.3) = 0.2654 kJ/kg ⋅ K &1 = m &2 = m & . We take the nozzle as the system, which is a Analysis (a) There is only one inlet and one exit, and thus m control volume. The energy balance for this steady-flow system can be expressed as E& − E& 1in424out 3

=



=0


E& in = E& out

m& (h1 + V12

/ 2) = h2 =

m& (h2 + V22 /2) V 2 − V12 h1 − 2

(since W& = Q& ≅ ∆pe ≅ 0)

230 kPa 627°C 60 m/s

Comb. gases

70 kPa 450°C

2

Then the exit velocity becomes V2 = 2c p (T1 − T2 ) + V12 ⎛ 1000 m 2 /s 2 = 2(1.15 kJ/kg ⋅ K)(627 − 450)K⎜⎜ ⎝ 1 kJ/kg = 641 m/s

⎞ ⎟ + (60 m/s) 2 ⎟ ⎠

(b) The decrease in exergy of combustion gases is simply the difference between the initial and final values of flow exergy, and is determined to be

ψ 1 −ψ 2 = wrev = h1 − h2 − ∆ke − ∆pe

0

+ T0 ( s 2 − s1 ) = c p (T1 − T2 ) + T0 ( s 2 − s1 ) − ∆ke

where

∆ke =

V22 − V12 (641 m/s) 2 − (60 m/s) 2 ⎛ 1 kJ/kg ⎞ = ⎜ ⎟ = 203.6 kJ/kg 2 2 ⎝ 1000 m 2 / s 2 ⎠

and s 2 − s1 = c p ln

T2 P − R ln 2 T1 P1

= (1.15 kJ/kg ⋅ K) ln = 0.06386 kJ/kg ⋅ K


Substituting, Decrease in exergy = ψ 1 − ψ 2 = (1.15 kJ/kg ⋅ K)(627 − 450)°C + (293 K)(0.06386 kJ/kg ⋅ K) − 203.6 kJ/kg = 18.7 kJ/kg


8-65

8-77 Steam is accelerated in an adiabatic nozzle. The exit velocity of the steam, the isentropic efficiency, and the exergy destroyed within the nozzle are to be determined. Assumptions 1 The nozzle operates steadily. 2 The changes in potential energies are negligible. Properties The properties of steam at the inlet and the exit of the nozzle are (Tables A-4 through A-6) P1 = 7 MPa ⎫ h1 = 3411.4 kJ/kg ⎬ T1 = 500°C ⎭ s1 = 6.8000 kJ/kg ⋅ K

7 MPa 500°C 70 m/s

P2 = 5 MPa ⎫ h2 = 3317.2 kJ/kg ⎬ T2 = 450°C ⎭ s 2 = 6.8210 kJ/kg ⋅ K

STEAM

5 MPa 450°C

P2 s = 5 MPa ⎫ ⎬ h2 s = 3302.0 kJ/kg ⎭

s 2 s = s1

Analysis (a) We take the nozzle to be the system, which is a control volume. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3


=


=0


E& in = E& out

m& (h1 + V12 / 2) = m& (h2 + V22 /2) (since W& = Q& ≅ ∆pe ≅ 0) 0 = h2 − h1 +

V 22 − V12 2

Then the exit velocity becomes

⎛ 1000 m 2 /s 2 V 2 = 2(h1 − h2 ) + V12 = 2(3411.4 − 3317.2) kJ/kg⎜ ⎜ 1 kJ/kg ⎝

⎞ ⎟ + (70 m/s) 2 = 439.6 m/s ⎟ ⎠

(b) The exit velocity for the isentropic case is determined from ⎛ 1000 m 2 /s 2 V 2 s = 2(h1 − h2 s ) + V12 = 2(3411.4 − 3302.0) kJ/kg⎜ ⎜ 1 kJ/kg ⎝

⎞ ⎟ + (70 m/s) 2 = 472.9 m/s ⎟ ⎠

Thus,

ηN =

V 22 / 2 V 22s / 2

=

(439.6 m/s) 2 / 2 (472.9 m/s) 2 / 2

= 86.4%

(c) The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0S gen where the entropy generation Sgen is determined from an entropy balance on the actual nozzle. It gives S& in − S& out 1424 3

+


S& gen {



m& s1 − m& s 2 + S& gen = 0 → S& gen = m& (s 2 − s1 ) or

s gen = s 2 − s1

Substituting, the exergy destruction in the nozzle on a unit mass basis is determined to be

x destroyed = T0 s gen = T0 ( s 2 − s1 ) = (298 K)(6.8210 − 6.8000)kJ/kg ⋅ K = 6.28 kJ/kg


8-66

8-78 Air is compressed in a steady-flow device isentropically. The work done, the exit exergy of compressed air, and the exergy of compressed air after it is cooled to ambient temperature are to be determined. Assumptions 1 Air is an ideal gas with constant specific heats at room temperature. 2 The process is given to be reversible and adiabatic, and thus isentropic. Therefore, isentropic relations of ideal gases apply. 3 The environment temperature and pressure are given to be 300 K and 100 kPa. 4 The kinetic and potential energies are negligible. Properties The gas constant of air is R = 0.287 kJ/kg.K (Table A-1). The constant pressure specific heat and specific heat ratio of air at room temperature are cp = 1.005 kJ/kg.K and k = 1.4 (Table A-2). Analysis (a) From the constant specific heats ideal gas isentropic relations,

⎛P T2 = T1 ⎜⎜ 2 ⎝ P1

⎞ ⎟⎟ ⎠

( k −1) / k

⎛ 1000 kPa ⎞ ⎟⎟ = (300 K )⎜⎜ ⎝ 100 kPa ⎠

= 579.2 K

For a steady-flow isentropic compression process, the work input is determined from wcomp,in

{

1 MPa s2 = s1

0.4 / 1.4

}

kRT1 (P2 P1 )(k −1) / k − 1 = k −1 (1.4)(0.287kJ/kg ⋅ K )(300K ) (1000/100) 0.4/1.4 − 1 = 1.4 − 1 = 280.5 kJ/kg

{

}

AIR

100 kPa 300 K

(b) The exergy of air at the compressor exit is simply the flow exergy at the exit state,

ψ 2 = h 2 − h 0 − T0 ( s 2 − s 0 )

0

V2 + 2 2

0 0

+ gz 2 (since the proccess 0 - 2 is isentropic)

= c p (T2 − T0 ) = (1.005 kJ/kg.K)(579.2 - 300)K = 280.6 kJ/kg

which is the same as the compressor work input. This is not surprising since the compression process is reversible. (c) The exergy of compressed air at 1 MPa after it is cooled to 300 K is again the flow exergy at that state, V2 ψ 3 = h3 − h0 − T0 ( s 3 − s 0 ) + 3 2 = c p (T3 − T0 )

0

0

+ gz 3

0

− T0 ( s 3 − s 0 ) (since T3 = T0 = 300 K)

= −T0 ( s 3 − s 0 )

where T s 3 − s 0 = c p ln 3 T0

0

− R ln

P3 P 1000 kPa = − R ln 3 = −(0.287 kJ/kg ⋅ K)ln = −0.661 kJ/kg.K P0 P0 100 kPa

Substituting,

ψ 3 = − (300 K)(−0.661 kJ / kg.K) = 198 kJ / kg Note that the exergy of compressed air decreases from 280.6 to 198 as it is cooled to ambient temperature.


8-67

8-79 A rigid tank initially contains saturated R-134a vapor. The tank is connected to a supply line, and R-134a is allowed to enter the tank. The mass of the R-134a that entered the tank and the exergy destroyed during this process are to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process, but it can be analyzed as a uniform-flow process since the state of fluid at the inlet remains constant. 2 Kinetic and potential energies are negligible. 3 There are no work interactions involved. 4 The direction of heat transfer is to the tank (will be verified). Properties The properties of refrigerant are (Tables A-11 through A-13)

v 1 = v g @ 1.2 MPa = 0.01672 m 3 / kg P1 = 1.2 MPa ⎫ ⎬ u1 = u g @ 1.2 MPa = 253.81 kJ/kg sat. vapor ⎭s =s 1 g @ 1.2 MPa = 0.91303 kJ/kg ⋅ K

R-134a

v 2 = v f @ 1.4 MPa = 0.0009166 m 3 / kg T2 = 1.4 MPa ⎫ ⎬ u 2 = u f @ 1.4 MPa = 125.94 kJ/kg sat. liquid ⎭s =s 2 f @ 1.4 MPa = 0.45315 kJ/kg ⋅ K Pi = 1.6 MPa ⎫ hi = 93.56 kJ/kg ⎬ Ti = 30°C ⎭ s i = 0.34554 kJ/kg ⋅ K

1.6 MPa 30°C

R-134a 0.1 m3 1.2 MPa Sat. vapor

Q

Analysis We take the tank as the system, which is a control volume. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances for this uniform-flow system can be expressed as

Mass balance:

min − m out = ∆msystem → mi = m 2 − m1 Energy balance: E −E 1in424out 3

=


∆E system 1 424 3


Qin + mi hi = m 2 u 2 − m1u1 (since W ≅ ke ≅ pe ≅ 0)

(a) The initial and the final masses in the tank are m1 =

V1 0.1 m 3 = = 5.983 kg v 1 0.01672 m 3 /kg

m2 =

V2 0.1 m 3 = = 109.10 kg v 2 0.0009166 m 3 /kg

Then from the mass balance mi = m 2 − m1 = 109.10 − 5.983 = 103.11 kg

The heat transfer during this process is determined from the energy balance to be Qin = − mi hi + m 2 u 2 − m1u1

= −(103.11 kg )(93.56 kJ/kg) + (109.10)(125.94 kJ/kg ) − (5.983 kg )(253.81 kJ/kg ) = 2573 kJ


8-68

(b) The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen . The entropy generation Sgen in this case is determined from an entropy balance on an extended system that includes the tank and its immediate surroundings so that the boundary temperature of the extended system is the surroundings temperature Tsurr at all times. It gives S in − S out 1424 3



Change in entropy

Qin + mi s i + S gen = ∆S tank = (m 2 s 2 − m1 s1 ) tank Substituting, the exergy destruction Tb,in S gen = m 2 s 2 − m1 s1 − mi s i −

Qin T0

is determined to be ⎡ Q ⎤ X destroyed = T0 S gen = T0 ⎢m 2 s 2 − m1 s1 − m i s i − in ⎥ T0 ⎦ ⎣ = (318 K)[109.10 × 0.45315 − 5.983 × 0.91303 − 103.11 × 0.34554 − (2573 kJ)/(318 K)] = 80.3 kJ


8-69

8-80 A rigid tank initially contains saturated liquid water. A valve at the bottom of the tank is opened, and half of mass in liquid form is withdrawn from the tank. The temperature in the tank is maintained constant. The amount of heat transfer, the reversible work, and the exergy destruction during this process are to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process, but it can be analyzed as a uniform-flow process since the state of fluid leaving the device remains constant. 2 Kinetic and potential energies are negligible. 3 There are no work interactions involved. 4 The direction of heat transfer is to the tank (will be verified). Properties The properties of water are (Tables A-4 through A-6) 3

v 1 = v f @170o C = 0.001114 m /kg T1 = 170°C ⎫ ⎬ u1 = u f @170o C = 718.20 kJ/kg sat. liquid ⎭ s1 = s f @170o C = 2.0417 kJ/kg ⋅ K Te = 170°C ⎫ he = h f @170o C = 719.08 kJ/kg ⎬ sat. liquid ⎭ s e = s f @170o C = 2.0417 kJ/kg ⋅ K

H2O 0.6 m3 170°C T = const.

Q

me

Analysis We take the tank as the system, which is a control volume since mass crosses the boundary. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances for this uniform-flow system can be expressed as

Mass balance:

min − m out = ∆msystem → m e = m1 − m 2 Energy balance: E −E 1in424out 3

=


∆E system 1 424 3


Qin = m e he + m 2 u 2 − m1u1 (since W ≅ ke ≅ pe ≅ 0)

The initial and the final masses in the tank are m1 =

0.6 m 3 V = = 538.47 kg v 1 0.001114 m 3 /kg

m2 =

1 1 m1 = (538.47 kg ) = 269.24 kg = m e 2 2

Now we determine the final internal energy and entropy,

v2 = x2 =

V m2

=

0.6 m 3 = 0.002229 m 3 /kg 269.24 kg

v 2 −v f v fg

=

0.002229 − 0.001114 = 0.004614 0.24260 − 0.001114

⎫ u 2 = u f + x 2 u fg = 718.20 + (0.004614 )(1857.5) = 726.77 kJ/kg ⎬ x 2 = 0.004614 ⎭ s 2 = s f + x 2 s fg = 2.0417 + (0.004614 )(4.6233) = 2.0630 kJ/kg ⋅ K

T2 = 170°C

The heat transfer during this process is determined by substituting these values into the energy balance equation, Qin = m e he + m 2 u 2 − m1u1 = (269.24 kg )(719.08 kJ/kg ) + (269.24 kg )(726.77 kJ/kg ) − (538.47 kg )(718.20 kJ/kg ) = 2545 kJ


8-70

(b) The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen . The entropy generation Sgen in this case is determined from an entropy balance on an extended system that includes the tank and the region between the tank and the source so that the boundary temperature of the extended system at the location of heat transfer is the source temperature Tsource at all times. It gives S in − S out 1424 3



Change in entropy

Qin − m e s e + S gen = ∆S tank = (m 2 s 2 − m1 s1 ) tank Tb,in S gen = m 2 s 2 − m1 s1 + m e s e −

Qin Tsource

Substituting, the exergy destruction is determined to be ⎡ Qin ⎤ X destroyed = T0 S gen = T0 ⎢m 2 s 2 − m1 s1 + m e s e − ⎥ Tsource ⎦ ⎣ = (298 K)[269.24 × 2.0630 − 538.47 × 2.0417 + 269.24 × 2.0417 − (2545 kJ)/(523 K)] = 141.2 kJ

For processes that involve no actual work, the reversible work output and exergy destruction are identical. Therefore,

X destroyed = Wrev,out − Wact,out → Wrev,out = X destroyed = 141.2 kJ


8-71

8-81E An insulated rigid tank equipped with an electric heater initially contains pressurized air. A valve is opened, and air is allowed to escape at constant temperature until the pressure inside drops to 20 psia. The amount of electrical work done and the exergy destroys are to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process, but it can be analyzed as a uniform-flow process since the exit temperature (and enthalpy) of air remains constant. 2 Kinetic and potential energies are negligible. 3 The tank is insulated and thus heat transfer is negligible. 4 Air is an ideal gas with variable specific heats. 5 The environment temperature is given to be 70°F. Properties The gas constant of air is R = 0.3704 psia.ft3/lbm.R (Table A-1E). The properties of air are (Table A-17E)

Te = 640 R ⎯ ⎯→ he = 153.09 Btu/lbm T1 = 640 R ⎯ ⎯→ u1 = 109.21 Btu/lbm

T2 = 640 R ⎯ ⎯→ u 2 = 109.21 Btu/lbm

Analysis We take the tank as the system, which is a control volume. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances for this uniform-flow system can be expressed as min − m out = ∆msystem → m e = m1 − m 2 Mass balance:


=


∆E system 1 424 3


We,in − m e he = m 2 u 2 − m1u1 (since Q ≅ ke ≅ pe ≅ 0) The initial and the final masses of air in the tank are m1 =

P1V (40 psia)(260 ft 3 ) = = 43.86 lbm RT1 (0.3704 psia ⋅ ft 3 /lbm ⋅ R)(640 R )

m2 =

P2V (20 psia)(260 ft 3 ) = = 21.93 lbm RT2 (0.3704 psia ⋅ ft 3 /lbm ⋅ R)(640 R )

AIR 260 ft3 40 psia 180°F

We

Then from the mass and energy balances, me = m1 − m 2 = 43.86 − 21.93 = 21.93 lbm

We,in = me he + m 2 u 2 − m1u1 = (21.93 lbm)(153.09 Btu/lbm) + (21.93 lbm)(109.21 Btu/lbm) − (43.86 lbm)(109.21 Btu/lbm) = 962 Btu (b) The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen where the entropy generation Sgen is determined from an entropy balance on the insulated tank. It gives S − S out 1in424 3



Change in entropy

− m e s e + S gen = ∆S tank = (m 2 s 2 − m1 s1 ) tank S gen = m 2 s 2 − m1 s1 + m e s e = m 2 s 2 − m1 s1 + (m1 − m 2 ) s e = m 2 ( s 2 − s e ) − m1 ( s1 − s e )

Assuming a constant average pressure of (40 + 20) / 2 = 30 psia for the exit stream, the entropy changes are determined to be 0

T P 20 psia = 0.02779 Btu/lbm ⋅ R s 2 − s e = c p ln 2 − R ln 2 = −(0.06855 Btu/lbm ⋅ R) ln 30 psia Te Pe 0

s1 − s e = c p ln

T1 P 40 psia − R ln 1 = −(0.06855 Btu/lbm ⋅ R) ln = −0.01972 Btu/lbm ⋅ R 30 psia Te Pe

Substituting, the exergy destruction is determined to be X destroyed = T0 S gen = T0 [m 2 ( s 2 − s e ) − m1 ( s1 − s e )] = (530 R)[(21.93 lbm)(0.02779 Btu/lbm ⋅ R) − (43.86 lbm)(−0.01972 Btu/lbm ⋅ R)] = 782 Btu PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

8-72

8-82 A cylinder initially contains helium gas at a specified pressure and temperature. A valve is opened, and helium is allowed to escape until its volume decreases by half. The work potential of the helium at the initial state and the exergy destroyed during the process are to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process, but it can be analyzed as a uniform-flow process by using constant average properties for the helium leaving the tank. 2 Kinetic and potential energies are negligible. 3 There are no work interactions involved other than boundary work. 4 The tank is insulated and thus heat transfer is negligible. 5 Helium is an ideal gas with constant specific heats. Properties The gas constant of helium is R = 2.0769 kPa.m3/kg.K = 2.0769 kJ/kg.K. The specific heats of helium are cp = 5.1926 kJ/kg.K and cv = 3.1156 kJ/kg.K (Table A-2). Analysis (a) From the ideal gas relation, the initial and the final masses in the cylinder are determined to be P1V (300 kPa)(0.1 m 3 ) = = 0.0493 kg RT1 (2.0769 kPa ⋅ m 3 /kg ⋅ K)(293 K )

m1 =

m e = m 2 = m1 / 2 = 0.0493 / 2 = 0.0247 kg

The work potential of helium at the initial state is simply the initial exergy of helium, and is determined from the closed-system exergy relation, Φ 1 = m1φ = m1 [(u1 − u 0 ) − T0 ( s1 − s 0 ) + P0 (v 1 − v 0 )]

HELIUM 300 kPa 0.1 m3 20°C

Q

where

v1 =

RT1 (2.0769 kPa ⋅ m 3 /kg ⋅ K)(293 K ) = = 2.0284 m 3 /kg P1 300 kPa

v0 =

RT0 (2.0769 kPa ⋅ m 3 /kg ⋅ K)(293 K ) = = 6.405 m 3 /kg P0 95 kPa

and s1 − s 0 = c p ln

T1 P − R ln 1 T0 P0

= (5.1926 kJ/kg ⋅ K) ln = −2.388 kJ/kg ⋅ K


Thus, Φ 1 = (0.0493 kg){(3.1156 kJ/kg ⋅ K)(20 − 20)°C − (293 K)(−2.388 kJ/kg ⋅ K) + (95 kPa)(2.0284 − 6.405)m 3 /kg[kJ/kPa ⋅ m 3 ]} = 14.0 kJ

(b) We take the cylinder as the system, which is a control volume. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances for this uniform-flow system can be expressed as Mass balance:

min − m out = ∆msystem → m e = m1 − m 2 Energy balance: E −E 1in424out 3


=

∆E system 1 424 3


Qin − m e he + W b,in = m 2 u 2 − m1u1 Combining the two relations gives


8-73

Qin = (m1 − m 2 )he + m 2 u 2 − m1u1 − W b,in = (m1 − m 2 )he + m 2 h2 − m1 h1 = (m1 − m 2 + m 2 − m1 )h1 =0 since the boundary work and ∆U combine into ∆H for constant pressure expansion and compression processes. The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen where the entropy generation Sgen can be determined from an entropy balance on the cylinder. Noting that the pressure and temperature of helium in the cylinder are maintained constant during this process and heat transfer is zero, it gives S − S out 1in424 3



Change in entropy

− m e s e + S gen = ∆S cylinder = (m 2 s 2 − m1 s1 ) cylinder S gen = m 2 s 2 − m1 s1 + m e s e = m 2 s 2 − m1 s1 + (m1 − m 2 ) s e = (m 2 − m1 + m1 − m 2 ) s1 =0

since the initial, final, and the exit states are identical and thus se = s2 = s1. Therefore, this discharge process is reversible, and

X destroyed = T0 Sgen = 0


8-74

8-83 A rigid tank initially contains saturated R-134a vapor at a specified pressure. The tank is connected to a supply line, and R-134a is allowed to enter the tank. The amount of heat transfer with the surroundings and the exergy destruction are to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process, but it can be analyzed as a uniform-flow process since the state of fluid at the inlet remains constant. 2 Kinetic and potential energies are negligible. 3 There are no work interactions involved. 4 The direction of heat transfer is from the tank (will be verified). Properties The properties of refrigerant are (Tables A-11 through A-13) u1 = u g @ 1 MPa = 250.68 kJ/kg P1 = 1 MPa ⎫ 1.4 MPa R-134a ⎬ s1 = s g @ 1 MPa = 0.91558 kJ/kg ⋅ K 60°C sat.vapor ⎭ v 1 = v g @ 1 MPa = 0.020313 m 3 / kg Pi = 1.4 MPa ⎫ hi = 285.47 kJ/kg ⎬ Ti = 60°C ⎭ s i = 0.93889 kJ/kg ⋅ K Analysis (a) We take the tank as the system, which is a control volume since mass crosses the boundary. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances for this uniform-flow system can be expressed as min − m out = ∆msystem → mi = m 2 − m1 Mass balance:


=


R-134a 0.2 m3 1 MPa Sat. vapor

Q

∆E system 1 424 3


mi hi − Qout = m 2 u 2 − m1u1 (since W ≅ ke ≅ pe ≅ 0) The initial and the final masses in the tank are 0 .2 m 3 V m1 = = = 9.846 kg v 1 0.020313 m 3 / kg m2 = m f + m g =

Vf

Vg

vg

=

0.1 m 3

+

0.1 m 3

= 111.93 + 5.983 = 117.91 kg 0.0008934 m 3 / kg 0.016715 m 3 / kg U 2 = m 2 u 2 = m f u f + m g u g = 111.93 × 116.70 + 5.983 × 253.81 = 14,581 kJ S 2 = m 2 s 2 = m f s f + m g s g = 111.93 × 0.42441 + 5.983 × 0.91303 = 52.967 kJ/K

vf

+

Then from the mass and energy balances, mi = m 2 − m1 = 117.91 − 9.846 = 108.06 kg The heat transfer during this process is determined from the energy balance to be Qout = mi hi − m 2 u 2 + m1u1 = 108.06 × 285.47 − 14,581 + 9.846 × 250.68 = 18,737 kJ (b) The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen . The entropy generation Sgen in this case is determined from an entropy balance on an extended system that includes the cylinder and its immediate surroundings so that the boundary temperature of the extended system is the surroundings temperature Tsurr at all times. It gives S − S out + S gen = ∆S system 1in424 3 { 1 424 3 Net entropy transfer by heat and mass

−

Entropy generation

Change in entropy

Qout + mi s i + S gen = ∆S tank = (m 2 s 2 − m1 s1 ) tank Tb,out S gen = m 2 s 2 − m1 s1 − mi s i +

Qout T0

Substituting, the exergy destruction is determined to be ⎡ Q ⎤ X destroyed = T0 S gen = T0 ⎢m 2 s 2 − m1 s1 − mi s i + out ⎥ T0 ⎦ ⎣ = (298 K)[52.967 − 9.846 × 0.91558 − 108.06 × 0.93889 + 18,737 / 298] = 1599 kJ PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

8-75

8-84 An insulated cylinder initially contains saturated liquid-vapor mixture of water. The cylinder is connected to a supply line, and the steam is allowed to enter the cylinder until all the liquid is vaporized. The amount of steam that entered the cylinder and the exergy destroyed are to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process, but it can be analyzed as a uniform-flow process since the state of fluid at the inlet remains constant. 2 The expansion process is quasi-equilibrium. 3 Kinetic and potential energies are negligible. 4 The device is insulated and thus heat transfer is negligible. Properties The properties of steam are (Tables A-4 through A-6) ⎫ h1 = h f + x1 h fg = 561.43 + 0.8667 × 2163.5 = 2436.5 kJ/kg ⎬ x1 = 13 / 15 = 0.8667 ⎭ s1 = s f + x1 s fg = 1.6716 + 0.8667 × 5.3200 = 6.2824 kJ/kg ⋅ K

P1 = 300 kPa

P2 = 300 kPa ⎫ h2 = h g @ 300 kPa = 2724.9 kJ/kg ⎬ sat.vapor ⎭ s 2 = s g @ 300 kPa = 6.9917 kJ/kg ⋅ K Pi = 2 MPa ⎫ hi = 3248.4 kJ/kg ⎬ Ti = 400°C ⎭ s i = 7.1292 kJ/kg ⋅ K

Analysis (a) We take the cylinder as the system, which is a control volume. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances for this unsteady-flow system can be expressed as

H 2O 300 kPa P = const.

2 MPa 400°C

min − m out = ∆msystem → mi = m 2 − m1

Mass balance: Energy balance:

E −E 1in424out 3

=


∆E system 1 424 3


mi hi = W b,out + m 2 u 2 − m1u1 (since Q ≅ ke ≅ pe ≅ 0) Combining the two relations gives 0 = W b,out − (m 2 − m1 )hi + m 2 u 2 − m1u1 0 = −(m 2 − m1 )hi + m 2 h2 − m1 h1

or,

since the boundary work and ∆U combine into ∆H for constant pressure expansion and compression processes. Solving for m2 and substituting, m2 =

Thus,

hi − h1 (3248.4 − 2436.5)kJ/kg (15 kg) = 23.27 kg m1 = (3248.4 − 2724.9)kJ/kg hi − h2

mi = m 2 − m1 = 23.27 − 15 = 8.27 kg

(b) The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen where the entropy generation Sgen is determined from an entropy balance on the insulated cylinder, S − S out 1in424 3



Change in entropy

mi s i + S gen = ∆S system = m 2 s 2 − m1 s1 S gen = m 2 s 2 − m1 s1 − mi s i

Substituting, the exergy destruction is determined to be X destroyed = T0 S gen = T0 [m 2 s 2 − m1 s1 − mi s i ] = (298 K)(23.27 × 6.9917 − 15 × 6.2824 − 8.27 × 7.1292) = 2832 kJ


8-76

8-85 Each member of a family of four takes a shower every day. The amount of exergy destroyed by this family per year is to be determined. Assumptions 1 Steady operating conditions exist. 2 The kinetic and potential energies are negligible. 3 Heat losses from the pipes, mixing section are negligible and thus Q& ≅ 0. 4 Showers operate at maximum flow conditions during the entire shower. 5 Each member of the household takes a shower every day. 6 Water is an incompressible substance with constant properties at room temperature. 7 The efficiency of the electric water heater is 100%. Properties The density and specific heat of water are at room temperature are ρ = 997 kg/m3 and c = 4.18 kJ/kg.°C (Table A-3). Analysis The mass flow rate of water at the shower head is

m& = ρV& = (0.997 kg/L)(10 L/min) = 9.97 kg/min The mass balance for the mixing chamber can be expressed in the rate form as

m& in − m& out = ∆m& system

0 (steady)

= 0 → m& in = m& out → m& 1 + m& 2 = m& 3

where the subscript 1 denotes the cold water stream, 2 the hot water stream, and 3 the mixture. The rate of entropy generation during this process can be determined by applying the rate form of the entropy balance on a system that includes the electric water heater and the mixing chamber (the T-elbow). Noting that there is no entropy transfer associated with work transfer (electricity) and there is no heat transfer, the entropy balance for this steadyflow system can be expressed as S& in − S& out 1424 3


+

S& gen {


= ∆S& system 0 (steady) 1442443 Rate of change of entropy

m& 1 s1 + m& 2 s 2 − m& 3 s 3 + S& gen = 0 (since Q = 0 and work is entropy free) S& gen = m& 3 s 3 − m& 1 s1 − m& 2 s 2

Noting from mass balance that m& 1 + m& 2 = m& 3 and s2 = s1 since hot water enters the system at the same temperature as the cold water, the rate of entropy generation is determined to be T S& gen = m& 3 s 3 − ( m& 1 + m& 2 ) s1 = m& 3 ( s 3 − s1 ) = m& 3 c p ln 3 T1 = (9.97 kg/min)(4.18 kJ/kg.K) ln

42 + 273 = 3.735 kJ/min.K 15 + 273

Noting that 4 people take a 6-min shower every day, the amount of entropy generated per year is S gen = ( S& gen ) ∆t ( No. of people)(No. of days) = (3.735 kJ/min.K)(6 min/person ⋅ day)(4 persons)(365 days/year) = 32,715 kJ/K (per year)

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X destroyed = T0 S gen = ( 298 K)(32,715 kJ/K ) = 9,749,000 kJ

Discussion The value above represents the exergy destroyed within the water heater and the T-elbow in the absence of any heat losses. It does not include the exergy destroyed as the shower water at 42°C is discarded or cooled to the outdoor temperature. Also, an entropy balance on the mixing chamber alone (hot water entering at 55°C instead of 15°C) will exclude the exergy destroyed within the water heater.


8-77

8-86 Liquid water is heated in a chamber by mixing it with superheated steam. For a specified mixing temperature, the mass flow rate of the steam and the rate of exergy destruction are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 There are no work interactions. Properties Noting that T < Tsat @ 200 kPa = 120.23°C, the cold water and the exit mixture streams exist as a compressed liquid, which can be approximated as a saturated liquid at the given temperature. From Tables A-4 through A-6, P1 = 200 kPa ⎫ h1 ≅ h f @15o C = 62.98 kJ/kg ⎬s ≅ s = 0.22447 kJ/kg ⋅ K T1 = 15°C f @15o C ⎭ 1 P2 = 200 kPa ⎫ h2 = 2870.4 kJ/kg ⎬ ⎭ s 2 = 7.5081 kJ/kg ⋅ K

T2 = 200°C

1

P3 = 200 kPa ⎫ h3 ≅ h f @ 80°C = 335.02 kJ/kg ⎬ T3 = 80°C ⎭ s 3 ≅ s f @80°C = 1.0756 kJ/kg ⋅ K

Analysis (a) We take the mixing chamber as the system, which is a control volume. The mass and energy balances for this steady-flow system can be expressed in the rate form as


Mass balance:

600 kJ/min

0 (steady)

15°C 4 kg/s

MIXING CHAMBER 200 kPa

80°C

3

2 200°C

=0 ⎯ ⎯→ m& 1 + m& 2 = m& 3

Energy balance: E& − E& 1in424out 3



=

=0


E& in = E& out m& h1 + m& 2 h2 = Q& out + m& 3 h3

Combining the two relations gives

Q& out = m& 1 h1 + m& 2 h2 − (m& 1 + m& 2 )h3 = m& 1 (h1 − h3 ) + m& 2 (h2 − h3 )

& 2 and substituting, the mass flow rate of the superheated steam is determined to be Solving for m

m& 2 = Also,

Q& out − m& 1 (h1 − h3 ) (600/60 kJ/s) − (4 kg/s )(62.98 − 335.02)kJ/kg = = 0.429 kg/s (2870.4 − 335.02)kJ/kg h2 − h3

m& 3 = m& 1 + m& 2 = 4 + 0.429 = 4.429 kg/s

(b) The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen where the entropy generation Sgen is determined from an entropy balance on an extended system that includes the mixing chamber and its immediate surroundings. It gives S& in − S& out 1424 3


m& 1 s1 + m& 2 s 2 − m& 3 s 3 −

+

S& gen {



Q& Q& out + S& gen = 0 → S& gen = m& 3 s 3 − m& 1 s1 − m& 2 s 2 + out T0 Tb,surr

Substituting, the exergy destruction is determined to be ⎞ ⎛ Q& X& destroyed = T0 S& gen = T0 ⎜ m& 3 s 3 − m& 2 s 2 − m& 1 s1 + out ⎟ ⎜ Tb, surr ⎟⎠ ⎝ = (298 K)(4.429 × 1.0756 − 0.429 × 7.5081 − 4 × 0.22447 + 10 / 298)kW/K = 202 kW


8-78

8-87 Air is preheated by hot exhaust gases in a cross-flow heat exchanger. The rate of heat transfer and the rate of exergy destruction in the heat exchanger are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 Fluid properties are constant. Properties The specific heats of air and combustion gases are given to be 1.005 and 1.10 kJ/kg.°C, respectively. The gas constant of air is R = 0.287 kJ/kg.K (Table A-1). Analysis We take the exhaust pipes as the system, which is a control volume. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3


=


=0


Air 101 kPa 30°C 0.5 m3/s

E& in = E& out m& h1 = Q& out + m& h2 (since ∆ke ≅ ∆pe ≅ 0) Q& out = m& C p (T1 − T2 )

Then the rate of heat transfer from the exhaust gases becomes

Exhaust gases 1.1 kg/s, 190°C

Q& = [m& c p (Tin − Tout )]gas. = (1.1 kg/s)(1.1 kJ/kg.°C)(240°C − 190°C) = 60.5 kW

The mass flow rate of air is m& =

(101 kPa)(0.5 m 3 /s) PV& = = 0.5807 kg/s RT (0.287 kPa.m 3 /kg.K) × 303 K

Noting that heat loss by exhaust gases is equal to the heat gain by the air, the air exit temperature becomes

Q& 60.5 kW Q& = m& C p (Tout − Tin ) air → Tout = Tin + = 30°C + = 133.7°C m& c p (0.5807 kg/s)(1.005 kJ/kg.°C)

[

]

The rate of entropy generation within the heat exchanger is determined by applying the rate form of the entropy balance on the entire heat exchanger: S& − S& out 1in424 3

+


S& gen {



m& 1 s1 + m& 3 s 3 − m& 2 s 2 − m& 3 s 4 + S& gen = 0 (since Q = 0) m& exhaust s1 + m& air s 3 − m& exhaust s 2 − m& air s 4 + S& gen = 0 S& gen = m& exhaust ( s 2 − s1 ) + m& air ( s 4 − s 3 ) Noting that the pressure of each fluid remains constant in the heat exchanger, the rate of entropy generation is

T T S& gen = m& exhaust c p ln 2 + m& air c p ln 4 T3 T1 = (1.1 kg/s)(1.1 kJ/kg.K)ln

190 + 273 133.7 + 273 + (0.5807 kg/s)(1.005 kJ/kg.K)ln 240 + 273 30 + 273

= 0.04765 kW/K

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X& destroyed = T0 S& gen = (303 K)(0.04765 kW/K ) = 14.4 kW


8-79

8-88 Water is heated by hot oil in a heat exchanger. The outlet temperature of the oil and the rate of exergy destruction within the heat exchanger are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 Fluid properties are constant. Properties The specific heats of water and oil are given to be 4.18 and 2.3 kJ/kg.°C, respectively. Analysis We take the cold water tubes as the system, which is a control volume. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

=



=0


Oil 170°C 10 kg/s

70°C

E& in = E& out Q& in + m& h1 = m& h2 (since ∆ke ≅ ∆pe ≅ 0) Q& in = m& c p (T2 − T1 )

Water 20°C 4.5 kg/s

(12 tube passes)

Then the rate of heat transfer to the cold water in this heat exchanger becomes Q& = [ m& c p (Tout − Tin )] water = ( 4.5 kg/s)(4.18 kJ/kg. °C)(70°C − 20°C) = 940.5 kW

Noting that heat gain by the water is equal to the heat loss by the oil, the outlet temperature of the hot water is determined from

Q& 940.5 kW = 170°C − = 129.1°C Q& = [m& c p (Tin − Tout )] oil → Tout = Tin − m& c p (10 kg/s)(2.3 kJ/kg.°C) (b) The rate of entropy generation within the heat exchanger is determined by applying the rate form of the entropy balance on the entire heat exchanger: S& − S& out 1in424 3


+

S& gen {



m& 1 s1 + m& 3 s 3 − m& 2 s 2 − m& 3 s 4 + S& gen = 0 (since Q = 0) m& water s1 + m& oil s 3 − m& water s 2 − m& oil s 4 + S& gen = 0 S& gen = m& water ( s 2 − s1 ) + m& oil ( s 4 − s 3 ) Noting that both fluid streams are liquids (incompressible substances), the rate of entropy generation is determined to be T T S& gen = m& water c p ln 2 + m& oil c p ln 4 T1 T3 = (4.5 kg/s)(4.18 kJ/kg.K) ln

70 + 273 129.1 + 273 + (10 kg/s)(2.3 kJ/kg.K) ln = 0.736 kW/K 20 + 273 170 + 273

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X& destroyed = T0 S& gen = ( 298 K)(0.736 kW/K ) = 219 kW


8-80

8-89E Steam is condensed by cooling water in a condenser. The rate of heat transfer and the rate of exergy destruction within the heat exchanger are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 Fluid properties are constant. 5 The temperature of the environment is 77°F. Properties The specific heat of water is 1.0 Btu/lbm.°F (Table A-3E). The enthalpy and entropy of vaporization of water at 120°F are 1025.2 Btu/lbm and sfg = 1.7686 Btu/lbm.R (Table A-4E). Analysis We take the tube-side of the heat exchanger where cold water is flowing as the system, which is a control volume. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

=



=0

Steam 120°F


E& in = E& out

73°F

Q& in + m& h1 = m& h2 (since ∆ke ≅ ∆pe ≅ 0) Q& in = m& c p (T2 − T1 )

Then the rate of heat transfer to the cold water in this heat exchanger becomes Q& = [m& c p (Tout − Tin )] water

60°F

= (115.3 lbm/s)(1.0 Btu/lbm.°F)(73°F − 60°F) = 1499 Btu/s

Noting that heat gain by the water is equal to the heat loss by the condensing steam, the rate of condensation of the steam in the heat exchanger is determined from

Water 120°F

Q& 1499 Btu/s ⎯→ m& steam = = = 1.462 lbm/s Q& = (m& h fg ) steam = ⎯ h fg 1025.2 Btu/lbm (b) The rate of entropy generation within the heat exchanger is determined by applying the rate form of the entropy balance on the entire heat exchanger: S& − S& out 1in424 3

+


S& gen {



m& 1 s1 + m& 3 s 3 − m& 2 s 2 − m& 4 s 4 + S& gen = 0 (since Q = 0) m& water s1 + m& steam s 3 − m& water s 2 − m& steam s 4 + S& gen = 0 S& gen = m& water ( s 2 − s1 ) + m& steam ( s 4 − s 3 ) Noting that water is an incompressible substance and steam changes from saturated vapor to saturated liquid, the rate of entropy generation is determined to be T T S& gen = m& water c p ln 2 + m& steam ( s f − s g ) = m& water c p ln 2 − m& steam s fg T1 T1 = (115.3 lbm/s)(1.0 Btu/lbm.R)ln

73 + 460 − (1.462 lbm/s)(1.7686 Btu/lbm.R) = 0.2613 Btu/s.R 60 + 460

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X& destroyed = T0 S& gen = (537 R)(0.2613 Btu/s.R ) = 140.3 Btu/s


8-81

8-90 Steam expands in a turbine, which is not insulated. The reversible power, the exergy destroyed, the second-law efficiency, and the possible increase in the turbine power if the turbine is well insulated are to be determined. Assumptions 1 Steady operating conditions exist. 2 Potential energy change is negligible. Analysis (a) The properties of the steam at the inlet and exit of the turbine are (Tables A-4 through A-6)

P1 = 9 MPa ⎫ h1 = 3634.1 kJ/kg ⎬ T1 = 600°C ⎭ s1 = 6.9605 kJ/kg.K P2 = 20 kPa ⎫ h2 = 2491.1 kJ/kg ⎬ x 2 = 0.95 ⎭ s 2 = 7.5535 kJ/kg.K

Steam 9 MPa 600°C, 60 m/s

The enthalpy at the dead state is

Turbine

T0 = 25°C ⎫ ⎬h0 = 104.83 kJ/kg x=0 ⎭

The mass flow rate of steam may be determined from an energy balance on the turbine ⎛ V2 m& ⎜ h1 + 1 ⎜ 2 ⎝ ⎡ (60 m/s) 2 ⎛ 1 kJ/kg m& ⎢3634.1 kJ/kg + ⎜ 2 ⎝ 1000 m 2 /s 2 ⎣⎢

Q

20 kPa 130 m/s x = 0.95

2 ⎞ ⎞ ⎛ ⎟ = m& ⎜ h2 + V 2 ⎟ + Q& out + W& a ⎟ ⎜ 2 ⎟⎠ ⎠ ⎝

⎡ (130 m/s) 2 ⎛ 1 kJ/kg ⎞⎤ ⎟⎥ = m& ⎢2491.1 kJ/kg + ⎜ 2 ⎠⎦⎥ ⎝ 1000 m 2 /s 2 ⎣⎢

⎞⎤ ⎟⎥ ⎠⎦⎥

+ 220 kW + 4500 kW ⎯ ⎯→ m& = 4.137 kg/s

The reversible power may be determined from ⎡ V 2 - V 22 ⎤ W& rev = m& ⎢h1 − h2 − T0 ( s1 − s 2 ) + 1 ⎥ 2 ⎦⎥ ⎣⎢ ⎡ (60 m/s) 2 − (130 m/s) 2 ⎛ 1 kJ/kg = (2.693) ⎢(3634.1 − 2491.1) − (298)(6.9605 - 7.5535) + ⎜ 2 ⎝ 1000 m 2 /s 2 ⎢⎣ = 5451 kW

⎞⎤ ⎟⎥ ⎠⎥⎦

(b) The exergy destroyed in the turbine is

X& dest = W& rev − W& a = 5451 − 4500 = 951 kW (c) The second-law efficiency is W&

4500 kW

η II = & a = = 0.826 Wrev 5451 kW (d) The energy of the steam at the turbine inlet in the given dead state is

Q& = m& (h1 − h0 ) = (4.137 kg/s)(3634.1 - 104.83)kJ/kg = 14,602 kW The fraction of energy at the turbine inlet that is converted to power is 4500 kW W& f = &a = = 0.3082 Q 14,602 kW

Assuming that the same fraction of heat loss from the turbine could have been converted to work, the possible increase in the power if the turbine is to be well-insulated becomes

W& increase = fQ& out = (0.3082)(220 kW) = 67.8 kW


8-82

8-91 Air is compressed in a compressor that is intentionally cooled. The actual and reversible power inputs, the second law efficiency, and the mass flow rate of cooling water are to be determined. Assumptions 1 Steady operating conditions exist. 2 Potential energy change is negligible. 3 Air is an ideal gas with constant specific heats.

900 kPa 60°C 80 m/s

Properties The gas constant of air is R = 0.287 kJ/kg.K and the specific heat of air at room is cp = 1.005 kJ/kg.K. the specific heat of water at room temperature is cw = 4.18 kJ/kg.K (Tables A-2, A-3).

Compressor

Analysis (a) The mass flow rate of air is P (100 kPa) m& = ρV&1 = 1 V&1 = (4.5 m 3 /s) = 5.351 kg/s (0.287 kJ/kg.K)(20 + 273 K) RT1

Q

Air 100 kPa 20°C

The power input for a reversible-isothermal process is given by P ⎛ 900 kPa ⎞ W& rev = m& RT1 ln 2 = (5.351 kg/s)(0.287 kJ/kg.K)(20 + 273 K)ln⎜ ⎟ = 988.8 kW P1 ⎝ 100 kPa ⎠

Given the isothermal efficiency, the actual power may be determined from W& 988.8 kW = 1413 kW W& actual = rev = 0.70 ηT

(b) The given isothermal efficiency is actually the second-law efficiency of the compressor

η II = η T = 0.70 (c) An energy balance on the compressor gives ⎡ V 2 − V22 ⎤ & Q& out = m& ⎢c p (T1 − T2 ) + 1 ⎥ + Wactual,in 2 ⎣⎢ ⎦⎥ ⎡ 0 − (80 m/s) 2 ⎛ 1 kJ/kg = (5.351 kg/s)⎢(1.005 kJ/kg.°C)(20 − 60)°C + ⎜ 2 ⎝ 1000 m 2 /s 2 ⎣ = 1181 kW

⎞⎤ ⎟⎥ + 1413 kW ⎠⎦

The mass flow rate of the cooling water is

m& w =

Q& out 1181 kW = = 28.25 kg/s c w ∆T (4.18 kJ/kg.°C)(10°C)


8-83

8-92 Water is heated in a chamber by mixing it with saturated steam. The temperature of the steam entering the chamber, the exergy destruction, and the second-law efficiency are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. 3 Heat loss from the chamber is negligible. Analysis (a) The properties of water are (Tables A-4 through A-6)

T1 = 15°C⎫ h1 = h0 = 62.98 kJ/kg ⎬ x1 = 0 ⎭ s1 = s 0 = 0.22447 kJ/kg.K T3 = 45°C⎫ h3 = 188.44 kJ/kg ⎬ x1 = 0 ⎭ s3 = 0.63862 kJ/kg.K

Water 15°C 4.6 kg/s

Mixing chamber Mixture 45°C

Sat. vap. 0.23 kg/s

An energy balance on the chamber gives m& 1h1 + m& 2h2 = m& 3h3 = (m& 1 + m& 2 )h3

(4.6 kg/s)(62.98 kJ/kg) + (0.23 kg/s)h2 = (4.6 + 0.23 kg/s)(188.44 kJ/kg) h2 = 2697.5 kJ/kg

The remaining properties of the saturated steam are h2 = 2697.5 kJ/kg ⎫ T2 = 114.3 °C ⎬ x2 = 1 ⎭ s 2 = 7.1907 kJ/kg.K

(b) The specific exergy of each stream is

ψ1 = 0 ψ 2 = h2 − h0 − T0 ( s2 − s0 ) = (2697.5 − 62.98)kJ/kg − (15 + 273 K)(7.1907 − 0.22447)kJ/kg.K = 628.28 kJ/kg

ψ 3 = h3 − h0 − T0 ( s3 − s0 ) = (188.44 − 62.98)kJ/kg − (15 + 273 K)(0.63862 − 0.22447)kJ/kg.K = 6.18 kJ/kg

The exergy destruction is determined from an exergy balance on the chamber to be X& dest = m& 1ψ 1 + m& 2ψ 2 − (m& 1 + m& 2 )ψ 3 = 0 + (0.23 kg/s)(628.28 kJ/kg) − (4.6 + 0.23 kg/s)(6.18 kJ/kg) = 114.7 kW

(c) The second-law efficiency for this mixing process may be determined from

ηII =

(m& 1 + m& 2 )ψ 3 (4.6 + 0.23 kg/s)(6.18 kJ/kg) = = 0.207 & & m1ψ 1 + m2ψ 2 0 + (0.23 kg/s)(628.28 kJ/kg)


8-84

8-93 An expression is to be derived for the work potential of the single-phase contents of a rigid adiabatic container when the initially empty container is filled through a single opening from a source of working fluid whose properties remain fixed. Analysis The conservation of mass principle for this system reduces to dm CV = m& i dt

where the subscript i stands for the inlet state. When the entropy generation is set to zero (for calculating work potential) and the combined first and second law is reduced to fit this system, it becomes d (U − T0 S ) W& rev = − + (h − T0 S ) i m& i dt

When these are combined, the result is d (U − T0 S ) dm CV + ( h − T0 S ) i W& rev = − dt dt

Recognizing that there is no initial mass in the system, integration of the above equation produces W rev = (h − T0 s ) i m 2 − m 2 (h2 − T0 s 2 ) W rev = (hi − h2 ) − T0 ( s i − s 2 ) m2

where the subscript 2 stands for the final state in the container.


8-85

Review Problems

8-94E The 2nd-law efficiency of a refrigerator and the refrigeration rate are given. The power input to the refrigerator is to be determined. Analysis From the definition of the second law efficiency, the COP of the refrigerator is determined to be COPR,rev =

η II =

1 1 = = 7.462 TH / TL − 1 550 / 485 − 1

90°F

COPR ⎯ ⎯→ COPR = η II COPR,rev = 0.28 × 7.462 = 2.089 COPR, rev

Thus the power input is

η II = 0.28 R 800 Btu/min

Q& L 1 hp 800 Btu/min ⎛ ⎞ W& in = = ⎟ = 9.03 hp ⎜ COPR 2.089 42.41 Btu/min ⎠ ⎝

25°F

8-95 Refrigerant-134a is expanded adiabatically in an expansion valve. The work potential of R-134a at the inlet, the exergy destruction, and the second-law efficiency are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. Analysis (a) The properties of the refrigerant at the inlet and exit of the valve and at dead state are (Tables A-11 through A13) P1 = 0.9 MPa ⎫ h1 = 93.57 kJ/kg ⎬ ⎭ s1 = 0.34751 kJ/kg.K P2 = 120 kPa ⎫ ⎬s 2 = 0.37614 kJ/kg.K h2 = h1 = 93.57 kJ/kg ⎭

T1 = 30°C

R-134a 0.9 MPa 30°C

120 kPa

P0 = 100 kPa ⎫ h0 = 272.17 kJ/kg ⎬ T0 = 20°C ⎭ s 0 = 1.0918 kJ/kg.K

The specific exergy of the refrigerant at the inlet and exit of the valve are

ψ 1 = h1 − h0 − T0 ( s1 − s 0 ) = (93.57 − 272.17)kJ/kg − (20 + 273.15 K)(0.34751 − 1.0918)kJ/kg ⋅ K = 39.59 kJ/kg

ψ 2 = h2 − h0 − T0 ( s 2 − s 0 ) = (93.57 − 272.17)kJ/kg − (20 + 273.15 K)(0.37614 − 1.0918 kJ/kg.K = 31.20 kJ/kg

(b) The exergy destruction is determined to be x dest = T0 ( s 2 − s1 ) = (20 + 273.15 K)(037614 − 0.34751)kJ/kg ⋅ K = 8.39 kJ/kg

(c) The second-law efficiency for this process may be determined from

η II =

ψ 2 31.20 kJ/kg = = 0.788 = 78.8% ψ 1 39.59 kJ/kg


8-86

8-96 Steam is accelerated in an adiabatic nozzle. The exit velocity, the rate of exergy destruction, and the second-law efficiency are to be determined. Assumptions 1 Steady operating conditions exist. 2 Potential energy changes are negligible. Analysis (a) The properties of the steam at the inlet and exit of the turbine and at the dead state are (Tables A-4 through A6) P1 = 3.5 MPa ⎫ h1 = 2978.4 kJ/kg ⎬ T1 = 300°C ⎭ s1 = 6.4484 kJ/kg.K

1.6 MPa 250°C V2

Steam 3.5 MPa 300°C

P2 = 1.6 kPa ⎫ h2 = 2919.9 kJ/kg ⎬ T2 = 250°C ⎭ s 2 = 6.6753 kJ/kg.K T0 = 18°C⎫ h0 = 75.54 kJ/kg ⎬ x=0 ⎭ s 0 = 0.2678 kJ/kg.K

The exit velocity is determined from an energy balance on the nozzle h1 + 2978.4 kJ/kg +

V12 V2 = h2 + 2 2 2

V22 ⎛ 1 kJ/kg ⎞ (0 m/s) 2 ⎛ 1 kJ/kg ⎞ = + 2919 . 9 kJ/kg ⎜ ⎟ ⎜ ⎟ 2 2 ⎝ 1000 m 2 /s 2 ⎠ ⎝ 1000 m 2 /s 2 ⎠ V 2 = 342.0 m/s

(b) The rate of exergy destruction is the exergy decrease of the steam in the nozzle ⎡ ⎤ V 2 − V12 X& dest = m& ⎢h2 − h1 + 2 − T0 ( s 2 − s1 ⎥ 2 ⎣⎢ ⎦⎥ ⎡ (342 m/s) 2 − 0 ⎛ 1 kJ/kg ⎜ ⎢(2919.9 − 2978.4)kJ/kg + = (0.4 kg/s) ⎢ 2 ⎝ 1000 m 2 /s 2 ⎢− (291 K )(6.6753 − 6.4484)kJ/kg.K ⎣

⎞⎤ ⎟⎥ ⎠⎥ ⎥ ⎦

= 26.41 kW

(c) The exergy of the refrigerant at the inlet is ⎡ ⎤ V2 X& 1 = m& ⎢h1 − h0 + 1 − T0 ( s1 − s0 ⎥ 2 ⎢⎣ ⎥⎦ = (0.4 kg/s)[(2978.4 − 75.54) kJ/kg + 0 − (291 K )(6.4484 − 0.2678)kJ/kg.K ] = 441.72 kW

The second-law efficiency for this device may be defined as the exergy output divided by the exergy input:

η II =

X& X& 2 26.41 kW = 1 − dest = 1 − = 0.940 & & 441.72 kW X1 X1


8-87

8-97 R-134a is expanded in an adiabatic process with an isentropic efficiency of 0.85. The second law efficiency is to be determined. Assumptions 1 Kinetic and potential energy changes are negligible. 2 The device is adiabatic and thus heat transfer is negligible. Analysis We take the R-134a 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 −E 1in424out 3


=

∆E system 1 424 3


− Wout = ∆U = m(u 2 − u1 )

T

From the R-134a tables (Tables A-11 through A-13),

1 1.6 MPa

3

v = 0.014362 m /kg P1 = 1600 kPa ⎫ 1 ⎬ u1 = 282.09 kJ/kg T1 = 80°C ⎭ s = 0.9875 kJ/kg ⋅ K 1

100 kPa 2s 2

P2 = 100 kPa ⎫ ⎬ u 2 s = 223.16 kJ/kg s 2 s = s1 ⎭

s

The actual work input is w a ,out = η T w s ,out = η T (u1 − u 2 s ) = (0.85)( 282.09 − 223.16) kJ/kg = 50.09 kJ/kg

The actual internal energy at the end of the expansion process is w a ,out = (u1 − u 2 ) ⎯ ⎯→ u 2 = u1 − w a ,out = 282.09 − 50.09 = 232.00 kJ/kg

Other actual properties at the final state are (Table A-13) P2 = 100 kPa

⎫ v 2 = 0.2139 m 3 /lbm ⎬ u 2 = 232.00 kJ/kg ⎭ s 2 = 1.0251 kJ/kg ⋅ K

The useful work is determined from wu = wa ,out − wsurr = wa ,out − P0 (v 2 − v 1 ) ⎛ 1 kJ ⎞ = 50.09 kJ/kg − (100 kPa )(0.2139 − 0.014362) m 3 /kg⎜ ⎟ ⎝ 1 kPa ⋅ m 3 ⎠ = 30.14 kJ/kg The exergy change between initial and final states is

φ1 − φ 2 = u1 − u 2 + P0 (v 1 − v 2 ) − T0 ( s1 − s 2 ) ⎛ 1 kJ ⎞ = (282.09 − 232.00)kJ/kg + (100 kPa )(0.014362 − 0.2139) m 3 /kg⎜ ⎟ ⎝ 1 kPa ⋅ m 3 ⎠ − (298 K )(0.9875 − 1.0251)kJ/kg ⋅ K = 41.34 kJ/kg


η II =

wu 30.14 kJ/kg = = 0.729 ∆φ 41.34 kJ/kg


8-88

8-98 Steam is condensed in a closed system at a constant pressure from a saturated vapor to a saturated liquid by rejecting heat to a thermal energy reservoir. The second law efficiency is to be determined. Assumptions 1 Kinetic and potential energy changes are negligible. Analysis We take the steam 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 −E 1in424out 3

=


∆E system 1 424 3


Steam 75 kPa Sat. vapor

Wb,in − Qout = ∆U = m(u 2 − u1 ) From the steam tables (Table A-5),

v 1 = v g = 2.2172 m 3 /kg P1 = 75 kPa ⎫ ⎬ u1 = u g = 2496.1 kJ/kg Sat. vapor ⎭ s1 = s g = 7.4558 kJ/kg ⋅ K

q

T

v 2 = v f = 0.001037 m 3 /kg P2 = 75 kPa ⎫ ⎬ u 2 = u f = 384.36 kJ/kg Sat. liquid ⎭ s 2 = s f = 1.2132 kJ/kg ⋅ K

2

75 kPa

1 s

The boundary work during this process is ⎛ 1 kJ ⎞ wb,in = P(v 1 − v 2 ) = (75 kPa )(2.2172 − 0.001037 ) m 3 /kg ⎜ ⎟ = 166.2 kJ/kg 3 ⎝ 1 kPa ⋅ m ⎠

The heat transfer is determined from the energy balance: q out = wb,in − (u 2 − u1 ) = 166.2 kJ/kg − (384.36 − 2496 .1) kJ/kg = 2278 kJ/kg

The exergy change between initial and final states is ⎛

φ1 − φ 2 = u1 − u 2 + P0 (v 1 − v 2 ) − T0 ( s1 − s 2 ) − q out ⎜⎜1 − ⎝

T0 TR

⎞ ⎟⎟ ⎠

⎛ 1 kJ ⎞ = (2496.1 − 384.36)kJ/kg + (100 kPa )(2.2172 − 0.001037) m 3 /kg⎜ ⎟ ⎝ 1 kPa ⋅ m 3 ⎠ 298 K ⎞ ⎛ − (298 K )(7.4558 − 1.2132)kJ/kg ⋅ K − (2278 kJ/kg)⎜1 − ⎟ ⎝ 310 K ⎠ = 384.9 kJ/kg The second law efficiency is then

η II =

wb,in ∆φ

=

166.2 kJ/kg = 0.432 = 43.2% 384.9 kJ/kg


8-89

8-99 R-134a is vaporized in a closed system at a constant pressure from a saturated liquid to a saturated vapor by transferring heat from a reservoir at two pressures. The pressure that is more effective from a second-law point of view is to be determined. Assumptions 1 Kinetic and potential energy changes are negligible. Analysis We take the R-134a 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 −E 1in424out 3


=

∆E system 1 424 3


R-134a 100 kPa sat. liquid

Qin − Wb,out = ∆U = m(u 2 − u1 ) Qin = Wb,out + ∆U Qin = ∆H = m(h2 − h1 )

q

At 100 kPa: T

From the R-134a tables (Table A-12), u fg @ 100 kPa = 197.98 kJ/kg h fg @ 100 kPa = 217.16 kJ/kg

1 100 kPa

s fg @ 100 kPa = 0.87995 kJ/kg ⋅ K

v fg @ 100 kPa = v g − v f = 0.19254 − 0.0007259 = 0.19181 m 3 /kg

2 s

The boundary work during this process is wb ,out = P (v 2 − v 1 ) = Pv

fg

⎛ 1 kJ ⎞ = (100 kPa )(0.19181) m 3 /kg ⎜ ⎟ = 19.18 kJ/kg ⎝ 1 kPa ⋅ m 3 ⎠

The useful work is determined from wu = wb,out − wsurr = P (v 2 − v 1 ) − P0 (v 2 − v 1 ) = 0 kJ/kg

since P = P0 = 100 kPa. The heat transfer from the energy balance is

q in = h fg = 217.16 kJ/kg The exergy change between initial and final states is ⎛

φ1 − φ 2 = u1 − u 2 + P0 (v 1 − v 2 ) − T0 ( s1 − s 2 ) + q in ⎜⎜1 − ⎝

⎛ T = −u fg − P0v fg + T0 s fg + q in ⎜⎜1 − 0 ⎝ TR

T0 TR

⎞ ⎟⎟ ⎠

⎞ ⎟⎟ ⎠

⎛ 1 kJ ⎞ = −197.98 kJ/kg − (100 kPa )(0.19181 m 3 /kg)⎜ ⎟ + (298 K )(0.87995 kJ/kg ⋅ K) ⎝ 1 kPa ⋅ m 3 ⎠ ⎛ 298 K ⎞ + (217.16 kJ/kg)⎜1 − ⎟ ⎝ 273 K ⎠ = 25.18 kJ/kg


η II =

wu 0 kJ/kg = =0 ∆φ 25.18 kJ/kg


8-90

At 200 kPa: u fg @ 200 kPa = 186.21 kJ/kg h fg @ 200 kPa = 206.03 kJ/kg s fg @ 200 kPa = 0.78316 kJ/kg ⋅ K

v fg @ 200 kPa = v g − v f = 0.099867 − 0.0007533 = 0.099114 m 3 /kg ⎛ 1 kJ ⎞ wb ,out = P (v 2 − v 1 ) = Pv fg = (200 kPa )(0.099114 ) m 3 /kg ⎜ ⎟ = 19.82 kJ/kg ⎝ 1 kPa ⋅ m 3 ⎠

wu = wb,out − wsurr = P(v 2 − v 1 ) − P0 (v 2 − v 1 ) ⎛ 1 kJ ⎞ = ( P − P0 )v fg = (200 − 100) kPa (0.099114) m 3 /kg⎜ ⎟ = 9.911 kJ/kg ⎝ 1 kPa ⋅ m 3 ⎠

q in = h fg = 206.03 kJ/kg ⎛

φ1 − φ 2 = u1 − u 2 + P0 (v 1 − v 2 ) − T0 ( s1 − s 2 ) + q in ⎜⎜1 − ⎝

= −u fg − P0v fg + T0 s fg

⎛ T + q in ⎜⎜1 − 0 ⎝ TR

T0 TR

⎞ ⎟⎟ ⎠

⎞ ⎟⎟ ⎠

⎛ 1 kJ ⎞ = −186.21 kJ/kg − (100 kPa )(0.099114 m 3 /kg)⎜ ⎟ + (298 K )(0.78316 kJ/kg ⋅ K) ⎝ 1 kPa ⋅ m 3 ⎠ ⎛ 298 K ⎞ + (206.03 kJ/kg)⎜1 − ⎟ ⎝ 273 K ⎠ = 18.39 kJ/kg

η II =

wu 9.911 kJ/kg = = 0.539 ∆φ 18.39 kJ/kg

The process at 200 kPa is more effective from a work production standpoint.


8-91

8-100 An electrical radiator is placed in a room and it is turned on for a period of time. The time period for which the heater was on, the exergy destruction, and the second-law efficiency are to be determined. Assumptions 1 Kinetic and potential energy changes are negligible. 2 Air is an ideal gas with constant specific heats. 3 The room is well-sealed. 4 Standard atmospheric pressure of 101.3 kPa is assumed. Properties The properties of air at room temperature are R = 0.287 kPa.m3/kg.K, cp = 1.005 kJ/kg.K, cv = 0.718 kJ/kg.K (Table A-2). The properties of oil are given to be ρ = 950 kg/m3, coil = 2.2 kJ/kg.K. Analysis (a) The masses of air and oil are P1V (101.3 kPa)(75 m 3 ) = = 94.88 kg RT1 (0.287 kPa ⋅ m 3 /kg ⋅ K)(6 + 273 K)

ma =

6°C

Room

Q

Radiator

moil = ρ oilV oil = (950 kg/m 3 )(0.050 m 3 ) = 47.50 kg An energy balance on the system can be used to determine time period for which the heater was kept on (W& in − Q& out )∆t = [mcv (T2 − T1 )]a + [mc(T2 − T1 )]oil

(2.4 − 0.75 kW) ∆t = [(94.88 kg)(0.718 kJ/kg.°C)(20 − 6)°C] + [(47.50 kg)(2.2 kJ/kg.°C)(60 − 6)°C] ∆t = 3988 s = 66.6 min

(b) The pressure of the air at the final state is Pa 2 =

m a RTa 2

V

=

(94.88 kg)(0.287 kPa ⋅ m 3 /kg ⋅ K)(20 + 273 K) 75 m 3

= 106.4 kPa

The amount of heat transfer to the surroundings is

Qout = Q& out ∆t = (0.75 kJ/s)(3988 s) = 2999 kJ The entropy generation is the sum of the entropy changes of air, oil, and the surroundings ⎡ T P ⎤ ∆S a = m ⎢c p ln 2 − R ln 2 ⎥ T P1 ⎦ 1 ⎣ ⎡ (20 + 273) K 106.4 kPa ⎤ = (94.88 kg) ⎢(1.005 kJ/kg.K)ln − (0.287 kJ/kg.K)ln ⎥ (6 + 273) K 101.3 kPa ⎦ ⎣ = 3.335 kJ/K T (60 + 273) K ∆S oil = mc ln 2 = (47.50 kg)(2.2 kJ/kg.K)ln = 18.49 kJ/K T1 (6 + 273) K ∆S surr =

Qout 2999 kJ = = 10.75 kJ/K Tsurr (6 + 273) K

S gen = ∆S a + ∆S oil + ∆S surr = 3.335 + 18.49 + 10.75 = 32.57 kJ/K

The exergy destruction is determined from

X dest = T0 S gen = (6 + 273 K)(32.57 kJ/K) = 9088 kJ = 9.09 MJ (c) The second-law efficiency may be defined in this case as the ratio of the exergy recovered to the exergy input. That is, X a , 2 = m[cv (T2 − T1 )] − T0 ∆S a

= (94.88 kg)[(0.718 kJ/kg.°C)(20 − 6)°C] − (6 + 273 K)(3.335 kJ/K) = 23.16 kJ

X oil, 2 = m[C (T2 − T1 )] − T0 ∆S a

= (47.50 kg)[(2.2 kJ/kg.°C)(60 − 6)°C] − (6 + 273 K)(18.49 kJ/K) = 484.5 kJ

η II =

X recovered X a ,2 + X oil,2 (23.16 + 484.5) kJ = = 0.0529 = 5.3% = X supplied (2.4 kJ/s)(3998 s) W& in ∆t


8-92

8-101 Hot exhaust gases leaving an internal combustion engine is to be used to obtain saturated steam in an adiabatic heat exchanger. The rate at which the steam is obtained, the rate of exergy destruction, and the second-law efficiency are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. 3 Air properties are used for exhaust gases. 4 Pressure drops in the heat exchanger are negligible. Properties The gas constant of air is R = 0.287 kJkg.K. The specific heat of air at the average temperature of exhaust gases (650 K) is cp = 1.063 kJ/kg.K (Table A-2). Analysis (a) We denote the inlet and exit states of exhaust gases by (1) and (2) and that of the water by (3) and (4). The properties of water are (Table A-4)

T3 = 20°C⎫ h3 = 83.91 kJ/kg ⎬ x3 = 0 ⎭ s3 = 0.29649 kJ/kg.K T4 = 200°C⎫ h4 = 2792.0 kJ/kg ⎬ x4 = 1 ⎭ s 4 = 6.4302 kJ/kg.K An energy balance on the heat exchanger gives

Exh. gas 400°C 150 kPa

Heat Exchanger

Sat. vap. 200°C

350°C Water 20°C

m& a h1 + m& wh3 = m& a h2 + m& wh4 m& a c p (T1 − T2 ) = m& w (h4 − h3 ) (0.8 kg/s)(1.063 kJ/kg°C)(400 − 350)°C = m& w (2792.0 − 83.91)kJ/kg m& w = 0.01570 kg/s

(b) The specific exergy changes of each stream as it flows in the heat exchanger is ∆s a = c p ln

T2 (350 + 273) K = (0.8 kg/s)(1.063 kJ/kg.K)ln = −0.08206 kJ/kg.K T1 (400 + 273) K

∆ψ a = c p (T2 − T1 ) − T0 ∆sa = (1.063 kJ/kg.°C)(350 - 400)°C − (20 + 273 K)(-0.08206 kJ/kg.K) = −29.106 kJ/kg

∆ψ w = h4 − h3 − T0 ( s4 − s3 ) = (2792.0 − 83.91)kJ/kg − (20 + 273 K)(6.4302 − 0.29649)kJ/kg.K = 910.913 kJ/kg

The exergy destruction is determined from an exergy balance on the heat exchanger to be − X& dest = m& a ∆ψ a + m& w∆ψ w = (0.8 kg/s)(-29.106 kJ/kg) + (0.01570 kg/s)(910.913) kJ/kg = −8.98 kW

or X& dest = 8.98 kW

(c) The second-law efficiency for a heat exchanger may be defined as the exergy increase of the cold fluid divided by the exergy decrease of the hot fluid. That is,

η II =

(0.01570 kg/s)(910.913 kJ/kg) m& w∆ψ w = = 0.614 − m& a ∆ψ a − (0.8 kg/s)(-29.106 kJ/kg)


8-93

8-102 The inner and outer surfaces of a window glass are maintained at specified temperatures. The amount of heat loss and the amount of exergy destruction in 5 h are to be determined Assumptions Steady operating conditions exist since the surface temperatures of the glass remain constant at the specified values. Analysis We take the glass to be the system, which is a closed system. The amount of heat loss is determined from

Q = Q& ∆t = (4.4 kJ/s)(5 × 3600 s) = 79,200 k J Glass

Under steady conditions, the rate form of the entropy balance for the glass simplifies to S& in − S& out 1424 3


+

S& gen {



Q& Q& in − out + S& gen,glass = 0 Tb,in Tb,out

10°C

3°C

4400 W 4400 W & − + S gen, wall = 0 → S& gen,glass = 0.3943 W/K 283 K 276 K

Then the amount of entropy generation over a period of 5 h becomes S gen,glass = S& gen,glass ∆t = (0.3943 W/K)(5 × 3600 s) = 7098 J/K

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen ,

X destroyed = T0 S gen = (278 K)(7.098 kJ/K) = 1973 kJ Discussion The total entropy generated during this process can be determined by applying the entropy balance on an extended system that includes the glass and its immediate surroundings on both sides so that the boundary temperature of the extended system is the room temperature on one side and the environment temperature on the other side at all times. Using this value of entropy generation will give the total exergy destroyed during the process, including the temperature gradient zones on both sides of the window.


8-94

8-103 Heat is transferred steadily to boiling water in the pan through its bottom. The inner and outer surface temperatures of the bottom of the pan are given. The rate of exergy destruction within the bottom plate is to be determined. Assumptions Steady operating conditions exist since the surface temperatures of the pan remain constant at the specified values. Analysis We take the bottom of the pan to be the system, which is a closed system. Under steady conditions, the rate form of the entropy balance for this system can be expressed as S& in − S& out 1424 3

+


S& gen {



Q& Q& in − out + S& gen,system = 0 Tb,in Tb,out 1100 W 1100 W & − + S gen,system = 0 378 K 377 K S& gen,system = 0.007719 W/K

104°C

1100 W 105°C

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X& destroyed = T0 S& gen = (298 K)(0.007719 W/K ) = 2.30 W

8-104 Elevation, base area, and the depth of a crater lake are given. The maximum amount of electricity that can be generated by a hydroelectric power plant 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 potential energy it possesses relative to the ground level,

Exergy = PE = mgh Therefore,

12 m

∫

∫

∫

dz

Exergy = PE = dPE = gz dm = gz ( ρAdz ) = ρAg

∫

z2

z1

zdz = ρAg ( z22 − z12 ) / 2

= 0.5(1000 kg/m3 )(2 × 10 4 m 2 )(9.81 m/s 2 )

(

)

⎛ 1 h ⎞⎛ 1 kJ/kg ⎞ × (152 m) − (140 m ) ⎜ ⎟⎜ ⎟ 2 2 ⎝ 3600 s ⎠⎝ 1000 m / s ⎠ 2

2

140 m

z

= 9.55 × 104 kWh


8-95

8-105 An electric resistance heater is immersed in water. The time it will take for the electric heater to raise the water temperature to a specified temperature, the minimum work input, and the exergy destroyed during this process are to be determined. Assumptions 1 Water is an incompressible substance with constant specific heats. 2 The energy stored in the container itself and the heater is negligible. 3 Heat loss from the container is negligible. 4 The environment temperature is given to be T0 = 20°C. Properties The specific heat of water at room temperature is c = 4.18 kJ/kg·°C (Table A-3). Analysis Taking the water in the container as the system, which is a closed system, the energy balance can be expressed as

E −E 1in424out 3

=


∆E system 1 424 3


We,in = (∆U ) water W& e,in ∆t = mc(T2 − T1 ) water Substituting,

Water 40 kg

Heater

(800 J/s)∆t = (40 kg)(4180 J/kg·°C)(80 - 20)°C Solving for ∆t gives ∆t = 12,540 s = 209 min = 3.48 h Again we take the water in the tank to be the system. Noting that no heat or mass crosses the boundaries of this system and the energy and entropy contents of the heater are negligible, the entropy balance for it can be expressed as S in − S out 1424 3



Change in entropy

0 + S gen = ∆S water Therefore, the entropy generated during this process is S gen = ∆S water = mc ln

T2 353 K = (40 kg )(4.18 kJ/kg ⋅ K ) ln = 31.15 kJ/K T1 293 K


X destroyed = T0 S gen = (293 K)(31.15 kJ/K) = 9127 kJ The actual work input for this process is Wact,in = W& act,in ∆t = (0.8 kJ/s)(12,540 s) = 10,032 kJ

Then the reversible (or minimum required )work input becomes

Wrev,in = Wact,in − X destroyed = 10,032 − 9127 = 906 kJ


8-96

8-106 A hot water pipe at a specified temperature is losing heat to the surrounding air at a specified rate. The rate at which the work potential is wasted during this process is to be determined. Assumptions Steady operating conditions exist. Analysis We take the air in the vicinity of the pipe (excluding the pipe) as our system, which is a closed system.. The system extends from the outer surface of the pipe to a distance at which the temperature drops to the surroundings temperature. In steady operation, the rate form of the entropy balance for this system can be expressed as S& in − S& out 1424 3


+

S& gen {


= ∆S& system 0 = 0 14243

80°C


Q& Q& in − out + S& gen,system = 0 Tb,in Tb,out

D = 5 cm L = 10 m

1175 W 1175 W & − + S gen,system = 0 → S& gen,system = 0.8980 W/K 353 K 278 K

Air, 5°C

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X& destroyed = T0 S& gen = ( 278 K)(0.8980 W/K ) = 250 W


8-97

8-107 Air expands in an adiabatic turbine from a specified state to another specified state. The second-law efficiency is to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 The device is adiabatic and thus heat transfer is negligible. 3 Air is an ideal gas with constant specific heats. 4 Kinetic and potential energy changes are negligible. Properties At the average temperature of (425 + 325)/2 = 375 K, the constant pressure specific heat of air is cp = 1.011 kJ/kg.K (Table A-2b). The gas constant of air is R = 0.287 kJ/kg.K (Table A-1). Analysis There is only one inlet and one exit, and thus m& 1 = m& 2 = m& . We take the turbine as the system, which is a control volume since mass crosses the boundary. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

=



=0


E& in = E& out

550 kPa 425 K

m& h1 = W& out + m& h2 W& out = m& (h1 − h2 )

Air

wout = c p (T1 − T2 ) Substituting,

wout = c p (T1 − T2 ) = (1.011 kJ/kg ⋅ K)(425 − 325)K = 101.1 kJ/kg

110 kPa 325 K

The entropy change of air is s 2 − s1 = c p ln

T2 P − R ln 2 T1 P1

= (1.011 kJ/kg ⋅ K) ln = 0.1907 kJ/kg ⋅ K


The maximum (reversible) work is the exergy difference between the inlet and exit states w rev,out = c p (T1 − T2 ) − T0 ( s1 − s 2 ) = wout − T0 ( s1 − s 2 ) = 101.1 kJ/kg − (298 K)(−0.1907 kJ/kg ⋅ K) = 157.9 kJ/kg


η II =

wout 101.1 kJ/kg = = 0.640 w rev,out 157.9 kJ/kg


8-98

8-108 Steam is accelerated in a nozzle. The actual and maximum outlet velocities are to be determined. Assumptions 1 The nozzle operates steadily. 2 The changes in potential energies are negligible. Properties The properties of steam at the inlet and the exit of the nozzle are (Tables A-4 through A-6) P1 = 300 kPa ⎫ h1 = 2761.2 kJ/kg ⎬ T1 = 150°C ⎭ s1 = 7.0792 kJ/kg ⋅ K P2 = 150 kPa

⎫ h2 = 2693.1 kJ/kg ⎬ x 2 = 1 (sat. vapor) ⎭ s 2 = 7.2231 kJ/kg ⋅ K

Analysis We take the nozzle to be the system, which is a control volume. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

=


∆E& system 0 (steady) 144 42444 3

=0


E& in = E& out m& (h1 + V12 / 2) = m& (h2 + V22 /2) V 22 − V12 = h1 − h2 = ∆ke actual 2

500 kPa 200°C 30 m/s

200 kPa sat. vapor

H2O

Substituting, ∆ke actual = h1 − h2 = 2761.2 − 2693.1 = 68.1 kJ/kg

The actual velocity at the exit is then V22 − V12 = ∆ke actual 2 ⎛ 1000 m 2 /s 2 V2 = V12 + 2∆ke actual = (45 m/s) 2 + 2(68.1 kJ/kg)⎜⎜ ⎝ 1 kJ/kg

⎞ ⎟ = 371.8 m/s ⎟ ⎠

The maximum kinetic energy change is determined from ∆ke max = h1 − h2 − T0 ( s1 − s 2 ) = 68.1 − (298)(7.0792 − 7.2231) = 111.0 kJ/kg

The maximum velocity at the exit is then V22,max − V12 2

= ∆ke max

⎛ 1000 m 2 /s 2 V2,max = V12 + 2∆ke max = (45 m/s) 2 + 2(111.0 kJ/kg)⎜ ⎜ 1 kJ/kg ⎝ = 473.3 m/s

⎞ ⎟ ⎟ ⎠


8-99

8-109E Steam is expanded in a two-stage turbine. Six percent of the inlet steam is bled for feedwater heating. The isentropic efficiencies for the two stages of the turbine are given. The second-law efficiency of the turbine is to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 The turbine is well-insulated, and there is no heat transfer from the turbine. Analysis There is one inlet and two exits. We take the turbine as the system, which is a control volume since mass crosses the boundary. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

=



=0


500 psia 600°F

E& in = E& out

m& 1 h1 = m& 2 h2 + m& 3 h3 + W& out W& out = m& 1 h1 − m& 2 h2 − m& 3 h3

Turbine 100 psia

wout = h1 − 0.06h2 − 0.94h3 wout = (h1 − h2 ) + 0.94(h2 − h3 )

The isentropic and actual enthalpies at three states are determined using steam tables as follows: P1 = 500 psia ⎫ h1 = 1298.6 Btu/lbm ⎬ T1 = 600°F ⎭ s1 = 1.5590 Btu/lbm ⋅ R

5 psia

T 1 500 psia 100 psia 5 psia

P2 = 100 psia s 2s

⎫ x 2 s = 0.9609 ⎬ = s1 = 1.5590 Btu/lbm ⋅ R ⎭ h2 s = 1152.7 Btu/lbm

η T ,1 =

2 3s 3

s

h1 − h2 ⎯ ⎯→ h2 = h1 − η T ,1 (h1 − h2 s ) = 1298.6 − (0.97)(1298.6 − 1152.7) = 1157.1 kJ/kg h1 − h2 s

⎫ x 2 = 0.9658 ⎬ h2 = 1157.1 Btu/lbm ⎭ s 2 = 1.5646 Btu/lbm ⋅ R P2 = 100 psia P3 = 5 psia

⎫ x 3s = 0.8265 ⎬ s 3 = s 2 = 1.5646 kJ/kg ⋅ K ⎭ h3s = 957.09 Btu/lbm

ηT , 2 =

h2 − h3 ⎯ ⎯→ h3 = h2 − ηT , 2 (h2 − h3s ) = 1157.1 − (0.95)(1157.1 − 957.09) = 967.09 kJ/kg h2 − h3s

⎫ x 3 = 0.8364 ⎬ h3 = 967.09 Btu/lbm ⎭ s 3 = 1.5807 Btu/lbm ⋅ R P3 = 5 psia

Substituting into the energy balance per unit mass flow at the inlet of the turbine, we obtain wout = (h1 − h2 ) + 0.94(h2 − h3 ) = (1298.6 − 1157.1) + 0.94(1157.1 − 967.09) = 320.1 Btu/lbm

The reversible work output per unit mass flow at the turbine inlet is w rev = h1 − h2 − T0 ( s1 − s 2 ) + 0.94[h2 − h3 − T0 ( s 2 − s 3 )]

= 1298.6 − 1157.1 − (537)(1.5590 − 1.5646) + 0.94[(1157.1 − 967.09 − (537)(1.5646 − 1.5807)] = 331.2 Btu/lbm


η II =

wout 320.1 Btu/lbm = = 0.966 wrev 331.2 Btu/lbm


8-100

8-110 A throttle valve is placed in the steam line supplying the turbine inlet in order to control an isentropic steam turbine. The second-law efficiency of this system when the valve is partially open to when it is fully open is to be compared. Assumptions 1 This is a steady-flow process since there is no change with time. 2 The turbine is well-insulated, and there is no heat transfer from the turbine. Analysis

Valve is fully open:

The properties of steam at various states are

T

6 MPa

P0 = 100 kPa ⎫ h0 ≅ h f @25°C = 104.8 kJ/kg ⎬ T1 = 25°C ⎭ s 0 ≅ s f @25°C = 0.3672 kJ/kg ⋅ K

1

3 MPa 2

70 kPa

P1 = P2 = 6 MPa ⎫ h1 = h2 = 3894.3 kJ/kg ⎬ T1 = T2 = 700°C ⎭ s1 = s 2 = 7.4247 kJ/kg ⋅ K

3

3p s

P3 = 70 kPa ⎫ x 3 = 0.9914 ⎬ s 2 = s1 ⎭ h3 = 2639.7 kJ/kg

The stream exergy at the turbine inlet is

ψ 1 = h1 − h0 − T0 ( s1 − s 0 ) = 3894.3 − 104.8 − (298)(7.4247 − 0.3672) = 1686 kJ/kg The second law efficiency of the entire system is then

η II =

wout h1 − h3 h − h3 = = 1 = 1.0 w rev h1 − h3 − T0 ( s1 − s 3 ) h1 − h3

since s1 = s3 for this system. Valve is partly open: P2 = 3 MPa

⎫ ⎬ s 2 = 7.7405 kJ/kg ⋅ K h2 = h1 = 3894.3 kJ/kg ⎭

(from EES)

P3 = 70 kPa ⎫ ⎬ h3 = 2760.8 kJ/kg (from EES) s3 = s 2 ⎭

ψ 2 = h2 − h0 − T0 ( s 2 − s 0 ) = 3894.3 − 104.8 − (298)(7.7405 − 0.3672) = 1592 kJ/kg η II =

wout h2 − h3 3894.3 − 2760.8 = 1.0 = = wrev h2 − h3 − T0 ( s 2 − s 3 ) 3894.3 − 2760.8 − (298)(7.7405 − 7.7405)


8-101

8-111 Two rigid tanks that contain water at different states are connected by a valve. The valve is opened and steam flows from tank A to tank B until the pressure in tank A drops to a specified value. Tank B loses heat to the surroundings. The final temperature in each tank and the work potential wasted during this process are to be determined. Assumptions 1 Tank A is insulated and thus heat transfer is negligible. 2 The water that remains in tank A undergoes a reversible adiabatic process. 3 The thermal energy stored in the tanks themselves is negligible. 4 The system is stationary and thus kinetic and potential energy changes are negligible. 5 There are no work interactions. Analysis (a) The steam in tank A undergoes a reversible, adiabatic process, and thus s2 = s1. From the steam tables (Tables A-4 through A-6), Tank A :

v 1, A = v f + x1v fg = 0.001084 + (0.8)(0.46242 − 0.001084 ) = 0.37015 m 3 /kg P1 = 400 kPa ⎫ ⎬ u1, A = u f + x1u fg = 604.22 + (0.8)(1948.9 ) = 2163.3 kJ/kg x1 = 0.8 ⎭ s = s + x s = 1.7765 + (0.8)(5.1191) = 5.8717 kJ/kg ⋅ K 1, A f 1 fg

T2, A = Tsat @300 kPa = 133.52 °C s 2, A − s f 5.8717 − 1.6717 P2 = 300 kPa ⎫ x = = 0.7895 ⎪ 2, A = 5.3200 s fg s 2 = s1 ⎬ (sat. mixture) ⎪⎭ v 2, A = v f + x 2, Av fg = 0.001073 + (0.7895)(0.60582 − 0.001073) = 0.47850 m 3 /kg u 2, A = u f + x 2, A u fg = 561.11 + (0.7895)(1982.1 kJ/kg ) = 2125.9 kJ/kg

Tank B :

v = 1.1989 m 3 /kg P1 = 200 kPa ⎫ 1, B ⎬ u1, B = 2731.4 kJ/kg T1 = 250°C ⎭ s1, B = 7.7100 kJ/kg ⋅ K

900 kJ A

V = 0.2 m3

The initial and the final masses in tank A are m1, A

and

steam P = 400 kPa x = 0.8

V 0.2 m 3 = A = = 0.5403 kg v 1, A 0.37015 m 3 /kg

m 2, A =

×

B m = 3 kg steam T = 250°C P = 200 kPa

VA 0.2m 3 = = 0.4180 kg v 2, A 0.479m 3 /kg

Thus, 0.540 - 0.418 = 0.122 kg of mass flows into tank B. Then,

m2, B = m1, B − 0122 . = 3 + 0122 . = 3.122 kg The final specific volume of steam in tank B is determined from v 2, B =

VB m 2, B

=

(m1v1 )B m 2, B

=

(3 kg )(1.1989 m 3 /kg ) = 1.152 m 3 /kg 3.122 m 3

We take the entire contents of both tanks as the system, which is a closed system. The energy balance for this stationary closed system can be expressed as E −E 1in424out 3


=

∆E system 1 424 3


− Qout = ∆U = (∆U ) A + (∆U ) B

(since W = KE = PE = 0)

− Qout = (m 2 u 2 − m1u1 ) A + (m 2 u 2 − m1u1 ) B Substituting,

{

}

− 900 = {(0.418)(2125.9 ) − (0.5403)(2163.3)} + (3.122 )u 2, B − (3)(2731.4 ) u 2, B = 2425.9 kJ/kg


8-102

Thus,

v 2, B = 1.152 m 3 /kg ⎫⎪ u 2, B

T2, B = 110.1 °C ⎬ = 2425.9 kJ/kg ⎪⎭ s 2, B = 6.9772 kJ/kg ⋅ K

(b) The total entropy generation during this process is determined by applying the entropy balance on an extended system that includes both tanks and their immediate surroundings so that the boundary temperature of the extended system is the temperature of the surroundings at all times. It gives S in − S out 1424 3


−


Change in entropy

Qout + S gen = ∆S A + ∆S B Tb,surr

Rearranging and substituting, the total entropy generated during this process is determined to be S gen = ∆S A + ∆S B +

Qout Q = (m 2 s 2 − m1 s1 ) A + (m 2 s 2 − m1 s1 ) B + out Tb,surr Tb,surr

= {(0.418)(5.8717 ) − (0.5403)(5.8717 )} + {(3.122 )(6.9772 ) − (3)(7.7100 )} +

900 kJ 273 K

= 1.234 kJ/K

The work potential wasted is equivalent to the exergy destroyed during a process, which can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen ,

X destroyed = T0 S gen = (273 K)(1.234 kJ/K) = 337 kJ


8-103

8-112E A cylinder initially filled with helium gas at a specified state is compressed polytropically to a specified temperature and pressure. The actual work consumed and the minimum useful work input needed are to be determined. Assumptions 1 Helium is an ideal gas with constant specific heats. 2 The cylinder is stationary and thus the kinetic and potential energy changes are negligible. 3 The thermal energy stored in the cylinder itself is negligible. 4 The compression or expansion process is quasi-equilibrium. 5 The environment temperature is 70°F. Properties The gas constant of helium is R = 2.6805 psia.ft3/lbm.R = 0.4961 Btu/lbm.R (Table A-1E). The specific heats of helium are cv = 0.753 and cp = 1.25 Btu/lbm.R (Table A-2E). Analysis (a) Helium at specified conditions can be treated as an ideal gas. The mass of helium is m=

P1V1 (40 psia )(8 ft 3 ) = = 0.2252 lbm RT1 (2.6805 psia ⋅ ft 3 /lbm ⋅ R )(530 R )

The exponent n and the boundary work for this polytropic process are determined to be P1V1 P2V 2 T P (780 R )(40 psia ) (8 ft 3 ) = 3.364 ft 3 = ⎯ ⎯→ V 2 = 2 1 V1 = (530 R )(140 psia ) T1 T2 T1 P2 P2V 2n

=

P1V1n

⎛P ⎯ ⎯→ ⎜⎜ 2 ⎝ P1

⎞ ⎛ V1 ⎟⎟ = ⎜⎜ ⎠ ⎝V 2

n

HELIUM 8 ft3 n PV = const

Q

n

⎞ ⎛ 140 ⎞ ⎛ 8 ⎞ ⎟⎟ ⎯ ⎯→ ⎜ ⎯→ n = 1.446 ⎟=⎜ ⎟ ⎯ ⎝ 40 ⎠ ⎝ 3.364 ⎠ ⎠

Then the boundary work for this polytropic process can be determined from

∫

2

Wb,in = − P dV = − 1

=−

P2V 2 − P1V1 mR(T2 − T1 ) =− 1− n 1− n

(0.2252 lbm )(0.4961 Btu/lbm ⋅ R )(780 − 530)R = 62.62 Btu 1 − 1.446

Also, ⎛ 1 Btu Wsurr,in = − P0 (V 2 − V1 ) = −(14.7 psia)(3.364 − 8)ft 3 ⎜ ⎜ 5.4039 psia ⋅ ft 3 ⎝

⎞ ⎟ = 12.61 Btu ⎟ ⎠

Thus, Wu,in = Wb,in − Wsurr,in = 62.62 − 12.61 = 50.0 Btu

(b) We take the helium in the cylinder as the system, which is a closed system. Taking the direction of heat transfer to be from the cylinder, the energy balance for this stationary closed system can be expressed as E −E 1in424out 3

=


∆E system 1 424 3


− Qout + W b,in = ∆U = m(u 2 − u1 ) − Qout = m(u 2 − u1 ) − W b,in Qout = W b,in − mcv (T2 − T1 )

Substituting, Qout = 62.62 Btu − (0.2252 lbm )(0.753 Btu/lbm ⋅ R )(780 − 530 )R = 20.69 Btu

The total entropy generation during this process is determined by applying the entropy balance on an extended system that includes the cylinder and its immediate surroundings so that the boundary temperature of the extended system is the temperature of the surroundings at all times. It gives S − S out 1in424 3


−


Change in entropy

Qout + S gen = ∆S sys Tb,surr


8-104

where the entropy change of helium is ⎛ T P ⎞ ∆S sys = ∆S helium = m⎜⎜ c p ,avg ln 2 − R ln 2 ⎟⎟ T1 P1 ⎠ ⎝ ⎡ 780 R 140 psia ⎤ = (0.2252 lbm) ⎢(1.25 Btu/lbm ⋅ R )ln − (0.4961 Btu/lbm ⋅ R )ln ⎥ 530 R 40 psia ⎦ ⎣ = −0.03201 Btu/R Rearranging and substituting, the total entropy generated during this process is determined to be S gen = ∆S helium +

Qout 20.69 Btu = (−0.03201 Btu/R) + = 0.007022 Btu/R T0 530 R

The work potential wasted is equivalent to the exergy destroyed during a process, which can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X destroyed = T0 S gen = (530 R) (0.007022 Btu/R ) = 3.722 Btu

The minimum work with which this process could be accomplished is the reversible work input, Wrev, in which can be determined directly from Wrev,in = Wu,in − X destroyed = 50.0 − 3.722 = 46.3 Btu

Discussion The reversible work input, which represents the minimum work input Wrev,in in this case can be determined from the exergy balance by setting the exergy destruction term equal to zero, X − X out 1in 4243



Change in exergy

Substituting the closed system exergy relation, the reversible work input during this process is determined to be Wrev = (U 2 − U 1 ) − T0 ( S 2 − S1 ) + P0 (V 2 − V1 ) = (0.2252 lbm)(0.753 Btu/lbm ⋅ R)(320 − 70)°F − (530 R)(−0.03201 Btu/R) + (14.7 psia)(3.364 − 8)ft 3 [Btu/5.4039 psia ⋅ ft 3 ] = 46.7 Btu

The slight difference is due to round-off error.


8-105

8-113 Steam expands in a two-stage adiabatic turbine from a specified state to specified pressure. Some steam is extracted at the end of the first stage. The wasted power potential is to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 The turbine is adiabatic and thus heat transfer is negligible. 4 The environment temperature is given to be T0 = 25°C. Analysis The wasted power potential is equivalent to the rate of exergy destruction during a process, which can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen .

The total rate of entropy generation during this process is determined by taking the entire turbine, which is a control volume, as the system and applying the entropy balance. Noting that this is a steady-flow process and there is no heat transfer, S& in − S& out 1424 3


+

S& gen {



m& 1 s1 − m& 2 s 2 − m& 3 s 3 + S& gen = 0 m& 1 s1 − 0.1m& 1 s 2 − 0.9m& 1 s 3 + S& gen = 0 → S& gen = m& 1 [0.9 s 3 + 0.1s 2 − s1 ]

and

X destroyed = T0 Sgen = T0m& 1[0.9s3 + 0.1s2 − s1]

9 MPa 500°C

From the steam tables (Tables A-4 through 6) P1 = 9 MPa ⎫ h1 = 3387.4 kJ / kg ⎬ T1 = 500°C ⎭ s1 = 6.6603 kJ / kg ⋅ K P2 = 1.4 MPa ⎫ ⎬ h2 s = 2882.4 kJ / kg s 2 s = s1 ⎭

STEAM 13.5 kg/s

STEAM 15 kg/s I

II

1.4 MPa

50 kPa

and,

ηT =

h1 − h2 ⎯ ⎯→ h2 = h1 − ηT (h1 − h2 s ) h1 − h2 s = 3387.4 − 0.88(3387.4 − 2882.4)

10%

90%

= 2943.0 kJ/kg

P2 = 1.4 MPa

⎫ ⎬ s 2 = 6.7776 kJ / kg ⋅ K h2 = 2943.0 kJ/kg ⎭ s 3s − s f 6.6603 − 1.0912 P3 = 50 kPa ⎫ x 3s = = = 0.8565 6.5019 s fg ⎬ s 3s = s1 ⎭ h = h + x h = 340.54 + 0.8565 × 2304.7 = 2314.6 kJ/kg 3s f 3 s fg

and

ηT =

h1 − h3 ⎯ ⎯→ h3 = h1 − ηT (h1 − h3s ) h1 − h3s = 3387.4 − 0.88(3387.4 − 2314.6) = 2443.3 kJ/kg

h3 − h f 2443.3 − 340.54 = = 0.9124 ⎫ x3 = 2304.7 h ⎬ fg h3 = 2443.3 kJ/kg ⎭ s 3 = s f + x 3 s fg = 1.0912 + 0.9124 × 6.5019 = 7.0235 kJ/kg ⋅ K P3 = 50 kPa

Substituting, the wasted work potential is determined to be X& destroyed = T0 S& gen = ( 298 K)(15 kg/s)(0.9 × 7.0235 + 0.1 × 6.7776 − 6.6603) kJ/kg = 1514 kW


8-106

8-114 Steam expands in a two-stage adiabatic turbine from a specified state to another specified state. Steam is reheated between the stages. For a given power output, the reversible power output and the rate of exergy destruction are to be determined. Assumptions 1 This is a steady-flow process since there is no change Heat with time. 2 Kinetic and potential energy changes are negligible. 3 The turbine is adiabatic and thus heat transfer is negligible. 4 The 2 MPa 2 MPa environment temperature is given to be T0 = 25°C. 350°C 500°C Properties From the steam tables (Tables A-4 through 6) P1 = 8 MPa ⎫ h1 = 3399.5 kJ / kg ⎬ T1 = 500°C ⎭ s1 = 6.7266 kJ / kg ⋅ K

Stage II

Stage I

5 MW

P2 = 2 MPa ⎫ h2 = 3137.7 kJ / kg ⎬ T2 = 350°C ⎭ s 2 = 6.9583 kJ / kg ⋅ K

30 kPa x = 97%

8 MPa 500°C

P3 = 2 MPa ⎫ h3 = 3468.3 kJ / kg ⎬ T3 = 500°C ⎭ s 3 = 7.4337 kJ / kg ⋅ K

P4 = 30 kPa ⎫ h4 = h f + x 4 h fg = 289.27 + 0.97 × 2335.3 = 2554.5 kJ/kg ⎬ x 4 = 0.97 ⎭ s 4 = s f + x 4 s fg = 0.9441 + 0.97 × 6.8234 = 7.5628 kJ/kg ⋅ K

Analysis We take the entire turbine, excluding the reheat section, as the system, which is a control volume. The energy balance for this steady-flow system can be expressed in the rate form as = ∆E& system 0 (steady) =0 E& − E& 1in424out 3 1442444 3 Rate of net energy transfer by heat, work, and mass


E& in = E& out

m& h1 + m& h3 = m& h2 + m& h4 + W& out W& out = m& [(h1 − h2 ) + (h3 − h4 )]

Substituting, the mass flow rate of the steam is determined from the steady-flow energy equation applied to the actual process, W& out 5000 kJ/s m& = = = 4.253 kg/s h1 − h2 + h3 − h4 (3399.5 − 3137.7 + 3468.3 − 2554.5)kJ/kg The reversible (or maximum) power output is determined from the rate form of the exergy balance applied on the turbine and setting the exergy destruction term equal to zero, − X& destroyed 0 (reversible) = ∆X& system 0 (steady) = 0 X& − X& out 1in 4243 144424443 1442443 Rate of net exergy transfer by heat, work, and mass

Rate of exergy destruction


X& in = X& out m& ψ1 + m& ψ 3 = m& ψ 2 + m& ψ 4 + W&rev,out W&rev,out = m& (ψ1 −ψ 2 ) + m& (ψ 3 −ψ 4 ) = m& [(h1 − h2 ) + T0 (s2 − s1) − ∆ke + m& [(h3 − h4 ) + T0 (s4 − s3 ) − ∆ke

0

− ∆pe 0 ]

0

− ∆pe 0 ]

Then the reversible power becomes W& rev,out = m& [h1 − h2 + h3 − h4 + T0 ( s 2 − s1 + s 4 − s 3 )] = (4.253 kg/s)[(3399.5 − 3137.7 + 3468.3 − 2554.5)kJ/kg + (298 K)(6.9583 − 6.7266 + 7.5628 − 7.4337)kJ/kg ⋅ K] = 5457 kW Then the rate of exergy destruction is determined from its definition, X& = W& − W& = 5457 − 5000 = 457 kW destroyed

rev,out

out


8-107

8-115 An insulated cylinder is divided into two parts. One side of the cylinder contains N2 gas and the other side contains He gas at different states. The final equilibrium temperature in the cylinder and the wasted work potential are to be determined for the cases of piston being fixed and moving freely. Assumptions 1 Both N2 and He are ideal gases with constant specific heats. 2 The energy stored in the container itself is negligible. 3 The cylinder is well-insulated and thus heat transfer is negligible. Properties The gas constants and the specific heats are R = 0.2968 kPa.m3/kg.K, cp = 1.039 kJ/kg·°C, and cv = 0.743 kJ/kg·°C for N2, and R = 2.0769 kPa.m3/kg.K, cp = 5.1926 kJ/kg·°C, and cv = 3.1156 kJ/kg·°C for He (Tables A-1 and A2). Analysis The mass of each gas in the cylinder is

(

) ( ) (500 kPa )(1 m ) = (2.0769 kPa ⋅ m /kg ⋅ K )(298 K ) = 0.8079 kg

⎛ PV ⎞ (500 kPa ) 1 m 3 = 4.772 kg m N 2 = ⎜⎜ 1 1 ⎟⎟ = 0.2968 kPa ⋅ m 3 /kg ⋅ K (353 K ) ⎝ RT1 ⎠ N 2 ⎛ PV ⎞ m He = ⎜⎜ 1 1 ⎟⎟ ⎝ RT1 ⎠ He

N2 1 m3 500 kPa 80°C

3

He 1 m3 500 kPa 25°C

3

Taking the entire contents of the cylinder as our system, the 1st law relation can be written as E −E 1in424out 3

=


∆E system 1 424 3


0 = ∆U = (∆U ) N 2 + (∆U )He 0 = [mcv (T2 − T1 )] N 2 + [mcv (T2 − T1 )] He Substituting,

(4.772 kg )(0.743 kJ/kg⋅ o C )(T f

)

(

)(

)

− 80 o C + (0.8079 kg ) 3.1156 kJ/kg ⋅ o C T f − 25 o C = 0

It gives Tf = 57.2°C where Tf is the final equilibrium temperature in the cylinder. The answer would be the same if the piston were not free to move since it would effect only pressure, and not the specific heats. (b) We take the entire cylinder as our system, which is a closed system. Noting that the cylinder is well-insulated and thus there is no heat transfer, the entropy balance for this closed system can be expressed as S in − S out 1424 3



Change in entropy

0 + S gen = ∆S N 2 + ∆S He But first we determine the final pressure in the cylinder: 4.772 kg 0.8079 kg ⎛m⎞ ⎛m⎞ N total = N N 2 + N He = ⎜ ⎟ + ⎜ ⎟ = + = 0.3724 kmol M M 28 kg/kmol 4 kg/kmol ⎝ ⎠ N 2 ⎝ ⎠ He P2 =

(

)

N total Ru T (0.3724 kmol) 8.314 kPa ⋅ m 3 /kmol ⋅ K (330.2 K ) = = 511.1 kPa V total 2 m3

Then, ⎛ T P ⎞ ∆S N 2 = m⎜⎜ c p ln 2 − R ln 2 ⎟⎟ T1 P1 ⎠ N ⎝ 2 ⎡ 330.2 K 511.1 kPa ⎤ = (4.772 kg )⎢(1.039 kJ/kg ⋅ K ) ln − (0.2968 kJ/kg ⋅ K ) ln ⎥ = −0.3628 kJ/K 353 K 500 kPa ⎦ ⎣ PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

8-108

⎛ T P ⎞ ∆S He = m⎜⎜ c p ln 2 − R ln 2 ⎟⎟ T1 P1 ⎠ He ⎝ ⎡ 330.2 K 511.1 kPa ⎤ = (0.8079 kg )⎢(5.1926 kJ/kg ⋅ K ) ln − (2.0769 kJ/kg ⋅ K ) ln ⎥ = 0.3931 kJ/K 298 K 500 kPa ⎦ ⎣ S gen = ∆S N 2 + ∆S He = −0.3628 + 0.3931 = 0.0303 kJ/K

The wasted work potential is equivalent to the exergy destroyed during a process, and it can be determined from an exergy balance or directly from its definition X destroyed = T0S gen ,

X destroyed = T0 S gen = (298 K)(0.0303 kJ/K) = 9.03 kJ If the piston were not free to move, we would still have T2 = 330.2 K but the volume of each gas would remain constant in this case: ⎛ T V ∆S N 2 = m⎜ cv ln 2 − R ln 2 ⎜ T1 V1 ⎝

0

⎞ ⎟ = (4.772 kg )(0.743 kJ/kg ⋅ K ) ln 330.2 K = −0.2371 kJ/K ⎟ 353 K ⎠ N2

⎛ T V ∆S He = m⎜ cv ln 2 − R ln 2 ⎜ T V1 1 ⎝

0

⎞ ⎟ = (0.8079 kg )(3.1156 kJ/kg ⋅ K ) ln 330.2 K = 0.258 kJ/K ⎟ 298 K ⎠ He

S gen = ∆S N 2 + ∆S He = −0.2371 + 0.258 = 0.02089 kJ/K and

X destroyed = T0 S gen = (298 K)(0.02089 kJ/K) = 6.23 kJ


8-109

8-116 An insulated cylinder is divided into two parts. One side of the cylinder contains N2 gas and the other side contains He gas at different states. The final equilibrium temperature in the cylinder and the wasted work potential are to be determined for the cases of piston being fixed and moving freely. √ Assumptions 1 Both N2 and He are ideal gases with constant specific heats. 2 The energy stored in the container itself, except the piston, is negligible. 3 The cylinder is well-insulated and thus heat transfer is negligible. 4 Initially, the piston is at the average temperature of the two gases. Properties The gas constants and the specific heats are R = 0.2968 kPa.m3/kg.K, cp = 1.039 kJ/kg·°C, and cv = 0.743 kJ/kg·°C for N2, and R = 2.0769 kPa.m3/kg.K, cp = 5.1926 kJ/kg·°C, and cv = 3.1156 kJ/kg·°C for He (Tables A-1 and A2). The specific heat of copper piston is c = 0.386 kJ/kg·°C (Table A-3). Analysis The mass of each gas in the cylinder is mN2

N2 1 m3 500 kPa 80°C

(

) ( ) (500 kPa )(1 m ) = (2.0769 kPa ⋅ m /kg ⋅ K )(353 K ) = 0.8079 kg

⎛ PV ⎞ (500 kPa ) 1 m 3 = ⎜⎜ 1 1 ⎟⎟ = = 4.772 kg 0.2968 kPa ⋅ m 3 /kg ⋅ K (353 K ) ⎝ RT1 ⎠ N 2 3

⎛ PV ⎞ m He = ⎜⎜ 1 1 ⎟⎟ ⎝ RT1 ⎠ He

3

Taking the entire contents of the cylinder as our system, the 1st law relation can be written as E −E 1in424out 3

He 1 m3 500 kPa 25°C

=


Copper

∆E system 1 424 3


0 = ∆U = (∆U ) N 2 + (∆U )He + (∆U )Cu

0 = [mcv (T2 − T1 )] N 2 + [mcv (T2 − T1 )] He + [mc(T2 − T1 )] Cu where T1, Cu = (80 + 25) / 2 = 52.5°C Substituting,

(4.772 kg )(0.743 kJ/kg⋅ o C)(T f

)

(

)(

)

− 80 o C + (0.8079 kg ) 3.1156 kJ/kg⋅ o C T f − 25 o C

(

)(

)

+ (5.0 kg ) 0.386 kJ/kg⋅ o C T f − 52.5 o C = 0

It gives Tf = 56.0°C where Tf is the final equilibrium temperature in the cylinder. The answer would be the same if the piston were not free to move since it would effect only pressure, and not the specific heats. (b) We take the entire cylinder as our system, which is a closed system. Noting that the cylinder is well-insulated and thus there is no heat transfer, the entropy balance for this closed system can be expressed as S in − S out 1424 3



Change in entropy

0 + S gen = ∆S N 2 + ∆S He + ∆S piston But first we determine the final pressure in the cylinder: 4.772 kg 0.8079 kg ⎛m⎞ ⎛m⎞ N total = N N 2 + N He = ⎜ ⎟ + ⎜ ⎟ = + = 0.3724 kmol ⎝ M ⎠ N 2 ⎝ M ⎠ He 28 kg/kmol 4 kg/kmol P2 =

N total Ru T

V total

=

(0.3724 kmol)(8.314 kPa ⋅ m 3 /kmol ⋅ K )(329 K ) = 509.4 kPa 2 m3


8-110

Then, ⎛ P ⎞ T ∆S N 2 = m⎜⎜ c p ln 2 − R ln 2 ⎟⎟ P1 ⎠ N T1 ⎝ 2 ⎡ 329 K 509.4 kPa ⎤ = (4.772 kg )⎢(1.039 kJ/kg ⋅ K )ln − (0.2968 kJ/kg ⋅ K ) ln ⎥ = −0.3749 kJ/K 353 K 500 kPa ⎦ ⎣ ⎛ P ⎞ T ∆S He = m⎜⎜ c p ln 2 − R ln 2 ⎟⎟ P1 ⎠ He T1 ⎝ ⎡ 329 K 509.4 kPa ⎤ = (0.8079 kg )⎢(5.1926 kJ/kg ⋅ K ) ln − (2.0769 kJ/kg ⋅ K ) ln ⎥ = 0.3845 kJ/K 353 K 500 kPa ⎦ ⎣ ⎛ T ⎞ 329 K = (5 kg )(0.386 kJ/kg ⋅ K ) ln = 0.021 kJ/K ∆S piston = ⎜⎜ mc ln 2 ⎟⎟ 325.5 K T1 ⎠ piston ⎝ S gen = ∆S N 2 + ∆S He + ∆S piston = −0.3749 + 0.3845 + 0.021 = 0.03047 kJ/K

The wasted work potential is equivalent to the exergy destroyed during a process, and it can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen ,

X destroyed = T0 S gen = (298 K)(0.03047 kJ/K) = 9.08 kJ If the piston were not free to move, we would still have T2 = 330.2 K but the volume of each gas would remain constant in this case: ⎛ V 0 ⎞⎟ T 329 K = (4.772 kg )(0.743 kJ/kg ⋅ K ) ln = −0.2492 kJ/K ∆S N 2 = m⎜ cv ln 2 − R ln 2 ⎜ 353 K V1 ⎟⎠ T1 ⎝ N2 0⎞ ⎛ V T ⎟ = (0.8079 kg )(3.1156 kJ/kg ⋅ K ) ln 329 K = 0.2494 kJ/K ∆S He = m⎜ cv ln 2 − R ln 2 ⎟ ⎜ V 353 K T 1 1 ⎠ He ⎝ S gen = ∆S N 2 + ∆S He + ∆S piston = −0.2492 + 0.2494 + 0.021 = 0.02104 kJ/K

and



8-111

8-117E Argon enters an adiabatic turbine at a specified state with a specified mass flow rate, and leaves at a specified pressure. The isentropic and second-law efficiencies of the turbine are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 The device is adiabatic and thus heat transfer is negligible. 4 Argon is an ideal gas with constant specific heats. Properties The specific heat ratio of argon is k = 1.667. The constant pressure specific heat of argon is cp = 0.1253 Btu/lbm.R. The gas constant is R = 0.04971 Btu/lbm.R (Table A-2E). &1 = m &2 = m & . We take the isentropic turbine as the system, which is Analysis There is only one inlet and one exit, and thus m a control volume since mass crosses the boundary. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3


=


=0

1


E& in = E& out

(since Q& ≅ ∆ke ≅ ∆pe ≅ 0)

m& h1 = W& s ,out + m& h2 s

95 hp

Ar

ηT

W& s ,out = m& (h1 − h2 s ) From the isentropic relations, ⎛P ⎞ T2 s = T1⎜⎜ 2 s ⎟⎟ ⎝ P1 ⎠

( k −1) / k

⎛ 30 psia ⎞ ⎟⎟ = (1960 R)⎜⎜ ⎝ 200 psia ⎠

2

0.667 / 1.667

= 917.5 R

Then the power output of the isentropic turbine becomes 1 hp ⎛ ⎞ W& s ,out = m& c p (T1 − T2 s ) = (40 lbm/min)(0.1253 Btu/lbm ⋅ R)(1960 − 917.5)R ⎜ ⎟ = 123.2 hp 42.41 Btu/min ⎝ ⎠

Then the isentropic efficiency of the turbine is determined from

ηT =

W& a ,out 95 hp = = 0.771 = 77.1% & 123 .2 hp W s ,out

(b) Using the steady-flow energy balance relation W& a ,out = m& c p (T1 − T2 ) above, the actual turbine exit temperature is determined to be T2 = T1 −

W& a ,out m& c p

= 1500 −

⎛ 42.41 Btu/min ⎞ 95 hp ⎟⎟ = 696.1°F = 1156.1 R ⎜ (40 lbm/min)(0.1253 Btu/lbm ⋅ R) ⎜⎝ 1 hp ⎠

The entropy generation during this process can be determined from an entropy balance on the turbine, S& in − S& out 1424 3

+


S& gen {



m& s1 − m& s 2 + S& gen = 0 S& gen = m& ( s 2 − s1 )

where s 2 − s1 = c p ln

T2 P − R ln 2 T1 P1

= (0.1253 Btu/lbm ⋅ R) ln

30 psia 1156.1 R − (0.04971 Btu/lbm ⋅ R) ln 1960 R 200 psia

= 0.02816 Btu/lbm.R


8-112

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X& destroyed = T0 S&gen = m& T0 ( s2 − s1 ) 1 hp ⎞ ⎛ = (40 lbm/min)(537 R)(0.02816 Btu/lbm ⋅ R)⎜ ⎟ ⎝ 42.41 Btu/min ⎠ = 14.3 hp

Then the reversible power and second-law efficiency become W& rev,out = W& a ,out + X& destroyed = 95 + 14.3 = 109.3 hp

and W&

95 hp

η II = & a,out = = 86.9% Wrev,out 109.3 hp


8-113

8-118 The feedwater of a steam power plant is preheated using steam extracted from the turbine. The ratio of the mass flow rates of the extracted steam and the feedwater are to be determined. Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential energy changes are negligible. 3 Heat loss from the device to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. Properties The properties of steam and feedwater are (Tables A-4 through A-6) P1 = 1.6 MPa ⎫ h1 = 2919.9 kJ/kg ⎬ T1 = 250°C ⎭ s1 = 6.6753 kJ/kg ⋅ K

1

Steam from turbine

h2 = h f @1.6 MPa = 858.44 kJ/kg P2 = 1.6 MPa ⎫ s ⎬ 2 = s f @1.6 MPa = 2.3435 kJ/kg ⋅ K sat. liquid ⎭ T = 201.4°C

1.6 MPa 250°C Feedwater 3

4 MPa 30°C

2

P3 = 4 MPa ⎫ h3 ≅ h f @30o C = 129.37 kJ/kg ⎬ T3 = 30°C ⎭ s 3 ≅ s f @30o C = 0.4355 kJ/kg ⋅ K

4

⎫ h4 ≅ h f @191.4°C = 814.78 kJ/kg ⎬ T4 = T2 − 10°C ≅ 191.4°C ⎭ s 4 ≅ s f @191.4°C = 2.2446 kJ/kg ⋅ K P4 = 4 MPa

Analysis (a) We take the heat exchanger as the system, which is a control volume. The mass and energy balances for this steady-flow system can be expressed in the rate form as follows: Mass balance (for each fluid stream):


0 (steady)

= 0 → m& in = m& out → m& 1 = m& 2 = m& s and m& 3 = m& 4 = m& fw

Energy balance (for the heat exchanger): E& − E& out = ∆E&system 0 (steady) 1in 424 3 1442443 Rate of net energy transfer by heat, work, and mass

2

sat. liquid

=0⎯ ⎯→ E&in = E& out


m& 1h1 + m& 3h3 = m& 2h2 + m& 4h4 (since Q& = W& = ∆ke ≅ ∆pe ≅ 0)

Combining the two,

m& s (h2 − h1 ) = m& fw (h3 − h4 )

& fw and substituting, Dividing by m m& s h − h4 (129.37 − 814.78)kJ/kg = 3 = = 0.3325 m& fw h2 − h1 (858.44 − 2919.9 )kJ/kg

(b) The entropy generation during this process per unit mass of feedwater can be determined from an entropy balance on the feedwater heater expressed in the rate form as S& − S& out + S& gen = ∆S& system 0 = 0 1in424 3 { 14243 Rate of net entropy transfer by heat and mass



m& 1 s1 − m& 2 s 2 + m& 3 s 3 − m& 4 s 4 + S& gen = 0 m& s ( s1 − s 2 ) + m& fw ( s 3 − s 4 ) + S& gen = 0 S& gen m& fw

=

m& s (s 2 − s1 ) + (s 4 − s3 ) = (0.3325)(2.3435 − 6.6753) + (2.2446 − 0.4355) = 0.3688 kJ/K ⋅ kg fw m& fw

Noting that this process involves no actual work, the reversible work and exergy destruction become equivalent since X destroyed = Wrev,out − Wact,out → Wrev,out = X destroyed . The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X destroyed = T0 S gen = ( 298 K)(0.3688 kJ/K ⋅ kg fw) = 109.9 kJ/kg feedwater


8-114

8-119 Problem 8-118 is reconsidered. The effect of the state of the steam at the inlet of the feedwater heater on the ratio of mass flow rates and the reversible power is to be investigated. Analysis Using EES, the problem is solved as follows: "Input Data" "Steam (let st=steam data):" Fluid$='Steam_IAPWS' T_st[1]=250 [C] {P_st[1]=1600 [kPa]} P_st[2] = P_st[1] x_st[2]=0 "saturated liquid, quality = 0%" T_st[2]=temperature(steam, P=P_st[2], x=x_st[2]) "Feedwater (let fw=feedwater data):" T_fw[1]=30 [C] P_fw[1]=4000 [kPa] P_fw[2]=P_fw[1] "assume no pressure drop for the feedwater" T_fw[2]=T_st[2]-10 "Surroundings:" T_o = 25 [C] P_o = 100 [kPa] "Assumed value for the surrroundings pressure" "Conservation of mass:" "There is one entrance, one exit for both the steam and feedwater." "Steam: m_dot_st[1] = m_dot_st[2]" "Feedwater: m_dot_fw[1] = m_dot_fw[2]" "Let m_ratio = m_dot_st/m_dot_fw" "Conservation of Energy:" "We write the conservation of energy for steady-flow control volume having two entrances and two exits with the above assumptions. Since neither of the flow rates is know or can be found, write the conservation of energy per unit mass of the feedwater." E_in - E_out =DELTAE_cv DELTAE_cv=0 "Steady-flow requirement" E_in = m_ratio*h_st[1] + h_fw[1] h_st[1]=enthalpy(Fluid$, T=T_st[1], P=P_st[1]) h_fw[1]=enthalpy(Fluid$,T=T_fw[1], P=P_fw[1]) E_out = m_ratio*h_st[2] + h_fw[2] h_fw[2]=enthalpy(Fluid$, T=T_fw[2], P=P_fw[2]) h_st[2]=enthalpy(Fluid$, x=x_st[2], P=P_st[2]) "The reversible work is given by Eq. 7-47, where the heat transfer is zero (the feedwater heater is adiabatic) and the Exergy destroyed is set equal to zero" W_rev = m_ratio*(Psi_st[1]-Psi_st[2]) +(Psi_fw[1]-Psi_fw[2]) Psi_st[1]=h_st[1]-h_st_o -(T_o + 273)*(s_st[1]-s_st_o) s_st[1]=entropy(Fluid$,T=T_st[1], P=P_st[1]) h_st_o=enthalpy(Fluid$, T=T_o, P=P_o) s_st_o=entropy(Fluid$, T=T_o, P=P_o) Psi_st[2]=h_st[2]-h_st_o -(T_o + 273)*(s_st[2]-s_st_o) s_st[2]=entropy(Fluid$,x=x_st[2], P=P_st[2]) Psi_fw[1]=h_fw[1]-h_fw_o -(T_o + 273)*(s_fw[1]-s_fw_o) h_fw_o=enthalpy(Fluid$, T=T_o, P=P_o) s_fw[1]=entropy(Fluid$,T=T_fw[1], P=P_fw[1]) s_fw_o=entropy(Fluid$, T=T_o, P=P_o) Psi_fw[2]=h_fw[2]-h_fw_o -(T_o + 273)*(s_fw[2]-s_fw_o) s_fw[2]=entropy(Fluid$,T=T_fw[2], P=P_fw[2]) PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

8-115

Pst,1 [kPa] 200 400 600 800 1000 1200 1400 1600 1800 2000

mratio [kg/kg] 0.1361 0.1843 0.2186 0.2466 0.271 0.293 0.3134 0.3325 0.3508 0.3683

Wrev [kJ/kg] 42.07 59.8 72.21 82.06 90.35 97.58 104 109.9 115.3 120.3

0.4

mratio [kg/kg]

0.35 0.3 0.25 0.2 0.15 0.1 200

400

600

800

1000

1200

1400

1600

1800

2000

Pst[1] [kPa]

130 120

Wrev [kJ/kg]

110 100 90 80 70 60 50 40 200

400

600

800

1000

1200

1400

1600

1800

2000

Pst[1] [kPa]


8-116

8-120 A 1-ton (1000 kg) of water is to be cooled in a tank by pouring ice into it. The final equilibrium temperature in the tank and the exergy destruction are to be determined. Assumptions 1 Thermal properties of the ice and water are constant. 2 Heat transfer to the water tank is negligible. 3 There is no stirring by hand or a mechanical device (it will add energy). Properties The specific heat of water at room temperature is c = 4.18 kJ/kg·°C, and the specific heat of ice at about 0°C is c = 2.11 kJ/kg·°C (Table A-3). The melting temperature and the heat of fusion of ice at 1 atm are 0°C and 333.7 kJ/kg.. Analysis (a) We take the ice and the water as the system, and disregard any heat transfer between the system and the surroundings. Then the energy balance for this process can be written as E −E 1in424out 3

=


ice -5°C 80 kg

∆E system 1 424 3


0 = ∆U 0 = ∆U ice + ∆U water [ mc(0 o C − T1 ) solid + mhif + mc (T2 −0 o C) liquid ] ice + [ mc (T2 − T1 )] water = 0

WATER 1 ton

Substituting, (80 kg){(2.11 kJ / kg⋅o C)[0 − (-5)]o C + 333.7 kJ / kg + (4.18 kJ / kg⋅o C)(T2 − 0)o C} + (1000 kg)(4.18 kJ / kg⋅o C)(T2 − 20)o C = 0 It gives

T2 = 12.42°C

which is the final equilibrium temperature in the tank. (b) We take the ice and the water as our system, which is a closed system .Considering that the tank is well-insulated and thus there is no heat transfer, the entropy balance for this closed system can be expressed as S in − S out 1424 3



Change in entropy

0 + S gen = ∆S ice + ∆S water where ⎛ T ⎞ 285.42 K ∆S water = ⎜⎜ mc ln 2 ⎟⎟ = (1000 kg )(4.18 kJ/kg ⋅ K )ln = −109.590 kJ/K T1 ⎠ water 293 K ⎝ ∆S ice = ∆S solid + ∆S melting + ∆S liquid ice

(

⎛⎛ Tmelting = ⎜ ⎜⎜ mc ln ⎜ T1 ⎝⎝

)

mhig ⎞ ⎛ T ⎟ + + ⎜⎜ mc ln 2 ⎟ T1 ⎠ solid Tmelting ⎝

⎞ ⎞ ⎟ ⎟⎟ ⎟ ⎠ liquid ⎠ ice

⎛ 273 K 333.7 kJ/kg 285.42 K ⎞ ⎟ = (80 kg )⎜⎜ (2.11 kJ/kg ⋅ K )ln + + (4.18 kJ/kg ⋅ K )ln 268 K 273 K 273 K ⎟⎠ ⎝ = 115.783 kJ/K

Then,

S gen = ∆S water + ∆S ice = −109.590 + 115.783 = 6.193 kJ/K


X destroyed = T0 S gen = (293 K)(6.193 kJ/K) = 1815 kJ


8-117

8-121 One ton of liquid water at 65°C is brought into a room. The final equilibrium temperature in the room and the entropy generated are to be determined. Assumptions 1 The room is well insulated and well sealed. 2 The thermal properties of water and air are constant at room temperature. 3 The system is stationary and thus the kinetic and potential energy changes are zero. 4 There are no work interactions involved. Properties The gas constant of air is R = 0.287 kPa.m3/kg.K (Table A-1). The constant volume specific heat of water at room temperature is cv = 0.718 kJ/kg⋅°C (Table A-2). The specific heat of water at room temperature is c = 4.18 kJ/kg⋅°C (Table A-3). Analysis The volume and the mass of the air in the room are

V = 3 × 4 × 7 = 84 m3 mair

3m×4m×7m

PV (100 kPa )(84 m 3 ) = 1 = = 101.3 kg RT1 (0.2870 kPa ⋅ m 3 /kg ⋅ K )(289 K )

ROOM 16°C 100 kPa

Taking the contents of the room, including the water, as our system, the energy balance can be written as E −E 1in424out 3

=


Heat

∆E system 1 424 3

Water 65°C


0 = ∆U = (∆U )water + (∆U )air or

[mc(T2 − T1 )]water + [mcv (T2 − T1 )]air

Substituting,

(1000 kg )(4.18 kJ/kg ⋅ °C)(T f

=0

)

(

)

− 65 °C + (101.3 kg )(0.718 kJ/kg ⋅ °C) T f − 16 °C = 0

It gives the final equilibrium temperature in the room to be Tf = 64.2°C (b) We again take the room and the water in it as the system, which is a closed system. Considering that the system is wellinsulated and no mass is entering and leaving, the entropy balance for this system can be expressed as S in − S out 1424 3



Change in entropy

0 + S gen = ∆S air + ∆S water where ∆S air = mcv ln ∆S water = mc ln

V T2 + mR ln 2 V1 T1

0

= (101.3 kg )(0.718 kJ/kg ⋅ K )ln

337.2 K = 11.21 kJ/K 289 K

T2 337.2 K = (1000 kg )(4.18 kJ/kg ⋅ K ) ln = −10.37 kJ/K T1 338 K

Substituting, the entropy generation is determined to be Sgen = 11.21 − 10.37 = 0.834 kJ/K The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X destroyed = T0 S gen = ( 283 K)(0.834 kJ/K) = 236 kJ

(c) The work potential (the maximum amount of work that can be produced) during a process is simply the reversible work output. Noting that the actual work for this process is zero, it becomes X destroyed = W rev,out − Wact,out → W rev,out = X destroyed = 236 kJ


8-118

8-122 An evacuated bottle is surrounded by atmospheric air. A valve is opened, and air is allowed to fill the bottle. The amount of heat transfer through the wall of the bottle when thermal and mechanical equilibrium is established and the amount of exergy destroyed are to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process, but it can be analyzed as a uniform-flow process since the state of fluid at the inlet remains constant. 2 Air is an ideal gas. 3 Kinetic and potential energies are negligible. 4 There are no work interactions involved. 5 The direction of heat transfer is to the air in the bottle (will be verified). Properties The gas constant of air is 0.287 kPa.m3/kg.K (Table A-1). Analysis We take the bottle as the system, which is a control volume since mass crosses the boundary. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances can be expressed as

Mass balance: min − m out = ∆msystem → mi = m 2

(since mout = minitial = 0)


=


100 kPa 17°C

∆E system 1 424 3


Qin + m i hi = m 2 u 2 (since W ≅ E out = E initial = ke ≅ pe ≅ 0)

Combining the two balances:

12 L Evacuated

Qin = m 2 (u 2 − hi )

where m2 =

(

)

(100 kPa ) 0.012 m3 P2V = = 0.0144 kg RT2 0.287 kPa ⋅ m3 /kg ⋅ K (290 K )

(

Table A -17 ⎯⎯ ⎯→ Ti = T2 = 290 K ⎯⎯

)

hi = 290.16 kJ/kg u2 = 206.91 kJ/kg

Substituting, →

Qin = (0.0144 kg)(206.91 - 290.16) kJ/kg = - 1.2 kJ

Qout = 1.2 kJ

Note that the negative sign for heat transfer indicates that the assumed direction is wrong. Therefore, we reversed the direction. The entropy generated during this process is determined by applying the entropy balance on an extended system that includes the bottle and its immediate surroundings so that the boundary temperature of the extended system is the temperature of the surroundings at all times. The entropy balance for it can be expressed as S in − S out 1424 3


mi s i −


Change in entropy

Qout + S gen = ∆S tank = m 2 s 2 − m1 s1 Tb,in

0

= m2 s 2

Therefore, the total entropy generated during this process is S gen = − m i s i + m 2 s 2 +

Qout = m 2 (s 2 − s i ) Tb,out

0

+

Qout Q 1.2 kJ = out = = 0.00415 kJ/K Tb,out Tsurr 290 K




8-119

8-123 Argon gas in a piston–cylinder device expands isothermally as a result of heat transfer from a furnace. The useful work output, the exergy destroyed, and the reversible work are to be determined. Assumptions 1 Argon at specified conditions can be treated as an ideal gas since it is well above its critical temperature of 151 K. 2 The kinetic and potential energies are negligible. Analysis We take the argon gas contained within the piston–cylinder device as the system. This is a closed system since no mass crosses the system boundary during the process. We note that heat is transferred to the system

from a source at 1200 K, but there is no heat exchange with the environment at 300 K. Also, the temperature of the system remains constant during the expansion process, and its volume doubles, that is, T2 = T1 and V2 = 2V1. (a) The only work interaction involved during this isothermal process is the quasi-equilibrium boundary work, which is determined from 2

∫

W = W b = PdV = P1V1 ln 1

V2 0.02 m 3 = (350 kPa)(0.01 m 3 )ln = 2.43 kPa ⋅ m 3 = 2.43 kJ V1 0.01 m 3

This is the total boundary work done by the argon gas. Part of this work is done against the atmospheric pressure P0 to push the air out of the way, and it cannot be used for any useful purpose. It is determined from

Wsurr = P0 (V 2 − V1 ) = (100 kPa)(0.02 − 0.01)m 3 = 1 kPa ⋅ m 3 = 1 kJ The useful work is the difference between these two: Wu = W − Wsurr = 2.43 − 1 = 1.43 kJ

That is, 1.43 kJ of the work done is available for creating a useful effect such as rotating a shaft. Also, the heat transfer from the furnace to the system is determined from an energy balance on the system to be E −E 1in424out 3

=


∆E system 1 424 3


Qin − Qb,out = ∆U = mc v ∆T = 0 Qin = Qb,out = 2.43 kJ

(b) The exergy destroyed during a process can be determined from an exergy balance, or directly from Xdestroyed = T0Sgen. We will use the second approach since it is usually easier. But first we determine the entropy generation by applying an entropy balance on an extended system (system + immediate surroundings), which includes the temperature gradient zone between the cylinder and the furnace so that the temperature at the boundary where heat transfer occurs is TR = 1200 K. This way, the entropy generation associated with the heat transfer is included. Also, the entropy change of the argon gas can be determined from Q/Tsys since its temperature remains constant. S in − S out 14243



Change in entropy

Q Q + S gen = ∆S sys = TR Tsys

Therefore, S gen =

Q Q 2.43 kJ 2.43 kJ − = − = 0.00405 kJ/K 400 K 1200 K Tsys TR

and X dest = T0 S gen = (300 K)(0.00405 kJ/K) = 1.22 kJ


8-120

(c) The reversible work, which represents the maximum useful work that could be produced Wrev,out, can be determined from the exergy balance by setting the exergy destruction equal to zero, X −X 1in424out 3

Net exergy transfer by heat, work,and mass


⎛ T0 ⎜1 − ⎜ T b ⎝

Change in exergy

⎞ ⎟Q − Wrev,out = X 2 − X 1 ⎟ ⎠ = (U 2 − U 1 ) + P0 (V 2 −V1 ) − T0 (S 2 − S1 ) = 0 + Wsurr − T0

Q Tsys

since ∆KE = ∆PE = 0 and ∆U = 0 (the change in internal energy of an ideal gas is zero during an isothermal process), and ∆Ssys = Q/Tsys for isothermal processes in the absence of any irreversibilities. Then, ⎛ T ⎞ Q − Wsurr + ⎜⎜1 − 0 ⎟⎟Q Tsys ⎝ TR ⎠ 2.43 kJ 300 K ⎞ ⎛ = (300 K) − (1 kJ) + ⎜1 − ⎟(2.43 kJ) 400 K ⎝ 1200 K ⎠ = 2.65 kJ

Wrev,out = T0

Therefore, the useful work output would be 2.65 kJ instead of 1.43 kJ if the process were executed in a totally reversible manner. Alternative Approach The reversible work could also be determined by applying the basics only, without resorting to exergy balance. This is done by replacing the irreversible portions of the process by reversible ones that create

the same effect on the system. The useful work output of this idealized process (between the actual end states) is the reversible work. The only irreversibility the actual process involves is the heat transfer between the system and the furnace through a finite temperature difference. This irreversibility can be eliminated by operating a reversible heat engine between the furnace at 1200 K and the surroundings at 300 K. When 2.43 kJ of heat is supplied to this heat engine, it produces a work output of ⎛ T WHE = η revQH = ⎜⎜1 − L ⎝ TH

⎞ 300 K ⎞ ⎛ ⎟⎟QH = ⎜1 − ⎟(2.43 kJ) = 1.82 kJ 1200 K⎠ ⎝ ⎠

The 2.43 kJ of heat that was transferred to the system from the source is now extracted from the surrounding air at 300 K by a reversible heat pump that requires a work input of WHP,in =

QH QH 2.43 kJ = = = 0.61 kJ COPHP TH /(TH − TL ) (400 K) /(400 − 300)K

Then the net work output of this reversible process (i.e., the reversible work) becomes W rev = Wu + W HE − W HP,in = 1.43 + 1.82 − 0.61 = 2.64 kJ

which is practically identical to the result obtained before. Also, the exergy destroyed is the difference between the reversible work and the useful work, and is determined to be X dest = W rev,out − W u,out = 2.65 − 1.43 = 1.22 kJ

which is identical to the result obtained before.


8-121

8-124 A heat engine operates between two constant-pressure cylinders filled with air at different temperatures. The maximum work that can be produced and the final temperatures of the cylinders are to be determined. Assumptions Air is an ideal gas with constant specific heats at room temperature. Properties The gas constant of air is 0.287 kPa.m3/kg.K (Table A-1). The constant pressure specific heat of air at room temperature is cp = 1.005 kJ/kg.K (Table A-2). Analysis For maximum power production, the entropy generation must be zero. We take the two cylinders (the heat source and heat sink) and the heat engine as the system. Noting that the system involves no heat and mass transfer and that the entropy change for cyclic devices is zero, the entropy balance can be expressed as

S in − S out 1424 3


+ S gen 0 = ∆S system 123 1 424 3 Entropy generation

0 + S gen

0

Change in entropy

= ∆S cylinder,source + ∆S cylinder,sink + ∆S heat engine

0

AIR 30 kg 900 K QH

∆S cylinder,source + ∆S cylinder,sink = 0 ⎛ ⎜ mc ln T2 − mR ln P2 ⎜ p T1 P1 ⎝ ln

0

⎛ ⎞ T P ⎟ + 0 + ⎜ mc p ln 2 − mR ln 2 ⎜ ⎟ T1 P1 ⎝ ⎠ source

0

⎞ ⎟ =0 ⎟ ⎠ sink

T2 T2 =0 ⎯ ⎯→ T22 = T1 AT1B T1 A T1B

where T1A and T1B are the initial temperatures of the source and the sink, respectively, and T2 is the common final temperature. Therefore, the final temperature of the tanks for maximum power production is

W

HE QL AIR 30 kg 300 K

T2 = T1 AT1B = (900 K)(300 K) = 519.6 K

The energy balance Ein − Eout = ∆Esystem for the source and sink can be expressed as follows: Source:

−Qsource,out + Wb,in = ∆U → Qsource,out = ∆H = mc p (T1A − T2 ) Qsource,out = mc p (T1A − T2 ) = (30 kg)(1.005 kJ/kg ⋅ K)(900 − 519.6)K = 11,469 kJ Sink:

Qsink,in − Wb,out = ∆U → Qsink,in = ∆H = mc p (T2 − T1 A ) Qsink,in = mc p (T2 − T1B ) = (30 kg)(1.005 kJ/kg ⋅ K)(519.6 − 300)K = 6621 kJ Then the work produced becomes W max,out = Q H − Q L = Qsource,out − Qsink,in = 11,469 − 6621 = 4847 kJ

Therefore, a maximum of 4847 kJ of work can be produced during this process


8-122

8-125 A heat engine operates between a nitrogen tank and an argon cylinder at different temperatures. The maximum work that can be produced and the final temperatures are to be determined. Assumptions Nitrogen and argon are ideal gases with constant specific heats at room temperature. Properties The constant volume specific heat of nitrogen at room temperature is cv = 0.743 kJ/kg.K. The constant pressure specific heat of argon at room temperature is cp = 0.5203 kJ/kg.K (Table A-2). Analysis For maximum power production, the entropy generation must be zero. We take the tank, the cylinder (the heat source and the heat sink) and the heat engine as the system. Noting that the system involves no heat and mass transfer and that the entropy change for cyclic devices is zero, the entropy balance can be expressed as

S in − S out 1424 3


+ S gen 0 = ∆S system 123 1 424 3 Entropy generation

0 + S gen

0

Change in entropy

= ∆S tank,source + ∆S cylinder,sink + ∆S heat engine

0

N2 20 kg 1000 K QH

(∆S ) source + (∆S ) sink = 0 ⎛ ⎜ mcv ln T2 − mR ln V 2 ⎜ T1 V1 ⎝

0

⎛ ⎞ T P ⎟ + 0 + ⎜ mc p ln 2 − mR ln 2 ⎜ ⎟ T P1 1 ⎝ ⎠ source

0

⎞ ⎟ =0 ⎟ ⎠ sink

Substituting, T2 T2 (20 kg)(0.743 kJ / kg ⋅ K) ln + (10 kg)(0.5203 kJ / kg ⋅ K) ln =0 1000 K 300 K

Solving for T2 yields T2 = 731.8 K

W

HE QL Ar 10 kg 300 K

where T2 is the common final temperature of the tanks for maximum power production. The energy balance E in − E out = ∆E system for the source and sink can be expressed as follows: Source: −Qsource, out = ∆U = mcv (T2 − T1 A ) → Qsource, out = mcv (T1 A − T 2 )

Qsource,out = mcv (T1 A − T2 ) = ( 20 kg)(0.743 kJ/kg ⋅ K)(1000 − 731.8)K = 3985 kJ

Sink:

Qsink,in − Wb,out = ∆U → Qsink,in = ∆H = mc p (T2 − T1 A ) Qsink,in = mcv (T2 − T1 A ) = (10 kg)(0.5203 kJ/kg ⋅ K)(731.8 − 300)K = 2247 kJ

Then the work produced becomes Wmax,out = Q H − Q L = Qsource,out − Qsink,in = 3985 − 2247 = 1739 kJ

Therefore, a maximum of 1739 kJ of work can be produced during this process


8-123

8-126 A rigid tank containing nitrogen is considered. Heat is now transferred to the nitrogen from a reservoir and nitrogen is allowed to escape until the mass of nitrogen becomes one-half of its initial mass. The change in the nitrogen's work potential is to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process. 2 Kinetic and potential energies are negligible. 3 There are no work interactions involved. 4 Nitrogen is an ideal gas with constant specific heats. Properties The properties of nitrogen at room temperature are cp = 1.039 kJ/kg⋅K, cv = 0.743 kJ/kg⋅K, and R = 0.2968 kJ/kg⋅K (Table A-2a). Analysis The initial and final masses in the tank are m1 =

(1200 kPa )(0.050 m 3 ) PV = = 0.690 kg RT1 (0.2968 kPa ⋅ m 3 /kg ⋅ K )(293 K )

m2 = me =

m1 0.690 kg = = 0.345 kg 2 2

Nitrogen 50 L 1200 kPa 20°C

Q

The final temperature in the tank is T2 =

(1200 kPa )(0.050 m 3 ) PV = = 586 K m 2 R (0.345 kg)(0.2968 kPa ⋅ m 3 /kg ⋅ K )

me

We take the tank as the system, which is a control volume since mass crosses the boundary. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances for this uniform-flow system can be expressed as Mass balance:

min − mout = ∆msystem → me = m 2 Energy balance: E −E 1in424out 3

=


∆E system 1 424 3


Qin − m e he = m 2 u 2 − m1u1 Qout = m e he + m 2 u 2 − m1u1

Using the average of the initial and final temperatures for the exiting nitrogen, Te = 0.5(T1 + T2 ) = 0.5((293 + 586) = 439.5 K this energy balance equation becomes Qout = me he + m 2 u 2 − m1u1 = me c p Te + m 2 cv T2 − m1cv T1 = (0.345)(1.039)(439.5) + (0.345)(0.743)(586) − (0.690)(0.743)(293) = 157.5 kJ The work potential associated with this process is equal to the exergy destroyed during the process. The exergy destruction during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen . The entropy generation Sgen in this case is determined from an entropy balance on the system: S in − S out 1424 3



Change in entropy

Qin − m e s e + S gen = ∆S tank = m 2 s 2 − m1 s1 TR S gen = m 2 s 2 − m1 s1 + m e s e −

Qin TR


8-124

Noting that pressures are same, rearranging and substituting gives S gen = m 2 s 2 − m1 s1 + m e s e −

Qin TR

= m 2 c p ln T2 − m1c p ln T1 + me c p ln Te −

Qin TR

= (0.345)(1.039) ln(586) − (0.690)(1.039) ln(293) + (0.345)(1.039) ln(439.5) −

157.5 773

= 0.190 kJ/K

Then, W rev = X destroyed = T0 S gen = ( 293 K)(0.190 kJ/K) = 55.7 kJ

Alternative More Accurate Solution

This problem may also be solved by considering the variation of gas temperature at the outlet of the tank. The mass and energy balances are dm dt d mu ) ( dm c v d (mT ) dm Q& = −h = − c pT dt dt dt dt

m& e = −

Combining these expressions and replacing T in the last term gives d (mT ) c p PV dm − Q& = c v dt Rm dt

Integrating this over the time required to release one-half the mass produces

Q = c v (m 2 T2 − m1T1 ) −

c p PV R

ln

m2 m1

The reduced combined first and second law becomes ⎛ T W& rev = Q& ⎜⎜1 − 0 ⎝ TR

⎞ d (U − T0 S ) dm ⎟⎟ − + ( h − T0 s ) dt dt ⎠

when the mass balance is substituted and the entropy generation is set to zero (for maximum work production). Expanding the system time derivative gives ⎛ T W& rev = Q& ⎜⎜1 − 0 ⎝ TR

⎞ d (mu − T0 ms ) dm ⎟⎟ − + ( h − T0 s ) dt dt ⎠

⎛ T = Q& ⎜⎜1 − 0 ⎝ TR

⎞ d (mu ) ds dm dm ⎟⎟ − + T0 m + T 0 s + ( h − T0 s ) dt dt dt dt ⎠

⎛ T = Q& ⎜⎜1 − 0 ⎝ TR

⎞ d (mu ) T dh dm ⎟⎟ − +h +m 0 dt dt T dt ⎠


8-125

Substituting Q& from the first law, dm ⎤⎛ T0 ⎡ d (mu ) ⎜1 − W& rev = ⎢ −h dt dt ⎥⎦⎜⎝ T R ⎣

⎞ ⎡ d (mu ) T0 dh dm ⎤ ⎟⎟ − ⎢ −h ⎥ + m T dt dt dt ⎦ ⎠ ⎣

T0 ⎡ d (mu ) dm dh ⎤ −h −m ⎥ T R ⎢⎣ dt dt dt ⎦ T ⎡ d (mT ) dm dT ⎤ = − 0 ⎢c v − c pT − mc p TR ⎣ dt dt dt ⎥⎦ =−

At any time, T=

PV mR

which further reduces this result to ⎛ c p dT R dP ⎞ T PV dm ⎟ + T0 m⎜⎜ − W& rev = 0 c p ⎟ TR mR dt T dt P dt ⎝ ⎠ When this integrated over the time to complete the process, the result is c p PV T0 c p PV m ln 2 + T0 TR R m1 R

⎛1 1 ⎞ ⎜⎜ − ⎟⎟ ⎝ T1 T2 ⎠ (1.039)(1200)(0.050) ⎛ 1 293 (1.039)(1200)(0.050) 1 1 ⎞ ln + (293) = − ⎟ ⎜ 773 0.2968 2 0.2968 ⎝ 293 586 ⎠ = 49.8 kJ

Wrev =


8-126

8-127 A rigid tank containing nitrogen is considered. Nitrogen is allowed to escape until the mass of nitrogen becomes onehalf of its initial mass. The change in the nitrogen's work potential is to be determined. Assumptions 1 This is an unsteady process since the conditions within the device are changing during the process. 2 Kinetic and potential energies are negligible. 3 There are no work interactions involved. 4 Nitrogen is an ideal gas with constant specific heats. Properties The properties of nitrogen at room temperature are cp = 1.039 kJ/kg⋅K, cv = 0.743 kJ/kg⋅K, k = 1.4, and R = 0.2968 kJ/kg⋅K (Table A-2a). Analysis The initial and final masses in the tank are m1 =

(1000 kPa )(0.100 m 3 ) PV = = 1.150 kg RT1 (0.2968 kPa ⋅ m 3 /kg ⋅ K )(293 K )

m 2 = me =

m1 1.150 kg = = 0.575 kg 2 2

Nitrogen 100 L 1000 kPa 20°C

me

We take the tank as the system, which is a control volume since mass crosses the boundary. Noting that the microscopic energies of flowing and nonflowing fluids are represented by enthalpy h and internal energy u, respectively, the mass and energy balances for this uniform-flow system can be expressed as Mass balance:

min − mout = ∆msystem → me = m 2 Energy balance: E −E 1in424out 3

=


∆E system 1 424 3


− m e he = m 2 u 2 − m1u1

Using the average of the initial and final temperatures for the exiting nitrogen, this energy balance equation becomes − m e he = m 2 u 2 − m1u1 − m e c p Te = m 2 cv T2 − m1 cv T1 − (0.575)(1.039)(0.5)(293 + T2 ) = (0.575)(0.743)T2 − (1.150)(0.743)(293) Solving for the final temperature, we get T2 = 224.3 K

The final pressure in the tank is P2 =

m 2 RT2

V

=

(0.575 kg)(0.2968 kPa ⋅ m 3 /kg ⋅ K )(224.3 K) 0.100 m 3

= 382.8 kPa

The average temperature and pressure for the exiting nitrogen is Te = 0.5(T1 + T2 ) = 0.5(293 + 224.3) = 258.7 K Pe = 0.5( P1 + P2 ) = 0.5(1000 + 382.8) = 691.4 kPa

The work potential associated with this process is equal to the exergy destroyed during the process. The exergy destruction during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen . The entropy generation Sgen in this case is determined from an entropy balance on the system: S − S out 1in424 3



Change in entropy

− m e s e + S gen = ∆S tank = m 2 s 2 − m1 s1 S gen = m 2 s 2 − m1 s1 + m e s e PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

8-127

Rearranging and substituting gives S gen = m 2 s 2 − m1 s1 + m e s e = m 2 (c p ln T2 − R ln P2 ) − m1 (c p ln T1 − R ln P1 ) + m e (c p ln Te − R ln Pe )

= (0.575)[1.039 ln(224.3) − (0.2968) ln(382.8)] − (1.15)[1.039 ln(293) − (0.2968) ln(1000)] + (0.575)[1.039 ln(258.7) − (0.2968) ln(691.4)] = 2.2188 − 4.4292 + 2.2032 = −0.007152 kJ/K Then,

Wrev = X destroyed = T0 S gen = (293 K)(−0.007152 kJ/K) = −2.10 kJ The entropy generation cannot be negative for a thermodynamically possible process. This result is probably due to using average temperature and pressure values for the exiting gas and using constant specific heats for nitrogen. This sensitivity occurs because the entropy generation is very small in this process.

Alternative More Accurate Solution

This problem may also be solved by considering the variation of gas temperature and pressure at the outlet of the tank. The mass balance in this case is m& e = −

dm dt

which when combined with the reduced first law gives d (mu ) dm =h dt dt

Using the specific heats and the ideal gas equation of state reduces this to cv

V dP R dt

= c pT

dm dt

which upon rearrangement and an additional use of ideal gas equation of state becomes

1 dP c p 1 dm = P dt c v m dt When this is integrated, the result is ⎛m P2 = P1 ⎜⎜ 2 ⎝ m1

k

⎞ ⎛1⎞ ⎟⎟ = 1000⎜ ⎟ ⎝2⎠ ⎠

1.4

= 378.9 kPa

The final temperature is then T2 =

P2V (378.9 kPa )(0.100 m 3 ) = = 222.0 K m 2 R (0.575 kg)(0.2968 kPa ⋅ m 3 /kg ⋅ K )

The process is then one of

mk = const or P

m k −1 = const T

The reduced combined first and second law becomes d (U − T0 S ) dm W& rev = − + ( h − T0 s ) dt dt

when the mass balance is substituted and the entropy generation is set to zero (for maximum work production). Replacing the enthalpy term with the first law result and canceling the common dU/dt term reduces this to PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

8-128

d ( ms ) dm W& rev = T0 − T0 s dt dt

Expanding the first derivative and canceling the common terms further reduces this to ds W& rev = T0 m dt

Letting a = P1 / m1k and b = T1 / m1k −1 , the pressure and temperature of the nitrogen in the system are related to the mass by P = am k and T = bm k −1

according to the first law. Then, dP = akm k −1 dm and dT = b(k − 1)m k − 2 dm

The entropy change relation then becomes ds = c p

[

]

dT dP dm −R = ( k − 1)c p − Rk T P m

Now, multiplying the combined first and second laws by dt and integrating the result gives 2

2

∫ ∫ [ = T [(k − 1)c − Rk ](m

]

W rev = T0 mds = T0 mds (k − 1)c p − Rk dm 1

0

1

p

2

− m1 )

= (293)[(1.4 − 1)(1.039) − (0.2968)(1.4)](0.575 − 1.15) = −0.0135 kJ

Once again the entropy generation is negative, which cannot be the case for a thermodynamically possible process. This is probably due to using constant specific heats for nitrogen. This sensitivity occurs because the entropy generation is very small in this process.


8-129

8-128 Steam is condensed by cooling water in the condenser of a power plant. The rate of condensation of steam and the rate of exergy destruction are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 Fluid properties are constant. Properties The enthalpy and entropy of vaporization of water at 45°C are hfg = 2394.0 kJ/kg and sfg = 7.5247 kJ/kg.K (Table A-4). The specific heat of water at room temperature is cp = 4.18 kJ/kg.°C (Table A-3). Analysis (a) We take the cold water tubes as the system, which is a control volume. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

=



=0

Steam 45°C


20°C


Then the heat transfer rate to the cooling water in the condenser becomes

12°C

Q& = [m& c p (Tout − Tin )] cooling water

Water

= (330 kg/s)(4.18 kJ/kg.°C)(20°C − 12°C) = 11,035 kJ/s

45°C

The rate of condensation of steam is determined to be

Q& 11,035 kJ/s Q& = (m& h fg ) steam ⎯ ⎯→ m& steam = = = 4.61 kg/s h fg 2394.0 kJ/kg (b) The rate of entropy generation within the condenser during this process can be determined by applying the rate form of the entropy balance on the entire condenser. Noting that the condenser is well-insulated and thus heat transfer is negligible, the entropy balance for this steady-flow system can be expressed as S& − S& out 1in424 3

+


S& gen {



m& 1 s1 + m& 3 s 3 − m& 2 s 2 − m& 4 s 4 + S& gen = 0 (since Q = 0) m& water s1 + m& steam s 3 − m& water s 2 − m& steam s 4 + S& gen = 0 S& gen = m& water ( s 2 − s1 ) + m& steam ( s 4 − s 3 ) Noting that water is an incompressible substance and steam changes from saturated vapor to saturated liquid, the rate of entropy generation is determined to be T T S& gen = m& water c p ln 2 + m& steam ( s f − s g ) = m& water c p ln 2 − m& steam s fg T1 T1 = (330 kg/s)(4.18 kJ/kg.K)ln

20 + 273 − (4.61 kg/s)(7.5247 kJ/kg.K) = 3.501 kW/K 12 + 273

Then the exergy destroyed can be determined directly from its definition X destroyed = T0 S gen to be X& destroyed = T0 S& gen = (285 K)(3.501 kW/K) = 998 kW


8-130

8-129 A system consisting of a compressor, a storage tank, and a turbine as shown in the figure is considered. The change in the exergy of the air in the tank and the work required to compress the air as the tank was being filled are to be determined. Assumptions 1 Changes in the kinetic and potential energies are negligible. 4 Air is an ideal gas with constant specific heats. Properties The properties of air at room temperature are R = 0.287 kPa⋅m3/kg⋅K, cp = 1.005 kJ/kg⋅K, cv = 0.718 kJ/kg⋅K, k = 1.4 (Table A-2a). Analysis The initial mass of air in the tank is m initial =

PinitialV (100 kPa)(5 × 10 5 m 3 ) = = 0.5946 × 10 6 kg RTinitial (0.287 kPa ⋅ m 3 /kg ⋅ K)(293 K)

and the final mass in the tank is m final =

PfinalV (600 kPa)(5 × 10 5 m 3 ) = = 3.568 × 10 6 kg RTfinal (0.287 kPa ⋅ m 3 /kg ⋅ K)(293 K)

Since the compressor operates as an isentropic device,

⎛P T2 = T1 ⎜⎜ 2 ⎝ P1

⎞ ⎟⎟ ⎠

( k −1) / k

The conservation of mass applied to the tank gives dm = m& in dt

while the first law gives d ( mu ) dm Q& = −h dt dt

Employing the ideal gas equation of state and using constant specific heats, expands this result to

Vc dP V dP Q& = v − c p T2 R dt RT dt Using the temperature relation across the compressor and multiplying by dt puts this result in the form ⎛P⎞ Vc Q& dt = v dP − c p T1 ⎜⎜ ⎟⎟ R ⎝ P1 ⎠

( k −1) / k

V RT

dP

When this integrated, it yields (i and f stand for initial and final states) ⎡ k c pV ⎢ ⎛⎜ P f ( P f − Pi ) − Q= Pf 2k − 1 R ⎢ ⎜⎝ Pi R ⎣

Vcv

=

⎞ ⎟ ⎟ ⎠

( k −1) / k

⎤ − Pi ⎥ ⎥ ⎦

0.4 / 1.4 ⎤ (5 × 10 5 )(0.718) (1.005)(5 × 10 5 ) ⎡ 1.4 ⎛ 600 ⎞ (600 − 100) − 600 − 100⎥ ⎢ ⎜ ⎟ 0.287 2(1.4) − 1 0.287 ⎝ 100 ⎠ ⎥⎦ ⎢⎣

= −6.017 × 10 8 kJ

The negative result show that heat is transferred from the tank. Applying the first law to the tank and compressor gives

(Q& − W& out )dt = d (mu ) − h1 dm which integrates to

Q − Wout = (m f u f − mi u i ) − h1 (m f − mi ) PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

8-131

Upon rearrangement, W out = Q + (c p − cv )T (m f − m i ) = −6.017 × 10 8 + (1.005 − 0.718)(293)[(3.568 − 0.5946) × 10 6 ] = −3.516 × 10 8 kJ

The negative sign shows that work is done on the compressor. When the combined first and second laws is reduced to fit the compressor and tank system and the mass balance incorporated, the result is ⎛ T W& rev = Q& ⎜⎜1 − 0 ⎝ TR

⎞ d (U − T0 S ) dm ⎟⎟ − + ( h − T0 s ) dt dt ⎠

which when integrated over the process becomes ⎛ T W rev = Q⎜⎜1 − 0 ⎝ TR ⎛ T = Q⎜⎜1 − 0 ⎝ TR

[

⎞ ⎟⎟ + m i [(u i − h1 ) − T0 ( s i − s1 )] − m f (u f − h1 ) − T0 ( s f − s1 ) ⎠

[

]

⎞ ⎟⎟ + m i Ti (c v − c p ) − m f ⎠

⎡ ⎛ Pf ⎢T f (c v − c p ) − T0 R ln⎜⎜ ⎢⎣ ⎝ Pi

]

⎞⎤ ⎟⎥ ⎟ ⎠⎥⎦

⎛ 293 ⎞ 6 = −6.017 × 10 8 ⎜1 − ⎟ + 0.5946 × 10 [(0.718 − 1.005)293] ⎝ 293 ⎠ 600 ⎤ ⎡ − 3.568 × 10 6 ⎢(0.718 − 1.005)293 + 293(0.287) ln 100 ⎥⎦ ⎣ = 7.875 × 10 8 kJ

This is the exergy change of the air stored in the tank.


8-132

8-130 The air stored in the tank of the system shown in the figure is released through the isentropic turbine. The work produced and the change in the exergy of the air in the tank are to be determined. Assumptions 1 Changes in the kinetic and potential energies are negligible. 4 Air is an ideal gas with constant specific heats. Properties The properties of air at room temperature are R = 0.287 kPa⋅m3/kg⋅K, cp = 1.005 kJ/kg⋅K, cv = 0.718 kJ/kg⋅K, k = 1.4 (Table A-2a). Analysis The initial mass of air in the tank is m initial =

PinitialV (600 kPa)(5 × 10 5 m 3 ) = = 3.568 × 10 6 kg RTinitial (0.287 kPa ⋅ m 3 /kg ⋅ K)(293 K)

and the final mass in the tank is

mfinal =

PfinalV (100 kPa)(5 × 105 m3 ) = = 0.5946 × 106 kg 3 RTfinal (0.287 kPa ⋅ m /kg ⋅ K)(293 K)

The conservation of mass is dm = m& in dt

while the first law gives d ( mu ) dm Q& = −h dt dt

Employing the ideal gas equation of state and using constant specific heats, expands this result to

Vc dP V dP − c pT Q& = v R dt RT dt cv − c p dP = V R dt dP = −V dt When this is integrated over the process, the result is (i and f stand for initial and final states) Q = −V ( P f − Pi ) = −5 × 10 5 (100 − 600) = 2.5 × 10 8 kJ

Applying the first law to the tank and compressor gives

(Q& − W& out )dt = d (mu ) − hdm which integrates to Q − Wout = (m f u f − mi u i ) + h(mi − m f ) − Wout = −Q + m f u f − m i u i + h(mi − m f ) Wout = Q − m f u f + mi u i − h(mi − m f ) = Q − m f c v T + mi c p T − c p T (mi − m f ) = 2.5 × 10 8 − (0.5946 × 10 6 )(0.718)(293) + (3.568 × 10 6 )(1.005)(293) − (1.005)(293)(3.568 × 10 6 − 0.5946 × 10 6 ) = 3.00 × 10 8 kJ This is the work output from the turbine. When the combined first and second laws is reduced to fit the turbine and tank system and the mass balance incorporated, the result is


8-133

⎛ T W& rev = Q& ⎜⎜1 − 0 ⎝ TR ⎛ T = Q& ⎜⎜1 − 0 ⎝ TR ⎛ T = Q& ⎜⎜1 − 0 ⎝ TR ⎛ T = Q& ⎜⎜1 − 0 ⎝ TR

⎞ d (U − T0 S ) dm ⎟⎟ − + ( h − T0 s ) dt dt ⎠ ⎞ d (u − T0 s ) dm dm ⎟⎟ − (u − T0 s ) −m + ( h − T0 s ) dt dt dt ⎠ ⎞ dm ds ⎟⎟ + (c p − c v )T + mT0 dt dt ⎠ ⎞ T dm ⎟⎟ + (c p − c v )T + V 0 ( P f − Pi ) dt T ⎠

where the last step uses entropy change equation. When this is integrated over the process it becomes ⎞ T ⎟⎟ + (c p − c v )T (m f − mi ) + V 0 ( P f − Pi ) T ⎠ ⎛ 293 ⎞ 6 5 293 = 3.00 × 10 8 ⎜1 − (100 − 600) ⎟ + (1.005 − 0.718)(293)(0.5946 − 3.568) × 10 + 5 × 10 293 293 ⎝ ⎠

⎛ T W rev = Q⎜⎜1 − 0 ⎝ TR

= 0 − 2.500 × 10 8 − 2.5 × 10 8 = −5.00 × 10 8 kJ This is the exergy change of the air in the storage tank.


8-134

8-131 A heat engine operates between a tank and a cylinder filled with air at different temperatures. The maximum work that can be produced and the final temperatures are to be determined. Assumptions Air is an ideal gas with constant specific heats at room temperature. Properties The specific heats of air are cv = 0.718 kJ/kg.K and cp = 1.005 kJ/kg.K (Table A-2). Analysis For maximum power production, the entropy generation must be zero. We take the tank, the cylinder (the heat source and the heat sink) and the heat engine as the system. Noting that the system involves no heat and mass transfer and that the entropy change for cyclic devices is zero, the entropy balance can be expressed as

S in − S out 1424 3


+ S gen 0 = ∆S system 123 1 424 3 Change in entropy

Entropy generation

0 + S gen

0

= ∆S tank,source + ∆S cylinder,sink + ∆S heat engine

0

Air 40 kg 600 K

(∆S ) source + (∆S ) sink = 0 ⎛ ⎜ mc v ln T2 − mR ln V 2 ⎜ V1 T1 ⎝

0

QH

⎞ ⎛ P T ⎟ + 0 + ⎜ mc p ln 2 − mR ln 2 ⎟ ⎜ P1 T 1 ⎠ source ⎝

cp T T T ⎛T ln 2 + ln 2 = 0 ⎯ ⎯→ 2 ⎜⎜ 2 T1 A cv T1B T1 A ⎝ T1B

k

0

⎞ ⎟ =0 ⎟ ⎠ sink

(

⎞ ⎟⎟ = 1 ⎯ ⎯→ T2 = T1 AT1kB ⎠

)

1 /( k +1)

where T1A and T1B are the initial temperatures of the source and the sink, respectively, and T2 is the common final temperature. Therefore, the final temperature of the tanks for maximum power production is

(

T2 = (600 K)(280 K)1.4

)

1 2.4

W

HE

QL Air 40 kg 280 K

= 384.7 K

Source: −Qsource,out = ∆U = mcv (T2 − T1 A ) → Qsource,out = mcv (T1 A − T2 ) Qsource,out = mcv (T1 A − T2 ) = ( 40 kg)(0.718 kJ/kg ⋅ K)(600 − 384.7)K = 6184 kJ

Sink:

Qsink,in − Wb,out = ∆U → Qsink,in = ∆H = mc p (T2 − T1 A ) Qsink,in = mcv (T2 − T1 A ) = (40 kg)(1.005 kJ/kg ⋅ K)(384.7 − 280)K = 4208 kJ

Then the work produced becomes W max,out = Q H − Q L = Qsource,out − Qsink,in = 6184 − 4208 = 1977 kJ

Therefore, a maximum of 1977 kJ of work can be produced during this process.


8-135

8-132E Large brass plates are heated in an oven at a rate of 300/min. The rate of heat transfer to the plates in the oven and the rate of exergy destruction associated with this heat transfer process are to be determined. Assumptions 1 The thermal properties of the plates are constant. 2 The changes in kinetic and potential energies are negligible. 3 The environment temperature is 75°F. Properties The density and specific heat of the brass are given to be ρ = 532.5 lbm/ft3 and cp = 0.091 Btu/lbm.°F. Analysis We take the plate to be the system. The energy balance for this closed system can be expressed as

E −E 1in424out 3

=


∆E system 1 424 3


Qin = ∆U plate = m(u 2 − u1 ) = mc(T2 − T1 ) The mass of each plate and the amount of heat transfer to each plate is

m = ρV = ρLA = (532.5 lbm/ft 3 )[(1.2 / 12 ft )(2 ft)(2 ft)] = 213 lbm Qin = mc(T2 − T1 ) = (213 lbm/plate)(0.091 Btu/lbm.°F)(1000 − 75)°F = 17,930 Btu/plate

Then the total rate of heat transfer to the plates becomes Q& total = n& plate Qin, per plate = (300 plates/min) × (17,930 Btu/plate ) = 5,379,000 Btu/min = 89,650 Btu/s

We again take a single plate as the system. The entropy generated during this process can be determined by applying an entropy balance on an extended system that includes the plate and its immediate surroundings so that the boundary temperature of the extended system is at 1300°F at all times: S − S out 1in424 3



Change in entropy

Q Qin + S gen = ∆S system → S gen = − in + ∆S system Tb Tb

where ∆S system = m( s 2 − s1 ) = mc avg ln S gen = −

T2 (1000 + 460) R = (213 lbm)(0.091 Btu/lbm.R) ln = 19.46 Btu/R Substituting, T1 (75 + 460) R

Qin 17,930 Btu + ∆S system = − + 19.46 Btu/R = 9.272 Btu/R (per plate) Tb 1300 + 460 R

Then the rate of entropy generation becomes S& gen = S gen n& ball = (9.272 Btu/R ⋅ plate)(300 plates/min) = 2781 Btu/min.R = 46.35 Btu/s.R

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X& destroyed = T0 S& gen = (535 R)(46.35 Btu/s.R) = 24,797 Btu/s


8-136

8-133 Long cylindrical steel rods are heat-treated in an oven. The rate of heat transfer to the rods in the oven and the rate of exergy destruction associated with this heat transfer process are to be determined. Assumptions 1 The thermal properties of the rods are constant. 2 The changes in kinetic and potential energies are negligible. 3 The environment temperature is 30°C. Properties The density and specific heat of the steel rods are given to be ρ = 7833 kg/m3 and cp = 0.465 kJ/kg.°C. Analysis Noting that the rods enter the oven at a velocity of 3 m/min and exit at the same velocity, we can say that a 3-m long section of the rod is heated in the oven in 1 min. Then the mass of the rod heated in 1 minute is

m = ρV = ρLA = ρL(πD 2 / 4 ) = (7833 kg / m 3 )(3 m)[π (01 . m) 2 / 4 ] = 184.6 kg We take the 3-m section of the rod in the oven as the system. The energy balance for this closed system can be expressed as E −E 1in424out 3

=


∆E system 1 424 3


Qin = ∆U rod = m(u 2 − u1 ) = mc(T2 − T1 )

Substituting, Qin = mc(T2 − T1 ) = (184.6 kg )(0.465 kJ/kg.°C)(700 − 30)°C = 57,512 kJ

Noting that this much heat is transferred in 1 min, the rate of heat transfer to the rod becomes

Q& in = Qin / ∆t = (57,512 kJ)/(1 min) = 57,512 kJ/min = 958.5 kW We again take the 3-m long section of the rod as the system The entropy generated during this process can be determined by applying an entropy balance on an extended system that includes the rod and its immediate surroundings so that the boundary temperature of the extended system is at 900°C at all times: S in − S out 1424 3



Change in entropy

Qin Q + S gen = ∆S system → S gen = − in + ∆S system Tb Tb

where ∆S system = m( s 2 − s1 ) = mc avg ln

T2 700 + 273 = (184.6 kg)(0.465 kJ/kg.K) ln = 100.1 kJ/K T1 30 + 273

Substituting, S gen = −

Qin 57,512 kJ + ∆S system = − + 100.1 kJ/K = 51.1 kJ/K Tb (900 + 273) R

Noting that this much entropy is generated in 1 min, the rate of entropy generation becomes S& gen =

S gen ∆t

=

51.1 kJ/K = 51.1 kJ/min.K = 0.852 kW/K 1 min

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X& destroyed = T0 S& gen = ( 298 K)(0.852 kW/K) = 254 kW


8-137

8-134 Water is heated in a heat exchanger by geothermal water. The rate of heat transfer to the water and the rate of exergy destruction within the heat exchanger are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 Fluid properties are constant. 5 The environment temperature is 25°C. Properties The specific heats of water and geothermal fluid are given to be 4.18 and 4.31 kJ/kg.°C, respectively. Analysis (a) We take the cold water tubes as the system, which is a control volume. The energy balance for this steady-flow system can be expressed in the rate form as E& − E& 1in424out 3

=



=0

60°C Brine 140°C



Water 25°C

Then the rate of heat transfer to the cold water in the heat exchanger becomes Q& in, water = [m& c p (Tout − Tin )] water = (0.4 kg/s)(4.18 kJ/kg. °C)(60°C − 25°C) = 58.52 kW

Noting that heat transfer to the cold water is equal to the heat loss from the geothermal water, the outlet temperature of the geothermal water is determined from Q& 58.52 kW Q& out = [m& c p (Tin − Tout )] geo ⎯ ⎯→ Tout = Tin − out = 140°C − = 94.7°C & mc p (0.3 kg/s)(4.31 kJ/kg.°C)

(b) The rate of entropy generation within the heat exchanger is determined by applying the rate form of the entropy balance on the entire heat exchanger: S& − S& out 1in424 3


+

S& gen {



m& 1 s1 + m& 3 s 3 − m& 2 s 2 − m& 4 s 4 + S& gen = 0 (since Q = 0) m& water s1 + m& geo s 3 − m& water s 2 − m& geo s 4 + S& gen = 0 S& gen = m& water ( s 2 − s1 ) + m& geo ( s 4 − s 3 ) Noting that both fresh and geothermal water are incompressible substances, the rate of entropy generation is determined to be T T S& gen = m& water c p ln 2 + m& geo c p ln 4 T3 T1

The exergy 60 + 273 94.7 + 273 = (0.4 kg/s)(4.18 kJ/kg.K)ln + (0.3 kg/s)(4.31 kJ/kg.K)ln = 0.0356 kW/K 25 + 273 140 + 273 destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X& destroyed = T0 S& gen = ( 298 K)(0.0356 kW/K) = 10.61 kW


8-138

8-135 A regenerator is considered to save heat during the cooling of milk in a dairy plant. The amounts of fuel and money such a generator will save per year and the rate of exergy destruction within the regenerator are to be determined. Assumptions 1 Steady operating conditions exist. 2 The properties of the milk are constant. 5 The environment temperature is 18°C. Properties The average density and specific heat of milk can be taken to be ρmilk ≅ ρ water = 1 kg/L and cp,milk= 3.79

kJ/kg.°C (Table A-3). Analysis The mass flow rate of the milk is

m& milk = ρV&milk = (1 kg/L)(12 L/s) = 12 kg/s = 43,200 kg/h Taking the pasteurizing section as the system, the energy balance for this steady-flow system can be expressed in the rate form as

E& − E& 1in424out 3


=


= 0 → E& in = E& out


Q& in + m& h1 = m& h2 (since ∆ke ≅ ∆pe ≅ 0) Q& in = m& milk c p (T2 − T1 ) Therefore, to heat the milk from 4 to 72°C as being done currently, heat must be transferred to the milk at a rate of Q& current = [ m& c p (Tpasturizat ion − Trefrigeration )] milk = (12 kg/s)(3.79 kJ/kg. °C)(72 − 4)°C = 3093 kJ/s

The proposed regenerator has an effectiveness of ε = 0.82, and thus it will save 82 percent of this energy. Therefore,

Q& saved = εQ& current = (0.82)(3093 kJ / s) = 2536 kJ / s Noting that the boiler has an efficiency of ηboiler = 0.82, the energy savings above correspond to fuel savings of Fuel Saved =

Q& saved

η boiler

=

(2536 kJ / s) (1therm) = 0.02931therm / s (0.82) (105,500 kJ)

Noting that 1 year = 365×24=8760 h and unit cost of natural gas is $1.04/therm, the annual fuel and money savings will be Fuel Saved = (0.02931 therms/s)(8760×3600 s) = 924,450 therms/yr Money saved = (Fuel saved)(Unit cost of fuel) = (924,450 therm/yr)($1.04/therm) = $961,430/y r

The rate of entropy generation during this process is determined by applying the rate form of the entropy balance on an extended system that includes the regenerator and the immediate surroundings so that the boundary temperature is the surroundings temperature, which we take to be the cold water temperature of 18°C.: S& in − S& out 1424 3


+

S& gen {


= ∆S& system 0 (steady) → S& gen = S& out − S& in 1442443 Rate of change of entropy

Disregarding entropy transfer associated with fuel flow, the only significant difference between the two cases is the reduction in the entropy transfer to water due to the reduction in heat transfer to water, and is determined to be Q& out, reduction Q& saved 2536 kJ/s = = = 8.715 kW/K S& gen, reduction = S& out, reduction = Tsurr Tsurr 18 + 273 S gen, reduction = S& gen, reduction ∆t = (8.715 kJ/s.K)(87 60 × 3600 s/year) = 2.75 × 10 8 kJ/K (per year)

The exergy destroyed during a process can be determined from an exergy balance or directly from its definition X destroyed = T0 S gen , X destroyed, reduction = T0 Sgen, reduction = ( 291 K)(2.75 × 108 kJ/K) = 8.00 × 1010 kJ (per year)

8-136 Exhaust gases are expanded in a turbine, which is not well-insulated. Tha actual and reversible power outputs, the exergy destroyed, and the second-law efficiency are to be determined. PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

8-139

Assumptions 1 Steady operating conditions exist. 2 Potential energy change is negligible. 3 Air is an ideal gas with constant specific heats. Properties The gas constant of air is R = 0.287 kJ/kg.K and the specific heat of air at the average temperature of (627+527)/2 = 577ºC = 850 K is cp = 1.11 kJ/kg.ºC (Table A-2). Analysis (a) The enthalpy and entropy changes of air across the turbine are

∆h = c p (T2 − T1 ) = (1.11 kJ/kg.°C)(527 − 627)°C = −111 kJ/kg ∆s = c p ln

T2 P − R ln 2 T1 P1

(527 + 273) K 500 kPa = (1.11 kJ/kg.K)ln − (0.287 kJ/kg.K) ln (627 + 273) K 1200 kPa = 0.1205 kJ/kg.K The actual and reversible power outputs from the turbine are

Exh. gas 627°C 1.2 MPa Turbine

Q

527°C 500 kPa

− W& a,out = m& ∆h + Q& out = (2.5 kg/s)(-111 kJ/kg) + 20 kW = −257.5 kW − W& rev,out = m& (∆h − T0 ∆s ) = (2.5 kg/s)(111 kJ/kg) − (25 + 273 K)(0.1205 kJ/kg.K) = −367.3 kW or W& a,out = 257.5 kW W& rev,out = 367.3 kW (b) The exergy destroyed in the turbine is

X& dest = W& rev − W& a = 367.3 − 257.5 = 109.8 kW (c) The second-law efficiency is

η II =

W& a 257.5 kW = = 0.701 = 70.1% W& rev 367.3 kW


8-140

8-137 Refrigerant-134a is compressed in an adiabatic compressor, whose second-law efficiency is given. The actual work input, the isentropic efficiency, and the exergy destruction are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. Analysis (a) The properties of the refrigerant at the inlet of the compressor are (Tables A-11 through A-13) Tsat@160 kPa = −15.60°C P1 = 160 kPa

⎫ h1 = 243.60 kJ/kg ⎬ T1 = (−15.60 + 3)°C⎭ s1 = 0.95153 kJ/kg.K

The enthalpy at the exit for if the process was isentropic is

1 MPa

Compressor

P2 = 1 MPa

⎫ ⎬h2 s = 282.41 kJ/kg s 2 = s1 = 0.95153 kJ/kg.K ⎭

The expressions for actual and reversible works are

R-134a 160 kPa

wa = h2 − h1 = (h2 − 243.60)kJ/kg wrev = h2 − h1 − T0 ( s 2 − s1 ) = (h2 − 243.60)kJ/kg − (25 + 273 K)(s 2 − 0.95153)kJ/kg.K

Substituting these into the expression for the second-law efficiency

η II =

wrev h − 243.60 − (298)(s 2 − 0.95153) ⎯ ⎯→ 0.80 = 2 wa h2 − 243.60

The exit pressure is given (1 MPa). We need one more property to fix the exit state. By a trial-error approach or using EES, we obtain the exit temperature to be 60ºC. The corresponding enthalpy and entropy values satisfying this equation are h2 = 293.36 kJ/kg s 2 = 0.98492 kJ/kg.K

Then, wa = h2 − h1 = 293.36 − 243.60 = 49.76kJ/kg wrev = h2 − h1 − T0 ( s2 − s1 ) = (293.36 − 243.60)kJ/kg − (25 + 273 K)(0.98492 − 0.9515)kJ/kg ⋅ K = 39.81 kJ/kg

(b) The isentropic efficiency is determined from its definition

ηs =

h2s − h1 (282.41 − 243.60)kJ/kg = = 0.780 h2 − h1 (293.36 − 243.60)kJ/kg

(b) The exergy destroyed in the compressor is x dest = wa − wrev = 49.76 − 39.81 = 9.95 kJ/kg


8-141

8-138 The isentropic efficiency of a water pump is specified. The actual power output, the rate of frictional heating, the exergy destruction, and the second-law efficiency are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are negligible. Analysis (a) Using saturated liquid properties at the given temperature for the inlet state (Table A-4) h = 125.82 kJ/kg T1 = 30°C⎫ s1 = 0.4367 kJ/kg.K ⎬ 1 x1 = 0 ⎭ v = 0.001004 m 3 /kg 1

Water 100 kPa 30°C 1.35 kg/s

4 MPa PUMP

The power input if the process was isentropic is W&s = m& v 1 ( P2 − P1 ) = (1.35 kg/s)(0.00 1004 m 3 /kg)(4000 − 100 ) kPa = 5.288 kW

Given the isentropic efficiency, the actual power may be determined to be

W& 5.288 kW = 7.554 kW W& a = s = 0.70 ηs (b) The difference between the actual and isentropic works is the frictional heating in the pump

Q& frictional = W& a − W& s = 7.554 − 5.288 = 2.266 kW (c) The enthalpy at the exit of the pump for the actual process can be determined from W&a = m& (h2 − h1 ) ⎯ ⎯→ 7.554 kW = (1.35 kg/s)(h2 − 125.82)kJ/kg ⎯ ⎯→ h2 = 131.42 kJ/kg

The entropy at the exit is P2 = 4 MPa

⎫ ⎬s 2 = 0.4423 kJ/kg.K h2 = 131.42 kJ/kg ⎭

The reversible power and the exergy destruction are W&rev = m& [h2 − h1 − T0 ( s2 − s1 )]

= (1.35 kg/s)[(131.42 − 125.82) kJ/kg − (20 + 273 K)(0.4423 − 0.4367)kJ/kg.K ] = 5.362 kW

X& dest = W& a − W& rev = 7.554 − 5.362 = 2.193 kW (d) The second-law efficiency is

η II =

W& rev 5.362 kW = = 0.710 7.554 kW W& a


8-142

8-139 Argon gas is expanded adiabatically in an expansion valve. The exergy of argon at the inlet, the exergy destruction, and the second-law efficiency are to be determined. Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential energy changes are zero. 3 Argon is an ideal gas with constant specific heats. Properties The properties of argon gas are R = 0.2081 kJ/kg.K, cp = 0.5203 kJ/kg.ºC (Table A-2). Analysis (a) The exergy of the argon at the inlet is x1 = h1 − h0 − T0 ( s1 − s 0 )

Argon 3.5 MPa 100°C

500 kPa

⎡ P ⎤ T = c p (T1 − T0 ) − T0 ⎢c p ln 1 − R ln 1 ⎥ P0 ⎦ T0 ⎣ 373 K 3500 kPa ⎤ ⎡ = (0.5203 kJ/kg.K)(100 − 25)°C − (298 K) ⎢(0.5203 kJ/kg.K)ln − (0.2081 kJ/kg.K)ln 298 K 100 kPa ⎥⎦ ⎣ = 224.7 kJ/kg

(b) Noting that the temperature remains constant in a throttling process of an ideal gas, the exergy destruction is determined from x dest = T0 s gen = T0 ( s 2 − s1 ) ⎛ P ⎞ ⎡ ⎛ 500 kPa ⎞⎤ = T0 ⎜⎜ − R ln 1 ⎟⎟ = (298 K) ⎢− (0.2081 kJ/kg.K)ln⎜ ⎟⎥ P0 ⎠ ⎝ 3500 kPa ⎠⎦ ⎣ ⎝ = 120.7 kJ/kg

(c) The second-law efficiency is

η II =

x1 − x dest (224.7 − 120.7)kJ/kg = = 0.463 224.7 kJ/kg x1


8-143

8-140 Heat is lost from the air flowing in a diffuser. The exit temperature, the rate of exergy destruction, and the second law efficiency are to be determined. Assumptions 1 Steady operating conditions exist. 2 Potential energy change is negligible. 3 Nitrogen is an ideal gas with variable specific heats. Properties The gas constant of nitrogen is R = 0.2968 kJ/kg.K. Analysis (a) For this problem, we use the properties from EES software. Remember that for an ideal gas, enthalpy is a function of temperature only whereas entropy is functions of both temperature and pressure. At the inlet of the diffuser and at the dead state, we have T1 = 110°C = 383 K ⎫ h1 = 88.39 kJ/kg ⎬ P1 = 100 kPa ⎭ s1 = 7.101 kJ/kg ⋅ K

q Nitrogen 100 kPa 110°C 205 m/s

T1 = 300 K ⎫ h0 = 1.93 kJ/kg ⎬ P1 = 100 kPa ⎭ s 0 = 6.846 kJ/kg ⋅ K

110 kPa 45 m/s

An energy balance on the diffuser gives V12 V2 = h2 + 2 + q out 2 2 2 (205 m/s) ⎛ 1 kJ/kg ⎞ (45 m/s) 2 ⎛ 1 kJ/kg ⎞ 88.39 kJ/kg + ⎜ ⎟ = h2 + ⎜ ⎟ + 2.5 kJ/kg 2 2 2 2 ⎝ 1000 m /s ⎠ ⎝ 1000 m 2 /s 2 ⎠ h1 +

⎯ ⎯→ h2 = 105.9 kJ/kg

The corresponding properties at the exit of the diffuser are h2 = 105.9 kJ/kg ⎫ T2 = 127°C = 400 K ⎬ P1 = 110 kPa ⎭ s 2 = 7.117 kJ/kg ⋅ K

(b) The mass flow rate of the nitrogen is determined to be m& = ρ 2 A2V2 =

P2 110 kPa A2V2 = (0.04 m 2 )(45 m/s) = 1.669 kg/s RT2 (0.2968 kJ/kg.K)(400 K)

The exergy destruction in the nozzle is the exergy difference between the inlet and exit of the diffuser ⎡ ⎤ V 2 − V22 − T0 ( s1 − s 2 )⎥ X& dest = m& ⎢h1 − h2 + 1 2 ⎣⎢ ⎦⎥ ⎡ (205 m/s) 2 − (45 m/s) 2 ⎛ 1 kJ/kg ⎜ ⎢(88.39 − 105.9)kJ/kg + = (1.669 kg/s) ⎢ 2 ⎝ 1000 m 2 /s 2 ⎢⎣− (300 K )(7.101 − 7.117)kJ/kg.K

⎞⎤ ⎟⎥ ⎠⎥ = 12.4 kW ⎥⎦

(c) The second-law efficiency for this device may be defined as the exergy output divided by the exergy input: ⎡ ⎤ V2 X& 1 = m& ⎢h1 − h0 + 1 − T0 ( s1 − s 0 ) ⎥ 2 ⎢⎣ ⎥⎦ ⎡ (205 m/s) 2 = (1.669 kg/s) ⎢(88.39 − 1.93) kJ/kg + 2 ⎣⎢

⎤ ⎛ 1 kJ/kg ⎞ ⎟ − (300 K )(7.101 − 6.846) kJ/kg.K ⎥ ⎜ 2 2 ⎝ 1000 m /s ⎠ ⎦⎥

= 51.96 kW X& X& 12.4 kW η II = 2 = 1 − dest = 1 − = 0.761 = 76.1% 51.96 kW X& 1 X& 1


8-144

8-141 Using an incompressible substance as an example, it is to be demonstrated if closed system and flow exergies can be negative. Analysis The availability of a closed system cannot be negative. However, the flow availability can be negative at low pressures. A closed system has zero availability at dead state, and positive availability at any other state since we can always produce work when there is a pressure or temperature differential.

To see that the flow availability can be negative, consider an incompressible substance. The flow availability can be written as

ψ = h − h0 + T0 (s − s0 ) = (u − u0 ) + v ( P − P0 ) + T0 (s − s0 ) = ξ + v ( P − P0 )

The closed system availability ξ is always positive or zero, and the flow availability can be negative when P << P0.

8-142 A relation for the second-law efficiency of a heat engine operating between a heat source and a heat sink at specified temperatures is to be obtained. Source Analysis The second-law efficiency is defined as the ratio of the availability recovered TH to availability supplied during a process. The work W produced is the availability recovered. The decrease in the availability of the heat supplied QH is the availability QH supplied or invested. Therefore,

η II =

W ⎛ T0 ⎜⎜1 − ⎝ TH

⎞ ⎛ T ⎟⎟Q H − ⎜⎜1 − 0 ⎠ ⎝ TL

W

HE ⎞ ⎟⎟(Q H − W ) ⎠

Note that the first term in the denominator is the availability of heat supplied to the heat engine whereas the second term is the availability of the heat rejected by the heat engine. The difference between the two is the availability consumed during the process.

QL TL Sink


8-145

8-143 Writing energy and entropy balances, a relation for the reversible work is to be obtained for a closed system that exchanges heat with surroundings at T0 in the amount of Q0 as well as a heat reservoir at temperature TR in the amount QR. Assumptions Kinetic and potential changes are negligible. Analysis We take the direction of heat transfers to be to the system (heat input) and the direction of work transfer to be from the system (work output). The result obtained is still general since quantities wit opposite directions can be handled the same way by using negative signs. The energy and entropy balances for this stationary closed system can be expressed as

⎯→ W = U 1 − U 2 + Q0 + Q R Energy balance: E in − E out = ∆E system → Q0 + Q R − W = U 2 − U 1 ⎯ Entropy balance: S in − S out + S gen = ∆S system → S gen = ( S 2 − S1 ) +

−Q R −Q0 + TR T0

(1) (2)

Solving for Q0 from (2) and substituting in (1) yields ⎛ T W = (U 1 − U 2 ) − T0 ( S1 − S 2 ) − Q R ⎜⎜1 − 0 ⎝ TR

Source TR

⎞ ⎟⎟ − T0 S gen ⎠

System

The useful work relation for a closed system is obtained from

QR

Wu = W − Wsurr ⎛ T = (U 1 − U 2 ) − T0 ( S1 − S 2 ) − Q R ⎜⎜1 − 0 ⎝ TR

⎞ ⎟⎟ − T0 S gen − P0 (V 2 −V1 ) ⎠

Then the reversible work relation is obtained by substituting Sgen = 0, ⎛ T W rev = (U 1 − U 2 ) − T0 ( S1 − S 2 ) + P0 (V1 −V 2 ) − Q R ⎜⎜1 − 0 ⎝ TR

⎞ ⎟⎟ ⎠

A positive result for Wrev indicates work output, and a negative result work input. Also, the QR is a positive quantity for heat transfer to the system, and a negative quantity for heat transfer from the system.


8-146

8-144 Writing energy and entropy balances, a relation for the reversible work is to be obtained for a steady-flow system that exchanges heat with surroundings at T0 at a rate of Q& 0 as well as a heat reservoir at temperature TR in the amount Q& . R

Analysis We take the direction of heat transfers to be to the system (heat input) and the direction of work transfer to be from the system (work output). The result obtained is still general since quantities wit opposite directions can be handled the same way by using negative signs. The energy and entropy balances for this stationary closed system can be expressed as

Energy balance: E& in − E& out = ∆E& system → E& in = E& out V2 V2 Q& 0 + Q& R − W& = ∑ m& e (he + e + gz e ) − ∑ m& i (hi + i + gz i ) 2 2 or

System

V2 V2 W& = ∑ m& i (hi + i + gz i ) − ∑ m& e (he + e + gz e ) + Q& 0 + Q& R (1) 2 2

Entropy balance: S& in − S& out + S& gen = ∆S& system = 0 S& gen = S& out − S& in − Q& R − Q0 + S&gen = ∑ m& e se − ∑ m& i si + TR T0

(2)

Solving for Q& 0 from (2) and substituting in (1) yields ⎛ T V2 V2 W& = ∑ m& i (hi + i + gz i − T0 s i ) − ∑ m& e (he + e + gz e − T0 s e ) − T0 S& gen − Q& R ⎜⎜1 − 0 2 2 ⎝ TR

⎞ ⎟⎟ ⎠

Then the reversible work relation is obtained by substituting Sgen = 0, ⎛ T V2 V2 W& rev = ∑ m& i (hi + i + gz i − T0 s i ) − ∑ m& e (he + e + gz e − T0 s e ) − Q& R ⎜⎜1 − 0 2 2 ⎝ TR

⎞ ⎟⎟ ⎠



8-147

8-145 Writing energy and entropy balances, a relation for the reversible work is to be obtained for a uniform-flow system that exchanges heat with surroundings at T0 in the amount of Q0 as well as a heat reservoir at temperature TR in the amount QR. Assumptions Kinetic and potential changes are negligible. Analysis We take the direction of heat transfers to be to the system (heat input) and the direction of work transfer to be from the system (work output). The result obtained is still general since quantities wit opposite directions can be handled the same way by using negative signs. The energy and entropy balances for this stationary closed system can be expressed as

Energy balance: E in − E out = ∆E system

Q0 + Q R − W = ∑ m e (he + or,

W = ∑ m i ( hi +

Ve2 V2 + gz e ) − ∑ mi (hi + i + gz i ) + (U 2 − U 1 ) cv 2 2

Vi 2 V2 + gz i ) − ∑ m e (he + e + gz e ) − (U 2 − U 1 ) cv + Q0 + Q R 2 2

(1)

Entropy balance: Sin − Sout + Sgen = ∆Ssystem Sgen = ( S2 − S1 )cv + ∑ me se − ∑ mi si +

−QR −Q0 + TR T0

(2)

Source TR System

Solving for Q0 from (2) and substituting in (1) yields Vi 2 V2 + gz i − T0 s i ) − ∑ m e (he + e + gz e − T0 s e ) 2 2 ⎛ T ⎞ +[(U 1 − U 2 ) − T0 ( S1 − S 2 )]cv − T0 S gen − Q R ⎜⎜1 − 0 ⎟⎟ ⎝ TR ⎠

Q

W = ∑ m i (hi +

me

The useful work relation for a closed system is obtained from Vi 2 V2 + gz i − T0 s i ) − ∑ m e (he + e + gz e − T0 s e ) 2 2 ⎛ T ⎞ +[(U 1 − U 2 ) − T0 ( S1 − S 2 )]cv − T0 S gen − Q R ⎜⎜1 − 0 ⎟⎟ − P0 (V 2 −V1 ) ⎝ TR ⎠

Wu = W − Wsurr = ∑ m i (hi +

Then the reversible work relation is obtained by substituting Sgen = 0, Vi 2 V2 + gz i − T0 s i ) − ∑ m e (he + e + gz e − T0 s e ) 2 2 ⎛ T ⎞ +[(U 1 − U 2 ) − T0 ( S1 − S 2 ) + P0 (V1 −V 2 )]cv − Q R ⎜⎜1 − 0 ⎟⎟ ⎝ TR ⎠

W rev = ∑ m i (hi +



8-148

Fundamentals of Engineering (FE) Exam Problems

8-146 Heat is lost through a plane wall steadily at a rate of 800 W. If the inner and outer surface temperatures of the wall are 20°C and 5°C, respectively, and the environment temperature is 0°C, the rate of exergy destruction within the wall is (a) 40 W

(b) 17,500 W

(c) 765 W

(d) 32,800 W

(e) 0 W

Answer (a) 40 W

Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). Q=800 "W" T1=20 "C" T2=5 "C" To=0 "C" "Entropy balance S_in - S_out + S_gen= DS_system for the wall for steady operation gives" Q/(T1+273)-Q/(T2+273)+S_gen=0 "W/K" X_dest=(To+273)*S_gen "W" "Some Wrong Solutions with Common Mistakes:" Q/T1-Q/T2+Sgen1=0; W1_Xdest=(To+273)*Sgen1 "Using C instead of K in Sgen" Sgen2=Q/((T1+T2)/2); W2_Xdest=(To+273)*Sgen2 "Using avegage temperature in C for Sgen" Sgen3=Q/((T1+T2)/2+273); W3_Xdest=(To+273)*Sgen3 "Using avegage temperature in K" W4_Xdest=To*S_gen "Using C for To"

8-147 Liquid water enters an adiabatic piping system at 15°C at a rate of 3 kg/s. It is observed that the water temperature rises by 0.3°C in the pipe due to friction. If the environment temperature is also 15°C, the rate of exergy destruction in the pipe is (a) 3.8 kW

(b) 24 kW

(c) 72 kW

(d) 98 kW

(e) 124 kW

Answer (a) 3.8 kW

Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). Cp=4.18 "kJ/kg.K" m=3 "kg/s" T1=15 "C" T2=15.3 "C" To=15 "C" S_gen=m*Cp*ln((T2+273)/(T1+273)) "kW/K" X_dest=(To+273)*S_gen "kW" "Some Wrong Solutions with Common Mistakes:" W1_Xdest=(To+273)*m*Cp*ln(T2/T1) "Using deg. C in Sgen" W2_Xdest=To*m*Cp*ln(T2/T1) "Using deg. C in Sgen and To" W3_Xdest=(To+273)*Cp*ln(T2/T1) "Not using mass flow rate with deg. C" W4_Xdest=(To+273)*Cp*ln((T2+273)/(T1+273)) "Not using mass flow rate with K"


8-149

8-148 A heat engine receives heat from a source at 1500 K at a rate of 600 kJ/s and rejects the waste heat to a sink at 300 K. If the power output of the engine is 400 kW, the second-law efficiency of this heat engine is (a) 42%

(b) 53%

(c) 83%

(d) 67%

(e) 80%

Answer (c) 83%

Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). Qin=600 "kJ/s" W=400 "kW" TL=300 "K" TH=1500 "K" Eta_rev=1-TL/TH Eta_th=W/Qin Eta_II=Eta_th/Eta_rev "Some Wrong Solutions with Common Mistakes:" W1_Eta_II=Eta_th1/Eta_rev; Eta_th1=1-W/Qin "Using wrong relation for thermal efficiency" W2_Eta_II=Eta_th "Taking second-law efficiency to be thermal efficiency" W3_Eta_II=Eta_rev "Taking second-law efficiency to be reversible efficiency" W4_Eta_II=Eta_th*Eta_rev "Multiplying thermal and reversible efficiencies instead of dividing"

8-149 A water reservoir contains 100 tons of water at an average elevation of 60 m. The maximum amount of electric power that can be generated from this water is (a) 8 kWh

(b) 16 kWh

(c) 1630 kWh

(d) 16,300 kWh

(e) 58,800 kWh

Answer (b) 16 kWh

Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). m=100000 "kg" h=60 "m" g=9.81 "m/s^2" "Maximum power is simply the potential energy change," W_max=m*g*h/1000 "kJ" W_max_kWh=W_max/3600 "kWh" "Some Wrong Solutions with Common Mistakes:" W1_Wmax =m*g*h/3600 "Not using the conversion factor 1000" W2_Wmax =m*g*h/1000 "Obtaining the result in kJ instead of kWh" W3_Wmax =m*g*h*3.6/1000 "Using worng conversion factor" W4_Wmax =m*h/3600"Not using g and the factor 1000 in calculations"


8-150

8-150 A house is maintained at 21°C in winter by electric resistance heaters. If the outdoor temperature is 9°C, the secondlaw efficiency of the resistance heaters is (a) 0%

(b) 4.1%

(c) 5.7%

(d) 25%

(e) 100%

Answer (b) 4.1%

Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). TL=9+273 "K" TH=21+273 "K" To=TL COP_rev=TH/(TH-TL) COP=1 Eta_II=COP/COP_rev "Some Wrong Solutions with Common Mistakes:" W1_Eta_II=COP/COP_rev1; COP_rev1=TL/(TH-TL) "Using wrong relation for COP_rev" W2_Eta_II=1-(TL-273)/(TH-273) "Taking second-law efficiency to be reversible thermal efficiency with C for temp" W3_Eta_II=COP_rev "Taking second-law efficiency to be reversible COP" W4_Eta_II=COP_rev2/COP; COP_rev2=(TL-273)/(TH-TL) "Using C in COP_rev relation instead of K, and reversing"

8-151 A 10-kg solid whose specific heat is 2.8 kJ/kg.°C is at a uniform temperature of -10°C. For an environment temperature of 25°C, the exergy content of this solid is (a) Less than zero

(b) 0 kJ

(c) 22.3 kJ

(d) 62.5 kJ

(e) 980 kJ

Answer (d) 62.5 kJ

Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). m=10 "kg" Cp=2.8 "kJ/kg.K" T1=-10+273 "K" To=25+273 "K" "Exergy content of a fixed mass is x1=u1-uo-To*(s1-so)+Po*(v1-vo)" ex=m*(Cp*(T1-To)-To*Cp*ln(T1/To)) "Some Wrong Solutions with Common Mistakes:" W1_ex=m*Cp*(To-T1) "Taking the energy content as the exergy content" W2_ex=m*(Cp*(T1-To)+To*Cp*ln(T1/To)) "Using + for the second term instead of -" W3_ex=Cp*(T1-To)-To*Cp*ln(T1/To) "Using exergy content per unit mass" W4_ex=0 "Taking the exergy content to be zero"


8-151

8-152 Keeping the limitations imposed by the second-law of thermodynamics in mind, choose the wrong statement below: (a) A heat engine cannot have a thermal efficiency of 100%. (b) For all reversible processes, the second-law efficiency is 100%. (c) The second-law efficiency of a heat engine cannot be greater than its thermal efficiency. (d) The second-law efficiency of a process is 100% if no entropy is generated during that process. (e) The coefficient of performance of a refrigerator can be greater than 1. Answer (c) The second-law efficiency of a heat engine cannot be greater than its thermal efficiency.

8-153 A furnace can supply heat steadily at a 1300 K at a rate of 500 kJ/s. The maximum amount of power that can be produced by using the heat supplied by this furnace in an environment at 300 K is (a) 115 kW

(b) 192 kW

(c) 385 kW

(d) 500 kW

(e) 650 kW

Answer (c) 385 kW

Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). Q_in=500 "kJ/s" TL=300 "K" TH=1300 "K" W_max=Q_in*(1-TL/TH) "kW" "Some Wrong Solutions with Common Mistakes:" W1_Wmax=W_max/2 "Taking half of Wmax" W2_Wmax=Q_in/(1-TL/TH) "Dividing by efficiency instead of multiplying by it" W3_Wmax =Q_in*TL/TH "Using wrong relation" W4_Wmax=Q_in "Assuming entire heat input is converted to work"


8-152

8-154 Air is throttled from 50°C and 800 kPa to a pressure of 200 kPa at a rate of 0.5 kg/s in an environment at 25°C. The change in kinetic energy is negligible, and no heat transfer occurs during the process. The power potential wasted during this process is (a) 0

(b) 0.20 kW

(c) 47 kW

(d) 59 kW

(e) 119 kW

Answer (d) 59 kW

Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). R=0.287 "kJ/kg.K" Cp=1.005 "kJ/kg.K" m=0.5 "kg/s" T1=50+273 "K" P1=800 "kPa" To=25 "C" P2=200 "kPa" "Temperature of an ideal gas remains constant during throttling since h=const and h=h(T)" T2=T1 ds=Cp*ln(T2/T1)-R*ln(P2/P1) X_dest=(To+273)*m*ds "kW" "Some Wrong Solutions with Common Mistakes:" W1_dest=0 "Assuming no loss" W2_dest=(To+273)*ds "Not using mass flow rate" W3_dest=To*m*ds "Using C for To instead of K" W4_dest=m*(P1-P2) "Using wrong relations"


8-153

8-155 Steam enters a turbine steadily at 4 MPa and 400°C and exits at 0.2 MPa and 150°C in an environment at 25°C. The decrease in the exergy of the steam as it flows through the turbine is (a) 58 kJ/kg

(b) 445 kJ/kg

(c) 458 kJ/kg

(d) 518 kJ/kg

(e) 597 kJ/kg

Answer (e) 597 kJ/kg

Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). P1=4000 "kPa" T1=400 "C" P2=200 "kPa" T2=150 "C" To=25 "C" h1=ENTHALPY(Steam_IAPWS,T=T1,P=P1) s1=ENTROPY(Steam_IAPWS,T=T1,P=P1) h2=ENTHALPY(Steam_IAPWS,T=T2,P=P2) s2=ENTROPY(Steam_IAPWS,T=T2,P=P2) "Exergy change of s fluid stream is Dx=h2-h1-To(s2-s1)" -Dx=h2-h1-(To+273)*(s2-s1) "Some Wrong Solutions with Common Mistakes:" -W1_Dx=0 "Assuming no exergy destruction" -W2_Dx=h2-h1 "Using enthalpy change" -W3_Dx=h2-h1-To*(s2-s1) "Using C for To instead of K" -W4_Dx=(h2+(T2+273)*s2)-(h1+(T1+273)*s1) "Using wrong relations for exergy"

8- 156 … 8- 160 Design and Essay Problems

KJ


Chapter 8 EXERGY – A MEASURE OF WORK POTENTIAL PROPRIETARY AND CONFIDENTIAL

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