CANKAYA UNIVERSITY

FACULTY OF ENGINEERING AND ARCHITECTURE

MECHANICAL ENGINEERING DEPARTMENT

ME 211 THERMODYNAMICS I

CHAPTER 3 EXAMPLES

SOLUTIONS

1) The average atmospheric pressure in Denver (elevation=1610 m) is 83.4 kPa.

Determine the temperature at which water in an uncovered pan will boil in

Denver.

?TTbar834.0p

bar834.0Pa10

bar1Pa1083.4 kPa 83.4 p

sat

5

3

From Table A-3:

C71.96Tbar9.0p

C50.93Tbar8.0p

o

o

C59.94z5.93T

0914.1z

5.9371.96

z

8.09.0

8.0834.0

o

0.8

0.834

0.9

z

96.71 93.5

p

T

2) A rigid tank with a volume of 2.5 m3 contains 5 kg of saturated liquid-vapor

mixture of water at 75oC. Now the water is slowly heated. Determine the quality

x at the initial state. Also, determine the temperature at which the liquid in the

tank is completely vaporized.

v = constant

kg/m5.0kg5

m5.2

m

Vv 3

3

1

T1 = 75oC

From Table A-2 @ T1 = 75oC:

vf = 1.025910-3

m3/kg

vg = 4.131 m3/kg

121.0xvxvv 1fg1f1

@ State 2:

v2 = v1 = 0.5 m3/kg

x2 = 1

/kgm5.0v1 x

/kgm 0.5 v v 3

g

2

3

12

From Table A-2 @ vg = 0.5 m3/kg:

C7.140TC140C150

C140T

kg/m5089.0kg/m3928.0

kg/m5089.0kg/m5.0 o

2oo

o

2

33

33

1

2

75oC

T2

T vapor

liquid

v

3) Consider a two phase mixture at 100oC with x3 = 0.9. Determine the specific

volume.

From Table A-2 @100oC:

kg/m673.1v

kg/m100435.1v

3

g

33

f

kg/m506.1v

kg/m)100435.1673.1(9.0kg/m100435.1v

)vv(xvv

3

3

3333

3

fgf3

Note: Once the quality x, is known, it can be applied to calculate v, h or s in the same

manner as above.

4) Refrigerant 134a with a quality of 0.4 and a temperature of 12oC is contained in

a rigid tank that has a volume of 0.17 m3. Find the mass of liquid present.

gfg mmm,xmm,v

Vm

fgf xvvv

From Table A-10 @12oC:

kg/m046.0v

kg/m000797.0v

3

g

3

f

kg/m0189.0v

kg/m)000797.0046.0(4.0kg/m000797.0vxvvv

3

1

33

1fgf1

Total mass kg9kg/m0189.0

m17.0

v

Vm

3

3

1

kg6.3)9)(4.0(xmmg

kg4.5kg6.3kg9mmm gf

vapor

liquid

T1 = 12oC

x1 = 0.4

V = 0.17 m3

5) Consider Refrigerant-22 at 12oC. It is specific internal energy 144.58 kJ/kg.

Determine phase description and enthalpy.

From Table A-7 @12oC:

kg/kJ38.230u

kg/kJ77.58u

g

f

So uf < u < ug saturated liquid vapor mixture.

5.0kg/kJ)77.5838.230(

kg/kJ)77.5858.144(

)uu(

)uu(x

fg

f

kg/kJ67.156h

kg/kJ)35.5999.253(5.0kg/kJ35.59xhhh fgf

6) Determine the temperature of water at a state of p = 0.5 MPa and h = 2890

kJ/kg.

P = 0.5 MPa = 500 kPa = 5bar

h = 2890 kJ/kg

From Table A-3 for p = 5 bar

h > hg so superheated vapor

From Table A-4 @ p = 5 bar:

C38.216z200T

37.16z

200240

z

4.28559.2939

4.28552890

o

h

2855.4

2890

2939.9

z

240 200 T

T (oC)

200

h (kJ/kg)

2855.4

240 2939.9

5

hg (kJ/kg) hf (kJ/kg) p (bar)

640.23 2748.7

7) Determine the missing properties and the phase descriptions in the following

table for water:

T, oC p, kPa h, kJ/kg x Phase description

(a) 200 0.7

(b) 140 1800

(c) 950 0.0

(d) 80 500

(e) 800 3161.7

(a)

C2.120TT7.0x

bar2kPa200po

sat

kg/kJ2046h)9.2201(7.07.504xhhh fgf saturated mixture

(b)

kPa3.361bar613.3pkg/kJ1800h

C140T o

564.0x

)kg/kJ7.2144(xkg/kJ13.589kg/kJ1800xhhh fgf

saturated mixture

(c)

0x

bar5.9kPa950psaturated liquid

p = 950 kPa = 0.95 MPa = 9.5 bar T = 177.65oC

h = hf = 752.82 kJ/kg

(d)

C9.151Tbar5pC80T

bar5MPa5.0kPa500po

sato

satTT compressed liquid

kg/kJ91.334hhC80T@f o

(e)

kg/kJ7.3161h

bar8MPa8.0kPa800p

g

gf

hh

kg/kJ1.2769h,kg/kJ11.721hbar8p

superheated vapor

C06.350Tkg/kJ7.3161h

bar8po

8) A frictionless piston-cylinder device initially contains 200 L of saturated liquid

refrigerant R-134a. The piston is free to move, and its mass is such that it

maintains a pressure of 800 kPa on the refrigerant. The refrigerant is now heated

until its temperature rises to 50 oC. Calculate the work done during this process.

9) A piston-cylinder device contains 50 kg of water at 150 kPa and 25 oC. The

cross-sectional area of the piston is 0.1 m2. Heat is now transferred to the water,

causing part of it to evaporate and expand. When the volume reaches 0.2 m3, the

piston reaches a linear spring whose spring constant is 100kN/m. More heat is

transferred to the water until the piston rises 20 cm more. Determine (a) the

final pressure and temperature and (b) the work done during this process. Also,

show the process on p-v diagram.

10) The radiator of a steam heating system has a volume of 20 L and is filled with

superheated vapor at 300 kPa and 250°C. At this moment both the inlet and

exit valves to the radiator are closed. Determine the amount of heat that will

be transferred to the room when the steam pressure drops to 100 kPa. Also,

show the process on a p-v diagram with respect to saturation lines.

11) Two kilograms of saturated liquid water at 50 kPa are heated slowly at constant

pressure. During the process 5876 kJ of heat are added. Find the final

temperature.

p1 = p2 = 50 kPa

m = 2 kg

State 1:

kg/kJ5.340hh0x

bar5.0kPa50pf1

1

1

Q – W = U2 – U1

112212

2

1

VpVp)VV(ppdVW

kg/kJ32785.3402

5876h

m

Qh

)hh(mHHUUVpVp)UU(WQ

12

121212112212

State 2:

kg/kJ3278h

bar5.0kPa50p

2

2

kg/kJ9.2645h,kg/kJ5.340hbar5.0p@ gf2

g2 hh superheated vapor

From Table A-4 T2 = 400oC

H2O

Q = 5876 kJ

12) A vessel having a volume of 5 m3 contains 0.05 m

3 of saturated liquid water and

4.95 m3 of saturated water vapor at 0.1 MPa. Heat is transferred until the vessel

is filled with saturated vapor. Determine the heat transfer for this process.

0PEKEUUWQ 121212

State 1:

kg/m694.1v,kg/m001043.0vbar1MPa1.0p 3

1g

3

1f1

kg/kJ1.2506u,kg/kJ36.417u 1g1f

kg94.47kg/m001043.0

m05.0

v

Vm

3

3

1f

f1f

kg92.2kg/m694.1

m95.4

v

Vm

3

3

1g

g

1g

kJ27326)kg/kJ1.2506)(kg92.2()kg/kJ36.417)(kg94.47()um()um(U 1gg1ff1

State 2:

kg92.2kg94.47mmm 1g1f

kg/m09831.0kg86.50

m5

m

Vv 3

3

2

kg/m09831.0v

1x

3

2

2We need to do interpolation

vg (m3/kg) p (bar)

0.09963 20

0.09831 ?

0.07998 25

vapor

liquid

H2O

)pressurefinal(bar3.20p

ug (kJ/kg) p (bar)

2600.3 20

? 20.3

2603.1 25

kg/kJ5.2600u 2

kJ132261)kg/kJ5.2600)(kg86.50(umU 222

kJ104935UUQ 1212

13) A cylinder fitted with a piston has a volume of 0.1 m3 and contains 0.5 kg of

steam at 0.4 MPa. Heat is transferred to the steam until the temperature is 300oC

while the pressure remains constant. Determine the work and heat transfer for

this process.

)vv(pm)VV(pdVppdVW 1212

2

1

2

1

12

)hh(mQ

)vpu()vpu(m)vv(mp)uu(mWUUQ

1212

1112221212121212

State 1:

MPa4.0pandkg/m2.05.0

1.0

m

Vv 1

311

From Table A-3 @ p1 = 0.4 Mpa = 4 bar:

vf = 0.001084 m3/kg, vg = 0.4625 m

3/kg

mixturevaporliquidvvv g1f

4311.0x

kg/m)001084.04625.0(xkg/m001084.0kg/m2.0vxvv

1

3

1

33

fg1f1

kg/kJ7.1524h

kg/kJ)74.6046.2738)(4311.0(kg/kJ74.604hhxhh

1

1fg1f1

State 2:

1

p

V

p2 = p1 = 0.4 MPa

p2 = 0.4 Mpa = 4 bar

T2 = 300oC

p2 = 0.4 Mpa = 4 bar sat

o

sat TT,C6.143T superheated vapor

Using Table A-4:

p = 3 bar

T (oC) h (kJ/kg)

280 3028.6

300 ?

320 3110.1

kg/kJ35.3069z6.3028h75.40z)6.30281.3110(

z

280320

280300

p = 5 bar

T (oC) h (kJ/kg)

280 3022.9

300 ?

320 3105.6

T

280

300

320

z

3105.6 3022.9 h

T

280

300

320

z

3110.1 3028.6 h

kg/kJ25.3064z9.3022h35.41z)9.30226.3105(

z

280320

280300

T = 300 oC

p (bar) h (kJ/kg)

3 3069.35

4 ?

5 3064.25

kg/kJ8.3066z25.3064h55.2z)25.306435.3069(

z

35

342

We do similar interpolation for v2 = 0.6548 m3/kg

kJ96.90m)2.06548.0)(kPa400)(kg5.0()vv(mpW

kJ1.771kg/kJ)7.15248.3066(kg5.0)hh(mQ

3

1212

1212

p

3

4

5

z

3069.35 3064.25 h

14) A frictionless piston is used to provide a constant pressure of 400 kPa in a

cylinder containing steam originally at 200oC with a volume of 2 m

3. Calculate

the final temperature if 3500 kJ of heat added.

State 1:

3

1

o

1

5

1

m2V

C200T

bar4Pa104kPa400p

From Table A-3:

p2 = 4 bar sat

o

sat TT,C6.143T superheated vapor

Using Table A-4 for C200Tandbar4p o

11 :

kg/kJ2860h,kg/kJ2647u,kg/m5342.0v 11

3

1

)VV(kPa400)VV(pdVppdVW 1212

2

1

2

1

12

kg744.3kg/m5342.0

m2

v

Vm

3

3

1

1

)uu(m)VV(kPa400kJ3500

)uu(m)UU(WQ

1212

1212

kg/kJ)2647u)(kg744.3(m)2v744.3(kPa400kJ3500 2

3

2

This problem requires trial error procedure:

1) Guess a value of v2

2) Calculate u2 from above equation

3) If this value checks with u2 from steam tables then the guess is correct one.

Steam

Q = 3500 kJ

A) 1) Let v2 = 1 m3/kg

2) kg/kJ)2647u)(kg744.3(m2)1)(744.3(kPa400kJ3500 2

3

kg/kJ3395u 2

3) From Table A-4 for:

kg/kJ2.3300ukg/m1v

bar4p23

2

2

These two u values are not the same. So revise value of v.

B) 1) Let v2 = 1.06 m3/kg

2) kg/kJ)2647u)(kg744.3(m2)06.1)(744.3(kPa400kJ3500 2

3

kg/kJ3372u 2

3) From Table A-4 for:

kg/kJ3372ukg/m06.1v

bar4p23

2

2

We now accept that u2 values are close enough. So temperature is T2 640oC.

Second Method:

)UU(WQ 12

)VV(pdVppdVW 12

2

1

2

1

)hh(mHHQ)UU()VV(p)UU(WQ 1212121212

kg/kJ3795kg/kJ2860kg744.3

kJ3500h

m

Qh 12

From Table A-4 for:

C641Tkg/kJ3795h

bar4po

2

2

2

15) Determine the enthalpy change h of nitrogen, in kJ/kg, as it is heated from 600

to 1000 K, using (a) the empirical specific heat equation as a function of

temperature (Table A-21), (b) the cP value at the average temperature (Table A-

20), and (c) the cP value at room temperature at 300 K.

a) 32

p

32pTTTK.kmol/kJ314.8cTTT

R

c

From Table A-21:

963 10632.0,10324.2,10208.1,675.3

449336

223

T

T

432

T

T

p

600100010632.04

1600100010324.2

3

1

600100010208.12

16001000675.3

K.kmol/kJ314.8h

T4

1T

3

1T

2

1TK.kmol/kJ314.8dTch

2

1

2

1

kmol

kJ12544h

kg

kJ8.447

kmol/kg01.28

kmol/kJ12544

M

hh

b) From Table A-20 at Tavg = 800 K, cp,avg = 1.121 kJ/kg.K

kg

kJ4.448hK)6001000(

K.kg

kJ121.1)TT(ch 12avg,p

c) Taking the cp at room temperature from Table A-20:

cp = 1.039 kJ/kg.K

kg

kJ6.415hK)6001000(

K.kg

kJ039.1)TT(ch 12p

16) A rigid tank has a volume of 400 cm3 contains air initially at 22

oC and 100 kPa.

A paddle wheel stirs the gas until the temperature is 428oC. During the process,

600 J of heat are transferred from air to surroundings. Calculate the work done

by the paddlewheel two ways:

(a) assuming constant specific heat

(b) assuming variable specific heat

(a) cv = constant

)TT(mcU

WQU

12v

K498C2252

42822T o

avg

From Table A-20:

K.kg/J742K.kg/kJ742.0cv

TR

pVMm

RT

pVmmRTpV

kg0004725.0

)K295)(K.kmol/kJ314.8(

cm10/m)cm400)(kPa100)(kmol/kg97.28(m

3633

J3.742)TT(mcQUQW 12v

(b) variable cv with temperature:

)TcTc(mWQ)TcTc(mU

WQU

11v22v11v22v

From Table A-20:

air

At T1 = 295 K cv1 = 0.718 kJ/kg.K = 718 J/kg.K

T2 = 701 K cv2 = 1.098 kJ/kg.K = 1098 J/kg.K

TR

pVMm

RT

pVmmRTpV

kg0004725.0

)K295)(K.kmol/kJ314.8(

cm10/m)cm400)(kPa100)(kmol/kg97.28(m

3633

J080.100)K295)(K.kg/J718)(kg0004725.0(UTmcU

J682.363)K701)(K.kg/J1098)(kg0004725.0(UTmcU

UUWQ

111v1

222v2

12

J602.863)080.100682.363(J600UUQW 12

17) A mass of 15 kg of air in a piston-cylinder device is heated from 25 to 77°C by

passing current through a resistance heater inside the cylinder. The pressure

inside the cylinder is held constant at 300 kPa during the process, and a heat

loss of 60 kJ occurs. Determine the electric energy supplied, in kWh.

18) Helium (He) gas initially at 2 bar and 200 K undergoes a polytropic process

with n = k to a final pressure of 14 bar. Determine work and heat transfer for the

process, each in kJ/kg of helium. Assume ideal gas behavior.

U W -Q

k1

)TT(mR

k1

)vpvp(m

v

dVmpdVW 121122

V

V

k

V

V

2

1

2

1

To find T2 use:

k

1k

1212 )p/p(TT

667.15.1

5.2

c

ck

v

p

K575667.1

1667.1

2 )1/14(200T

kJ7.1167667.11

)200575)(003.4/314.8(

m

W

m

W)TT(c

m

Wuu

m

QW)uu(mQ 12v1212

Rcc vp

1k

Rc

c

R1k

c

R1

c

c

v

vvv

p

0k1

)TT(R)TT(

1k

R

m

Q 1212

2

1 V

p

He

14

2

PVk = c

19) Determine the specific volume of superheated water vapor at 1.6 MPa and 225 oC, using (a) the ideal-gas equation, (b) the generalized compressibility chart,

and (c) the steam tables. Also determine the error involved in the first two cases.

K.kg/kJ4615.0kmol/kg02.18

K.kmol/kJ314.8

M

RR

MPa09.22bar9.220p

K3.647T

c

c

a) ideal gas law:

kg

m14364.0

kPa1600

)K498)(K.kg/kJ4615.0(

p

RTv

3

b) compressibility chart, see Figure A-1, Figure A-2, Figure A-3:

935.0Z

769.0K3.647

K498

T

TT

072.0MPa09.22

MPa6.1

p

pp

c

R

c

R

%710014364.0

14364.01343.0error

kg/m13430.0)14364.0(935.0p

RTZv 3

c) using superheated table:

13287.0vC225T

bar16MPa6.1p

o

20)

21 )

21) Air is confined to one side of a rigid container divided by a partition, as shown in

Figure below. The other side is initially evacuated. The air is initially at =5 bar , and

=500 K, and . When the partition is removed, the air expands to fill the

entire chamber. Measurements show that and . Assuming the air

behaves as an ideal gas, determine (a) the final temperature, in K, and (b) the heat

transfer, kJ.

solution