Heat and Mass Transfer Fundamentals and Applications

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.

10-1

Solutions Manual for

Heat and Mass Transfer: Fundamentals & Applications Fourth Edition

Yunus A. Cengel & Afshin J. Ghajar

McGraw-Hill, 2011

Chapter 10

BOILING AND CONDENSATION

PROPRIETARY AND CONFIDENTIAL

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10-16

10-24 Water is boiled at Tsat = 90°C in a brass heating element. The surface temperature of the heater is to be determined.

Assumptions 1 Steady operating conditions exist. 2 Heat losses from the heater and the boiler are negligible.

Properties The properties of water at the saturation temperature of 90°C are (Tables 10-1 and A-9)

Heating element

Water, 90°C

96.1PrCJ/kg 4206cN/m 0608.0

skg/m 10315.0kg/m 4235.0J/kg 102283kg/m 3.96533

33

=°⋅==

⋅×==

×==−

l

pl

lv

fgl h

σµρ

ρqmin

Also, 0.0060 and n = 1.0 for the boiling of water on a brass heating (Table 10-3). =sfC

Analysis The minimum heat flux is determined from

24/1

23

4/1

2min

W/m715,13)4235.03.965(

)4235.03.965)(81.9)(0608.0()102283)(4235.0(09.0

)()(

09.0

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

−×=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

−=

vl

vlfgv

ghq

ρρ

ρρσρ&

The surface temperature can be determined from Rohsenow equation to be

C92.3°=

⎟⎟⎠

⎞⎜⎜⎝

⎛

×

−⎥⎦

⎤⎢⎣

⎡××=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −⎥⎦

⎤⎢⎣

⎡ −=

−

s

s

nlfgsf

slpvlfgl

T

T

hC

TTcghq

3

3

1/2332

3sat,

2/1

nucleate

96.1)102283(0060.0)90(4206

0608.00.4235)-9.81(965.3

)10)(228310315.0( W/m715,13

Pr

)()(σ

ρρµ&


10-17

10-25 Water is boiled at 1 atm pressure and thus at a saturation (or boiling) temperature of Tsat = 100°C by a horizontal nickel plated copper heating element. The maximum (critical) heat flux and the temperature jump of the wire when the operating point jumps from nucleate boiling to film boiling regime are to be determined. Assumptions 1 Steady operating conditions exist. 2 Heat losses from the boiler are negligible. Properties The properties of water at the saturation temperature of 100°C are (Tables 10-1 and A-9)

75.1PrN/m 0589.0kg/m 5978.0

kg/m 9.9573

3

===

=

l

v

l

σρ

ρ

CJ/kg 4217

m/skg 10282.0

J/kg 1022573

3

°⋅=⋅×=

×=−

pl

l

fg

c

h

µ

Also, 0.0060 and n = 1.0 for the boiling of water on a nickel plated surface (Table 10-3 ). Note that we expressed the properties in units specified under Eqs. 10-2 and 10-3 in connection with their definitions in order to avoid unit manipulations. The vapor properties at the anticipated film temperature of T

=sfC

f = (Ts+Tsat )/2 of 1000°C (will be checked) (Table A-16)

skg/m 10762.4

CJ/kg 2471

C W/m1362.0kg/m 1725.0

5

3

⋅×=

°⋅=

°⋅==

−v

pv

v

vc

k µρ P = 1 atm

Ts

Heating element

Water, 100°C Analysis (a) For a horizontal heating element, the coefficient Ccr is determined from Table 10-4 to be qmax

1269.0)7986.0(12.0*12.0

1.2< 7986.00589.0

)5978.09.957(81.9)002.0()(

*

25.025.0

2/12/1

===

=⎟⎠⎞

⎜⎝⎛ −

=⎟⎠

⎞⎜⎝

⎛ −=

−−LC

gLL

cr

vl

σρρ

Then the maximum or critical heat flux is determined from

2 W/m1,074,000=−×××=

−=4/123

4/12max

)]5978.09.957()5978.0(81.90589.0)[102257(1269.0

)]([ vlvfgcr ghCq ρρρσ&

The Rohsenow relation which gives the nucleate boiling heat flux for a specified surface temperature can also be used to determine the surface temperature when the heat flux is given. Substituting the maximum heat flux into the Rohsenow relation together with other properties gives

3sat,

2/1

nucleatePr

)()(⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −⎥⎦

⎤⎢⎣

⎡ −=

nlfgsf

slpvlfgl

hC

TTcghq

σρρ

µ&

3

3

1/233

75.1)102257(0060.0)100(4217

0589.00.5978)-9.81(957.9)10)(225710282.0(000,074,1 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

×

−⎥⎦⎤

⎢⎣⎡××= − sT

It gives C109.1°=sT(b) Heat transfer in the film boiling region can be expressed as

)(43)(

)()](4.0)[(

62.043 4

sat4

sat

4/1

sat

3

radfilmtotal TTTTTTD

TTchgkqqq ss

sv

satspvfgvlvv −+−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−+−=+= εσ

µ

ρρρ&&&

Substituting,

[ ]44428

4/1

5

33

)273100()273()K W/m1067.5)(3.0(43)100(

)100)(004.0)(10762.4()]100(24714.0102257)[1725.09.957)(1725.0()1362.0(81.9

62.0000,074,1

+−+⋅×+−×

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−×

−×+×−=

−

−

ss

s

s

TT

TT

Solving for the surface temperature gives Ts = 2200°C. Therefore, the temperature jump of the wire when the operating point jumps from nucleate boiling to film boiling is Temperature jump: C2091°=−=−=∆ 1092200crit,film s, sTTT

Note that the film temperature is (2200+100)/2=1150°C, which is close enough to the assumed value of 1000°C for the evaluation of vapor paroperties.


10-18

10-26 Prob. 10-25 is reconsidered. The effects of the local atmospheric pressure and the emissivity of the wire on the critical heat flux and the temperature rise of wire are to be investigated.

Analysis The problem is solved using EES, and the solution is given below.

"GIVEN" L=0.3 [m] D=0.004 [m] epsilon=0.3 P=101.3 [kPa] "PROPERTIES" Fluid$='steam_IAPWS' T_sat=temperature(Fluid$, P=P, x=1) rho_l=density(Fluid$, T=T_sat, x=0) rho_v=density(Fluid$, T=T_sat, x=1) sigma=SurfaceTension(Fluid$, T=T_sat) mu_l=Viscosity(Fluid$,T=T_sat, x=0) Pr_l=Prandtl(Fluid$, T=T_sat, P=P+1) c_l=CP(Fluid$, T=T_sat, x=0)*Convert(kJ/kg-C, J/kg-C) h_f=enthalpy(Fluid$, T=T_sat, x=0) h_g=enthalpy(Fluid$, T=T_sat, x=1) h_fg=(h_g-h_f)*Convert(kJ/kg, J/kg) C_sf=0.0060 "from Table 10-3 of the text" n=1 "from Table 10-3 of the text" T_vapor=1000-273 "[C], assumed vapor temperature in the film boiling region" rho_v_f=density(Fluid$, T=T_vapor, P=P) "f stands for film" c_v_f=CP(Fluid$, T=T_vapor, P=P)*Convert(kJ/kg-C, J/kg-C) k_v_f=Conductivity(Fluid$, T=T_vapor, P=P) mu_v_f=Viscosity(Fluid$,T=T_vapor, P=P) g=9.81 [m/s^2] “gravitational acceleraton" sigma_rad=5.67E-8 [W/m^2-K^4] “Stefan-Boltzmann constant" "ANALYSIS" "(a)" "C_cr is to be determined from Table 10-4 of the text" C_cr=0.12*L_star^(-0.25) L_star=D/2*((g*(rho_l-rho_v))/sigma)^0.5 q_dot_max=C_cr*h_fg*(sigma*g*rho_v^2*(rho_l-rho_v))^0.25 q_dot_nucleate=q_dot_max q_dot_nucleate=mu_l*h_fg*(((g*(rho_l-rho_v))/sigma)^0.5)*((c_l*(T_s_crit-T_sat))/(C_sf*h_fg*Pr_l^n))^3 "(b)" q_dot_total=q_dot_film+3/4*q_dot_rad "Heat transfer in the film boiling region" q_dot_total=q_dot_nucleate q_dot_film=0.62*((g*k_v_f^3*rho_v_f*(rho_l-rho_v_f)*(h_fg+0.4*c_v_f*(T_s_film-T_sat)))/(mu_v_f*D*(T_s_film-T_sat)))^0.25*(T_s_film-T_sat) q_dot_rad=epsilon*sigma_rad*((T_s_film+273)^4-(T_sat+273)^4) DELTAT=T_s_film-T_s_crit


10-19

P [kPa]

maxq& [kW/m2]

∆T [C]

70 71.65 73.29 74.94 76.59 78.24 79.88 81.53 83.18 84.83 86.47 88.12 89.77 91.42 93.06 94.71 96.36 98.01 99.65 101.3

925656 934417 943050 951559 959948 968222 976385 984439 992389 1000237 1007987 1015642 1023205 1030677 1038062 1045363 1052581 1059719 1066778 1073762

2073 2079 2084 2090 2096 2101 2107 2112 2117 2122 2127 2132 2137 2142 2146 2151 2155 2160 2164 2168

70 75 80 85 90 95 100 105920000

940000

960000

980000

1000000

1.020x106

1.040x106

1.060x106

1.080x106

2080

2100

2120

2140

2160

2180

P [kPa]

qm

ax [

W/m

2]

∆T

[C

]

ε maxq&

[kW/m2] ∆T [C]

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762 1073762

2760 2535 2380 2262 2168 2091 2025 1967 1917 1872 1831 1794 1761 1730 1701 1674 1649 1626 1604

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11600

1800

2000

2200

2400

2600

2800

5.0x105

7.5x105

1.0x106

1.3x106

1.5x106

ε

∆T

[C

]

qm

ax [

W/m

2]


10-20

10-27 Water is boiled at sea level (1 atm pressure) and thus at a saturation (or boiling) temperature of Tsat = 100°C in a teflon-pitted stainless steel pan placed on an electric burner. The water level drops by 10 cm in 30 min during boiling. The inner surface temperature of the pan is to be determined.

Assumptions 1 Steady operating conditions exist. 2 Heat losses from the pan are negligible. 3 The boiling regime is nucleate boiling (this assumption will be checked later).


75.1PrN/m 0589.0

kg/m 60.0

kg/m 9.9573

3

===

=

l

v

l

σρ

ρ

P = 1 atm

Ts

100°C Water

CJ/kg 4217m/skg 10282.0

J/kg 1022573

3

°⋅=⋅×=

×=−

pl

l

fg

c

h

µ Heating

Also, 0.0058 and n = 1.0 for the boiling of water on a teflon-pitted stainless steel surface (Table 10-3). Note that we expressed the properties in units specified under Eq. 10-2 connection with their definitions in order to avoid unit manipulations.

=sfC

Analysis The rate of heat transfer to the water and the heat flux are

22

222

evap

23evap

evap

W/m240,200=)m 42 W)/(0.0317547(/

m 03142.04/m) 20.0(4/

kW 547.7kJ/kg) kg/s)(2257 03344.0(

kg/s 003344.0s 6015

m) 0.10 /4m) 0.2()(kg/m 9.957(

==

===

===

=×

××=

∆∆

=∆

=

s

s

fg

AQq

DA

hmQtV

tm

m

&&

&&

&

ππ

πρ

The Rohsenow relation which gives the nucleate boiling heat flux for a specified surface temperature can also be used to determine the surface temperature when the heat flux is given. Assuming nucleate boiling, the temperature of the inner surface of the pan is determined from Rohsenow relation to be

3sat,

2/1

nucleatePr

)()(⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −⎥⎦

⎤⎢⎣

⎡ −=

nlfgsf

slpvlfgl

hC

TTcghq

σρρ

µ&

3

3

1/233

75.1)102257(0058.0)100(4217

0589.00.60)9.8(957.9)10)(225710282.0(200,240 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

×

−⎥⎦

⎤⎢⎣

⎡ −××= − sT

It gives

Ts = 105.3°C

which is in the nucleate boiling range (5 to 30°C above surface temperature). Therefore, the nucleate boiling assumption is valid.


10-21

10-28 Water is boiled at sea level (1 atm pressure) and thus at a saturation (or boiling) temperature of Tsat = 100°C in a polished copper pan placed on an electric burner. The water level drops by 10 cm in 30 min during boiling. The inner surface temperature of the pan is to be determined.

Assumptions 1 Steady operating conditions exist. 2 Heat losses from the pan are negligible. 3 The boiling regime is nucleate boiling (this assumption will be checked later).


75.1PrN/m 0589.0

kg/m 60.0

kg/m 9.9573

3

===

=

l

v

l

σρ

ρ

P = 1 atm

Ts

100°C Water

CJ/kg 4217m/skg 10282.0

J/kg 1022573

3

°⋅=⋅×=

×=−

pl

l

fg

c

h

µ Heating

Also, 0.0130 and n = 1.0 for the boiling of water on a copper surface (Table 10-3). Note that we expressed the properties in units specified under Eq. 10-2 connection with their definitions in order to avoid unit manipulations.

=sfC

Analysis The rate of heat transfer to the water and the heat flux are

22

222

evap

23evap

evap

W/m240,200=)m 42 W)/(0.0317547(/

m 03142.04/m) 20.0(4/

kW 547.7kJ/kg) kg/s)(2257 03344.0(

kg/s 003344.0s 6015

m) 0.10 /4m) 0.2()(kg/m 9.957(

==

===

===

=×

××=

∆∆

=∆

=

s

s

fg

AQq

DA

hmQtt

mm

&&

&&

&

ππ

πρ V

The Rohsenow relation which gives the nucleate boiling heat flux for a specified surface temperature can also be used to determine the surface temperature when the heat flux is given. Assuming nucleate boiling, the temperature of the inner surface of the pan is determined from Rohsenow relation to be

3sat,

2/1

nucleatePr

)()(⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −⎥⎦

⎤⎢⎣

⎡ −=

nlfgsf

slpvlfgl

hC

TTcghq

σρρ

µ&

3

3

1/233

75.1)102257(0130.0)100(4217

0589.00.60)9.8(957.9)10)(225710282.0(200,240 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

×

−⎥⎦

⎤⎢⎣

⎡ −××= − sT

It gives

Ts = 111.9°C

which is in the nucleate boiling range (5 to 30°C above surface temperature). Therefore, the nucleate boiling assumption is valid.


10-22

10-29 Boiling experiments are conducted by heating water at 1 atm pressure with an electric resistance wire, and measuring the power consumed by the wire as well as temperatures. The boiling heat transfer coefficient is to be determined.

Assumptions 1 Steady operating conditions exist. 2 Heat losses from the water are negligible. 1 atmAnalysis The heat transfer area of the heater wire is

Ts=130°C

Heating wire, 3.8 kW

2m 003142.0m) m)(0.50 002.0( === ππDLAs

Noting that 3800 W of electric power is consumed when the heater surface temperature is 130°C, the boiling heat transfer coefficient is determined from Newton’s law of cooling to be

C W/m40,320 2 °⋅=°−

=−

=→−=C)100)(130m (0.003142

W3800 )(

)(2

satsat TTA

QhTThAQss

ss

&&

10-30 Water is boiled at Tsat = 120°C in a mechanically polished stainless steel pressure cooker whose inner surface temperature is maintained at Ts = 128°C. The boiling heat transfer coefficient is to be determined.



44.1PrCJ/kg 4244N/m 0550.0

skg/m 10232.0kg/m 121.1J/kg 102203kg/m 4.94333

33

=°⋅==

⋅×==

×==−

l

pl

lv

fgl

c

h

σµρ

ρ

128°C

120°CWater

Also, 0.0130 and n = 1.0 for the boiling of water on a mechanically polished stainless steel surface (Table 10-3). Note that we expressed the properties in units specified under Eq. 10-2 in connection with their definitions in order to avoid unit manipulations.

=sfC

Heating

Analysis The excess temperature in this case is C8120128sat °=−=−=∆ TTT s which is relatively low (less than 30°C). Therefore, nucleate boiling will occur. The heat flux in this case can be determined from Rohsenow relation to be

2

3

3

1/233

3sat,

2/1

nucleate

W/m900,116

44.1)102203(0130.0)120128(4244

0550.01.121)-9.81(943.4)10)(220310232.0(

Pr

)()(

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

×−

⎥⎦⎤

⎢⎣⎡××=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −⎥⎦

⎤⎢⎣

⎡ −=

−

nlfgsf

slpvlfgl hC

TTcghq

σρρ

µ&

The boiling heat transfer coefficient is

CkW/m 14.6 2 ⋅=°⋅=°−

=−

=⎯→⎯−= C W/m610,14C)120128(

W/m900,116)( 22

sat

nucleatesatnucleate TT

qhTThq

ss

&&


10-34

10-42 Water is boiled at Tsat = 100°C by a spherical platinum heating element immersed in water. The surface temperature is Ts = 350°C. The rate of heat transfer is to be determined.


Properties The properties of water at the saturation temperature of 100°C are (Table A-9)

3

3

kg/m 9.957

J/kg 102257

=

×=

l

fgh

ρ

The properties of water vapor at (350+100)/2 = 225°C are (Table A-16)

100°CWater

350°C

C W/m03581.0

CJ/kg 1951skg/m 10749.1

kg/m 444.05

3

°⋅=

°⋅=⋅×=

=−

v

pv

v

v

k

cµ

ρ

Analysis The film boiling occurs since the temperature difference between the surface and the fluid. The heat flux in this case can be determined from

[ ]

[ ]

2

4/1

5

33

sat

4/1

sat

sat3

film

W/m207,25

)100350()100350)(15.0)(10749.1(

)100350)(1951(4.0102257)444.09.957)(444.0()03581.0)(81.9(67.0

)()(

)(4.0)(67.0

=

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−×

−+×−=

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−+−=

−

TTTTD

TTchgkq s

sv

spvfgvlvv

µ

ρρρ&

The radiation heat transfer is

[ ] 24484sat

4rad W/m745)273100()273350()1067.5)(10.0()( =+−+×=−= −TTq sεσ&

The total heat flux is

2radfilmtotal W/m766,25)745(

43207,25

43

=+=+= qqq &&&

Then the total rate of heat transfer becomes

W1821=== ) W/m766,25()15.0( 22totaltotal πqAQ &&


10-35

Condensation Heat Transfer

10-43C Condensation is a vapor-to-liquid phase change process. It occurs when the temperature of a vapor is reduced below its saturation temperature Tsat. This is usually done by bringing the vapor into contact with a solid surface whose temperature Ts is below the saturation temperature Tsat of the vapor.

10-44C In film condensation, the condensate wets the surface and forms a liquid film on the surface which slides down under the influence of gravity. The thickness of the liquid film increases in the flow direction as more vapor condenses on the film. This is how condensation normally occurs in practice. In dropwise condensation, the condensed vapor forms droplets on the surface instead of a continuous film, and the surface is covered by countless droplets of varying diameters. Dropwise condensation is a much more effective mechanism of heat transfer.

10-45C The presence of noncondensable gases in the vapor has a detrimental effect on condensation heat transfer. Even small amounts of a noncondensable gas in the vapor cause significant drops in heat transfer coefficient during condensation.

10-46C The modified latent heat of vaporization is the amount of heat released as a unit mass of vapor condenses at a specified temperature, plus the amount of heat released as the condensate is cooled further to some average temperature between T

*fgh

sat and . It is defined as where csT )(68.0 sat*

splfgfg TTchh −+= pl is the specific heat of the liquid at the average film temperature.

10-47C During film condensation on a vertical plate, heat flux at the top will be higher since the thickness of the film at the top, and thus its thermal resistance, is lower.

10-48C The condensation heat transfer coefficient for the tubes will be the highest for the case of horizontal side by side (case b) since (1) for long tubes, the horizontal position gives the highest heat transfer coefficients, and (2) for tubes in a vertical tier, the average thickness of the liquid film at the lower tubes is much larger as a result of condensate falling on top of them from the tubes directly above, and thus the average heat transfer coefficient at the lower tubes in such arrangements is smaller.

10-49C In condensate flow, the wetted perimeter is defined as the length of the surface-condensate interface at a cross-section of condensate flow. It differs from the ordinary perimeter in that the latter refers to the entire circumference of the condensate at some cross-section.


10-36

10-50 The necessary surface temperature of the plate used to condensate saturated water vapor at a desired condensation rate is to be determined.

Assumptions 1 Steady operating condition exists. 2 The plate surface has uniform temperature. 3 The film temperature is 90°C.

Properties The properties of water at the saturation temperature of 100°C are hfg = 2257 kJ/kg (Table A-2) and ρv = 0.5978 kg/m3 (Table A-9). The properties of liquid water at the film temperature of Tf = (Tsat + Ts)/2 = 90°C are, from Table A-9,

ρl = 965.3 kg/m3 cpl = 4206 J/kg·K

µl = 0.315 × 10−3 kg/m·s kl = 0.675 W/m·K

νl = µl / ρl = 0.326 × 10−6 m2/s

Analysis The calculation of the modified latent heat of vaporization requires the knowledge of the Ts. Hence, we assume Ts = 80°C, and iterate the solution, if necessary, until good agreement with the calculated value of Ts is achieved:

J/kg 102314

)80100)(4206(68.0102257

)(68.0

3

3

sat

×=

−+×=

−+=∗splfgfg TTchh

The Reynolds number is

3.406)skg/m 10315.0)(m 5.0(

)kg/s 016.0(44Re 3 =⋅×

==−

lpmµ&

which is between 30 and 1800, and thus the flow is wavy-laminar. The heat transfer coefficient is

K W/m7558

)/sm 10326.0(m/s 81.9

2.5)3.406(08.1)K W/m675.0)(3.406(

2.5Re08.1Re

2

3/1

226

2

22.1

3/1

222.1 wavyvert,

⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

×−

⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−==

−

l

l gkhh

ν

Hence, the surface temperature can be calculated using

→ ∗=− fgss hmTThA &)( sats

fgs hA

hmTT

∗

−=&

sat

C80.4°=⋅

×−°= 22

3

)m 5.0)(K W/m7558()J/kg 102314)(kg/s 016.0(C100sT

Discussion The assumed Ts = 80°C is good, thus the solution does not require iteration.


10-37

10-51 The local heat transfer coefficients at the middle and at the bottom of a vertical plate undergoing film condensation are to be determined.

Assumptions 1 Steady operating condition exists. 2 The plate surface has uniform temperature. 3 The flow is laminar.


ρl = 965.3 kg/m3 cpl = 4206 J/kg·K

µl = 0.315 × 10−3 kg/m·s kl = 0.675 W/m·K

νl = µl / ρl = 0.326 × 10−6 m2/s

Analysis The modified latent heat of vaporization is

J/kg 102314

)80100)(4206(68.0102257

)(68.0

3

3

sat

×=

−+×=


The local heat transfer coefficient can be calculated using

KW/m14008

)80100)(10315.0(4)675.0)(102314)(5978.03.965)(3.965)(81.9(

)(4)(

24/1

4/1

3

33

4/1

sat

3

⋅⎟⎠⎞

⎜⎝⎛=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−××−

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=

−

∗

x

x

xTTkhg

hsl

lfgvllx µ

ρρρ

The local heat transfer coefficient at the middle of the plate (x = 0.1 m) is

K W/m7130 2 ⋅=⋅⎟⎠⎞

⎜⎝⎛=⋅⎟

⎠⎞

⎜⎝⎛= KW/m

1.014008KW/m14008 2

4/12

4/1

xhx

The local heat transfer coefficient at the bottom of the plate (x = 0.2 m) is

K W/m5990 2 ⋅=⋅⎟⎠⎞

⎜⎝⎛=⋅⎟

⎠⎞

⎜⎝⎛= KW/m

2.014008KW/m14008 2

4/12

4/1

xhx

Discussion The assumption that the flow is laminar is verified to be appropriate:

18001765990

675.0)10315.0(3)3.965)(81.9(4

34

Re3

23

23

2

2<=⎟

⎠⎞

⎜⎝⎛

×=⎟⎟

⎠

⎞⎜⎜⎝

⎛≅

−=Lx

l

l

l

hkg

µρ


10-38

10-52 The hydraulic diameter Dh for all 4 cases are expressed in terms of the boundary layer thickness δ as follows:

(a) Vertical plate: δδ 444===

ww

pA

D ch

(b) Tilted plate: δδ 444===

ww

pA

D ch

(c)Vertical cylinder: δπ

δπ 444===

DD

pA

D ch

(d) Horizontal cylinder: δδ 4

2)2(44===

LL

pA

D ch

(e) Sphere: δπ

δπ 444===

DD

pA

D ch

Therefore, the Reynolds number for all 5 cases can be expressed as

l

ll

l

llh

l

llc

l

VVDp

VAp

mµδρ

µρ

µρ

µ444Re ====

&


10-39

10-53 Saturated steam condenses outside of vertical tube. The rate of heat transfer to the coolant, the rate of condensation and the thickness of the condensate layer at the bottom are to be determined.

Assumptions 1 Steady operating conditions exist. 2 The tube is isothermal. 3 The tube can be treated as a vertical plate. 4 The condensate flow is wavy-laminar over the entire tube (this assumption will be verified). 5 Nusselt’s analysis can be used to determine the thickness of the condensate film layer. 6 The density of vapor is much smaller than the density of liquid,

. lv ρρ <<

Properties The properties of water at the saturation temperature of 30°C are hfg = 2431×103 J/kg and ρv = 0.03 kg/m3. The properties of liquid water at the film temperature of (30 + 20)/2 = 25°C are (Table A-9), =+= 2/)( sat sf TTT

C W/m607.0

CJ/kg 4180/sm10894.0/

skg/m10891.0

kg/m 0.997

26

3

3

°⋅=

°⋅=×==

⋅×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ

Analysis (a)The modified latent heat of vaporization is

J/kg 102459=C0)2C(30J/kg 41800.68+J/kg 102431

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh

Steam30°C

Condensate L = 2 m

D = 4 cm

20°C

Assuming wavy-laminar flow, the Reynolds number is determined from

3.157)s/m 10894.0(

m/s 8.9)J/kg 102459)(skg/m 10891.0(

C)2030(C) W/m607.0(m) 2(70.381.4

)(70.381.4ReRe

82.03/1

226

2

33

820.03/1

2*sat

wavyvertical,

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

××⋅×

°−×°⋅××+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+==

−−

lfgl

sl gh

TTLkνµ

which is between 30 and 1800, and thus our assumption of wavy laminar flow is verified. Then the condensation heat transfer coefficient is determined to be

C W/m4302)/sm 10894.0(

m/s 8.92.5)3.157(08.1C) W/m607.0(3.157

2.5Re08.1Re

23/1

226

2

22.1

3/1

222.1wavyvertical,

°⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

×−

°⋅×=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−==

−

l

l gkhh

ν

The heat transfer surface area of the tube is . Then the rate of heat transfer during this condensation process becomes

2m 2513.0m) m)(2 04.0( === ππDLAs

W10,811=°−°⋅=−= C)2030)(m 2513.0)(C W/m4302()( 22sat ss TThAQ&

(b) The rate of condensation of steam is determined from

kg/s 104.40 3-×=×

==J/kg 102459

J/s 811,103*oncondensati

fghQ

m&

&

(c) Combining equations llL hk /=δ and Lhh )3/4(= , the thickness of the liquid film at the bottom of the tube is determined to be

mm 0.2=×°⋅

°⋅== m 100.188=

C) W/m4302(3C) W/m607.0(4

34 3-

2hkl

Lδ


10-40

10-54 Saturated steam at atmospheric pressure thus at a saturation temperature of Tsat = 100°C condenses on a vertical plate which is maintained at 90°C by circulating cooling water through the other side. The rate of heat transfer to the plate and the rate of condensation of steam are to be determined.

Assumptions 1 Steady operating conditions exist. 2 The plate is isothermal. 3 The condensate flow is wavy-laminar over the entire plate (this assumption will be verified). 4 The density of vapor is much smaller than the density of liquid, lv ρρ << .

Properties The properties of water at the saturation temperature of 100°C are hfg = 2257×103 J/kg and ρv = 0.60 kg/m3. The properties of liquid water at the film temperature of =+= 2/)( sat sf TTT (100 + 90)/2 = 95°C are (Table A-9),

C W/m677.0

CJ/kg 4212/sm10309.0/

skg/m10297.0

kg/m 5.961

26

3

3

°⋅=

°⋅=×==

⋅×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ

m&

90°C

10 m 1 atmSteam

2 mAnalysis The modified latent heat of vaporization is

J/kg 102,286=C90)C(100J/kg 42120.68+J/kg 102257

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh


0.798)s/m 10309.0(

m/s 8.9)J/kg 102286)(skg/m 10297.0(

C)90100(C) W/m677.0(m) 2(70.381.4

)(70.381.4ReRe

82.03/1

226

2

33

820.03/1

2*sat

wavyvertical,

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

××⋅×

°−×°⋅××+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+==

−−

lfgl

sl gh

TTLkνµ


C W/m6757)/sm 10309.0(

m/s 81.92.5)798(08.1C) W/m677.0(798

2.5Re08.1Re

23/1

226

2

22.1

3/1

222.1wavyvertical,

°⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

×−

°⋅×=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−==

−

l

l gkhh

ν

The heat transfer surface area of the plate is

2m 20m) m)(10 2( ==×= LWAs

Then the rate of heat transfer during this condensation process becomes

kW 1352==°−°⋅=−= W1,351,500C)90100)(m 20)(C W/m6757()( 22sat ss TThAQ&


kg/s 0.591=×

==J/kg 102286J/s 500,351,1

3*oncondensatifgh

Qm&

&


10-41

10-55 Saturated steam at a saturation temperature of Tsat = 100°C condenses on a plate which is tilted 60° from the vertical and maintained at 90°C by circulating cooling water through the other side. The rate of heat transfer to the plate and the rate of condensation of the steam are to be determined.



C W/m677.0

CJ/kg 4212/sm10309.0/

skg/m10297.0

kg/m 5.961

26

3

3

°⋅=

°⋅=×==

⋅×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ


J/kg 102,286=

C90)C(100J/kg 42120.68+J/kg 102257

)(68.0

3

3

sat*

×

°−°⋅××=

−+= splfgfg TTchh

Assuming wavy-laminar flow, the Reynolds number is determined from the vertical plate relation by replacing g by g cosθ where θ = 60° to be

m&2 m

90°C

1 atmSteam

60°

10 m

5.660)s/m 10309.0(

60cos)m/s 8.9()J/kg 102286)(skg/m 10297.0(

C)90100(C) W/m677.0(m) 2(70.381.4

60cos)(70.381.4ReRe

82.03/1

226

2

33

820.03/1

2*sat

wavytilted,

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

××⋅×

°−×°⋅××+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+==

−−

lfgl

sl gh

TTLkνµ

which is between 30 and 1800, and thus our assumption of wavy laminar flow is verified. Then the condensation heat transfer coefficient is determined from

C W/m5593)/sm 10309.0(

60cos)m/s 8.9(2.5)5.660(08.1C) W/m677.0(5.660

cos2.5Re08.1

Re

23/1

226

2

22.1

3/1

222.1wavytilted,

°⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

×−

°⋅×=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−==

−

l

l gkhh

νθ

The heat transfer surface area of the plate is . 2m 20m) m)(10 2( ==×= LWAs


kW 1119==°−°⋅=−= W600,118,1C)90100)(m 20)(C W/m5593()( 22sat ss TThAQ&


kg/s 0.489=×

==J/kg 102286J/s 600,118,1

3*oncondensatifgh

Qm&

&

Discussion Using the heat transfer coefficient determined in the previous problem for the vertical plate, we could also determine the heat transfer coefficient from . It would give 5682 W/m4/1

vertinclined )(cosθhh = 2⋅°C, which is 1.6% different than the value determined above.


10-42

10-56 There is film condensation on the outer surfaces of N horizontal tubes arranged in a vertical tier. The value of N for which the average heat transfer coefficient for the entire tier be equal to half of the value for a single horizontal tube is to be determined.

Assumptions Steady operating conditions exist.

Analysis The relation between the heat transfer coefficients for the two cases is given to be

4/1

tube1 ,horizontal tubesN ,horizontal

N

hh =

Therefore,

16=⎯→⎯== NNh

h 1

21

4/1 tube1 ,horizontal

tubesN ,horizontal


10-43

10-57 Saturated steam at a saturation temperature of Tsat = 50°C condenses on the outer surfaces of a tube bank with 33 tubes in each column maintained at 20°C. The average heat transfer coefficient and the rate of condensation of steam on the tubes are to be determined.

Assumptions 1 Steady operating conditions exist. 2 The tubes are isothermal.


C W/m623.0

CJ/kg 4178/sm10724.0/

skg/m10720.0

kg/m 0.994

26

3

3

°⋅=

°⋅=×==

⋅×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρSteam 50°C

20°C

Condensate flow

33 tubes in a column

Analysis (a) The modified latent heat of vaporization is

J/kg 102468=

C0)2C(50J/kg 41780.68+J/kg 102383

)(68.0

3

3

sat*

×

°−°⋅××=

−+= splfgfg TTchh

The heat transfer coefficient for condensation on a single horizontal tube is

C W/m8425

m) C(0.015)2050(s)kg/m 10720.0()C W/m623.0)(J/kg 102468)(kg/m 08.0994)(kg/m 994)(m/s 8.9(

729.0

)()(

729.0

2

4/1

3

33332

4/1

sat

3*

horizontal

°⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µ

ρρρ

Then the average heat transfer coefficient for a 33-tube high vertical tier becomes

C W/m3515 2 °⋅=°⋅== C) W/m8425(33

11 24/1 tube1 horiz,4/1 tubesN horiz, h

Nh

The surface area for all 33 tubes per unit length is

2total m 1.555= m) m)(1 015.0(33ππ == DLNAs


W000,164C)2050)(m 555.1)(C W/m3515()( 22sat =°−°⋅=−= ss TThAQ&


kg/s 0.0664=×

==J/kg 102468J/s 000,164

3*oncondensatifgh

Qm

&&


10-44

10-58 Saturated steam at a saturation temperature of Tsat = 55°C condenses on the outer surface of a vertical tube which is maintained at 45°C. The required tube length to condense steam at a rate of 10 kg/h is to be determined.

Assumptions 1 Steady operating conditions exist. 2 The tube is isothermal. 3 The vertical tube can be treated as a vertical plate. 4 The density of vapor is much smaller than the density of liquid, lv ρρ << .


C W/m644.0

CJ/kg 4181/sm10554.0/

skg/m10547.0

kg/m 1.988

26

3

3

°⋅=

°⋅=×==

⋅×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ


J/kg 102399=C)45C(55J/kg 41810.68+J/kg 102371

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh

Steam55°C

Condensate Ltube = ?

D = 3 cm

45°C

The Reynolds number is determined from its definition to be

5.215)skg/m 10m)(0.547 (0.03

kg/s) 3600/10(44Re3

=⋅×

==−πµ lp

m&

which is between 30 and 1800. Therefore the condensate flow is wavy laminar, and the condensation heat transfer coefficient is determined from

C W/m5644)/sm 10554.0(

m/s 8.92.5)5.215(08.1C) W/m644.0(5.215

2.5Re08.1Re

23/1

226

2

22.1

3/1

222.1wavyvertical,

°⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

×−

°⋅×=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−==

−

l

l gkhh

ν

The rate of heat transfer during this condensation process is

W664,6)J/kg 10kg/s)(2399 3600/10( 3* =×== fghmQ &&

Heat transfer can also be expressed as

))(()( satsat sss TTDLhTThAQ −=−= π&

Then the required length of the tube becomes

m 1.21=°−°⋅

=−

=C)4555)(m 03.0()C W/m5844(

W6664))((

2

sat ππ sTTDhQL&


10-45

10-59 Saturated steam at a saturation temperature of Tsat = 55°C condenses on the outer surface of a horizontal tube which is maintained at 45°C. The required tube length to condense steam at a rate of 10 kg/h is to be determined.

Assumptions 1 Steady operating conditions exist. 2 The tube is isothermal.


C W/m644.0

CJ/kg 4181/sm10554.0/

skg/m10547.0

kg/m 1.988

26

3

3

°⋅=

°⋅=×==

⋅×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ Steam 55°C

Ltube = ?

Cooling water

45°C

Analysis The modified latent heat of vaporization is Condensate

J/kg 102399=C)45C(55J/kg 41810.68+J/kg 102371

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh

Noting that the tube is horizontal, the condensation heat transfer coefficient is determined from

C. W/m135,10

m) C(0.03)4555(s)kg/m 10547.0()C W/m644.0)(J/kg 102399)(kg/m 10.01.988)(kg/m 1.988)(m/s 8.9(

729.0

)()(

729.0

2

4/1

3

33332

4/1

sat

3*

horizontal

°=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µ

ρρρ

The rate of heat transfer during this condensation process is

W664,6)J/kg 10kg/s)(2399 3600/10( 3* =×== fghmQ &&

Heat transfer can also be expressed as

))(()( satsat sss TTDLhTThAQ −=−= π&

Then the required length of the tube becomes

m 0.70=°−°⋅

=−

=C)4555)(m 03.0()C W/m135,10(

W6664))((

2

sat ππ sTTDhQL&


10-46

10-60 Saturated steam at a saturation temperature of Tsat = 100°C condenses on a plate which is tilted 30° from the vertical and maintained at 80°C by circulating cooling water through the other side. The rate of heat transfer to the plate and the rate of condensation of the steam are to be determined.



C W/m675.0

CJ/kg 4206/sm10326.0/

skg/m10315.0

kg/m 3.965

26

3

3

°⋅=

°⋅=×==

⋅×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ


J/kg 102,314=C0)8C(100J/kg 42060.68+J/kg 102257

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh

2 mInclined plate 80°C

Condensate 30°

Vapor 100°C

Assuming wavy-laminar flow, the Reynolds number is determined from the vertical plate relation by replacing g by θcosg where θ = 30° to be

1237)s/m 10326.0(

30cos)m/s 8.9()J/kg 102314)(skg/m 10315.0(

C)80100(C) W/m675.0(m) 2(70.381.4

cos)(70.381.4ReRe

82.03/1

226

2

33

820.03/1

2*sat

wavytilted,

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

××⋅×

°−×°⋅××+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+==

−−

lfgl

sl gh

TTLkν

θµ


C W/m5623 2 °⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

×−

°⋅×=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−==

−

3/1

226

2

22.1

3/1

222.1wavytilted,

)/sm 10326.0(30cos)m/s 8.9(

2.5)1237(08.1C) W/m675.0(1237

cos2.5Re08.1

Re

l

l gkhh

νθ

The heat transfer surface area of the plate is: 2m 4m) m)(2 2( ==×= LwA .


W449,900C)80100)(m 4)(C W/m5623()( 22sat =°−°⋅=−= sTThAQ&


kg/s 0.194=×

==J/kg 102314J/s 900,449

3*oncondensatifgh

Qm&

&

Discussion We could also determine the heat transfer coefficient from . 4/1vertinclined )(cosθhh =


10-47

10-61 Prob. 10-60 is reconsidered. The effects of plate temperature and the angle of the plate from the vertical on the average heat transfer coefficient and the rate at which the condensate drips off are to be investigated.


"GIVEN" T_sat=100 [C] L=2 [m] theta=30 [degrees] T_s=80 [C]

"PROPERTIES" Fluid$='steam_IAPWS' T_f=1/2*(T_sat+T_s) P_sat=pressure(Fluid$, T=T_sat, x=1) rho_l=density(Fluid$, T=T_f, x=0) mu_l=Viscosity(Fluid$,T=T_f, x=0) nu_l=mu_l/rho_l c_l=CP(Fluid$, T=T_f, x=0)*Convert(kJ/kg-C, J/kg-C) k_l=Conductivity(Fluid$, T=T_f, P=P_sat+1) h_f=enthalpy(Fluid$, T=T_sat, x=0) h_g=enthalpy(Fluid$, T=T_sat, x=1) h_fg=(h_g-h_f)*Convert(kJ/kg, J/kg) g=9.8 [m/s^2]

"ANALYSIS" "(a)" h_fg_star=h_fg+0.68*c_l*(T_sat-T_s) Re=(4.81+(3.7*L*k_l*(T_sat-T_s))/(mu_l*h_fg_star)*((g*Cos(theta))/nu_l^2)^(1/3))^0.820 h=(Re*k_l)/(1.08*Re^1.22-5.2)*((g*Cos(theta))/nu_l^2)^(1/3) Q_dot=h*A*(T_sat-T_s) A=L^2 "(b)" m_dot_cond=Q_dot/h_fg_star

Ts [C]

h [W/m2.C]

condm& [kg/s]

40 42.5 45 47.5 50 52.5 55 57.5 60 62.5 65 67.5 70 72.5 75 77.5 80 82.5 85 87.5 90

4212 4272 4334 4398 4464 4533 4605 4680 4759 4841 4929 5023 5123 5231 5350 5480 5626 5791 5984 6213 6500

0.4165 0.406 0.3951 0.3839 0.3722 0.3601 0.3476 0.3346 0.3212 0.3073 0.2929 0.2779 0.2625 0.2465 0.2298 0.2125 0.1945 0.1758 0.1561 0.1355 0.1138

40 50 60 70 80 904000

4500

5000

5500

6000

6500

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Ts [C]

h [

W/m

2-C

]

mc

on

d

[kg

/s]


10-48

θ [degrees]

h [W/m2.C]

condm& [kg/s]

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

5851 5849 5842 5831 5816 5796 5771 5742 5708 5670 5626 5577 5522 5462 5396 5323 5243 5156 5061 4957 4842

0.2023 0.2022 0.202 0.2016 0.2011 0.2004 0.1996 0.1986 0.1974 0.196 0.1945 0.1928 0.1909 0.1889 0.1866 0.1841 0.1813 0.1783 0.175 0.1714 0.1674

0 10 20 30 40 50 604800

5000

5200

5400

5600

5800

6000

0.165

0.17

0.175

0.18

0.185

0.19

0.195

0.2

0.205

θ [degrees]h

[W

/m2-C

]

mc

on

d [k

g/s

]


10-49

10-62 Saturated ammonia vapor at a saturation temperature of Tsat = 10°C condenses on the outer surface of a horizontal tube which is maintained at -10°C. The rate of heat transfer from the ammonia and the rate of condensation of ammonia are to be determined.


Properties The properties of ammonia at the saturation temperature of 10°C are hfg = 1226×103 J/kg and ρv = 4.870 kg/m3 (Table A-11). The properties of liquid ammonia at the film temperature of =+= 2/)( sat sf TTT (10 + (-10))/2 = 0°C are (Table A-11),

Ammonia 10°C -10°C

Dtube = 4 cm Ltube = 15 m

C W/m5390.0

CJ/kg 4617/sm102969.0

skg/m10896.1

kg/m 6.638

26

4

3

°⋅=

°⋅=×=

⋅×=

=

−

−

l

pl

l

l

l

k

cν

µ

ρ


J/kg 101288=C)]10(C[10J/kg 46170.68+J/kg 101226

)(68.033

sat*

×°−−°⋅××=

−+= splfgfg TTchhCondensate


C. W/m7390

m) C(0.02)]10(10[s)kg/m 10896.1()C W/m5390.0)(J/kg 101288)(kg/m 870.4)(638.6kg/m 6.638)(m/s 81.9(

729.0

)()(

729.0

2

4/1

4

33332

4/1

sat

3*

horizontal

°=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µ

ρρρ

The heat transfer surface area of the tube is

2m 1.885= m) m)(15 04.0(ππ == DLAs


W278,600=°−−°=−= C)]10(10)[m 885.1)(C. W/m7390()( 22sat ss TThAQ&

(b) The rate of condensation of ammonia is determined from

kg/s 0.216=×

==J/kg 101288J/s 600,278

3*oncondensatifgh

Qm&

&


10-50

10-63 The rate of condensation and the heat transfer rate for a vertical pipe, with specified surface temperature, are to be determined.

Assumptions 1 Steady operating condition exists. 2 The surface has uniform temperature. 3 The pipe can be treated as a vertical plate. 4 The condensate flow is wavy-laminar over the entire tube (this assumption will be verified). 5 Nusselt’s analysis can be used to determine the thickness of the condensate film layer. 6 The density of vapor is much smaller than the density of liquid, ρv << ρl.


ρl = 965.3 kg/m3 cpl = 4206 J/kg·K

µl = 0.315 × 10−3 kg/m·s kl = 0.675 W/m·K

νl = µl / ρl = 0.326 × 10−6 m2/s


J/kg 102314

)80100)(4206(68.0102257

)(68.0

3

3

sat

×=

−+×=



7.729)10326.0(

81.9)102314)(10315.0(

)80100)(675.0)(1(70.381.4

)(70.381.4Re

820.03/1

2633

820.03/1

2sat

wavyvert,

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

×××

−+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

−−

∗lfgl

sl gh

TTLkνµ


K W/m6633

)/sm 10326.0(m/s 81.9

2.5)7.729(08.1)K W/m675.0)(7.729(

2.5Re08.1Re

2

3/1

226

2

22.1

3/1

222.1 wavyvert,

⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

×−

⋅=⎟

⎟⎠

⎞⎜⎜⎝

⎛

−==

−l

l gkhh

ν


W104.168 4×=

−⋅=

−=

K )80100)(K W/m6633)(m 1)(m 1.0(

)(2

sat

π

π sTTDLhQ&

The rate of condensation of steam is determined from

kg/s 0.018=×

×==

∗ J/kg 102314W101684

3

4

oncondensati .

hQmfg

&&

Discussion Combining equations LlL hk /=δ and Lhh )3/4(= , the thickness of the liquid film at the bottom of the tube is determined to be

mm 100mm 136.0)K W/m6633(3)K W/m675.0(4

34

2 <<=⋅⋅

==hkl

Lδ

Since δL << D, the pipe can be treated as a vertical plate.


10-51

10-64 Saturated steam at a pressure of 4.25 kPa and thus at a saturation temperature of Tsat = 30°C (Table A-9) condenses on the outer surfaces of 144 horizontal tubes arranged in a 12×12 square array maintained at 20°C by circulating cooling water. The rate of heat transfer to the cooling water and the rate of condensation of steam on the tubes are to be determined.


Properties The properties of water at the saturation temperature of 30°C are hfg = 2431×103 J/kg and ρv = 0.03 kg/m3. The properties of liquid water at the film temperature of

(30 + 20)/2 = 25°C are (Table A-9), =+= 2/)( sat sf TTT

C W/m607.0

CJ/kg 4180/sm10894.0/

skg/m10891.0

kg/m 0.997

26

3

3

°⋅=

°⋅=×==

⋅×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ


J/kg 102,459=C0)2C(30J/kg 41800.68+J/kg 102431

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh

L = 8 m

n = 144 tubes 20°C

P = 4.25 kPa

Saturated steam

Cooling water


C. W/m8674

m) C(0.03)2030(s)kg/m 10891.0()C W/m607.0)(J/kg 102459)(kg/m 03.0997)(kg/m 997)(m/s 8.9(

729.0

)()(

729.0

2

4/1

3

33332

4/1

sat

3*

horizontal

°=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µ

ρρρ

Then the average heat transfer coefficient for a 10-pipe high vertical tier becomes

C W/m4661C) W/m8674(12

11 224/1 tube1 horiz,4/1 tubesN horiz, °⋅=°⋅== h

Nh

The surface area for all 144 tubes is



kW 5060==°−°=−= W,060,0005C)2030)(m 6.108)(C. W/m4661()( 22sat ss TThAQ&


kg/s 2.06=×

==J/kg 102459J/s 000,060,5

3*oncondensatifgh

Qm&

&


10-52

10-65 Prob. 10-64 is reconsidered. The effect of the condenser pressure on the rate of heat transfer and the rate of condensation of the steam is to be investigated.


"GIVEN" P_sat=4.25 [kPa] n_tube=144 N=12 L=8 [m] D=0.03 [m] T_s=20 [C] "PROPERTIES" Fluid$='steam_IAPWS' T_sat=temperature(Fluid$, P=P_sat, x=1) T_f=1/2*(T_sat+T_s) h_f=enthalpy(Fluid$, T=T_sat, x=0) h_g=enthalpy(Fluid$, T=T_sat, x=1) h_fg=(h_g-h_f)*Convert(kJ/kg, J/kg) rho_v=density(Fluid$, T=T_sat, x=1) rho_l=density(Fluid$, T=T_f, x=0) mu_l=Viscosity(Fluid$,T=T_f, x=0) nu_l=mu_l/rho_l c_l=CP(Fluid$, T=T_f, x=0)*Convert(kJ/kg-C, J/kg-C) k_l=Conductivity(Fluid$, T=T_f, P=P_sat+1) g=9.8 [m/s^2] "ANALYSIS" h_fg_star=h_fg+0.68*c_l*(T_sat-T_s) h_1tube=0.729*((g*rho_l*(rho_l-rho_v)*h_fg_star*k_l^3)/(mu_l*(T_sat-T_s)*D))^0.25 h=1/N^0.25*h_1tube Q_dot=h*A*(T_sat-T_s) A=n_tube*pi*D*L m_dot_cond=Q_dot/h_fg_star

3 5 7 9 11 13 15

4.0x106

6.0x106

8.0x106

1.0x107

1.2x107

1.4x107

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Psat [kPa]

Q

[W]

mco

nd

[kg

/s]

Psat [kPa]

Q& [W]

condm& [kg/s]

3 4 5 6 7 8 9 10 11 12 13 14 15

2523819 4642913 6185636 7426047 8473057 9383654 10192213 10921191 11586164 12198509 12766595 13296988 13794833

1.028 1.889 2.515 3.017 3.441 3.809 4.136 4.43 4.698 4.945 5.174 5.387 5.587


10-53

10-66E Saturated steam at a saturation temperature of Tsat = 95°F condenses on the outer surfaces of horizontal pipes which are maintained at 65°F by circulating cooling water. The rate of heat transfer to the cooling water and the rate of condensation per unit length of a single horizontal pipe are to be determined.

Assumptions 1 Steady operating conditions exist. 2 The pipe is isothermal. 3 There is no interference between the pipes (no drip of the condensate from one tube to another).

Properties The properties of water at the saturation temperature of 95°F are hfg = 1040 Btu/lbm and ρv = 0.0025 lbm/ft3. The properties of liquid water at the film temperature of =+= 2/)( sat sf TTT (95 + 65)/2 = 80°F are (Table A-9E),

FftBtu/h 352.0

FBtu/lbm 999.0/hft 03335.0/

hlbm/ft 075.2slbm/ft 10764.5

lbm/ft 22.62

2

4

3

°⋅⋅=

°⋅===

⋅=⋅×=

=−

l

pl

lll

l

l

k

cρµν

µ

ρ


Btu/lbm 1060=

F)65F)(95Btu/lbm 999.0(0.68+Btu/lbm 1040

)(68.0 sat*

°−°⋅×=

−+= splfgfg TTchh

Steam95°F

................... 65°F

Condensate flow

Noting that we have condensation on a horizontal tube, the heat transfer coefficient is determined from

FftBtu/h 1501

ft) F(0.8/12)6595)(hlbm/ft 075.2](s) 3600h/ 1[()FftBtu/h 352.0)(Btu/lbm 1060)(lbm/ft 0025.022.62)(lbm/ft 22.62)(ft/s 2.32(729.0

)()(

729.0

2

4/1

2

3332

4/1

sat

3*

horiz

°⋅⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅

°⋅⋅−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

DTTkhg

hhsl

lfgvll

µρρρ

The heat transfer surface area of the tube per unit length is

2ft 2094.0ft) ft)(1 12/8.0( === ππDLAs


Btu/h 9431=°−°⋅⋅=−= F)6595)(ft 2094.0)(FftBtu/h 1501()( 22sat ss TThAQ&


lbm/h 8.90===Btu/lbm 1060

Btu/h 9431*oncondensatifgh

Qm&

&


10-54

10-67E Saturated steam at a saturation temperature of Tsat = 95°F condenses on the outer surfaces of 20 horizontal pipes which are maintained at 65°F by circulating cooling water and arranged in a rectangular array of 4 pipes high and 5 pipes wide. The rate of heat transfer to the cooling water and the rate of condensation per unit length of the pipes are to be determined.

Assumptions 1 Steady operating conditions exist. 2 The pipes are isothermal.


FftBtu/h 352.0

FBtu/lbm 999.0/hft 03335.0/


lbm/ft 22.62

2

4

3

°⋅⋅=

°⋅===

⋅=⋅×=

=−

l

pl

lll

l

l

k

cρµν

µ

ρ


Btu/lbm 1060=

F)65F)(95Btu/lbm 999.0(0.68+Btu/lbm 1040

)(68.0 sat*

°−°⋅×=

−+= splfgfg TTchh

Steam95°F

................... 65°F

Condensate flow

Noting that we have condensation on a horizontal tube, the heat transfer coefficient is determined from

FftBtu/h 1501


)()(

729.0

2

4/1

2

3332

4/1

sat

3*

horiz

°⋅⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅

°⋅⋅−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

DTTkhg

hhsl

lfgvll

µ

ρρρ


FftBtu/h 1061F)ftBtu/h 1501(4

11 224/1 tube1 horiz,4/1 tubesN horiz, °⋅⋅=°⋅⋅== h

Nh

The surface area for all 32 pipes per unit length of the pipes is

2total ft 6.702= ft) ft)(1 12/8.0(32ππ == DLNAs


Btu/h 213,300=°−°⋅=−= F)6595)(ft 702.6)(FBtu/h.ft 1061()( 22sat ss TThAQ&


lbm/h 201.3===Btu/lbm 1060

Btu/h 300,213*oncondensatifgh

Qm&

&


10-55

10-68 Saturated refrigerant-134a vapor at a saturation temperature of Tsat = 30°C condenses inside a horizontal tube which is maintained at 20°C. The fraction of the refrigerant that will condense at the end of the tube is to be determined.

Assumptions 1 Steady operating conditions exist. 2 The tube is isothermal. 3 The vapor velocity is low so that Revapor < 35,000.

Properties The properties of refrigerant-134a at the saturation temperature of 30°C are hfg = 173.1×103 J/kg and ρv = 37.53 kg/m3. The properties of liquid R-134a at the film temperature of =+= 2/)( sat sf TTT (30 + 20)/2 = 25°C are (Table A-10)

C W/m.08325.0

CJ/kg. 1427/sm101667.0/

kg/m.s 102.012

kg/m 1207

26

4

3

°=

°=×==

×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρR-134a 30°C 20°C

Dtube = 0.6 cmLtube = 5 m

Analysis The heat transfer coefficient for condensation inside horizontal tubes is

Condensate

C W/m2.509

C0)2C)(30J/kg 1427(83+J/kg 101.173

C0)2s)(30kg/m 10012.2(C) W/m08325.0)(kg/m )53.371207)(kg/m 1207)(m/s 81.9(

555.0

)(83

)()(

555.0

2

4/13

4

3332

4/1

satsat

3

internal

°⋅=

⎥⎦

⎤⎟⎠⎞

⎜⎝⎛ °−°⋅××

⎢⎢⎣

⎡

°−⋅×

°⋅−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+

−−

==

−

splfgsl

lvll TTchTT

kghh

µρρρ

The heat transfer surface area of the pipe is

2m 09425.0m) m)(5 006.0( === ππDLAs


W9.479C)2030)(m 09425.0)(C W/m2.509()( 22sat =°−°⋅=−= ss TThAQ&

The modified latent heat of vaporization in this case is, as indicated in the h relation,

J/kg 10178.5C0)2C)(30J/kg 1427(83+J/kg 101.173)(

83 33

sat* ×=°−°⋅×=−+= splfgfg TTchh

Then the rate of condensation becomes

kg/min 0.1613=kg/s 0.002689J/kg 10178.5

J/s 9.4793*oncondensati =

×==

fghQm&

&

Therefore, the fraction of the refrigerant that will condense at the end of the tube is

6.45%)(or 0.0645===kg/min 5.2

kg/min 0.1613condensedFraction total

condensed

mm&

&

Discussions Note that we used the modified hfg* instead of just hfg in heat transfer calculations to account for heat transfer

due to the cooling of the condensate below the saturation temperature.


10-56

10-69 Saturated refrigerant-134a vapor condenses inside a horizontal tube maintained at a uniform temperature. The fraction of the refrigerant that will condense at the end of the tube is to be determined.


Properties The properties of refrigerant-134a at the saturation temperature of 30°C are hfg = 173.1×103 J/kg and ρv = 37.53 kg/m3. The properties of liquid R-134a at the film temperature of (30 + 20)/2 = 25°C are (Table A-10) =+= 2/)( sat sf TTT

C W/m.08325.0

CJ/kg. 1427/sm101667.0/

kg/m.s 102.012

kg/m 1207

26

4

3

°=

°=×==

×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρR-134a 30°C 20°C

Dtube=0.6 cmLtube = 10 m

Analysis The heat transfer coefficient for condensation inside horizontal tubes is

Condensate

C W/m2.509

C0)2C)(30J/kg 1427(83+J/kg 101.173

C0)2s)(30kg/m 10012.2(C) W/m08325.0)(kg/m )53.371207)(kg/m 1207)(m/s 81.9(

555.0

)(83

)()(

555.0

2

4/13

4

3332

4/1

satsat

3

internal

°⋅=

⎥⎦

⎤⎟⎠⎞

⎜⎝⎛ °−°⋅××

⎢⎢⎣

⎡

°−⋅×

°⋅−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+

−−

==

−

splfgsl

lvll TTchTT

kghh

µρρρ


2m 1885.0m) m)(10 006.0( === ππDLAs


W8.959C)2030)(m 1885.0)(C W/m2.509()( 22sat =°−°⋅=−= ss TThAQ&

The modified latent heat of vaporization in this case is, as indicated in the h relation,

J/kg 10178.5C0)2C)(30J/kg 1427(83+J/kg 101.173)(

83 33

sat* ×=°−°⋅×=−+= splfgfg TTchh


kg/min 0.3226=kg/s 0.005377J/kg 10178.5

J/s 8.9593*oncondensati =

×==

fghQm&

&

Therefore, the fraction of the refrigerant that will condense at the end of the tube is

12.9%)(or 0.129===kg/min 5.2

kg/min 0.3226condensedFraction total

condensed

mm&

&


10-57

10-70 Prob. 10-68 is reconsidered. The the fraction of the refrigerant condensed at the end of the tube as a function of the temperature of the saturated R-134a vapor is to be plotted.


"GIVEN" T_sat=30 [C] L=5 [m] D=0.006 [m] T_s=20 [C] m_dot_total=2.5 [kg/min] "PROPERTIES" Fluid$='R134a' T_f=1/2*(T_sat+T_s) h_f=enthalpy(Fluid$, T=T_sat, x=0) h_g=enthalpy(Fluid$, T=T_sat, x=1) h_fg=(h_g-h_f)*Convert(kJ/kg, J/kg) rho_v=density(Fluid$, T=T_sat, x=1) rho_l=density(Fluid$, T=T_f, x=0) mu_l=Viscosity(Fluid$,T=T_f, x=0) c_l=CP(Fluid$, T=T_f, x=0)*Convert(kJ/kg-C, J/kg-C) k_l=Conductivity(Fluid$, T=T_f, P=P_sat+1) P_sat=pressure(Fluid$, T=T_sat, x=0) g=9.81 [m/s^2] "ANALYSIS" h=0.555*((g*rho_l*(rho_l-rho_v)*k_l^3)/(mu_l*(T_sat-T_s))*(h_fg+3/8*c_l*(T_sat-T_s)))^0.25 Q_dot=h*A*(T_sat-T_s) A=pi*D*L h_fg_star=h_fg+3/8*c_l*(T_sat-T_s) m_dot_cond=Q_dot/h_fg_star*Convert(kg/s, kg/min) f_condensed=m_dot_cond/m_dot_total*Convert(,%)

Tsat [C]

fcondensed [%]

25 26.25 27.5 28.75 30 31.25 32.5 33.75 35 36.25 37.5 38.75 40 41.25 42.5 43.75 45 46.25 47.5 48.75 50

3.871 4.577 5.249 5.893 6.516 7.12 7.707 8.282 8.843 9.395 9.936 10.47 10.99 11.51 12.02 12.53 13.03 13.53 14.02 14.51 15

25 30 35 40 45 502

4

6

8

10

12

14

16

Tsat [C]

f co

nd

en

se

d [

%]


10-58

10-71 Saturated steam at a saturation temperature of Tsat = 100°C condenses on the outer surfaces of a tube bank maintained at 80°C. The rate of condensation of steam on the tubes are to be determined.


Properties The properties of water at the saturation temperature of 100°C are hfg = 2257×103 J/kg and ρv = 0.5978 kg/m3 (Table A-9). The properties of liquid water at the film temperature of =+= 2/)( sat sf TTT (100 + 80)/2 = 90°C are (Table A-9),

C W/m675.0

CJ/kg 4206skg/m10315.0

kg/m 3.9653

3

°⋅=

°⋅=

⋅×=

=−

l

pl

l

l

k

cµ

ρ Steam 100°C

80°C

Condensate flow

4 × 4 array of tubes


J/kg 102314=

C80)C(100J/kg 42060.68+J/kg 102357

)(68.0

3

3

sat*

×

°−°⋅××=

−+= splfgfg TTchh


C W/m8736

m) C(0.05)80100(s)kg/m 10315.0()C W/m675.0)(J/kg 102314)(kg/m 5978.03.965)(kg/m 3.965)(m/s 8.9(

729.0

)()(

729.0

2

4/1

3

33332

4/1

sat

3*

horizontal

°⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µ

ρρρ


C W/m6177C) W/m8736(4

11 224/1 tube1 horiz,4/1 tubesN horiz, °⋅=°⋅== h

Nh






kg/h 966==×

== kg/s 0.2684J/kg 102314J/s 000,621

3*oncondensatifgh

Qm

&&


10-59

10-72 Saturated ammonia at a saturation temperature of Tsat = 30°C condenses on vertical plates which are maintained at 10°C. The average heat transfer coefficient and the rate of condensation of ammonia are to be determined.


Properties The properties of ammonia at the saturation temperature of 30°C are hfg = 1144×103 J/kg and ρv = 9.055 kg/m3. The properties of liquid ammonia at the film temperature of (30 + 10)/2 = 20°C are (Table A-11), =+= 2/)( sat sf TTT

C W/m4927.0

CJ/kg 4745/sm10489.2/

skg/m10519.1

kg/m 2.610

27

4

3

°⋅=

°⋅=×==

⋅×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ

m&

10°C

25 cm Ammonia 30°C

Analysis The modified latent heat of vaporization is 10 cm

J/kg 101209=

C10)C(30J/kg 47450.68+J/kg 101144

)(68.0

3

3

sat*

×

°−°⋅××=

−+= splfgfg TTchh


0.307)s/m 10489.2(

m/s 8.9)J/kg 101209)(skg/m 10519.1(

C)1030(C) W/m4927.0(m) 1.0(70.381.4

)(70.381.4ReRe

82.03/1

227

2

34

820.03/1

2*sat

wavyvertical,

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

××⋅×

°−×°⋅××+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+==

−−

lfgl

sl gh

TTLkνµ


C W/m7032 2 °⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

×−

°⋅×=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−==

−

3/1

227

2

22.1

3/1

222.1wavyvertical,

)/sm 10489.2(m/s 8.9

2.5)307(08.1C) W/m4927.0(307

2.5Re08.1Re

l

l gkhh

ν

The total heat transfer surface area of the plates is

2m 75.0m) m)(0.10 25.0(30 ==×= LWAs




kg/s 0.0872=×

==J/kg 101209J/s 480,105

3*oncondensatifgh

Qm

&&


10-60

10-73 Saturated water vapor at a pressure of 12.4 kPa condenses on a rectangular array of 100 horizontal tubes at 30°C. The condensation rate per unit length of is to be determined.

Assumptions 1 Steady operating condition exists. 2 The tube surfaces are isothermal.

Properties The properties of water at the saturation temperature of 50°C corresponding to 12.4 kPa are hfg = 2383 kJ/kg and ρv = 0.0831 kg/m3 (Table A-9). The properties of liquid water at the film temperature of Tf = (Tsat + Ts)/2 = 40°C are, from Table A-9,

ρl = 992.1 kg/m3 cpl = 4179 J/kg·K

µl = 0.653 × 10−3 kg/m·s kl = 0.631 W/m·K


J/kg 102440

)3050)(4179(68.0102383

)(68.0

3

3

sat

×=

−+×=



KW/m250,11

)008.0)(3050)(10653.0()631.0)(102440)(0831.01.992)(1.992)(81.9(729.0

)()(

729.0

2

4/1

3

33

4/1

sat

3

tube1 horiz,

⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−××−

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=

−

∗

DTTkhg

hsl

lfgvll

µρρρ


KW/m7523)KW/m250,11(5

11 224/1 tube1 horiz,4/1 tubes horiz, ⋅=⋅=== h

Nhh N

The rate of heat transfer per unit length during this condensation process becomes

W/m107813

K )3050)(K W/m7523)(m 008.0()100(

)(/

5

2sattotal

×=

−⋅=

−=

.

TTDhNLQ s

π

π&

The rate of condensation per unit length is

mkg/s 0.155 ⋅=×

×==

∗ J/kg 102440W/m10781.3/

3

5oncondensati

hLQ

Lm

fg

&&

Discussion Therefore, water vapor condenses at a rate of 155 g/s per meter length of the tubes.


10-61

10-74 Saturated water vapor at a pressure of 12.4 kPa condenses on an array of 100 horizontal tubes at 30°C. The condensation rates for (a) a rectangular array of 5 tubes high and 20 tubes wide and (b) a square array of 10 tubes high and 10 tubes wide are to be determined. Assumptions 1 Steady operating condition exists. 2 The tube surfaces are isothermal. Properties The properties of water at the saturation temperature of 50°C corresponding to 12.4 kPa are hfg = 2383 kJ/kg and ρv = 0.0831 kg/m3 (Table A-9). The properties of liquid water at the film temperature of Tf = (Tsat + Ts)/2 = 40°C are, from Table A-9, ρl = 992.1 kg/m3 cpl = 4179 J/kg·K µl = 0.653 × 10−3 kg/m·s kl = 0.631 W/m·K Analysis The modified latent heat of vaporization is

J/kg 102440

)3050)(4179(68.0102383

)(68.0

3

3

sat

×=

−+×=



KW/m250,11

)008.0)(3050)(10653.0()631.0)(102440)(0831.01.992)(1.992)(81.9(729.0

)()(

729.0

2

4/1

3

33

4/1

sat

3

tube1 horiz,

⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−××−

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=

−

∗

DTTkhg

hsl

lfgvll

µρρρ

(a) For a rectangular array, the average heat transfer coefficient for a 5-tube high vertical tier becomes

KW/m7523)KW/m250,11(5


Nhh N

The rate of heat transfer during this condensation process becomes

W107813K )3050)(K W/m7523)(m 1)(m 008.0()100()( 52sattotal ×=−⋅=−= .TTDLhNQ s ππ&

The rate of condensation is

kg/s 0.155=×

×==

∗ J/kg 102440W10781.3

3

5

oncondensati

hQmfg

&& (rectangular array)

(b) For a square array, the average heat transfer coefficient for a 10-tube high vertical tier becomes

KW/m6326)KW/m250,11(10


Nhh N


W101803K )3050)(K W/m6326)(m 1)(m 008.0()100()( 52sattotal ×=−⋅=−= .TTDLhNQ s ππ&

The rate of condensation is

kg/s 0.130=×

×==

∗ J/kg 102440W10180.3

3

5

oncondensati

hQmfg

&& (square array)

Discussion The condensation rate of the rectangular array tube bank is about 20% higher than that of the square array tube bank:

19.1kg/s 130.0kg/s 155.0

)square()rrectangula(

oncondensati

oncondensati ==m

m&

&


10-62

Special Topic: Non-Boiling Two-Phase Flow Heat Transfer

10-75 The flow quality of a non-boiling two-phase flow in a tube with is to be determined. 300/ =gl mm &&

Assumptions 1 Steady operating condition exists. 2 Two-phase flow is non-boiling and it does not involve phase change. 3 Fluid properties are constant.

Analysis The flow quality is given as

gl

g

mmm

x&&

&

+=

Hence, the equation can be rearranged as

1/

1/)(

/+

=+

=+

=glggl

gg

gl

g

mmmmmmm

mmm

x&&&&&

&&

&&

&

Thus, the flow quality is

0.00332=+

=1300

1x

Discussion The flow quality is a dimensionless parameter.

10-76 The flow quality and the mass flow rates of the gas and the liquid for a non-boiling two-phase flow, where Vsl = 3Vsg, are to be determined.


Properties The densities of the gas and liquid are given to be ρg = 8.5 kg/m3 and ρl = 855 kg/m3, respectively.

Analysis The mass flow rate of gas can be calculated using

kg/s 0.0556==

==

4m) 1020()m/s 8.0)(kg/m 5.8(

42

3

2

.

DVAVm sggcsggg

π

πρρ&

Then, the mass flow rate of liquid is (with Vsl = 3Vsg)

kg/s 16.8==

==

4m) 1020()m/s 8.0)(kg/m 855(3

43

23

2

.

DVAVm sglcslll

π

πρρ&

Thus, the flow quality is

0.00330=+

=+

=0556.08.16

0556.0

gl

g

mmm

x&&

&

Discussion The total mass flow rate of gas and liquid for this two-phase flow is simply

kg/s 86.16kg/s 0556.0kg/s 8.16tot =+=+= gl mmm &&&


10-63

10-77 The mass flow rate of air and the superficial velocities of air and engine oil for a non-boiling two-phase flow in a tube are to be determined.


Properties The densities of air and engine oil at the bulk mean temperature Tb = 140°C are ρg = 0.8542 kg/m3 (Table A-15) and ρl = 816.8 kg/m3 (Table A-13), respectively.

Analysis The flow quality is given as

1/

1/)(

/+

=+

=+

=glggl

gg

gl

g

mmmmmmm

mmm

x&&&&&

&&

&&

&

or

xm

m

g

l 11=+&

& →

xx

mm

g

l −=

1&

&

With known liquid (engine oil) mass flow rate and flow quality, the gas (air) mass flow rate is determined using

kg/s0.00189=×−

×=

−=

−

−

)kg/s 9.0(101.21

101.21 3

3

lg mx

xm &&

From the gas and liquid mass flow rates, the superficial gas and liquid velocities can be calculated:

m/s 4.51==== 232 )m 025.0()kg/m 8542.0(kg/s) 00189.0(44

ππρρ D

mA

mV

g

g

g

gsg

&&

m/s 2.25==== 232 )m 025.0()kg/m 8.816()kg/s 9.0(44

ππρρ Dm

Am

Vl

l

l

lsl

&&

Discussion The superficial velocity of air is twice that of the engine oil.


10-64

10-78 Starting with the two-phase non-boiling heat transfer correlation, the expression that is appropriate for the case when only water is flowing in the tube is to be determined.


Analysis The two-phase non-boiling heat transfer correlation is given as

( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎠⎞

⎜⎝⎛

−+=

25.0*25.025.04.01.0 1

155.01 I

PrPr

FF

xxFhh

g

l

l

g

p

ppltp µ

µ

where

22

1

)()(

tan2)1(⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−+−= −

gl

lggp gD

VVF

ρρρ

παα and

gl

g

mmm

x&&

&

+=

For the situation when the air flow is shut off and only water is flowing in the pipe, we have and 0=gm& 0=α . Hence, we get

and 10)01( =+−=pF 0=x

Thus, the two-phase non-boiling heat transfer correlation becomes

( ) [ ] llg

l

l

glαtp hhI

PrPr

hh =+=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛

−+== 01)1(

111

01055.01)1(

25.0*25.025.04.01.0

0, µµ

The liquid phase heat transfer coefficient is calculated using:

14.0

3/15/4 PrRe027.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=

s

lllll D

kh

µµ

where the in situ liquid Reynolds number is

D

mD

mD

m

l

l

l

l

l

ll πµµπµαπ

&&& 401

414

Re =−

=−

=

Therefore, we have

14.0

3/15/40, PrRe027.0 ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛===

s

lllllαtp D

khh

µµ

Discussion When only water is flowing in the tube, the two-phase non-boiling heat transfer correlation is reduced to a familiar equation for internal forced convection.


10-65

10-79 Air-water slug flows through a 25.4-mm diameter horizontal tube in microgravity condition. Using the non-boiling two-phase heat transfer correlation, the two-phase heat transfer coefficient (htp) is to be determined Assumptions 1 Steady operating condition exists. 2 Two-phase flow is non-boiling and it does not involve phase change. 3 Fluid properties are constant. Properties The properties of water (liquid) are given to be µl = 85.5×10-5 kg/m·s, µs = 73.9×10-5 kg/m·s, ρl = 997 kg/m3, kl = 0.613 W/m·K, and Prl = 5.0. The properties of air (gas) are given to be µg = 18.5×10-6 kg/m·s, ρg = 1.16 kg/m3, and Prg = 0.71. Analysis From the superficial gas and liquid velocities, and void fraction, the gas and liquid velocities can be calculated as

m/s 11.127.0m/s 3.0

===αsg

gV

V

m/s 745.027.01m/s 544.0

1=

−=

−=

αsl

lV

V

The gas and liquid mass flow rates are calculated as

kg/s 1076.14

m) 02540()m/s 3.0)(kg/m 16.1(4

42

32

−×====.DVAVm sggcsggg ππρρ&

kg/s 275.04

m) 02540()m/s 544.0)(kg/m 997(4

23

2====

.DVAVm sllcslll ππρρ&

Using the gas and liquid mass flow rates, the quality is determined to be

44

41040.6

1076.1275.01076.1 −

−

−

×=×+

×=

+=

gl

g

mmm

x&&

&

The flow pattern factor (Fp) can be calculated using

730.0)kg/m 16.1kg/m 997()m 0254.0)(m/s 81.9(

)m/s 745.0m/s 11.1)(kg/m 16.1(tan2)27.0()27.01(

)()(

tan2)1(

2

332

231

22

1

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−+−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−+−=

−

−

π

ρρ

ρ

παα

gl

lggp gD

VVF


K W/m2995skg/m109.73skg/m105.85

m 0254.0K W/m613.0)0.5()18870(027.0

PrRe027.0

214.0

5

53/15/4

14.03/15/4

⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅×

⋅×⎟⎠⎞

⎜⎝⎛ ⋅

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

−

s

lllll D

kh

µµ


18870)m 0254.0)(skg/m 105.85(27.01

)kg/s 275.0(414

Re5

=⋅×−

=−

=−πµαπ D

m

l

ll

&

The inclination factor (I *) has a value of one for horizontal tube (θ = 0). Thus, using the general two-phase heat transfer correlation, the value for htp is estimated to be

( )

9369.0skg/m105.18skg/m105.85

0.571.0

730.0730.01

1040.611040.655.01)730.0(

11

55.01

25.0

6

525.04.01.0

4

4

25.0*25.025.04.01.0

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅×⋅×

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛

×−×

+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎠⎞

⎜⎝⎛−

+=

−

−

−

−

IPrPr

FF

xxF

hh

g

l

l

g

p

pp

l

tp

µµ

or K W/m2810 2 ⋅=⋅== )K W/m2995(9369.09369.0 2

ltp hh

Discussion The non-boiling two-phase heat transfer coefficient (htp) is about 7% lower than the liquid phase heat transfer coefficient (hl).


10-66

10-80 Air-water mixture is flowing in a 5° inclined tube with diameter of 25.4 mm, and the mixture superficial gas and liquid velocities are 1 m/s and 2 m/s, respectively. The two-phase heat transfer coefficient (htp) is to be determined. Assumptions 1 Steady operating condition exists. 2 Two-phase flow is non-boiling and it does not involve phase change. 3 Fluid properties are constant. Properties The properties of water (liquid) at bulk mean temperature Tb = (Ti + Te)/2 = 45°C are, from Table A-9, µl = 0.596×10-3 kg/m·s, ρl = 990.1 kg/m3, kl = 0.637 W/m·K, and Prl = 3.91. The properties of air (gas) at bulk mean temperature Tb = 45°C are, from Table A-15, µg = 1.941×10-5 kg/m·s, ρg = 1.109 kg/m3, and Prg = 0.7241. Also, at Ts = 80°C we get µs = 0.355×10-3 kg/m·s from Table A-9. Analysis From the superficial gas and liquid velocities, and void fraction, the gas and liquid velocities can be calculated as

m/s 030.333.0

m/s 1===

αsg

gV

V m/s 985.233.01

m/s 21

=−

=−

=α

sll

VV


kg/s 10619.54

m) 02540()m/s 1)(kg/m 109.1(4

42

32


kg/s 003.14

m) 02540()m/s 2)(kg/m 1.990(4

23

2====



44

410599.5

10619.5003.110619.5 −

−

−

×=×+

×=

+=

gl

g

mmm

x&&

&


670.0)kg/m 109.1kg/m 1.990()m 0254.0)(m/s 81.9(

)m/s 985.2m/s 03.3)(kg/m 109.1(tan2)33.0()33.01(

)()(

tan2)1(

2

332

231

22

1

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−+−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−+−=

−

−

π

ρρ

ρ

παα

gl

lggp gD

VVF

The inclination factor (I *) for θ = 5° is calculated to be

023.95sinN/m 068.0

)m 0254.0)(m/s 81.9)(kg/m 109.1kg/m 1.990(1sin

)(1

22332

=°−

+=−

+=∗ θσ

ρρ gDI gl


K W/m11754skg/m10355.0skg/m10596.0

m 0254.0K W/m637.0)91.3()103100(027.0

PrRe027.0

214.0

3

33/15/4

14.03/15/4

⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅×

⋅×⎟⎠⎞

⎜⎝⎛ ⋅

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

−

s

lllll D

kh

µµ


103100)m 0254.0)(skg/m 10596.0(33.01

)kg/s 003.1(414

Re3

=⋅×−

=−

=−πµαπ D

m

l

ll

&

Thus, using the general two-phase heat transfer correlation, the value for htp is estimated to be

( )

021.1)023.9(941.1

6.5991.3

7241.0670.0

670.0110599.51

10599.555.01)670.0(

11

55.01

25.025.025.04.01.0

4

4

25.0*25.025.04.01.0

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛

×−×

+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎠⎞

⎜⎝⎛−

+=

−

−

IPrPr

FF

xxF

hh

g

l

l

g

p

pp

l

tp

µµ

or K W/m12,000 2 ⋅=⋅== )K W/m11754(021.1021.1 2ltp hh

Discussion The inclination factor is I* = 1 when the tube is at horizontal position, since sin(0°) = 0. The two-phase heat transfer coefficient for horizontal tube would be htp = 10300 W/m2·K, which is about 14% lower than that of 5° inclined tube.


10-67

10-81 Mixture of petroleum and natural gas is being transported in a pipeline that is located in a terrain that caused it to have an average inclination angle of 10°. The two-phase heat transfer coefficient is to be determined. Assumptions 1 Steady operating condition exists. 2 Two-phase flow is non-boiling and it does not involve phase change. 3 Fluid properties are constant. Properties The properties of petroleum (liquid) are given to be µl = 297.5×10-4 kg/m·s, µs = 238×10-4 kg/m·s, ρl = 853 kg/m3, kl = 0.163 W/m·K, σ = 0.020 N/m, and Prl = 405. The properties of natural gas are given to be µg = 9.225×10-6 kg/m·s, ρg = 9.0 kg/m3, and Prg = 0.80. Analysis From the gas and liquid mass flow rates, the superficial gas and liquid velocities can be calculated:

m/s 748.0)m 102.0()kg/m 0.9(

kg/s) 055.0(44232 ====

ππρρ D

mA

mV

g

g

g

gsg

&&

m/s 296.2)m 102.0()kg/m 853(

)kg/s 16(44232 ====

ππρρ Dm

Am

Vl

l

l

lsl

&&

Using the superficial velocities and void fraction, the gas and liquid velocities are found to be

m/s 400.322.0

m/s 748.0===

αsg

gV

V m/s 944.222.01m/s 296.2

1=

−=

−=

αsl

lV

V


310426.3055.016

055.0 −×=+

=+

=gl

g

mmm

x&&

&


7802.0)kg/m 0.9kg/m 853)(m 102.0)(m/s 81.9(

)m/s 944.2m/s 40.3)(kg/m 0.9(tan2)22.0()22.01(

)()(

tan2)1(

2

332

231

22

1

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−+−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−+−=

−

−

π

ρρ

ρ

παα

gl

lggp gD

VVF


K W/m1.419skg/m10238skg/m105.297

m 102.0K W/m163.0)405()7601(027.0

PrRe027.0

214.0

4

43/15/4

14.03/15/4

⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅×

⋅×⎟⎠⎞

⎜⎝⎛ ⋅

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

−

s

lllll D

kh

µµ


7601)m 102.0)(skg/m 105.297(22.01

)kg/s 16(414

Re4

=⋅×−

=−

=−πµαπ D

m

l

ll

&

The inclination factor (I *) for θ = 10° is calculated to be

9.74810sinN/m .0200

)m 102.0)(m/s 81.9)(kg/m 0.9kg/m 853(1sin

)(1

22332

=°−

+=−

+=∗ θσ

ρρ gDI gl


( )

999.1)9.748(225.9

29750405

80.07802.0

7802.0110426.31

10426.355.01)7802.0(

11

55.01

25.025.025.04.01.0

3

3

25.0*25.025.04.01.0

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛

×−×

+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎠⎞

⎜⎝⎛−

+=

−

−

IPrPr

FF

xxF

hh

g

l

l

g

p

pp

l

tp

µµ

or K W/m838 2 ⋅=⋅== )K W/m1.419(999.1999.1 2ltp hh

Discussion Since Vg > Vl, there will be slippage between the gas and liquid phases. When Vg ≠ Vl, slippage between the gas and liquid phases exists. When Vg = Vl, slippage between the gas and liquid phases is negligible, and the flow is called homogeneous two-phase flow.


10-68

10-82 Air-water mixture is flowing in a tube with diameter of 25.4 mm, and the mixture superficial gas and liquid velocities are 1 m/s and 2 m/s, respectively. The two-phase heat transfer coefficient (htp) for (a) horizontal tubee (θ = 0°) and (b) vertical tube (θ = 90°), are to be determined and compared.


Properties The properties of water (liquid) at bulk mean temperature Tb = (Ti + Te)/2 = 45°C are, from Table A-9, µl = 0.596×10-3 kg/m·s, ρl = 990.1 kg/m3, kl = 0.637 W/m·K, and Prl = 3.91. The properties of air (gas) at bulk mean temperature Tb = 45°C are, from Table A-15, µg = 1.941×10-5 kg/m·s, ρg = 1.109 kg/m3, and Prg = 0.7241. Also, at Ts = 80°C we get µs = 0.355×10-3 kg/m·s from Table A-9.

Analysis From the superficial gas and liquid velocities, and void fraction, the gas and liquid velocities can be calculated as

m/s 030.333.0

m/s 1===

αsg

gV

V

m/s 985.233.01

m/s 21

=−

=−

=α

sll

VV


kg/s 10619.54

m) 02540()m/s 1)(kg/m 109.1(4

42

32


kg/s 003.14

m) 02540()m/s 2)(kg/m 1.990(4

23

2====



44

410599.5

10619.5003.110619.5 −

−

−

×=×+

×=

+=

gl

g

mmm

x&&

&


670.0

)kg/m 109.1kg/m 1.990()m 0254.0)(m/s 81.9()m/s 985.2m/s 03.3)(kg/m 109.1(tan2)33.0()33.01(

)()(

tan2)1(

2

332

231

22

1

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

+−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−+−=

−

−

π

ρρρ

παα

gl

lggp gD

VVF


K W/m11754

skg/m10355.0skg/m10596.0

m 0254.0K W/m637.0)91.3()103100(027.0

PrRe027.0

2

14.0

3

33/15/4

14.03/15/4

⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅×

⋅×⎟⎠⎞

⎜⎝⎛ ⋅

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=

−

−

s

lllll D

kh

µµ


103100)m 0254.0)(skg/m 10596.0(33.01

)kg/s 003.1(414

Re3

=⋅×−

=−

=−πµαπ D

m

l

ll

&


10-69

(a) The inclination factor (I *) for horizontal tube (θ = 0°) is . Thus, using the general two-phase heat transfer correlation, the value for h

1=∗Itp is estimated to be

( )

8727.0

941.16.59

91.37241.0

670.0670.01

10599.5110599.555.01)670.0(

11

55.01

25.025.04.01.0

4

4

25.0*25.025.04.01.0

horiz ,

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛

×−

×+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎠⎞

⎜⎝⎛

−+=

−

−

IPrPr

FF

xxF

hh

g

l

l

g

p

pp

l

tp

µµ

or

K W/m10,300 2 ⋅=⋅== )K W/m11754(8727.087271.0 2horiz , ltp hh

(b) The inclination factor (I *) for vertical tube (θ = 90°) is calculated to be

05.93

90sinN/m 068.0

)m 0254.0)(m/s 81.9)(kg/m 109.1kg/m 1.990(1

sin)(

1

2233

2

=

°−

+=

−+=∗ θ

σρρ gD

I gl


( )

( )

30.1

05.93941.1

6.5991.3

7241.0670.0

670.0110599.51

10599.555.01)670.0(

11

55.01

25.025.025.04.01.0

4

4

25.0*25.025.04.01.0

vert,

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛

×−

×+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎠⎞

⎜⎝⎛

−+=

−

−

IPrPr

FF

xxF

hh

g

l

l

g

p

pp

l

tp

µµ

or

K W/m15,300 2 ⋅=⋅== )K W/m11754(30.130.1 2 vert, ltp hh

Discussion The two-phase heat transfer coefficient of vertical pipe is about 49% higher than that of horizontal pipe:

49.1300,10300,15

horiz ,

vert, ==tp

tp

hh


10-70

Review Problems

10-83 The initial boiling heat transfer coefficient and the total heat transfer coefficient, when a heated steel rod was submerged in a water bath, are to be determined.

Assumptions 1 Steady operating condition exists. 2 The steel rod has uniform initial surface temperature.

Properties The properties of water at the saturation temperature of 100°C are hfg = 2257 kJ/kg (Table A-2) and ρl = 957.9 kg/m3 (Table A-9). The properties of vapor at the film temperature of Tf = (Tsat + Ts)/2 = 300°C are, from Table A-16,

ρv = 0.3831 kg/m3 cpv = 1997 J/kg·K

µv = 2.045 × 10−5 kg/m·s kv = 0.04345 W/m·K

Analysis The excess temperature in this case is ∆T = Ts − Tsat = 400°C, which is much larger than 30°C for water from Fig. 10-6. Therefore, film boiling will occur. The film boiling heat flux in this case can be determined from

24

4/1

5

33

sat

4/1

sat

sat3

filmfilm

W/m10476.6

)400()400)(02.0)(10045.2(

)]400)(1997(4.0102257)[3831.09.957)(3831.0()04345.0(81.962.0

)()(

)](4.0)[(

×=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

×

+×−=

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−+−=

−

TTTTD

TTchgkCq s

sv

spvfgvlvv

µ

ρρρ&

Using the Newton’s law of cooling, the boiling heat transfer coefficient is

→ )( satfilmfilm TThq s −=&sat

filmfilm TT

qh

s −=

&

K W/m162 2 ⋅=−

×=

K )100500( W/m10476.6 24

filmh

The radiation heat transfer coefficient can be determined using

→ )()( satrad4

sat4

rad TThTTq ss −=−= εσ&sat

4sat

4

rad)(

TTTT

hs

s

−−

=εσ

K W/m08.43K )100500(

K )373773)(K W/m1067.5)(9.0()( 2444428

sat

4sat

4

rad ⋅=−

−⋅×=

−−

=−

TTTT

hs

sεσ

Then, the total heat transfer coefficient can be determined using

radfilmtotal 43 qqq &&& += → )(

43)()( satradsatfilmsattotal TThTThTTh sss −+−=−

or

K W/m194 2 ⋅=

⋅+⋅=

+=

)K W/m08.43(43K W/m162

43

22

radfilmtotal hhh

Discussion The boiling heat transfer coefficient (hfilm) is 3.76 times the radiation heat transfer coefficient (hrad).


10-71

10-84 The boiling heat transfer coefficient and the total heat transfer coefficient for water being boiled by a cylindrical metal rod are to be determined.

Assumptions 1 Steady operating condition exists. 2 Heat losses from the boiler are negligible.


ρv = 0.3831 kg/m3

cpv = 1997 J/kg·K

µv = 2.045 × 10−5 kg/m·s

kv = 0.04345 W/m·K

Analysis The excess temperature in this case is ∆T = Ts − Tsat = 400°C, which is much larger than 30°C for water from Fig 10-6. Therefore, film boiling will occur. The film boiling heat flux in this case can be determined from

25

4/1

5

33

sat

4/1

sat

sat3

filmfilm

W/m10152.1

)400()400)(002.0)(10045.2(

)]400)(1997(4.0102257)[3831.09.957)(3831.0()04345.0(81.962.0

)()(

)](4.0)[(

×=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

×

+×−=

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−+−=

−

TTTTD

TTchgkCq s

sv

spvfgvlvv

µρρρ

&



filmfilm TT

qh

s −=

&

K W/m288 2 ⋅=−

×=

K )100500( W/m10152.1 25

filmh


→ )()( satrad4

sat4


4sat

4

rad)(

TTTT

hs

s

−−

=εσ

K W/m93.23K )100500(

K )373773)(K W/m1067.5)(5.0()( 2444428

sat

4sat

4

rad ⋅=−

−⋅×=

−−

=−

TTTT

hs

sεσ




or

K W/m306 2 ⋅=

⋅+⋅=

+=

)K W/m93.23(43K W/m288

43

22

radfilmtotal hhh

Discussion The boiling heat transfer coefficient (hfilm) is about 12 times the radiation heat transfer coefficient (hrad).


10-72

10-85 Saturated ammonia vapor is condensed as it flows through a tube. With a given vapor flow rate at the exit, the flow rate of the vapor at the inlet is to be determined.

Assumptions 1 Steady operating condition exists. 2 The tube surfaces are isothermal. 3 The Reynolds number of the vapor at the inlet is less than 35,000 (this assumption will be verified).

Properties The properties of ammonia at the saturation temperature of 25°C corresponding to 1003 kPa are hfg = 1166 kJ/kg, ρv = 7.809 kg/m3, and µv = 1.037 × 10−5 kg/m·s (Table A-11). The properties of liquid ammonia at the film temperature of Tf = (Tsat + Ts)/2 = 15°C are, from Table A-11,

ρl = 617.5 kg/m3

cpl = 4709 J/kg·K

µl = 1.606 × 10−4 kg/m·s

kl = 0.5042 W/m·K

Analysis The modified latent heat of vaporization for film condensation inside horizontal tube is

J/kg 101201

)525)(4709(83

101166

)(83

3

3

sat

×=

−+×=


Assuming Revapor < 35,000 and the heat transfer coefficient for film condensation inside a horizontal tube can be determined using

KW/m5091

)025.0)(525)(10606.1()101201()5042.0)(809.75.617)(5.617)(81.9(555.0

)()(

555.0

2

4/1

4

33

4/1

sat

3

internal

⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−×

×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

∗

DTThkg

hhsl

fglvll

µρρρ


W3998

K )525)(K W/m5091)(m 5.0)(m 025.0(

)(2

sat

=−⋅=

−=

π

π sTTDLhQ&

Then, the rate of condensation can be calculated as

kg/s 00333.0J/kg 101201

W39983oncondensati =

×==

∗

hQmfg

&&

Applying the conservation of mass, the flow rate of vapor leaving the tube can be determined as

oncondensatioutlet ,inlet , mmm vv &&& +=

kg/s 0.00533=+= kg/s 00333.0kg/s 002.0inlet ,vm&

Discussion The Reynolds number associated with film condensation inside a horizontal tube is

000,35200,26)skg/m 10037.1)(m 025.0(

)kg/s 00533.0(44Re 5

inlet ,

inletvapor <=

⋅×==⎟⎟

⎠

⎞⎜⎜⎝

⎛=

−πµπµρ

v

v

v

vv

DmDV &

Thus, the Revapor < 35,000 assumption is appropriate for this problem.


10-73

10-86 Saturated ammonia at a saturation temperature of Tsat = 25°C condenses on the outer surface of vertical tube which is maintained at 15°C by circulating cooling water. The rate of heat transfer to the coolant and the rate of condensation of ammonia are to be determined.

Assumptions 1 Steady operating conditions exist. 2 The tube is isothermal. 3 The tube can be treated as a vertical plate. 4 The condensate flow is turbulent over the entire tube (this assumption will be verified). 5 The density of vapor is much smaller than the density of liquid, lv ρρ << .

Properties The properties of ammonia at the saturation temperature of 25°C are hfg = 1166×103 J/kg and ρv = 7.809 kg/m3. The properties of liquid ammonia at the film temperature of (25 + 15)/2 = 20°C are (Table A-11), =+= 2/)( sat sf TTT

463.1PrC W/m4927.0

CJ/kg 4745/sm102489.0/

skg/m 10519.1

kg/m 2.610

26

4-

3

=°⋅=

°⋅=×==

⋅×=

=

−

l

l

pl

lll

l

l

k

cρµν

µ

ρ


J/kg 101198=C)15C(25J/kg 47450.68+J/kg 101166

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh

Ammonia25°C

Condensate Ltube = 2 m

D =3.2 cm

15°C

Assuming turbulent flow, the Reynolds number is determined from

2142

253)463.1(151)s/m 102489.0(

81.9J/kg) 1098kg/m.s)(11 10519.1(

)1525()463.1(4927.020690.0

253Pr151)(Pr0690.0

ReRe

3/4

5.03/1

22634

5.0

3/4

5.03/1

2*

5.0

turbvertical,

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+−⎟

⎟⎠

⎞⎜⎜⎝

⎛

×××

−××=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+−⎟

⎟⎠

⎞⎜⎜⎝

⎛−==

−−

lfgl

ssatl

vg

hTTLk

µ

which is greater than 1800, and thus our assumption of turbulent flow is verified. Then the condensation heat transfer coefficient is determined from

C W/m4873)/sm 102489.0(

m/s 81.9)2532142(463.1588750

C) W/m4927.0(2142

)253(RePr588750Re

23/1

226

2

75.05.0

3/1

275.05.0turbulentvertical,

°⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

×−×+

°⋅×=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+==

−−

−l

l gkhh

ν

The heat transfer surface area of the tube is . Then the rate of heat transfer during this condensation process becomes

2m 2011.0m) m)(2 032.0( === ππDLAs

W9800=°−°⋅=−= C)1525)(m 2011.0)(C W/m4873()( 22sat ss TThAQ&

(b) The rate of condensation of ammonia is determined from

kg/s 108.180 3-×=×

==J/kg 101198

J/s 98003*oncondensati

fghQ

m&

&


10-74

10-87 Water is boiled at Tsat = 120°C in a mechanically polished stainless steel pressure cooker whose inner surface temperature is maintained at Ts = 130°C. The time it will take for the tank to empty is to be determined.



44.1PrCJ/kg 4244N/m 0550.0

skg/m 10232.0kg/m 121.1J/kg 102203kg/m 4.94333

33

=°⋅==

⋅×==

×==−

l

pl

lv

fgl

c

h

σµρ

ρ

130°C

120°CWater

Also, 0.0130 and n = 1.0 for the boiling of water on a mechanically polished stainless steel surface (Table 10-3). Note that we expressed the properties in units specified under Eq. 10-2 in connection with their definitions in order to avoid unit manipulations.

=sfC

Heating

Analysis The excess temperature in this case is C10120130sat °=−=−=∆ TTT s which is relatively low (less than 30°C). Therefore, nucleate boiling will occur. The heat flux in this case can be determined from Rohsenow relation to be

2

3

3

1/233

3sat,

2/1

nucleate

W/m400,228

44.1)102203(0130.0)120130(4244

0550.01.121)-9.8(943.4

)10)(220310232.0(

Pr

)()(

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

×

−⎥⎦

⎤⎢⎣

⎡××=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −⎥⎦

⎤⎢⎣

⎡ −=

−

nlfgsf

slpvlfgl

hC

TTcghq

σρρ

µ&

The rate of heat transfer is

W7174) W/m400,228(m) 20.0(41 22

nucleate === πqAQ &&

The rate of evaporation is

kg/s 003256.0kJ/kg 102203

W71743evap =

×==

fghQm&

&

Noting that the tank is half-filled, the mass of the water and the time it will take for the tank to empty are

[ ] kg 446.4m) 30.0(4/m) 20.0()kg/m 4.943(21

21 23 =×== πρ Vlm

min 22.8===== s 1365kg/s 003256.0

kg 446.4

evapevap m

mmt&

&


10-75

10-88 Saturated ammonia vapor at a saturation temperature of Tsat = 25°C condenses on the outer surfaces of a tube bank in which cooling water flows. The rate of condensation of ammonia, the overall heat transfer coefficient, and the tube length are to be determined.

Assumptions 1 Steady operating conditions exist. 2 The tubes are isothermal. 3 The thermal resistance of the tube walls is negligible.

Properties The properties of ammonia at the saturation temperature of 25°C are hfg = 1166×103 J/kg and ρv = 7.809 kg/m3 (Table A-11). We assume that the tube temperature is 20°C. Then, the properties of liquid ammonia at the film temperature of (25 + 20)/2 = 22.5°C are (Table A-11) =+= 2/)( sat sf TTT

C W/m4869.0

CJ/kg 4765skg/m10479.1

kg/m 5.6064

3

°⋅=

°⋅=⋅×=

=−

l

pl

l

l

k

cµ

ρ

The water properties at the average temperature of (14+17)/2 = 15.5°C are (Table A-9)

7.98PrC W/m590.0

skg/m10124.1

CJ/kg 4185kg/m 0.999

3

3

=°⋅=

⋅×=

°⋅==

−

k

c p

µ

ρ


J/kg 101182=

C)20C(25J/kg 47650.68+J/kg 101166

)(68.0

3

3

sat*

×

°−°⋅××=

−+= splfgfg TTchh


C W/m9280

m) C(0.025)2025(s)kg/m 10479.1()C W/m4869.0)(J/kg 101182)(kg/m 809.7)(606.5kg/m 5.606)(m/s 8.9(

729.0

)()(

729.0

2

4/1

4

33332

4/1

sat

3*

horizontal

°⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µ

ρρρ


C W/m6562C) W/m9280(4

11 224/1 tube1 horiz,4/1 tubesN horiz, °⋅=°⋅=== h

Nhho

The rate of heat transfer in the condenser is

W10970.1)1417)(CJ/kg 4185)(kg/s 69.15()(

kg/s 69.15)m/s 2()m 025.0)(25.0()kg/m 999(16165

inout

23

×=−°⋅=−=

===

TTcmQ

Am

p

c

&&

& πρ V


kg/s 0.167=×

×==

J/kg 101182 W10970.1

3

5

*condfgh

Qm

&&


10-76

(b) For the calculation of the heat transfer coefficient on the inner surfaces of the tubes, we first determine the Reynolds number

440,44skg/m 101.124

)kg/m m)(999.0 m/s)(0.025 2(Re

3-

3=

⋅×==

µρVD

which is greater than 10,000. Therefore, the flow is turbulent. Assuming fully developed flow, the Nusselt number and the heat transfer coefficient are determined to be

9.275)98.7()440,44(023.0PrRe023.0 4.08.04.08.0 ===Nu

C W/m6511)9.275(m 0.025

C) W/m590.0( 2 °⋅=°⋅

== NuDkhi

Let us check if the assumed value for the rube temperature was reasonable

C3.20

)25)(6562()5.15)(6511(

tube

tubetube

°=−=−

∆=∆

TTT

ThTh ooii

which is sufficiently close to the assumed value of 20°C. Disregarding thermal resistance of the tube walls, the overall heat transfer coefficient is determined from

C W/m3268 2 °⋅=⎟⎠⎞

⎜⎝⎛ +=⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

−− 11

65621

6511111

oi hhU

(c) The tube length may be determined from

m 5.05=

⎥⎦⎤

⎢⎣⎡ +−⋅=×

∆=

L

L

TUAQ

)1714(2125m) (0.025C)(16) W/m3268( W10970.1 25 π

&


10-77

10-89 Steam at a saturation temperature of Tsat = 40°C condenses on the outside of a thin horizontal tube. Heat is transferred to the cooling water that enters the tube at 10°C and exits at 30°C. The rate of condensation of steam, the average overall heat transfer coefficient, and the tube length are to be determined. Assumptions 1 Steady operating conditions exist. 2 The tube can be taken to be isothermal at the bulk mean fluid temperature in the evaluation of the condensation heat transfer coefficient. 3 Liquid flow through the tube is fully developed. 4 The thickness and the thermal resistance of the tube is negligible. Properties The properties of water at the saturation temperature of 40°C are hfg = 2407×103 J/kg and ρv = 0.05 kg/m3. The properties of liquid water at the film temperature of =+= 2/)( sat sf TTT (40+20)/2

= 30°C and at the bulk fluid temperature of (10 + 30)/2 = 20°C are (Table A-9),

=+= 2/)( outin TTTb

5.42=PrC W/m615.0

CJ/kg 4178/sm10801.0/

skg/m10798.0

kg/m 0.996

26

3

3

°⋅=

°⋅=×==

⋅×=

=

°

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ

:C30At

7.01PrC W/m598.0

CJ/kg 4182/sm10004.1/

skg/m10002.1

kg/m 0.998

26

3

3

=°⋅=

°⋅=×==

⋅×=

=

°

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ

:C20At

Steam 40°C

10°C

Cooling water 30°C

Condensate

Analysis The mass flow rate of water and the rate of heat transfer to the water are

W118,000=C)10C)(30J/kg kg/s)(4182 411.1()(

kg/s 411.1]4/m) 03.0(m/s)[ )(2kg/m 998( 23water

°−°⋅=−=

===

inoutp

c

TTcmQ

VAm&&

& πρ

The modified latent heat of vaporization is J/kg 102464=C0)2C(40J/kg 41820.68+J/kg 102407)(68.0 33

sat* ×°−°⋅××=−+= splfgfg TTchh


C W/m7572m) C(0.03)2040(s)kg/m 10798.0(

)C W/m615.0)(J/kg 102464)(kg/m 05.0996)(kg/m 996)(m/s 8.9(729.0

)()(

729.0

24/1

3

33332

4/1

sat

3*

horizontal

°⋅=⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvllo µ

ρρρ

The average heat transfer coefficient for flow inside the tube is determined as follows:

C W/m6618m 0.03

332.0C) W/m598.0(Nu0.332)01.7()761,59(023.0PrRe023.0Nu

761,59101.004

m) m/s)(0.03 2(Re

2

4.08.04.08.0

6-avg

°⋅=×°⋅

==

===

=×

==

Dkh

DV

i

ν

Noting that the thermal resistance of the tube is negligible, the overall heat transfer coefficient becomes

C. W/m3531 2 °=+

=+

=7572/16618/1

1/1/1

1

oi hhU

The logarithmic mean temperature difference is:

C20.18)10/30ln(

1030)/ln(lm °=

−=

∆∆∆−∆

=∆oi

ei

TTTT

T

The tube length is determined from

m 19.5=°°⋅

=∆

=→∆=C)20.18)(m 03.0()C W/m3531(

W000,118)(

2lm

lmππ TDU

QLTUAQ s

&&

Note that the flow is turbulent, and thus the entry length in this case is 10D = 0.3 m is much shorter than the total tube length. This verifies our assumption of fully developed flow.


10-78

10-90 Water is boiled at Tsat = 100°C by a spherical platinum heating element immersed in water. The surface temperature is Ts = 350°C. The boiling heat transfer coefficient is to be determined.


Properties The properties of water at the saturation temperature of 100°C are (Table A-9)

3

3

kg/m 9.957

J/kg 102257

=

×=

l

fgh

ρ

100°CWater

350°CThe properties of water vapor at (350+100)/2 = 225°C are (Table A-16)

C W/m03581.0

CJ/kg 1951skg/m 10749.1

kg/m 444.05

3

°⋅=

°⋅=⋅×=

=−

v

pv

v

v

k

cµ

ρ

Analysis The film boiling occurs since the temperature difference between the surface and the fluid. The heat flux in this case can be determined from

[ ]

[ ]2

4/1

5

33

sat

4/1

sat

sat3

film

W/m062,32

)100350()100350)(08.0)(10749.1(

)100350)(1951(4.0102257)444.09.957)(444.0()03581.0)(81.9(67.0

)()(

)(4.0)(67.0

=

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−×−+×−

=

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−+−=

−

TTTTD

TTchgkq s

sv

spvfgvlvv

µρρρ

&

The boiling heat transfer coefficient is

C W/m128 2 ⋅=°−

=−

=⎯→⎯−=C)100350(

W/m062,32)(2

sat

filmsatfilm TT

qhTThq

ss

&&


10-79

10-91 There is film condensation on the outer surfaces of 8 horizontal tubes arranged in a horizontal or vertical tier. The ratio of the condensation rate for the cases of the tubes being arranged in a horizontal tier versus in a vertical tier is to be determined.

Assumptions Steady operating conditions exist.

Horizontal tier Analysis The heat transfer coefficients for the two cases are related to the heat transfer coefficient on a single horizontal tube by

Horizontal tier:

tube1 ,horizontal tubesN of tier horizontal hh =

Vertical tier

Vertical tier:


tubesN of tier verticalN

hh =

Therefore,

1.68=8=

/

Ratio

1/4

4/1


tube1 ,horizontal

tubesN of tier vertical

tubesN of tier horizontal

tubesN of tier vertical

tubesN of tier horizontal

N

Nh

hh

hm

m

=

=

=

=&

&


10-80

10-92E Saturated steam at a saturation pressure of 0.95 psia and thus at a saturation temperature of Tsat = 100°F (Table A-9E) condenses on the outer surfaces of 100 horizontal tubes which are maintained at 80°F by circulating cooling water and arranged in a 10 × 10 square array. The rate of heat transfer to the cooling water and the rate of condensation of steam are to be determined.


Properties The properties of water at the saturation temperature of 100°F are hfg = 1037 Btu/lbm and ρv = 0.00286 lbm/ft3. The properties of liquid water at the film temperature of

(100 + 80)/2 = 90°F are (Table A-9E), =+= 2/)( sat sf TTT

FftBtu/h 358.0

FBtu/lbm 999.0/hft 02965.0/


lbm/ft 12.62

2

4

3

°⋅⋅=

°⋅===

⋅=⋅×=

=−

l

pl

lll

l

l

k

cρµν

µ

ρ


Btu/lbm 1051=

F)80F)(100Btu/lbm 999.0(0.68+Btu/lbm 1037

)(68.0 sat*

°−°⋅×=

−+= splfgfg TTchh

L = 15 ft

n = 100 tubes 80°F

P = 0.95 psia

Saturated steam

Cooling water


FftBtu/h 1562

ft) F(1.2/12)80100)(hlbm/ft 842.1](s) 3600h/ 1[()FftBtu/h 358.0)(Btu/lbm 1051)(lbm/ft 00286.012.62)(lbm/ft 12.62)(ft/s 2.32(

729.0

)()(

729.0

2

4/1

2

3332

4/1

sat

3*

horiz

°⋅⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅

°⋅⋅−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

DTTkhg

hhsl

lfgvll

µ

ρρρ


FftBtu/h 3.878F)ftBtu/h 1562(10


Nh


2total ft 471.2= ft) ft)(15 12/2.1(100ππ == DLNAs


Btu/h 8,278,000=°−°⋅=−= F)80100)(ft 2.471)(FBtu/h.ft 3.878()( 22sat ss TThAQ&


lbm/h 7879===Btu/lbm 1051

Btu/h 000,278,8*oncondensatifgh

Qm&

&


10-81

10-93E Saturated steam at a saturation pressure of 0.95 psia and thus at a saturation temperature of Tsat = 100°F (Table A-9E) condenses on the outer surfaces of 100 horizontal tubes which are maintained at 80°F by circulating cooling water and arranged in a 10 × 10 square array. The rate of heat transfer to the cooling water and the rate of condensation of steam are to be determined.


Properties The properties of water at the saturation temperature of 100°F are hfg = 1037 Btu/lbm and ρv = 0.00286 lbm/ft3. The properties of liquid water at the film temperature of

(100 + 80)/2 = 90°F are (Table A-9E), =+= 2/)( sat sf TTT

FftBtu/h 358.0

FBtu/lbm 999.0/hft 02965.0/


lbm/ft 12.62

2

4

3

°⋅⋅=

°⋅===

⋅=⋅×=

=−

l

pl

lll

l

l

k

cρµν

µ

ρ


Btu/lbm 1051=

F)80F)(100Btu/lbm 999.0(0.68+Btu/lbm 1037

)(68.0 sat*

°−°⋅×=

−+= splfgfg TTchh

L = 15 ft

n = 100 tubes 80°F

P = 0.95 psia

Saturated steam

Cooling water


FftBtu/h 1375


)()(

729.0

2

4/1

2

3332

4/1

sat

3*

horiz

°⋅⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅

°⋅⋅−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

DTTkhg

hhsl

lfgvll

µρρρ


FftBtu/h 0.773F)ftBtu/h 1375(10


Nh


2total ft 785.4= ft) ft)(15 12/2(100ππ == DLNAs


Btu/h 12,142,000=°−°⋅=−= F)80100)(ft 4.785)(FBtu/h.ft 0.773()( 22sat ss TThAQ&


lbm/h 11,560===Btu/lbm 1051

Btu/h 000,142,12*oncondensatifgh

Qm&

&


10-82

10-94 Water is boiled at Tsat = 100°C by a chemically etched stainless steel electric heater whose surface temperature is maintained at Ts = 115°C. The rate of heat transfer to the water, the rate of evaporation of water, and the maximum rate of evaporation are to be determined.



Water, 100°C

115°C

Steam 100°C

75.1PrN/m 0589.0

kg/m 60.0

kg/m 9.9573

3

===

=

l

v

l

σρ

ρ

CJ/kg 4217m/skg 10282.0

J/kg 1022573

3

°⋅=⋅×=

×=−

pl

l

fg

c

h

µ

Also, 0.0130 and n = 1.0 for the boiling of water on a chemically etched stainless steel surface (Table 10-3). Note that we expressed the properties in units specified under Eq. 10-2 in connection with their definitions in order to avoid unit manipulations.

=sfC

Analysis (a) The excess temperature in this case is C15100115sat °=−=−=∆ TTT s which is relatively low (less than 30°C). Therefore, nucleate boiling will occur. The heat flux in this case can be determined from Rohsenow relation to be

2

3

3

1/233

3sat,

2/1

nucleate

W/m900,474

75.1)102257(0130.0)100115(4217

0589.00.60)9.8(957.9

)10)(225710282.0(

Pr

)()(

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

×

−⎥⎦

⎤⎢⎣

⎡ −××=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −⎥⎦

⎤⎢⎣

⎡ −=

−

nlfgsf

slpvlfgl

hC

TTcghq

σρρ

µ&

The surface area of the bottom of the heater is . 2m 005027.0m) m)(0.8 002.0( === ππDLAs

Then the rate of heat transfer during nucleate boiling becomes

W2387=== ) W/m900,474)(m 005027.0( 22nucleateboiling qAQ s &

&

The rate of evaporation of water is determined from

kg/h 3.81 = kg/s 101.058 3−×=×

==J/kg 102257

J/s 23873

boilingnevaporatio

fghQ

m&

&

(b) For a horizontal heating wire, the coefficient Ccr is determined from Table 10-4 to be

151.0)399.0(12.0*12.0

1.2 < 399.00589.0

60.09.957(8.9)001.0()(

*

25.025.0

2/12/1

===

=⎟⎠

⎞⎜⎝

⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

−−LC

gLL

cr

vl

σρρ

Then the maximum or critical heat flux is determined from

2kW/m 1280==

−×××=−=2

4/1234/12max

W/m1,280,000

)]60.09.957()6.0(8.90589.0)[102257(151.0)]([ vlvfgcr ghCq ρρρσ&


10-83

10-95E Steam at a saturation temperature of Tsat = 100°F condenses on a vertical plate which is maintained at 80°F. The rate of heat transfer to the plate and the rate of condensation of steam per ft width of the plate are to be determined.



FftBtu/h 358.0

FBtu/lbm 999.0/hft 02965.0/


lbm/ft 12.62

2

4

3

°⋅⋅=

°⋅===

⋅=⋅×=

=−

l

pl

lll

l

l

k

cρµν

µ

ρ

Condensate

80°F

Steam 100°F

Analysis The modified latent heat of vaporization is 4 ft

Btu/lbm 1051=

F)80F)(100Btu/lbm 999.0(0.68+Btu/lbm 1037

)(68.0 sat*

°−°⋅×=

−+= splfgfg TTchh


145h) 1(

s) 3600()h/ft 02965.0(

ft/s 2.32)Btu/lbm 1051)(hlbm/ft 842.1(

F)80100(F)ftBtu/h 358.0(ft) 4(70.381.4

)(70.381.4ReRe

82.03/1

2

2

22

2

820.03/1

2*sat

wavyvertical,

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅°−×°⋅⋅××

+=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+==

lfgl

sl gh

TTLkνµ


FftBtu/h 875h) 1(

s) 3600()h/ft 02965.0(

ft/s 2.322.5)145(08.1

F)ftBtu/h 358.0(145

2.5Re08.1Re

23/1

2

2

22

2

22.1

3/1

222.1wavyvertical,

°⋅⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

−

°⋅⋅×=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−==

l

l gkhh

ν

The heat transfer surface area of the plate is

2ft 4ft) ft)(1 4( ==×= LWAs


Btu/h 70,000=°−°⋅⋅=−= F)80100)(ft 4)(FftBtu/h 875()( 22sat ss TThAQ&


lbm/h 66.6===Btu/lbm 1051

Btu/h 000,70*oncondensatifgh

Qm&

&


10-84

10-96 Saturated refrigerant-134a vapor condenses on the outside of a horizontal tube maintained at a specified temperature. The rate of condensation of the refrigerant is to be determined.


Properties The properties of refrigerant-134a at the saturation temperature of 35°C are hfg = 168.2×103 J/kg and ρv = 43.41 kg/m3. The properties of liquid R-134a at the film temperature of (35 + 25)/2 = 30°C are (Table A-10), =+= 2/)( sat sf TTT

C W/m.0808.0

CJ/kg. 1448kg/m.s 10888.1

kg/m 11884

3

°=

°=×=

=−

l

pl

l

l

k

cµ

ρ R-134a 35°C 25°C

Dtube = 1.2 cm Ltube = 8 m


J/kg 10178.0=C)25C(35J/kg 14480.68+J/kg 102.168

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchhCondensate


C W/m1988

m) C(0.012)2535(s)kg/m 10888.1()C W/m0808.0)(J/kg 100.178)(kg/m 41.43)(1188kg/m 1188)(m/s 81.9(729.0

)()(

729.0

2

4/1

4

33332

4/1

sat

3*

horizontal

°⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µρρρ


2m 3016.0m) m)(8 012.0( === ππDLAs


W5996C)2535)(m 3016.0)(C W/m1988()( 22sat =°−°⋅=−= ss TThAQ&


kg/min 2.02==×

== kg/s 0.03368J/kg 10178.0

J/s 59963*oncondensati

fghQm&

&


10-85

10-97 Saturated refrigerant-134a vapor condenses on the outside of a horizontal tube maintained at a specified temperature. The rate of condensation of the refrigerant is to be determined.


Properties The properties of refrigerant-134a at the saturation temperature of 35°C are hfg = 168.2×103 J/kg and ρv = 43.41 kg/m3. The properties of liquid R-134a at the film temperature of =+= 2/)( sat sf TTT (35 + 25)/2 = 30°C are (Table A-10),

C W/m.0808.0

CJ/kg. 1448/sm101590.0/

kg/m.s 10888.1

kg/m 1188

26

4

3

°=

°=×==

×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ R-134a 35°C 25°C

Dtube = 3 cm Ltube = 8 m


Condensate

J/kg 10178.0=C)25C(35J/kg 14480.68+J/kg 102.168

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh


C W/m1581

m) C(0.03)2535(s)kg/m 10888.1()C W/m0808.0)(J/kg 100.178)(kg/m 41.43)(1188kg/m 1188)(m/s 81.9(729.0

)()(

729.0

2

4/1

4

33332

4/1

sat

3*

horizontal

°⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µρρρ


2m 7540.0m) m)(8 03.0( === ππDLAs




kg/min 4.02==×

== kg/s 0.06697J/kg 10178.0

J/s 920,113*oncondensati

fghQm&

&


10-86

10-98 Saturated steam at 270.1 kPa pressure and thus at a saturation temperature of Tsat = 130°C (Table A-9) condenses inside a horizontal tube which is maintained at 110°C. The average heat transfer coefficient and the rate of condensation of steam are to be determined.



C W/m.683.0

CJ/kg. 4244/sm10246.0/

kg/m.s10232.0

kg/m 4.943

26

3

3

°=

°=×==

×=

=

−

−

l

pl

lll

l

l

k

cρµν

µ

ρ

Analysis The condensation heat transfer coefficient is determined from

C W/m3345 2 °⋅=

⎥⎦

⎤⎟⎠⎞

⎜⎝⎛ °−°⋅××

⎢⎢⎣

⎡

°−⋅×

°⋅−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+

−−

==

−

4/13

3

3332

4/1

satsat

3

internal

C110)C)(130kJ/kg 4244(83+J/kg 102174

C110)s)(130kg/m 10232.0(C) W/m683.0)(kg/m )50.14.943)(kg/m 4.943)(m/s 8.9(555.0

)(83

)()(

555.0 splfgsl

lvll TTchTT

kghh

µρρρ

Steam 270.1 kPa 110°C

Dtube = 2.5 cm Ltube = 10 m

Condensate


2m 7854.0m) m)(10 025.0( === ππDLAs




kg/s 0.0242=×

==J/kg 102174

J/s 543,523oncondensati

fghQ

m&

&


10-87

10-99 Steam is generated by an electrical heater rod with a rating of 1.5 kW. The heater surface temperature and the rate of steam generation are to be determined.

Assumptions 1 Steady operating conditions exist. 2 The heater surfaces are isothermal.

Properties The latent of vaporization of water at 1 atm (101.3 kPa) (Tsat = 100°C) is hfg = 2257 kJ/kg (Table A-9).

Analysis The tube surface temperature is determined from

[ ]

C97.8 °=−−=

−−=

−=

s

ss

ss

ss

TTT

TTDLTTQ

TThAQ

)100(m) m)(0.25 0.015()100(56.5 W1500

))(()(56.5

)(

3sat

3sat

sat

π

π&

&


kg/h 2.39=×=×

== − kg/s 10646.6J/kg 102257

J/s 1500 43oncondensati

fghQm&

&

10-100 Saturated steam is generated at 100°C in a horizontal heat exchanger. The tube surface temperature is to be determined.


Properties The latent of vaporization of water at 100°C is hfg = 2257 kJ/kg (Table A-9).

Analysis The rate of heat transfer is

W101.881J/kg) 10kg/s)(2257 60/50( 63oncondensati ×=×== fghmQ &&

Substituting the relation for the heat transfer coefficient, the tube surface temperature is determined from

[ ]

C110.2 °=−−=×

−−=

−=

s

ss

ss

ss

TTT

TTDLTTQ

TThAQ

)100(m) m)(2 0.05(100)100(56.5 W10881.1

))(100()(56.5

)(

36sat

3sat

sat

π

π&

&


10-88

10-101 Saturated steam condenses on a suspended silver sphere which is initially at 25°C. The time needed for the temperature of the sphere to rise to 50°C and the amount of steam condenses are to be determined.

Assumptions 1 The temperature of the sphere changes uniformly and thus the lumped system analysis is applicable. 2 The average condensation heat transfer coefficient evaluated for the average temperature can be used for the entire process. 3 Constant properties at room temperature can be used for the silver ball.

Properties The properties of water at the saturation temperature of 100°C are hfg = 2257×103 J/kg and ρv = 0.60 kg/m3. The properties of the silver ball at room temperature and the properties of liquid water at the average film temperature of

(100 + 37.5)/2 = 69°C ≈ 70°C are (Tables A-3 and A-9), =+= 2/)( ,sat avgsf TTT

C W/m429

CJ/kg 235/sm10174

kg/m 500,1026

3

°⋅=

°⋅=×=

=−

l

p

k

cα

ρ

:BallSilver

C W/m663.0

CJ/kg 4190skg/m10404.0

kg/m 5.9773

3

°⋅=

°⋅=⋅×=

=−

l

pl

l

l

k

cµ

ρ

: WaterLiquid

Silver sphere Steam

100°C1.2 cm

Ti = 25°C Analysis The modified latent heat of vaporization is

J/kg 102435=C)5.37C(100J/kg 41900.68+J/kg 102257

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh


C W/m9892

m) C(0.012)5.37100(s)kg/m 10404.0()C W/m663.0)(J/kg 102435)(kg/m 60.05.977)(kg/m 5.977)(m/s 8.9(813.0

)()(

813.0

2

4/1

3

33332

4/1

sat

3*

sph

°⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µρρρ

The characteristic length and the Biot number for the lumped system analysis is (see Chap. 4)

1.0 0461.0

)C W/m429()m 002.0)(C W/m9892(

m 002.06

m 012.06

6/

2

2

3

<=°⋅

°⋅==

=====

khL

Bi

DD

DA

L

c

cππV

The lumped system analysis is applicable since Bi < 0.1. Then the time needed for the temperature of the sphere to rise from 25 to 50°C is determined to be

s 0.202=⎯→⎯=

−−

⎯→⎯=−−

=°⋅

°⋅===

−−

∞

∞ teeTTTtT

Lch

chA

b

tbt

i

cpp

s

1002510050)(

s 004.2m) C)(0.002J/kg 235)(kg/m (10,500

C W/m9892

004.2

1-3

2

ρρ V

The total heat transfer to the ball and the amount of steam that condenses become

kg/s 102.29 5-×=×

==

=°−°⋅=−=

====

J/kg 102435J/s 81.55

J 81.55C)2550)(CJ/kg 235)(kg 009500.0(])([

kg 009500.06

m) 012.0()kg/m 500,10(6

3*oncondensati

sphere

33

3

sphere

fg

ip

hQm

TtTmcQ

Dm

&&

ππρρV


10-89

10-102 Steam at a saturation temperature of Tsat = 100°C condenses on a suspended silver sphere which is initially at 25°C. The time needed for the temperature of the sphere to rise to 50°C and the amount of steam condenses during this process are to be determined.

Assumptions 1 The temperature of the sphere changes uniformly and thus the lumped system analysis is applicable. 2 The average condensation heat transfer coefficient evaluated for the average temperature can be used for the entire process. 3 Constant properties at room temperature can be used for the silver ball.

Properties The properties of water at the saturation temperature of 100°C are hfg = 2257×103 J/kg and ρv = 0.60 kg/m3. The properties of the silver ball at room temperature and the properties of liquid water at the average film temperature of

(100 + 37.5)/2 = 69°C ≈ 70°C 70°C are (Tables A-3 and A-9), =+= 2/)( ,sat avgsf TTT

C W/m429

CJ/kg 235/sm10174

kg/m 500,1026

3

°⋅=

°⋅=×=

=−

l

p

k

cα

ρ

:BallSilver

C W/m663.0

CJ/kg 4190skg/m10404.0

kg/m 5.9773

3

°⋅=

°⋅=⋅×=

=−

l

pl

l

l

k

cµ

ρ

: WaterLiquid

Silver sphere Steam

100°C 3 cm

Ti = 25°C Analysis The modified latent heat of vaporization is

J/kg 102435=C)5.37C(100J/kg 41900.68+J/kg 102257

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh


C W/m7867

m) C(0.03)5.37100(s)kg/m 10404.0()C W/m663.0)(J/kg 102435)(kg/m 60.05.977)(kg/m 5.977)(m/s 8.9(813.0

)()(

813.0

2

4/1

3

33332

4/1

sat

3*

sph

°⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µρρρ

The characteristic length and the Biot number for the lumped system analysis is (see Chap. 4)

1.0 092.0

)C W/m429()m 005.0)(C W/m7867(

m 005.06

m 03.06

6/

2

2

3

<=°⋅

°⋅==

=====

khL

Bi

DD

DA

L

c

sc

ππV

The lumped system analysis is applicable since Bi < 0.1. Then the time needed for the temperature of the sphere to rise from 25 to 50°C is determined to be

s 0.636=⎯→⎯=

−−

⎯→⎯=−−

=°⋅

°⋅===

−−

∞

∞ teeTTTtT

Lch

chA

b

tbt

i

cpp

s

1002510050)(

s 6376.0m) C)(0.005J/kg 235)(kg/m (10,500

C W/m7867

6376.0

1-3

2

ρρ V

The total heat transfer to the ball and the amount of steam that condenses become

kg/s 103.57 4-×=×

==

=°−°⋅=−=

====

J/kg 102435J/s 5.869

J 5.869C)2550)(CJ/kg 235)(kg 148.0(])([

kg 148.06

m) 03.0()kg/m 500,10(6

3*oncondensati

sphere

33

3

sphere

fg

ip

hQm

TtTmcQ

Dm

&&

ππρρV


10-90

10-103 Ammonia is liquefied in a horizontal condenser at 37°C by a coolant at 20°C. The average value of overall heat transfer coefficient and the tube length are to be determined. Assumptions 1 Steady operating conditions exist. 2 The tubes are isothermal. 3 The thermal resistance of the tube walls is negligible. Properties The properties of ammonia at the saturation temperature of 310 K (37°C) are hfg = 1113×103 J/kg and ρv = 11.09 kg/m3 (Table A-11). We assume a tube outer surface temperature of 31°C. The properties of liquid ammonia at the film temperature of (37 + 31)/2 = 34°C are (Table A-11) =+= 2/)( sat sf TTT

C W/m4602.0

CJ/kg 4867skg/m10303.1

kg/m 0.5894

3

°⋅=

°⋅=⋅×=

=−

l

pl

l

l

k

cµ

ρ

The thermal conductivity of copper is 401 W/m⋅°C (Table A-3). Analysis (a) The modified latent heat of vaporization is

J/kg 101133=C)31C(37J/kg 48670.68+J/kg 101113

)(68.033

sat*

×°−°⋅××=

−+= splfgfg TTchh


C W/m7693

m) C(0.038)3137(s)kg/m 10303.1()C W/m4602.0)(J/kg 101133)(kg/m 09.11)(589.0kg/m 0.589)(m/s 8.9(

729.0

)()(

729.0

2

4/1

4

33332

4/1

sat

3*

horizontal

°⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

°−⋅×

°⋅×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

DTTkhg

hhsl

lfgvll

µ

ρρρ

Noting that there are two 2-pipe high, two 3-pipe high, and one 4-pipe high vertical tiers in the tube-layout, the average heat transfer coefficient is to be determined as follows

C W/m590814

544045845664694413222413222

C W/m5440C) W/m7693(4

11

C W/m5845C) W/m7693(3

11

C W/m6469C) W/m7693(2

11

2321

224/1 tube1 horiz,4/11



°⋅=×+×+×

=×+×+××+×+×

=

°⋅=°⋅==

°⋅=°⋅==

°⋅=°⋅==

hhhh

hN

h

hN

h

hN

h

o

Let us check if the assumed value for the rube temperature was reasonable

C1.31)37()038.0()5908()20()030.0()4000( tubetubetube °=⎯→⎯−=−

∆=∆

TTLTL

TAhTAh oooiii

ππ

which is very close to the assumed value of 31°C. The overall heat transfer coefficient based on the outer surface is determined from

C W/m2012 2 °⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛++

×=⎟⎟

⎠

⎞⎜⎜⎝

⎛++=

−− 11

59081

)401(2)0.3/8.3ln(038.0

4000030.0038.01

2)/ln(

o

ioo

ii

oo hk

DDDhD

DU

(b) The rate of heat transfer is

W10833.2J/kg) 10kg/s)(1133 3600/900( 53*oncondensati ×=×== fghmQ &&

Then the tube length may be determined from

m 4.96=⎯→⎯−⋅=×

∆=

LL

TAUQ oo

)2037(m) (0.038C)(14) W/m2012( W10833.2 25 π

&


10-91

10-104 The initial boiling heat transfer coefficient and the total heat transfer coefficient, when heated steel ball bearings are submerged in a water bath, are to be determined.

Assumptions 1 Steady operating condition exists. 2 The steel ball bearings have uniform initial surface temperature.


ρv = 0.3262 kg/m3 cpv = 2066 J/kg·K

µv = 2.446 × 10−5 kg/m·s kv = 0.05467 W/m·K

Analysis The excess temperature in this case is ∆T = Ts − Tsat = 600°C, which is much larger than 30°C for water in Fig. 10-6. Therefore, film boiling will occur. The film boiling heat flux in this case can be determined from

25

4/1

5

33

sat

4/1

sat

sat3

filmfilm

W/m10052.1

)600()600)(02.0)(10446.2(

)]600)(2066(4.0102257)[3262.09.957)(3262.0()05467.0(81.967.0

)()(

)](4.0)[(

×=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

×

+×−=

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−+−=

−

TTTTD

TTchgkCq s

sv

spvfgvlvv

µρρρ

&



filmfilm TT

qh

s −=

&

K W/m175 2 ⋅=−

×=

K )100700( W/m10052.1 25

filmh


→ )()( satrad4

sat4


4sat

4

rad)(

TTTT

hs

s

−−

=εσ

K W/m15.62K )100700(

K )373973)(K W/m1067.5)(75.0()( 2444428

sat

4sat

4

rad ⋅=−

−⋅×=

−−

=−

TTTT

hs

sεσ




or

K W/m222 2 ⋅=

⋅+⋅=

+=

)K W/m15.62(43K W/m175

43

22

radfilmtotal hhh

Discussion The boiling heat transfer coefficient (hfilm) is 2.82 times the radiation heat transfer coefficient (hrad).


10-92

10-105 Saturated refrigerant-134a vapor is condensed as it is flowing through a tube. With a given vapor flow rate at the entrance, the flow rate of the vapor at the exit is to be determined.

Assumptions 1 Steady operating condition exists. 2 The tube surfaces are isothermal.

Properties The properties of refrigerant-134a at the saturation temperature of 35°C corresponding to 888 kPa are hfg = 168.2 kJ/kg, ρv = 43.41 kg/m3, and µv = 1.327 × 10−5 kg/m·s (Table A-10). The properties of liquid refrigerant-134a at the film temperature of Tf = (Tsat + Ts)/2 = 25°C are, from Table A-10,

ρl = 1207 kg/m3 cpl = 1427 J/kg·K

µl = 2.012 × 10−4 kg/m·s kl = 0.0833 W/m·K

Analysis The modified latent heat of vaporization for film condensation inside horizontal tube is

J/kg 109.178

)1535)(1427(83102.168

)(83

3

3

sat

×=

−+×=


The Reynolds number associated with film condensation inside a horizontal tube is

000,35000,24)skg/m 10327.1)(m 012.0(

)kg/s 003.0(44Re 5

inlet ,

inletvapor <=

⋅×==⎟⎟

⎠

⎞⎜⎜⎝

⎛=

−πµπµρ

v

v

v

vv

DmDV &

Hence, the heat transfer coefficient for film condensation inside a horizontal tube can be determined using

KW/m1293

)012.0)(1535)(10012.2()109.178()0833.0)(41.431207)(1207)(81.9(555.0

)()(

555.0

2

4/1

4

33

4/1

sat

3

internal

⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−×

×−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−==

−

∗

DTThkg

hhsl

fglvll

µρρρ


W7.243

K )1535)(K W/m1293)(m 25.0)(m 012.0(

)(2

sat

=−⋅=

−=

π

π sTTDLhQ&

Then, the rate of condensation can be calculated as

kg/s 00136.0J/kg 109.178

W7.2433oncondensati =

×==

∗

hQmfg

&&

Applying the conservation of mass, the flow rate of vapor leaving the tube can be determined as

→ oncondensatioutlet ,inlet , mmm vv &&& += oncondensatiinlet ,outlet , mmm vv &&& −=

kg/s 0.00164=−= kg/s 00136.0kg/s 003.0outlet ,vm&

Discussion About 45% of the refrigerant-134a vapor that entered the tube is condensed inside it.


10-93

Fundamentals of Engineering (FE) Exam Problems

10-106 Heat transfer coefficients for a vapor condensing on a surface can be increased by promoting

(a) film condensation (b) dropwise condensation (c) rolling action (d) none of them

Answer (b) dropwise condensation

10-107 At a distance x down a vertical, isothermal flat plate on which a saturated vapor is condensing in a continuous film, the thickness of the liquid condensate layer is δ. The heat transfer coefficient at this location on the plate is given by

(a) (b) fδ/lk hδ (c) fghδ (d) ghδ (e) none of them

Answer (a) δ/lk


10-94

10-108 Steam condenses at 50ºC on a tube bank consisting of 20 tubes arranged in a rectangular array of 4 tubes high and 5 tubes wide. Each tube has a diameter of 3 cm and a length of 5 m and the outer surfaces of the tubes are maintained at 30ºC. The rate of condensation of steam is

(a) 0.12 kg/s (b) 0.28 kg/s (c) 0.31 kg/s (d) 0.45 kg/s (e) 0.62 kg/s

(For water, use ρl = 992.1 kg/m3, µl = 0.653×10-3 kg/m⋅s, kl = 0.631 W/m⋅°C, cpl = 4179 J/kg⋅°C, hfg @ Tsat = 2383 kJ/kg)

Answer (c) 0.31 kg/s

Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen.

T_sat=50 [C] T_s=30 [C] D=0.03 [m] L=5 [m] N=4 N_total=5*N h_fg=2383E3 [J/kg] "at 50 C from Table A-9" rho_v=0.0831 [kg/m^3] "at 50 C from Table A-9" "The properties of water at (50+30)/2=40 C are (Table A-9)" rho_l=992.1 [kg/m^3] mu_l=0.653E-3 [kg/m-s] c_p_l=4179 [J/kg-C] k_l=0.631 [W/m-C] g=9.81 [m/s^2] h_fg_star=h_fg+0.68*c_p_l*(T_sat-T_s) h_1tube=0.729*((g*rho_l*(rho_l-rho_v)*h_fg_star*k_l^3)/(mu_l*(T_sat-T_s)*D*N))^0.25 h_Ntubes=1/N^0.25*h_1tube A_s=N_total*pi*D*L Q_dot=h_Ntubes*A_s*(T_sat-T_s) m_dot_cond=Q_dot/h_fg_star


10-95

10-109 Saturated water vapor is condensing on a 0.5 m2 vertical flat plate in a continuous film with an average heat transfer coefficient of 5 kW/m2⋅K. The temperature of the water is 80oC (hfg = 2309 kJ/kg) and the temperature of the plate is 60oC. The rate at which condensate is being formed is

(a) 0.022 kg/s (b) 0.048 kg/s (c) 0.077 kg/s (d) 0.16 kg/s (e) 0.32 kg/s

Answer (a) 0.022 kg/s


hfg=2309 [kJ/kg] dT=20 [C] A=0.5 [m^2] h=5 [kJ/m^2-K-s] mdot=h*A*dT/hfg

10-110 An air conditioner condenser in an automobile consists of 2 m2 of tubular heat exchange area whose surface temperature is 30oC. Saturated refrigerant 134a vapor at 50oC (hfg = 152 kJ/kg) condenses on these tubes. What heat transfer coefficient must exist between the tube surface and condensing vapor to produce 1.5 kg/min of condensate?

(a) 95 W/m2⋅K (b) 640 W/m2⋅K (c) 727 W/m2⋅K (d) 799 W/m2⋅K (e) 960 W/m2⋅K

Answer (a) 95 W/m2⋅K


hfg=152000 [J/kg] dT=20 [C] A=2 [m^2] mdot=(1.5/60) [kg/s] Q=mdot*hfg Q=h*A*dT

10-111 When boiling a saturated liquid, one must be careful while increasing the heat flux to avoid “burnout.” Burnout occurs when the boiling transitions from _____ boiling.

(a) convection to nucleate (b) convection to film (c) film to nucleate

(d) nucleate to film (e) none of them

Answer (d) nucleate to film


10-96

10-112 Steam condenses at 50ºC on a 1.2-m-high and 2.4-m-wide vertical plate that is maintained at 30ºC. The condensation heat transfer coefficient is

(a) 4260 W/m2⋅ºC (b) 4780 W/m2⋅ºC (c) 5510 W/m2⋅ºC (d) 6260 W/m2⋅ºC (e) 6940 W/m2⋅ºC


Answer (b) 4780 W/m2⋅ºC


T_sat=50 [C] T_s=30 [C] L=1.2 [m] w=2.4 [m] h_fg=2383E3 [J/kg] "at 50 C from Table A-9" "The properties of water at (50+30)/2=40 C are (Table A-9)" rho_l=992.1 [kg/m^3] mu_l=0.653E-3 [kg/m-s] nu_l=mu_l/rho_l c_p_l=4179 [J/kg-C] k_l=0.631 [W/m-C] g=9.81 [m/s^2] h_fg_star=h_fg+0.68*c_p_l*(T_sat-T_s) Re=(4.81+(3.70*L*k_l*(T_sat-T_s))/(mu_l*h_fg_star)*(g/nu_l^2)^(1/3))^0.820 "Re is between 30 and 1800, and therefore there is wavy laminar flow" h=(Re*k_l)/(1.08*Re^1.22-5.2)*(g/nu_l^2)^(1/3)


10-97

10-113 Steam condenses at 50ºC on the outer surface of a horizontal tube with an outer diameter of 6 cm. The outer surface of the tube is maintained at 30ºC. The condensation heat transfer coefficient is

(a) 5493 W/m2⋅ºC (b) 5921 W/m2⋅ºC (c) 6796 W/m2⋅ºC (d) 7040 W/m2⋅ºC (e) 7350 W/m2⋅ºC


Answer (c) 6796 W/m2⋅ºC


T_sat=50 [C] T_s=30 [C] D=0.06 [m] h_fg=2383E3 [J/kg] "at 50 C from Table A-9" rho_v=0.0831 [kg/m^3] "The properties of water at (50+30)/2=40 C are (Table A-9)" rho_l=992.1 [kg/m^3] mu_l=0.653E-3 [kg/m-s] c_p_l=4179 [J/kg-C] k_l=0.631 [W/m-C] g=9.81 [m/s^2] h_fg_star=h_fg+0.68*c_p_l*(T_sat-T_s) h=0.729*((g*rho_l*(rho_l-rho_v)*h_fg_star*k_l^3)/(mu_l*(T_sat-T_s)*D))^0.25

10-114 When a saturated vapor condenses on a vertical, isothermal flat plate in a continuous film, the rate of heat transfer is proportional to

(a) (b) (c) (d) 4/1sat )( TTs −

2/1sat )( TTs −

4/3sat )( TTs − )( satTTs − (e) 3/2

sat )( TTs −

Answer (c) 4/3sat )( TTs −

10-115 ... 10-120 Design and Essay Problems

Heat and Mass Transfer Fundamentals and Applications

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