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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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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.
10-16
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&
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.
10-17
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.
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.
10-18
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
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.
10-19
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]
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.
10-20
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).
Properties The properties of water at the saturation temperature of 100°C are (Tables 10-1 and A-9)
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.
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.
10-21
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).
Properties The properties of water at the saturation temperature of 100°C are (Tables 10-1 and A-9)
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.
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.
10-22
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.
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 120°C are (Tables 10-1 and A-9)
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
&&
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.
10-34
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.
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 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 &&
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.
10-35
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.
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.
10-36
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.
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.
10-37
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.
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 modified latent heat of vaporization is
J/kg 102314
)80100)(4206(68.0102257
)(68.0
3
3
sat
×=
−+×=
−+=∗splfgfg TTchh
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
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.
10-38
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 ====
&
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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
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.
10-40
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
Assuming wavy-laminar flow, the Reynolds number is determined from
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
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/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&
(b) The rate of condensation of steam is determined from
kg/s 0.591=×
==J/kg 102286J/s 500,351,1
3*oncondensatifgh
Qm&
&
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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.
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
Analysis The modified latent heat of vaporization is
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
Then the rate of heat transfer during this condensation process becomes
kW 1119==°−°⋅=−= W600,118,1C)90100)(m 20)(C W/m5593()( 22sat ss TThAQ&
(b) The rate of condensation of steam is determined from
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.
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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
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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.
Properties The properties of water at the saturation temperature of 50°C are hfg = 2383×103 J/kg and ρv = 0.0831 kg/m3. The properties of liquid water at the film temperature of =+= 2/)( sat sf TTT (50 + 20)/2 = 35°C are (Table A-9),
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
Then the rate of heat transfer during this condensation process becomes
W000,164C)2050)(m 555.1)(C W/m3515()( 22sat =°−°⋅=−= ss TThAQ&
(b) The rate of condensation of steam is determined from
kg/s 0.0664=×
==J/kg 102468J/s 000,164
3*oncondensatifgh
Qm
&&
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.
10-44
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 << .
Properties The properties of water at the saturation temperature of 55°C are hfg = 2371×103 J/kg and ρv = 0.1045 kg/m3. The properties of liquid water at the film temperature of =+= 2/)( sat sf TTT (55 + 45)/2 = 50°C are (Table A-9),
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
Analysis The modified latent heat of vaporization is
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&
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.
10-45
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.
Properties The properties of water at the saturation temperature of 55°C are hfg = 2371×103 J/kg and ρv = 0.1045 kg/m3. The properties of liquid water at the film temperature of =+= 2/)( sat sf TTT (55 + 45)/2 = 50°C are (Table A-9),
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&
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.
10-46
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.
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 + 80)/2 = 90°C are (Table A-9),
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
Analysis The modified latent heat of vaporization is
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
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/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 .
Then the rate of heat transfer during this condensation process becomes
W449,900C)80100)(m 4)(C W/m5623()( 22sat =°−°⋅=−= sTThAQ&
(b) The rate of condensation of steam is determined from
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 =
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.
10-47
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.
Analysis The problem is solved using EES, and the solution is given below.
"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]
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.
10-48
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
]
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.
10-49
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.
Assumptions 1 Steady operating conditions exist. 2 The tube is isothermal.
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
Analysis The modified latent heat of vaporization is
J/kg 101288=C)]10(C[10J/kg 46170.68+J/kg 101226
)(68.033
sat*
×°−−°⋅××=
−+= splfgfg TTchhCondensate
Noting that the tube is horizontal, the condensation heat transfer coefficient is determined from
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
Then the rate of heat transfer during this condensation process becomes
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&
&
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.
10-50
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.
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 modified latent heat of vaporization is
J/kg 102314
)80100)(4206(68.0102257
)(68.0
3
3
sat
×=
−+×=
−+=∗splfgfg TTchh
Assuming wavy-laminar flow, the Reynolds number is determined from
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
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
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
Then the rate of heat transfer during this condensation process becomes
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.
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.
10-51
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.
Assumptions 1 Steady operating conditions exist. 2 The tubes are isothermal.
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 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
The heat transfer coefficient for condensation on a single horizontal tube is
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
2total m 108.6= m) m)(8 03.0(144ππ == DLNAs
Then the rate of heat transfer during this condensation process becomes
kW 5060==°−°=−= W,060,0005C)2030)(m 6.108)(C. W/m4661()( 22sat ss TThAQ&
(b) The rate of condensation of steam is determined from
kg/s 2.06=×
==J/kg 102459J/s 000,060,5
3*oncondensatifgh
Qm&
&
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.
10-52
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.
Analysis The problem is solved using EES, and the solution is given below.
"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
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.
10-53
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
Analysis The modified latent heat of vaporization is
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
Then the rate of heat transfer during this condensation process becomes
Btu/h 9431=°−°⋅⋅=−= F)6595)(ft 2094.0)(FftBtu/h 1501()( 22sat ss TThAQ&
(b) The rate of condensation of steam is determined from
lbm/h 8.90===Btu/lbm 1060
Btu/h 9431*oncondensatifgh
Qm&
&
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.
10-54
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.
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
Analysis The modified latent heat of vaporization is
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
µ
ρρρ
Then the average heat transfer coefficient for a 4-pipe high vertical tier becomes
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
Then the rate of heat transfer during this condensation process becomes
Btu/h 213,300=°−°⋅=−= F)6595)(ft 702.6)(FBtu/h.ft 1061()( 22sat ss TThAQ&
(b) The rate of condensation of steam is determined from
lbm/h 201.3===Btu/lbm 1060
Btu/h 300,213*oncondensatifgh
Qm&
&
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.
10-55
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
Then the rate of heat transfer during this condensation process becomes
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.
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.
10-56
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.
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 (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
The heat transfer surface area of the pipe is
2m 1885.0m) m)(10 006.0( === ππDLAs
Then the rate of heat transfer during this condensation process becomes
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
Then the rate of condensation becomes
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&
&
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.
10-57
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.
Analysis The problem is solved using EES, and the solution is given below.
"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 [
%]
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.
10-58
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.
Assumptions 1 Steady operating conditions exist. 2 The tubes are isothermal.
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
Analysis (a) The modified latent heat of vaporization is
J/kg 102314=
C80)C(100J/kg 42060.68+J/kg 102357
)(68.0
3
3
sat*
×
°−°⋅××=
−+= splfgfg TTchh
The heat transfer coefficient for condensation on a single horizontal tube is
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
Then the average heat transfer coefficient for a 4-pipe high vertical tier becomes
C W/m6177C) W/m8736(4
11 224/1 tube1 horiz,4/1 tubesN horiz, °⋅=°⋅== h
Nh
The surface area for all 16 tubes is
2total m 5.027= m) m)(2 05.0(16ππ == DLNAs
Then the rate of heat transfer during this condensation process becomes
W000,621C)80100)(m 027.5)(C W/m6177()( 22sat =°−°⋅=−= ss TThAQ&
The rate of condensation of steam is determined from
kg/h 966==×
== kg/s 0.2684J/kg 102314J/s 000,621
3*oncondensatifgh
Qm
&&
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.
10-59
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.
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 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
Assuming wavy-laminar flow, the Reynolds number is determined from
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
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/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
Then the rate of heat transfer during this condensation process becomes
W480,105C)1030)(m 75.0)(C W/m7032()( 22sat =°−°⋅=−= ss TThAQ&
(b) The rate of condensation of steam is determined from
kg/s 0.0872=×
==J/kg 101209J/s 480,105
3*oncondensatifgh
Qm
&&
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.
10-60
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
Analysis The modified latent heat of vaporization is
J/kg 102440
)3050)(4179(68.0102383
)(68.0
3
3
sat
×=
−+×=
−+=∗splfgfg TTchh
The heat transfer coefficient for condensation on a single horizontal tube is
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
Then the average heat transfer coefficient for a 5-tube high vertical tier becomes
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.
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.
10-61
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
×=
−+×=
−+=∗splfgfg TTchh
The heat transfer coefficient for condensation on a single horizontal tube is
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
11 224/1 tube1 horiz,4/1 tubes horiz, ⋅=⋅=== h
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
11 224/1 tube1 horiz,4/1 tubes horiz, ⋅=⋅=== h
Nhh N
The rate of heat transfer during this condensation process becomes
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&
&
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.
10-62
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.
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 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 &&&
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.
10-63
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.
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 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.
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.
10-64
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.
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 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.
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.
10-65
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
The liquid phase heat transfer coefficient is calculated using:
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
where the in situ liquid Reynolds number is
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).
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.
10-66
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
The gas and liquid mass flow rates are calculated as
kg/s 10619.54
m) 02540()m/s 1)(kg/m 109.1(4
42
32
−×====.DVAVm sggcsggg ππρρ&
kg/s 003.14
m) 02540()m/s 2)(kg/m 1.990(4
23
2====
.DVAVm sllcslll ππρρ&
Using the gas and liquid mass flow rates, the quality is determined to be
44
410599.5
10619.5003.110619.5 −
−
−
×=×+
×=
+=
gl
g
mmm
x&&
&
The flow pattern factor (Fp) can be calculated using
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
The liquid phase heat transfer coefficient is calculated using:
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
where the in situ liquid Reynolds number is
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.
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.
10-67
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
Using the gas and liquid mass flow rates, the quality is determined to be
310426.3055.016
055.0 −×=+
=+
=gl
g
mmm
x&&
&
The flow pattern factor (Fp) can be calculated using
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
The liquid phase heat transfer coefficient is calculated using:
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
where the in situ liquid Reynolds number is
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
Thus, using the general two-phase heat transfer correlation, the value for htp is estimated to be
( )
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.
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.
10-68
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.
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
The gas and liquid mass flow rates are calculated as
kg/s 10619.54
m) 02540()m/s 1)(kg/m 109.1(4
42
32
−×====.DVAVm sggcsggg ππρρ&
kg/s 003.14
m) 02540()m/s 2)(kg/m 1.990(4
23
2====
.DVAVm sllcslll ππρρ&
Using the gas and liquid mass flow rates, the quality is determined to be
44
410599.5
10619.5003.110619.5 −
−
−
×=×+
×=
+=
gl
g
mmm
x&&
&
The flow pattern factor (Fp) can be calculated using
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
The liquid phase heat transfer coefficient is calculated using:
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
where the in situ liquid Reynolds number is
103100)m 0254.0)(skg/m 10596.0(33.01
)kg/s 003.1(414
Re3
=⋅×−
=−
=−πµαπ D
m
l
ll
&
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.
10-69
(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
Thus, using the general two-phase heat transfer correlation, the value for htp is estimated to be
( )
( )
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
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.
10-70
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).
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.
10-71
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.
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
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
&
Using the Newton’s law of cooling, the boiling heat transfer coefficient is
→ )( satfilmfilm TThq s −=&sat
filmfilm TT
qh
s −=
&
K W/m288 2 ⋅=−
×=
K )100500( W/m10152.1 25
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/m93.23K )100500(
K )373773)(K W/m1067.5)(5.0()( 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/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).
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.
10-72
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
×=
−+×=
−+=∗splfgfg TTchh
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
The rate of heat transfer during this condensation process becomes
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.
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.
10-73
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
Analysis (a) The modified latent heat of vaporization is
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&
&
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.
10-74
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.
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 120°C are (Tables 10-1 and A-9)
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&
&
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.
10-75
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
Analysis (a) The modified latent heat of vaporization is
J/kg 101182=
C)20C(25J/kg 47650.68+J/kg 101166
)(68.0
3
3
sat*
×
°−°⋅××=
−+= splfgfg TTchh
The heat transfer coefficient for condensation on a single horizontal tube is
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
Then the average heat transfer coefficient for a 4-pipe high vertical tier becomes
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
Then the rate of condensation becomes
kg/s 0.167=×
×==
J/kg 101182 W10970.1
3
5
*condfgh
Qm
&&
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.
10-76
(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
&
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.
10-77
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
The heat transfer coefficient for condensation on a single horizontal tube is
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.
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.
10-78
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.
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 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
&&
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.
10-79
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:
4/1 tube1 ,horizontal
tubesN of tier verticalN
hh =
Therefore,
1.68=8=
/
Ratio
1/4
4/1
4/1 tube1 ,horizontal
tube1 ,horizontal
tubesN of tier vertical
tubesN of tier horizontal
tubesN of tier vertical
tubesN of tier horizontal
N
Nh
hh
hm
m
=
=
=
=&
&
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.
10-80
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.
Assumptions 1 Steady operating conditions exist. 2 The tubes are isothermal.
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/
hlbm/ft 842.1slbm/ft 10117.5
lbm/ft 12.62
2
4
3
°⋅⋅=
°⋅===
⋅=⋅×=
=−
l
pl
lll
l
l
k
cρµν
µ
ρ
Analysis (a) The modified latent heat of vaporization is
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
The heat transfer coefficient for condensation on a single horizontal tube is
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
Then the average heat transfer coefficient for a 4-tube high vertical tier becomes
FftBtu/h 3.878F)ftBtu/h 1562(10
11 224/1 tube1 horiz,4/1 tubesN horiz, °⋅⋅=°⋅⋅== h
Nh
The surface area for all 100 tubes is
2total ft 471.2= ft) ft)(15 12/2.1(100ππ == DLNAs
Then the rate of heat transfer during this condensation process becomes
Btu/h 8,278,000=°−°⋅=−= F)80100)(ft 2.471)(FBtu/h.ft 3.878()( 22sat ss TThAQ&
(b) The rate of condensation of steam is determined from
lbm/h 7879===Btu/lbm 1051
Btu/h 000,278,8*oncondensatifgh
Qm&
&
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.
10-81
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.
Assumptions 1 Steady operating conditions exist. 2 The tubes are isothermal.
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/
hlbm/ft 842.1slbm/ft 10117.5
lbm/ft 12.62
2
4
3
°⋅⋅=
°⋅===
⋅=⋅×=
=−
l
pl
lll
l
l
k
cρµν
µ
ρ
Analysis (a) The modified latent heat of vaporization is
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
The heat transfer coefficient for condensation on a single horizontal tube is
FftBtu/h 1375
ft) F(2.0/12)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
µρρρ
Then the average heat transfer coefficient for a 4-tube high vertical tier becomes
FftBtu/h 0.773F)ftBtu/h 1375(10
11 224/1 tube1 horiz,4/1 tubesN horiz, °⋅⋅=°⋅⋅== h
Nh
The surface area for all 100 tubes is
2total ft 785.4= ft) ft)(15 12/2(100ππ == DLNAs
Then the rate of heat transfer during this condensation process becomes
Btu/h 12,142,000=°−°⋅=−= F)80100)(ft 4.785)(FBtu/h.ft 0.773()( 22sat ss TThAQ&
(b) The rate of condensation of steam is determined from
lbm/h 11,560===Btu/lbm 1051
Btu/h 000,142,12*oncondensatifgh
Qm&
&
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.
10-82
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.
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 100°C are (Tables 10-1 and A-9)
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&
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.
10-83
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.
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°F are hfg = 1037 Btu/lbm and ρv = 0.00286 lbm/ft3. The properties of liquid water at the film temperature of =+= 2/)( sat sf TTT (100 + 80)/2 = 90°F are (Table A-9E),
FftBtu/h 358.0
FBtu/lbm 999.0/hft 02965.0/
hlbm/ft 842.1slbm/ft 10117.5
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
Assuming wavy-laminar flow, the Reynolds number is determined from
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
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
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
Then the rate of heat transfer during this condensation process becomes
Btu/h 70,000=°−°⋅⋅=−= F)80100)(ft 4)(FftBtu/h 875()( 22sat ss TThAQ&
The rate of condensation of steam is determined from
lbm/h 66.6===Btu/lbm 1051
Btu/h 000,70*oncondensatifgh
Qm&
&
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.
10-84
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.
Assumptions 1 Steady operating conditions exist. 2 The tube is isothermal.
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
Analysis The modified latent heat of vaporization is
J/kg 10178.0=C)25C(35J/kg 14480.68+J/kg 102.168
)(68.033
sat*
×°−°⋅××=
−+= splfgfg TTchhCondensate
The heat transfer coefficient for condensation on a single horizontal tube is
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
The heat transfer surface area of the pipe is
2m 3016.0m) m)(8 012.0( === ππDLAs
Then the rate of heat transfer during this condensation process becomes
W5996C)2535)(m 3016.0)(C W/m1988()( 22sat =°−°⋅=−= ss TThAQ&
The rate of condensation of steam is determined from
kg/min 2.02==×
== kg/s 0.03368J/kg 10178.0
J/s 59963*oncondensati
fghQm&
&
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.
10-85
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.
Assumptions 1 Steady operating conditions exist. 2 The tube is isothermal.
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
Analysis The modified latent heat of vaporization is
Condensate
J/kg 10178.0=C)25C(35J/kg 14480.68+J/kg 102.168
)(68.033
sat*
×°−°⋅××=
−+= splfgfg TTchh
The heat transfer coefficient for condensation on a single horizontal tube is
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
The heat transfer surface area of the pipe is
2m 7540.0m) m)(8 03.0( === ππDLAs
Then the rate of heat transfer during this condensation process becomes
W920,11C)2535)(m 7540.0)(C W/m1581()( 22sat =°−°⋅=−= ss TThAQ&
The rate of condensation of steam is determined from
kg/min 4.02==×
== kg/s 0.06697J/kg 10178.0
J/s 920,113*oncondensati
fghQm&
&
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.
10-86
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.
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 water at the saturation temperature of 130°C are hfg = 2174×103 J/kg and ρv = 1.50 kg/m3. The properties of liquid water at the film temperature of =+= 2/)( sat sf TTT (130 + 110)/2 = 120°C are (Table A-9),
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
The heat transfer surface area of the pipe is
2m 7854.0m) m)(10 025.0( === ππDLAs
Then the rate of heat transfer during this condensation process becomes
W543,52C)110130)(m 7854.0)(C W/m3345()( 22sat =°−°⋅=−= ss TThAQ&
The rate of condensation of steam is determined from
kg/s 0.0242=×
==J/kg 102174
J/s 543,523oncondensati
fghQ
m&
&
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.
10-87
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&
&
The rate of condensation of steam is determined from
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.
Assumptions 1 Steady operating conditions exist. 2 The tubes are isothermal.
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&
&
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.
10-88
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
Noting that the tube is horizontal, the condensation heat transfer coefficient is determined from
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
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.
10-89
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
Noting that the tube is horizontal, the condensation heat transfer coefficient is determined from
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
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.
10-90
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
The heat transfer coefficient for condensation on a single horizontal tube is
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
224/1 tube1 horiz,4/12
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
&
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.
10-91
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.
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 = 400°C are, from Table A-16,
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
&
Using the Newton’s law of cooling, the boiling heat transfer coefficient is
→ )( satfilmfilm TThq s −=&sat
filmfilm TT
qh
s −=
&
K W/m175 2 ⋅=−
×=
K )100700( W/m10052.1 25
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/m15.62K )100700(
K )373973)(K W/m1067.5)(75.0()( 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/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).
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.
10-92
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
×=
−+×=
−+=∗splfgfg TTchh
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
The rate of heat transfer during this condensation process becomes
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.
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.
10-93
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
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.
10-94
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
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.
10-95
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
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen.
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
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen.
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
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.
10-96
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
(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 (b) 4780 W/m2⋅ºC
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] 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)
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.
10-97
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
(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) 6796 W/m2⋅ºC
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=0.06 [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 −
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