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7-21
7-31 Ambient air flows over parallel plates of a solar collector that is maintained at a specified temperature. The rates of
convection heat transfer from the first and third plate are to be determined.
Assumptions 1 Steady operating conditions exist. 2 The
critical Reynolds number is Recr = 5105. 3 Radiation effects
are negligible. 4 Atmospheric pressure is taken 1 atm.
Properties The properties of air at the film temperature of
(15+10)/2=12.5C are (Table A-15)
7330.0Pr
/sm 10448.1
C W/m.02458.0
25-
=
=
=
k
Analysis (a) The critical length of the plate is first
determined to be
/s)m 10448.1)(105(
Re 255
cr
−
W109=−=−=
C10))(15m C)(4. W/m469.5()(
22
TThAQ
s
(b) Repeating the calculations for the second and third plates,
5
2
m) m/s)(2 2(
VL
Alternative solution for part (b)
(b) The average heat transfer coefficient for the combined first and second plates is determined as
4 m
1 m
V, T
7-23
7-32 Hydrogen gas flows in parallel over the upper and lower surfaces of a flat plate. The local convection heat
transfer coefficient and the localtotal convection heat flux along the plate are to be evaluated.
Assumptions 1 Steady operating conditions exist. 2 Surface temperature is uniform over the entire plate. 3Local atmospheric
pressure is 1 atm. 4 The critical Reynolds number is Recr = 5105. 5 Heat transfer by radiation is negligible. 7 Flow is laminar
(this assumption will be verified).
Analysis For laminar flow, the relation for local Nusselt number along a flat plate is
3/15.0
xPrRe332.0Nu x
x
k
xh ==
The total local convection heat flux at the plate upper and lower surfaces is
)(2 sxx TThq −=
The problem is solved using EES, and the solution is given below:
"GIVEN"
T_infinity=120 [C]
T_s=30 [C]
V=2.5 [m/s]
"PROPERTIES"
T_film=1/2*(T_s+T_infinity)
Fluid$='hydrogen'
k=Conductivity(Fluid$, T=T_film, P=101.3)
Pr=Prandtl(Fluid$, T=T_film, P=101.3)
rho=Density(Fluid$, T=T_film, P=101.3)
mu=Viscosity(Fluid$, T=T_film, P=101.3)
nu=mu/rho
"ANALYSIS"
Re_x=V*x/nu
Nusselt_x=0.332*Re_x^0.5*Pr^(1/3)
h_x=Nusselt_x*k/x
q_dot_x=2*h_x*(T_infinity-T_s)
x Rex hx q
̇x
[m] [W/m2∙K] [W/m2]
0.2 3512 17.79 3202
0.4 7023 12.58 2264
0.6 10535 10.27 1849
0.8 14047 8.895 1601
1.0 17558 7.956 1432
1.2 21070 7.263 1307
1.4 24582 6.724 1210
1.6 28093 6.290 1132
1.8 31605 5.930 1067
2.0 35117 5.626 1013
2.2 38628 5.364 965.5
2.4 42140 5.136 924.4
2.6 45652 4.934 888.1
2.8 49163 4.755 855.8
0 0.5 1 1.5 2 2.5 3
500
1000
1500
2000
2500
3000
3500
qx [W/m2]
7-24
0 0.5 1 1.5 2 2.5 3
2
4
6
8
10
12
14
16
18
20
hx [W/m2·K]
7-25
7-33 CO2 and H2 as ideal gases flow in parallel over a flat plate. The local Reynolds number, local Nusselt number,
and local convection heat transfer coefficient along the plate are to be evaluated.
Assumptions 1 Steady operating conditions exist. 2Surface temperature is uniform over the entire plate. 3 Local atmospheric
pressure is 1 atm. 4 The critical Reynolds number is Recr = 5 105. 5 Flow is laminar (this assumption will be verified).
Analysis For laminar flow, the relation for local Nusselt number along a flat plate is
3/15.0
x
k
xh ==
The problem is solved using EES, and the solution is given below:
"GIVEN"
V=1 [m/s]
T_s=20 [C]
7-26
CO2 gas H2 gas
x [m] Rex,CO2 Nux,CO2 hx,CO2 Rex,H2 Nux,H2 hx,H2
[W/m2∙K] [W/m2∙K]
0.2 28552 51.4 3.741 2143 13.8 11.40
0.4 57104 72.69 2.645 4287 19.52 8.062
0.6 85656 89.03 2.160 6430 23.91 6.583
0.8 114207 102.8 1.871 8574 27.61 5.701
1.0 142759 114.9 1.673 10717 30.87 5.099
1.2 171311 125.9 1.527 12861 33.81 4.655
1.4 199863 136.0 1.414 15004 36.52 4.309
1.6 228415 145.4 1.323 17147 39.05 4.031
1.8 256967 154.2 1.247 19291 41.41 3.800
2.0 285518 162.5 1.183 21434 43.65 3.605
2.2 314070 170.5 1.128 23578 45.78 3.438
2.4 342622 178.1 1.080 25721 47.82 3.291
2.6 371174 185.3 1.038 27865 49.77 3.162
2.8 399726 192.3 0.9999 30008 51.65 3.047
3.0 428278 199.1 0.9660 32151 53.46 2.944
0 0.5 1 1.5 2 2.5 3
0
100000
200000
300000
400000
500000
x [m]
Rex
CO2 gas
H2 gas
0 0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
x [m]
hx [W/m2·K]
CO2 gas
H2 gas
0 0.5 1 1.5 2 2.5 3
0
40
80
120
160
200
x [m]
Nux
CO2 gas
H2 gas
7-27
7-34 Air is blown over an aluminum plate mounted on an array of power transistors. The number of transistors that can be
placed on this plate is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The critical Reynolds number is Recr = 5105. 3 Radiation effects are
negligible 4 Heat transfer from the back side of the plate is negligible. 5 Air is an ideal gas with constant properties. 6 The
local atmospheric pressure is 1 atm.
Properties The properties of air at the film temperature of (Ts + T)/2 = (65+35)/2 = 50C are (Table A-15)
7228.0Pr
/sm 10798.1
C W/m.02735.0
25-
=
=
=
k
Analysis The Reynolds number is
m) m/s)(0.25 (4
VL
Transistors
Air
V = 4 m/s
T = 35C
Ts=65C
7-28
7-35 Air is blown over an aluminum plate mounted on an array of power transistors. The number of transistors that can be
placed on this plate is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The critical Reynolds number is Recr = 5105. 3 Radiation effects are
negligible 4 Heat transfer from the backside of the plate is negligible. 5 Air is an ideal gas with constant properties. 6 The
local atmospheric pressure is 1 atm.
Properties The properties of air at 1 atm and the film temperature of (Ts + T)/2 = (65+35)/2 = 50C are (Table A-15)
7228.0Pr
/sm 10798.1
C W/m.02735.0
25-
=
=
=
k
Note that the atmospheric pressure will only affect the kinematic
viscosity. The atmospheric pressure in atm is
atm 823.0
kPa 101.325
atm 1
kPa) 4.83(==P
The kinematic viscosity at this atmospheric pressure will be
/sm 10184.2823.0/) /sm 10798.1( 2525 −− ==
Analysis The Reynolds number is
4
m) m/s)(0.25 (4
VL
Transistors
Air
V = 4 m/s
T = 35C
L=25 cm
Ts=65C
7-29
7-36 Two ASTM B98 copper-silicon bolts are embedded at 0.25 m and 1.75 m from the plate’s leading edge. The
maximum use temperature for the bolt is 149°C. The air flows in parallel over the plate’s upper surface. The local heat fluxes
at the locations where the bolts are embedded are to be determined.
Assumptions1 The flow is steady and incompressible. 2 Uniform surface temperature. 3 Edge effects of plate are negligible.
4 The critical Reynolds number is Recr = 5 × 105.
Properties The properties of air at the film temperature of Tf = (Ts + T∞)/2 = (149 + 250)°C/2 ≈ 200°C are (Table A–15): Pr =
0.6974, k = 0.03779 W/m∙K, andν = 3.455 × 10−5 m2/s
Analysis The location where the critical Reynolds number occurs is
𝑥𝑐𝑟 =Recr𝜈
7-30
7-37E A refrigeration truck is traveling at 70 mph. The average temperature of the outer surface of the refrigeration
compartment of the truck is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The critical Reynolds number is Recr = 5105. 3 Radiation effects are
negligible. 4 Air is an ideal gas with constant properties. 5 The local atmospheric pressure is 1 atm.
Properties Assuming the film temperature to be approximately 80F, the
properties of air at this temperature and 1 atm are (Table A-15E)
7290.0Pr
/sft 10697.1
FBtu/h.ft. 01481.0
24-
=
=
=
k
Analysis The Reynolds number is
7
ft) ft/s](20 /3600)528070[
VL
Air
V = 70 mph
T = 80F
L = 20 ft
Refrigeration
truck
7-32
7-39 Air flows in parallel over a flat plate. The distance from the plate’s leading edge where the critical Reynolds
number is reached is to be determined. The local convection heat transfer coefficient along the plate is to be evaluated.
Assumptions 1 Steady operating conditions exist. 2Surface temperature is uniform over the entire plate. 3 Local atmospheric
pressure is 1 atm. 4 The critical Reynolds number is Recr = 5105.
Properties The kinematic viscosity of air at Tf = (120°C + 20°C)/2 = 70°C isν = 1.995 10−5 m2/s (Table A-15).
Analysis The distance from the plate’s leading edge when Recr = 5× 105 is
cr
−
)/sm 10995.1)(105(
Re 255
cr
7-33
x [m] Rex hx [W/m2∙K]
0.20 70161 11.34
0.25 87701 10.14
0.30 105241 9.260
0.40 140321 8.019
0.50 175402 7.172
0.60 210482 6.548
0.80 280643 5.670
1.0 350803 5.072
1.2 420964 4.630
1.424 499544 4.250
1.425 500000 19.41
1.8 631446 18.53
2.0 701607 18.14
2.2 771768 17.80
2.4 841928 17.49
2.7 947169 17.08
3.0 1.052E+06 16.73
0 0.5 1 1.5 2 2.5 3
2
4
6
8
10
12
14
16
18
20
22
x [m]
hx [W/m2·K]hx [W/m2·K]hx [W/m2·K]hx [W/m2·K]hx [W/m2·K]hx [W/m2·K]hx [W/m2·K]hx [W/m2·K]hx [W/m2·K]
Laminar
Turbulent
7-34
7-40 To prevent local hot spots on a machine surface from causing thermal burns, the thickness of an insulation to
cover the machine surface is to be determined.
Assumptions 1 Steady operating conditions exist. 2 One-dimensional heat conduction through the plate. 3 Thermal
conductivities of plate and insulationare constant. 4 Uniform surface temperature. 5 Local atmospheric pressure is 1 atm. 6
The critical Reynolds number is Recr = 5105.
PropertiesThe thermal conductivities of the aluminum and the insulation are given to bekal= 237 W/mK andkins = 0.06
W/mK, respectively. The thermal contact conductance at the interface is given as hc = 3000 W/m2K.
AnalysisFrom Chapter 3, the thermal resistances of different layers are
L
al
7-35
Re_x=V*x/nu
Nusselt_lam=0.332*Re_x^0.5*Pr^(1/3) "Laminar flow Nusselt_x"
Nusselt_turb=0.0296*Re_x^0.8*Pr^(1/3) "Turbulent flow Nusselt_x"
Nusselt_x=if(Re_x,5e5,Nusselt_lam,Nusselt_turb,Nusselt_turb)
h_x=Nusselt_x*k/x
"Heat balance at the outer surface"
q_dot_cond=(T_s_i-T_s_o)/(L_al/k_al+1/h_c+L_ins/k_ins)
q_dot_conv=(T_s_o-T_infinity)/(1/h_x)
q_dot_cond=q_dot_conv
x [m] Rex hx [W/m2∙K] Lins [m]
0.10 59585 19.25 0.009331
0.15 89378 15.71 0.01143
0.22 131087 12.98 0.01385
0.30 178756 11.11 0.01618
0.40 238341 9.623 0.01868
0.50 297926 8.607 0.02089
0.60 357511 7.857 0.02289
0.70 417096 7.275 0.02472
0.8391 499979 6.644 0.02707
0.8392 500039 30.36 0.005908
0.90 536267 29.94 0.005991
1.0 595852 29.31 0.006120
1.1 655437 28.76 0.006238
1.2 715022 28.26 0.006348
1.3 774607 27.81 0.006450
1.4 834192 27.4 0.006547
1.5 893778 27.03 0.006638
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
-30
-20
-10
0
10
20
30
40
Lins [m]
hx [W/m2·K]
hx
Lins
Laminar Turbulent
7-36
7-41 Air is flowing in parallel over a stationary thin flat plate: (a) the average friction coefficient, (b) the average convection
heat transfer coefficient, and (c) the average convection heat transfer coefficient using the modified Reynolds analogy are to
be determined.
Assumptions 1 Steady operating conditions exist. 2 Properties are constant. 3 The edge effects are negligible. 4 The critical
Reynolds number is Recr = 5105.
Properties The properties of air (1 atm) at the Tf = (20°C + 180°C)/2 = 100°C are given in Table A-15: k = 0.03095 W/m∙K,
= 2.306 10−5 m2/s, and Pr = 0.7111.
Analysis (a) The Reynolds at the trailing edge of the plate is
6
)m 5.0)(m/s 50(
VL
7-38
7-43 A 5-m long strip of sheet metal is being transported on a conveyor, while the coating on the upper surface is being cured
by infrared lamps. The surface temperature of the sheet metal is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Heat conduction through the sheet metal is negligible. 3 Thermal
properties are constant. 4 The surrounding ambient air is at 1 atm. 5 The critical Reynolds number is Recr = 5105.
Properties The properties of air at 80°C are (Table A-15)
k = 0.02953 W/m∙K
= 2.097 10−5 m2/s
Pr = 0.7154
Analysis The Reynolds number for L = 5 m is
6
)m 5)(m/s 5(
VL
7-39
7-44 Prob. 7-43 is reconsidered. The effect of the sheet metal velocity on its surface temperature is to be evaluated.
Assumptions 1 Steady operating conditions exist. 2 Heat conduction through the sheet metal is negligible. 3 The surrounding
ambient air is at 1 atm. 4 The critical Reynolds number is Recr = 5105. 5 Flow is combined laminar and turbulent (this
assumption will be verified).
Analysis For combined laminar and turbulent flow, the relation for Nusselt number is
3/18.0 Pr)871Re037.0(Nu −== L
hL
V [m/s] ReL Ts [°C]
3 620478 195.8
4 893402 162.3
5 1.181E+06 138.9
6 1.476E+06 122.2
8 2.081E+06 100.4
10 2.694E+06 86.94
12 3.312E+06 77.79
14 3.934E+06 71.16
16 4.558E+06 66.13
18 5.184E+06 62.17
20 5.811E+06 58.96
22 6.440E+06 56.32
24 7.069E+06 54.09
26 7.699E+06 52.18
28 8.329E+06 50.54
30 8.960E+06 49.10
temperature decreases with increasing velocity. For the sheet metal to maintain a surface temperature above 100°C, the
velocity should not go below 8 m/s.As shown in the table above, between 3 and 30 m/s, the Reynolds number is 5 105<
ReL< 107.Thus, the flow is combined laminar and turbulent.
0 5 10 15 20 25 30
40
60
80
100
120
140
160
180
200
V [m/s]
Ts [°C]
7-40
7-45 The top surface of a hot block is to be cooled by forced air. The rate of heat transfer is to be determined for two cases.
Assumptions 1 Steady operating conditions exist. 2 The critical Reynolds number is Recr = 5105. 3 Radiation effects are
negligible. 4 Air is an ideal gas with constant properties.
Properties The atmospheric pressure in atm is
atm 823.0
kPa 101.325
atm 1
kPa) 4.83(==P
For an ideal gas, the thermal conductivity and the Prandtl number are
independent of pressure, but the kinematic viscosity is inversely proportional
to the pressure. With these considerations, the properties of air at 0.823 atm
and at the film temperature of (120+30)/2=75C are (Table A-15)
7166.0Pr
/sm 102.486=823.0/)/sm 10046.2(/
C W/m.02917.0
25-25
1@
=
==
=
−
atmatm P
k
Analysis (a) If the air flows parallel to the 8 m side, the Reynolds number in this case becomes
6
m) m/s)(8 6(
VL
kW 18.10==−=−=
W100,18C30))(120m C)(20. W/m05.10()(
22
ss
TThAQ
(b) If the air flows parallel to the 2.5 m side, the Reynolds number is
5
m) m/s)(2.5 6(
VL
Air
V = 6 m/s
T = 30C
L
Ts = 120C
