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Copyright ©2020 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
Solutions Manual for
Heat and Mass Transfer: Fundamentals & Applications
6th Edition
Yunus A. Çengel, Afshin J. Ghajar
McGraw-Hill Education, 2020
Chapter 7
EXTERNAL FORCED CONVECTION
PROPRIETARY AND CONFIDENTIAL
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7-2
Drag Force and Heat Transfer in External Flow
7-1C The velocity of the fluid relative to the immersed solid body sufficiently far away from a body is called the free-stream
7-2C The force a flowing fluid exerts on a body in the flow direction is called drag. Drag is caused by friction between the
7-3C The force a flowing fluid exerts on a body in the normal direction to flow that tend to move the body in that direction is
7-4C When the drag force FD, the upstream velocityV, and the fluid density
are measured during flow over a body, the drag
coefficient can be determined from
AV
F
CD
D2
2
1
=
where A is ordinarily the frontal area (the area projected on a plane normal to the direction of flow) of the body.
7-6C The part of drag that is due directly to wall shear stress
w is called the skin friction dragFD, friction since it is caused by
7-7C A body is said to be streamlined if a conscious effort is made to align its shape with the anticipated streamlines in the
7-8C As a result of streamlining, (a) friction drag increases, (b) pressure drag decreases, and (c) total drag decreases at high
7-3
7-10C At sufficiently high velocities, the fluid stream detaches itself from the surface of the body. This is called separation.
7-4
Flow over Flat Plates
7-14 Air is flowing over a long flat plate with a specified velocity. The distance from the leading edge of the plate where the
flow becomes turbulent, and the thickness of the boundary layer at that location are to be determined.
Assumptions 1The flow is steady and incompressible. 2 The critical Reynolds number is Recr = 5105. 3 Air is an ideal gas.
4 The surface of the plate is smooth.
Properties The density and kinematic viscosity of air at 1 atm and 25C are = 1.184 kg/m3 and = 1.562×10–5 m2/s (Table
A-15).
AnalysisThe distance from the leading edge of the plate where the flow becomes turbulent is the distance xcrwhere the
7-5
7-15 Water is flowing over a long flat plate with a specified velocity. The distance from the leading edge of the plate where
the flow becomes turbulent, and the thickness of the boundary layer at that location are to be determined.
Assumptions 1The flow is steady and incompressible. 2 The critical Reynolds number is Recr = 5105. 3The surface of the
plate is smooth.
Properties The density and dynamic viscosity of water at 1 atm and 25C are = 997 kg/m3 and = 0.891×10–3 kg/ms
(Table A-9).
AnalysisThe distance from the leading edge of the plate where the flow becomes turbulent is the distance xcrwhere the
Reynolds number becomes equal to the critical Reynolds number,
cm 5.6==
==
→=
−
m 0.056
m/s) )(8kg/m (997
)105)(skg/m 10891.0(
Re
Re
3
53
V
x
Vx
cr
cr
cr
cr
The thickness of the boundary layer at that location is obtained by substituting this value of x into the laminar boundary layer
thickness relation,
mm 0.4 m 00040.0
)10(5
m) 056.0(5
Re
5
Re
5
2/152/12/1 ==
==→=
cr
cr
cr
x
cr
x
x
Therefore, the flow becomes turbulent after about 5 cm from the leading edge of the plate, and the thickness of the boundary
layer at that location is 0.4 mm.
Discussion When the flow becomes turbulent, the boundary layer thickness starts to increase, and the value of its thickness
can be determined from the boundary layer thickness relation for turbulent flow.
V
xcr
7-7
7-17 Air flows over a plate. Various quantities are to be determined at x = 0.3 m.
Assumptions 1 The flow is steady and incompressible. 2 The critical Reynolds number is Recr = 5105. 3 Air is an ideal gas.
4 The plate is smooth. 5 Edge effects are negligible and the upper surface of the plate is considered.
Properties The properties of air at the film temperature of Tf = (Ts + T)/2 = (65+15)/2 = 40C are (Table A-15)
7255.0Pr , skg/m 10918.1 ,K W/m02662.0K J/kg 1007 ,kg/m 127.1 53 ===== −
kc p
AnalysisThe critical length of the plate is first determined to be
m 84.2
)kg/m m/s)(1.127 3(
s)kg/m 10918.1)(105(
Re
3
55
cr
cr =
==
−
V
x
Thus flow at x = 0.3m is in the laminar region.
The calculations at x = 0.3 m are
883,52
skg/m 10918.1
)kg/m m)(1.127 m/s)(0.3 3(
Re 5
3
=
== −
Vx
x
(a) Hydrodynamic boundary layer thickness, Eq. 7-12a:
0.00641m===
883,52
m) 0.3(91.4
Re
91.4
x
x
(b) Local friction coefficient, Eq. 7-12b:
0.0029=== −− 2/12/1
,)883,52(664.0Re664.0 xxf
C
(c) Average friction coefficient, Eq. 7-14:
0.0058=== 2/12/1 883,52
33.1
Re
33.1
x
f
C
(d) Total drag force due to friction, Eq. 7-1:
N 0.0026=== 2
m/s) )(3kg/m 127.1(
)m 3.03.0)(0058.0(
2
23
2
2
V
ACF sff
(e) Local convection heat transfer coefficient, Eq. 7-19:
6.68)7255.0()883,52(332.0PrRe332.0Nu 3/12/13/12/1
xx ===
K W/m6.09 2=
== )6.68(
m 3.0
W/m02662.0
Nu x
K
x
k
hx
(f) Average convection heat transfer coefficient, Eq. 7-21:
2.137Nu2)7255.0()883,52(664.0PrRe664.0Nu x
3/12/13/12/1 ====
K W/m12.2 2==
== x
h
K
x
k
h2)2.137(
m 3.0
W/m02662.0
Nu x
(g) Rate of convective heat transfer, Eq. 7-9:
𝑄
̇= ℎ𝐴𝑠(𝑇𝑠−𝑇∞)= (1.22 W
m2∙K)(0.3 ×0.3 m2)(65−15)℃ = 𝟓𝟒.𝟗 𝐖
Ts = 65C
Air
V = 3 m/s
T = 15C
7-8
7-18 Hot engine oil flows over a flat plate. The total drag force and the rate of heat transfer per unit width of the plate are to
be determined.
Assumptions 1 Steady operating conditions exist. 2 The critical Reynolds number is Recr = 5105. 3 Radiation effects are
negligible.
Properties The properties of engine oil at the film temperature of (Ts + T)/2 = (85+35)/2 =60C are (Table A-13)
= 863.9 kg/m3 = 8.565 10-5 m2/s
k = 0.1404 W/mK Pr = 1080
Analysis Noting that L = 10 m, the Reynolds number at the end of
the plate is
Ts = 35C
Oil
V = 2.5 m/s
T = 85C
7-10
7-20E Prob. 7-19E is reconsidered. The local friction and heat transfer coefficients along the plate are to be plotted
against the distance from the leading edge.
Analysis The problem is solved using EES, and the solution is given below.
"GIVEN"
T_air=60 [F]
x=10 [ft]
Vel=7 [ft/s]
"PROPERTIES"
Fluid$='air'
k=Conductivity(Fluid$, T=T_air)
Pr=Prandtl(Fluid$, T=T_air)
rho=Density(Fluid$, T=T_air, P=14.7)
mu=Viscosity(Fluid$, T=T_air)*Convert(lbm/ft-h, lbm/ft-s)
nu=mu/rho
"ANALYSIS"
Re_x=(Vel*x)/nu
"Reynolds number is calculated to be smaller than the critical Re number. The flow is laminar."
Nusselt_x=0.332*Re_x^0.5*Pr^(1/3)
h_x=k/x*Nusselt_x
C_f_x=0.664/Re_x^0.5
x
[ft]
hx
[Btu/h.ft2.F]
Cf,x
0.1
2.848
0.01
0.2
2.014
0.007071
0.3
1.644
0.005774
0.4
1.424
0.005
0.5
1.273
0.004472
0.6
1.163
0.004083
0.7
1.076
0.00378
0.8
1.007
0.003536
0.9
0.9492
0.003333
1
0.9005
0.003162
…
…
…
…
…
…
9.1
0.2985
0.001048
9.2
0.2969
0.001043
9.3
0.2953
0.001037
9.4
0.2937
0.001031
9.5
0.2922
0.001026
9.6
0.2906
0.001021
9.7
0.2891
0.001015
9.8
0.2877
0.00101
9.9
0.2862
0.001005
10
0.2848
0.001
0 2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
0
0.002
0.004
0.006
0.008
0.01
0.012
x [ft]
hx [Btu/h-ft2-F]
Cf,x
hx
Cf,x
7-11
7-21 Laminar flow of a fluid over a flat plate is considered. The change in the drag force and the rate of heat transfer are to be
determined when the free-stream velocity of the fluid is doubled.
Analysis For the laminar flow of a fluid over a flat plate maintained at a constant temperature the drag force is given by
5.0
5.0
2/3
2
5.0
1
2
5.0
1
5.0
2
1
664.0
2
33.1
get werelation,number Reynolds ngSubstituti
2
Re
33.1
Therefore
Re
33.1
where
2
L
AV
V
A
VL
F
V
AF
C
V
ACF
ssD
sD
fsfD
=
=
=
==
When the free-stream velocity of the fluid is doubled, the new value of the drag force on the plate becomes
5.0
5.0
2/3
2
5.0
2)2(664.0
2
)2(
)2(
33.1
L
AV
V
A
LV
FssD
=
=
The ratio of drag forces corresponding to V and 2V is
3/2
2== 2/3
2/3
1
2)2(
V
V
F
F
D
D
We repeat similar calculations for heat transfer rate ratio corresponding to V and 2V
( )
)(Pr0.664=
)(Pr664.0=
)(PrRe664.0)()(
3/1
5.05.0
0.5
3/1
5.0
3/15.0
1
−
−
−
=−
=−=
TTA
L
k
V
TTA
VL
L
k
TTA
L
k
TTANu
L
k
TThAQ
ss
ss
ssssss
When the free-stream velocity of the fluid is doubled, the new value of the heat transfer rate between the fluid and the plate
becomes
)(Pr)0.664(2 3/1
5.05.0
0.5
2
−= TTA
L
k
VQ ss
Then the ratio is
=2=
)(2 0.5
0.5
0.5
1
2 2
V
V
Q
Q=
V
L
7-12
7-22 The ratio of the average convection heat transfer coefficient (h) to the local convection heat transfer coefficient (hx) is to
be determined from a given correlation.
Assumptions1 Steady operating conditions exist. 2 Properties are constant.
Analysis From the given correlation in the form of local Nusselt number, the local convection heat transfer coefficient is
3/18.0 PrRe035.0Nu xx =
3/18.0 PrRe035.0Nu xxx x
k
x
k
or
2.02.03/1
8.0
Pr035.0 −− =
=Cxx
V
khx
where
3/1
8.0
Pr035.0
=
V
kC
At x = L, the local convection heat transfer coefficient is
2.0−
==CLhLx
. The average convection heat transfer coefficient over
the entire plate length is
2.08.0
0
2.0
0
25.125.1
1−− ==== CLL
L
C
dxx
L
C
dxh
L
h
LL
x
Taking the ratio of h to hx at x = L, we get
1.25== −
−
=2.0
2.0
25.1
CL
CL
h
h
Lx
Discussion For constant properties, it should be noted that
25.1Nu/Nu =
=Lx
.
7-14
7-24 Water flows over a large plate. The rate of heat transfer per unit width of the plate is to be determined.
Assumptions 1 Steady operating conditions exist. 2 The critical Reynolds number is Recr = 5105. 3 Radiation effects are
negligible.
Properties The properties of water at the film temperature of (Ts + T)/2 = (10+43.3)/2 = 27C are (Table A-9)
85.5Pr
skg//m 10854.0
C W/m.610.0
kg/m 6.996
3
3
=
=
=
=
−
k
Analysis(a)The Reynolds number is
5
3
)kg/m m)(996.6 m/s)(1.0 3.0(
VL
Ts = 10C
Water
V =30 cm/s
T =43.3C
L = 1 m
7-16
7-26 Hot carbon dioxide exhaust gas is being cooled by flat plates, (a) the local convection heat transfer coefficient at 1 m
from the leading edge, (b) the average convection heat transfer coefficient over the entire plate, and (c) the total heat flux
transfer to the plate are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Surface temperature is uniform throughout the plate. 3 Thermal
properties are constant. 4 The critical Reynolds number is Recr = 5105. 5 Heat transfer by radiation is negligible.
Properties The properties of CO2 at Tf = (220°C + 80°C)/2 = 150°C are k = 0.02652 W/m∙K,
= 1.627 10−5 m2/s, Pr =
0.7445 (from Table A-16).
Analysis (a) The Reynolds number at x = 1 m is
5
)m 1)(m/s 3(
Vx
7-17
7-27 An ASTM B152 copper plate is cooled by air at 20°C. The average convection heat transfer rate from the plate
needed to keep the surface temperature from going above 260°C is to be determined.
Assumptions 1 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 = (260 + 20)°C/2 = 140°C are (Table A-15):Pr =
0.7041, k = 0.03374 W/m∙K, andν = 2.745 × 10−5 m2/s
Analysis The Reynolds number at the end of the plate is
7-20
7-30 Hot engine oil is flowing in parallel over a flat plate, the local convection heat transfer coefficient at 0.2 m from the
leading edge and the average convection heat transfer coefficient over the entire plate are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Surface temperature is uniform throughout the plate. 3 Thermal
properties are constant. 4 The critical Reynolds number is Recr = 5105.
Properties The properties of engine oil at Tf = (150°C + 50°C)/2 = 100°C are k = 0.1367 W/m∙K,
= 2.046 10−5 m2/s, Pr =
279.1 (from Table A-13).
Analysis (a) The Reynolds number at x = 0.2 m is
4
)m 2.0)(m/s 2(
Vx
)m 5.0(
(b) Using the Churchill and Ozoe (1973) relation for Nusselt number, the local convection heat transfer coefficient at x = 0.2
m from the leading edge is
2/13/1
RePr3387.0
xh
2/13/1
RePr3387.0
k
