6-41
V
dT
2
−
6-43
V
dT
2
−
6-44
6-72E Glycerin is flowing over a flat plate. The velocity and thermal boundary layer thicknesses are to be determined.
Assumptions 1 Isothermal condition exists between the flat plate and fluid flow. 2 Properties are constant.
Properties The properties of glycerin at 50°F are
= 0.03594 ft2/s and Pr = 34561 (Table A-13E).
Analysis The Reynolds number at x = 0.5 ft is
5
)ft 5.0)(ft/s 6(
Vx
6-45
6-73 Water is flowing between two parallel flat plates. The distances from the entrance at which the velocity and thermal
boundary layers meet are to be determined.
Assumptions 1 Isothermal condition exists between the flat plates and fluid flow. 2 Properties are constant.
Properties The properties of water at 20°C are
= 998.0 kg/m3,
= 1.002 10−3 kg/m∙s and Pr = 7.01 (Table A-9).
Analysis The kinematic viscosity for water at 20°C is
skg/m 10002.1 26
3−
−
6-46
Air (1 atm)
Liq. water
Isobutane
Engine oil
Mercury
0.301
6-74E The
t
/
ratios for different fluids in laminar boundary layer flow over a flat plate are to be determined.
Assumptions 1 Isothermal condition exists between the flat plate and fluid flow. 2 Properties are constant.
Properties The Prandtl numbers for the different fluids at 50°F are listed in the following table:
Fluid
Table
Pr
Air (1 atm)
A-15E
0.7336
Liq. water
A-9E
9.44
Isobutane
A-13E
4.114
Engine oil
A-13E
22963
Mercury
A-14E
0.02737
Analysis The velocity and thermal boundary layers for laminar flow can be related using
3/1
6-48
x (m)
Rex
t
0.00
0
0
0
0.10
17628
0.0038
0.0042
0.20
35255
0.0053
0.0059
0.30
52883
0.0065
0.0073
0.40
70511
0.0075
0.0084
0.50
88139
0.0084
0.0094
0.60
105766
0.0092
0.0103
0.70
123394
0.0100
0.0111
0.80
141022
0.0107
0.0119
0.90
158650
0.0113
0.0126
1.00
176277
0.0119
0.0133
1.10
193905
0.0125
0.0139
1.20
211533
0.0130
0.0145
1.30
229161
0.0136
0.0151
1.40
246788
0.0141
0.0157
1.50
264416
0.0146
0.0162
1.60
282044
0.0151
0.0168
1.70
299672
0.0155
0.0173
1.80
317299
0.0160
0.0178
1.90
334927
0.0164
0.0183
2.00
352555
0.0168
0.0187
2.10
370182
0.0173
0.0192
2.20
387810
0.0177
0.0197
2.30
405438
0.0181
0.0201
2.40
423066
0.0184
0.0205
2.50
440693
0.0188
0.0210
2.60
458321
0.0192
0.0214
2.70
475949
0.0196
0.0218
2.80
493577
0.0199
0.0222
2.81
495339
0.0200
0.0222
2.82
497102
0.0200
0.0223
2.83
498865
0.0200
0.0223
6-50
x (m)
Rex
t
0.000
0.000
0
0
0.005
22789
0.0002
0.0001
0.010
45579
0.0002
0.0001
0.015
68368
0.0003
0.0002
0.020
91158
0.0003
0.0002
0.025
113947
0.0004
0.0002
0.030
136737
0.0004
0.0002
0.035
159526
0.0004
0.0003
0.040
182315
0.0005
0.0003
0.045
205105
0.0005
0.0003
0.050
227894
0.0005
0.0003
0.055
250684
0.0005
0.0003
0.060
273473
0.0006
0.0004
0.065
296263
0.0006
0.0004
0.070
319052
0.0006
0.0004
0.075
341842
0.0006
0.0004
0.080
364631
0.0007
0.0004
0.085
387420
0.0007
0.0004
0.090
410210
0.0007
0.0004
0.095
432999
0.0007
0.0004
0.100
455789
0.0007
0.0005
0.105
478578
0.0008
0.0005
0.110
501368
0.0008
0.0005
6-52
6-79 For mercury flowing over a flat plate, the effect of plate location on the velocity and thermal boundary layer
thicknesses is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Properties are constant.
Properties The properties of mercury at 0°C are ν = 1.241 10−7 m2/s, and Pr = 0.0289 (Table A-14).
Analysis The Reynolds number at x = 0.5 m is
55
)m 5.0)(m/s 1.0(Vx
“GIVEN”
nu=1.241e-7 [m^2/s]
Pr=0.0289
“ANALYSIS”
x [m] δ [m] δt [m]
0.0001 0.0000547 0.0001782
0.001 0.0001730 0.0005636
0.01 0.0005470 0.001782
0.02 0.0007735 0.002521
0.03 0.0009474 0.003087
0.04 0.001094 0.003565
0.05 0.001223 0.003986
0.10 0.001730 0.005636
0.15 0.002118 0.006903
0.20 0.002446 0.007971
0.25 0.002735 0.008912
0.30 0.002996 0.009763
0.35 0.003236 0.01054
0.40 0.003459 0.01127
0.50 0.003868 0.01260
0 0.1 0.2 0.3 0.4 0.5
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
x [m]
d, dt [m]
Pr = 0.289 < 1
d
dt
6-53
6-80 For water vapor flowing over a flat plate, the effect of plate location on the velocity and thermal boundary layer
thicknesses is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Properties are constant.
Properties The properties of water vapor at 0°C and 1 atm are ν = 1.114 10−5 m2/s, and Pr = 1.0033 (Table A-16).
Analysis The Reynolds number at x = 0.5 m is
55
)m 5.0)(m/s0 1(
Vx
“GIVEN”
V=10 [m/s]
“PROPERTIES”
Pr=1.0033
“ANALYSIS”
x [m] δ [m] δt [m]
0.0001 0.00005182 0.00005177
0.001 0.0001639 0.0001637
0.01 0.0005182 0.0005177
0.02 0.0007329 0.0007321
0.03 0.0008976 0.0008966
0.04 0.001036 0.001035
0.05 0.001159 0.001158
0.10 0.001639 0.001637
0.15 0.002007 0.002005
0.20 0.002318 0.002315
0.25 0.002591 0.002588
0.30 0.002838 0.002835
0.35 0.003066 0.003063
0.40 0.003278 0.003274
0.50 0.003664 0.003660
0 0.1 0.2 0.3 0.4 0.5
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
x [m]
d, dt [m]
Pr » 1
ddt
»
6-54
6-81 A laminar ideal gas flows over a flat plate. Using the given
3/12/1 PrRe332.0Nu xx =
expression, the formulation for
local convection heat transfer coefficient,
m
xxTVCh )]/([=
, is to be determined.
Assumptions 1 Isothermal condition exists between the flat plate and fluid flow. 2 Gas behaves as ideal gas. 3 Flow is
laminar.
Analysis Using the definitions for Nusselt, Prandtl, and Reynolds numbers, we have
m
xxTVCh )]/([=
xhx
cp
Vx
m
xxTVCh )]/([=
6-55
6-82 For air flowing over a flat plate, the convection heat transfer coefficients and the Nusselt numbers at x = 0.5 m and 0.75
m are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Properties are constant. 3 Edge effects are negligible.
Properties The properties of air at 40°C are k = 0.02662 W/m∙K, ν = 1.702 10−5 m2/s, and Pr = 0.7255 (Table A-15).
Analysis The Reynolds number at x = 0.75 m is
55
)m 75.0)(m/s3(
Vx
6-56
6-83 Cold gas flows in parallel over the surface of an ASTM A240 410S stainless steel plate. The total heat flux on the
plate surface necessary to maintain the surface temperature at the minimum suitable temperature set by the ASME Code for
Process Piping is to be determined.
Assumptions1 Steady operating conditions exist. 2 Uniform surface temperature. 3 Edge effects of plate are negligible.
Properties The gas properties are given as cp = 1.002 kJ/kg·K, k = 0.02057 W/m·K, μ = 1.527×10−5 kg/m·s, and
ρ = 1.514 kg/m3.
Analysis The Reynolds number at x = 1 m is
6-57
6-84 An ASME SB-96 copper-silicon plate is heated by hot air at 200°C flowing in parallel over its surface. The ASME
Boiler and Pressure Vessel Code limits the maximum operating temperature of the plate at 93°C. The variation of the local
heat flux on the plate surface for 0 <x ≤ 1 m is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Uniform surface temperature. 3 Edge effects of plate are negligible.
Properties The air properties are given as cp = 1.016 kJ/kg·K, k = 0.03419 W/m·K, μ = 2.371×10−5 kg/m·s, and
ρ = 0.8412 kg/m3.
Analysis The Reynolds number at x = 1 m is
Using the q
̇s (x) function, the results for the variation of the local heat flux on the plate surface for 0 <x ≤ 1 m are calculated.
They are tabulated and plotted as follows:
x [m]
q
̇s (x) [W/m2]
x [m]
q
̇s (x) [W/m2]
0.01
0.02
0.04
0.06
0.08
0.1
0.2
0.3
5574
3942
2787
2276
1971
1763
1246
1018
0.4
0.5
0.6
0.7
0.8
0.9
1.0
881.4
788.3
719.7
666.3
623.2
587.6
557.4
0 0.2 0.4 0.6 0.8 1
0
1000
2000
3000
4000
5000
6000
x [m]
qs (x) [W/m2]
6-59
00.2 0.4 0.6 0.8 1
0
0.005
0.01
0.015
0.02
dt [m]
0 0.005 0.01 0.015 0.02
0
2000
4000
6000
8000
10000
qs [W/m2]
