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7/25/2019 Heat transfer_Internal Flow
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x
y
Inlet region Filled region
Entry length
Fully developed
flow
h
δ
U U U U(x) (x)1 2
(x)2
> U(x)1
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Rr
z
v (r)z
vzmax
z
vz
vφ φvr
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T
wT TwT T
TT
Fully developedThermal entrance region
w
(r,0) T(r,0)T(r,0)
r
(r,0)
w> T(r,0) qw
δ
δΤ
Τ 0y=r −r
r 0
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Æ Æ
Æ Æ
Æ Æ
0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1
Unheated wall Heated wall
Temperature profile Velocity
profile (unchanged)Temperature
profile
Velocity profile
(fully developed)
Æ Æ
Æ
Æ
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x
δ Tδ
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h
xxf d t
f,dh
Æ
dx
m
pv
Tm mT +dTm
pv +d(pv)
q Pdxconvw
=q
x
Inlet Outlet
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T
x
T
T
T i
w
w− Tm
T (x)m∆
Τ
x
Entrance region
Fully developed region
T (x)
w
m
T (x)
Tw− Tm
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Æ
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Æ
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Heated zone
x
r x
r 0
qw
is the wall heat flux
(does not vary with ) x
δ
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Simultaneouslydeveloping flow
Thermallydeveloping flow
Constant wall heat flux
Constant surface temperature
Pr = 0.7
NuD
4.36 f,d(t)
3.66 f,d(t)1
10
20
100
0.005 0.01 0.05 0.1 0.5 1
Æ
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Cross section
Constant q "s Constant T w
NuD
=(hDn)/ k b
a
4.36 3.66−
a
b
1.0
a
b
1.43 3.73 3.08
a
b
2.0 4.12 3.39
b
3.0 4.79 3.96
" 4.0 5.33
8.0 6.49
8.23 7.54α
3.00 2.35
a
"
(parallel plate)
4.44
5.60
3.61 2.98
Æ
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