internal flow bl
TRANSCRIPT
8/19/2019 Internal Flow BL
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Temperature field in duct flow pipe
2
21
x
T k
r
T r
r r k
r
T v
x
T uC p
∂
∂+⎟⎟
⎠
⎞⎜⎜
⎝
⎛ ⎟
⎠
⎞⎜⎝
⎛
∂
∂
∂
∂=⎟
⎠
⎞⎜⎝
⎛
∂
∂+
∂
∂ ρ
Neglect axial conduction
02
2
2
2
→∂
∂>>
∂
∂
x
T
r
T
For fully developed laminar flow v=0
)(r
T r
r r x
T u
r
T r
r r
k
x
T uC p
∂
∂
∂
∂=
∂
∂
⎟
⎠
⎞⎜
⎝
⎛
∂
∂
∂
∂=
∂
∂
α
ρ
8/19/2019 Internal Flow BL
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Define mixed mean temperature
mw
w
m
r
m
T T T T
rudr A
udA A
u
rdr r T r u AV
T o
−−=
==
=
∫∫∫
θ
π
π
211
2)()(1
0
Thermal layer is fully developed when θ is
independent of x for any set of B.C. θ is f(r)only
Laminar Duct Flow
8/19/2019 Internal Flow BL
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i.e
oo
mw
w
oomw
o
r r r
k T T
T T
r r r
k
T T
Aq
h
onlyr
r
and x
∂
∂=
−
−
∂
∂−=
−
=
=
=∂
∂
θ
θ θ
θ
)(
)(
0
but θ ≠ θ(x) => h ≠ h(x) and h is defined at thewall r=r o => h =constant ( uniform for fully
developed flow)
Laminar Duct Flow
8/19/2019 Internal Flow BL
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( )
( )dx
dT
dx
dT
x
T
dx
dT
dx
dT
T T
T T
dx
dT
dx
dT T T dx
d T T
dx
dT
dx
dT
T T
T T
T T
x
x
mw
mw
mw
ww
mw
ww
mw
mw
w
θ θ
θ
+−=∂∂
⇒=⎟ ⎠
⎞⎜⎝
⎛ −⎟⎟
⎠
⎞⎜⎜⎝
⎛
−
−−−
⇒=⎟⎟
⎠
⎞⎜⎜
⎝
⎛
−−+⎟
⎠
⎞⎜
⎝
⎛ −
−
⇒=⎟⎟ ⎠
⎞
⎜⎜⎝
⎛
−
−
∂
∂
⇒=∂∂
1
0
011
0
0
Laminar Duct Flow
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 5/23
Two cases:
A- Constant heat flux (qw/A)=constant
.
.
.
(2)
w
w m
w
w m oo
w m
w m
q A
h const T T
q A k
const r T T r
r
T T const
dT dT or uniform
dx dx
θ
⎛ ⎞⎜ ⎟⎝ ⎠ = =
−
⎛ ⎞⎜ ⎟ − ∂⎝ ⎠ = =∂−
⇒ − =
= =
Laminar Duct Flow
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 6/23
But, from equation (2)
Then usedx
dT
dx
T
x
T mw =∂
=
∂
∂
symmetry
r
T r
xT T r r
r
r
V
u
r
T r
r r x
T u
wo
o
m
00
)(
12
1
2
=
∂
∂=
==
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −=
⎟⎟ ⎠
⎞
⎜⎜⎝
⎛ ⎟ ⎠
⎞⎜⎝
⎛
∂
∂
∂
∂=
∂
∂α
Laminar Duct Flow
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 7/23
Case 2:
B- Constant temperature Tw =const.
⎟ ⎠
⎞⎜⎝
⎛
∂
∂
∂
∂=
∂
∂⎟⎟ ⎠
⎞⎜⎜⎝
⎛
−
−⇒
⎟ ⎠
⎞⎜⎝
⎛
∂
∂
∂
∂=
∂
∂⇒
∂
∂=
∂
∂
=∂
∂
+−=∂∂
r
T r
r r x
r T
T T
T T u
r
T r
r r x
T u
x
T
x
T
x
T
dxdT
dxdT
xT
w
mw
w
m
m
w
mw
α
α θ
θ
θ θ
)(
)1(
0
)1(
L
Laminar Duct Flow
w
w m
T T
T T θ
−=
−
8/19/2019 Internal Flow BL
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Iterate for solution, use θ(r) of qw= const, integrate
to get new θ(r) and so on.
Nu =3.66
xc
mm
m
m
mw
m
oo
mw
eT T dx
dT c
dx
dT const T T h
dx
dT r cvr
A
q
T T h A
q
10
)()(
2
)(
1
3
−≈⇒=−
=−
=
−=
π ρ π
Laminar Duct Flow
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 9/23
Example
• The sketch shows a cast ironwater pipe running outdoors.The pipe is 50 mm I.D. In a 5
mm wall thickness, and carriesa constant water flow rate of0.65kg/sec. The water enters theoutdoor section of the pipe at
x=0 with a bulk temperatureTv=15oC. The ambient airsurrounding the pipe is at –20oCand the outside heat transfer
coefficient between the pipeand air is 250 W/m2C. Ice startsforming at the pipe wall at adistance of x1.
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 10/23
Example
Assuming steady state, fully developed conditions
and constant water properties:
a) Find the inside heat transfer coefficient betweenwater and pipe inner wall area for x<x1 (no ice).
b) Find the overall heat transfer coefficient U based
on the pipe inner wall area x<x1.c) Sketch the temperature profile across the pipe for
x<x1. Show water, pipe wall, and air temperature.
d) Calculate the distance x1, and corresponding localwater bulk temperature T1 at the point where ice
begins to form.
8/19/2019 Internal Flow BL
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Example
e) Sketch the ice thickness profile along the pipe wall
for x>x1.
f) At x2>x1, water bulk temperature is 3C. Estimatethe steady state ice thickness.
g) Sketch the temperature profile across the pipe at
point x2. Show all temperatures.h) Sketch the bulk temperature distribution between
x=0 and x2. Clearly show that what happens at x1.
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 12/23
Solution
a)
C mW h
d
Nuk
hk
hd
Nu
Nu
Turbulent AV m
d
m
D
d m
A
d mVd
D
o
ooo
2
31
8.0
2
/6.1041
Pr Re023.0
11823Re
4
4
Re
=
=⇒=
=
=⇐=
====
ρ
µ π µ
π µ µ
ρ
b) Overall heat transfer coefficient
∑== th
i
RUA
1
8/19/2019 Internal Flow BL
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Solution
K mW U
d h
d
k
r r
d
hU
Ld A Ld A
Ah
A
kL
r r A
h A R
AU
AhkL
r r
Ah R
i
oout
ii
oi
ii
ooii
out out
ii
oi
ii
th
ii
out out
i
o
ii
th
2/02.198
0033.000009.000096.02
ln11
2
ln111
1
2
ln1
=⇒
++=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
++=⇒
==
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡++==
++=
∑
∑
π
π π
π
π
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 14/23
Solution
c)
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 15/23
d) @ x=x1, Ti=0C
Solution
C xT T xT R R
R xT
R xT xT fluxheat
R RT xT
W K m R
W K m R
W K m R x x for
o
iiwo
i
b
i
ib
wo
i
o
w
ii
6.5)())(()(
)()()(
/0033.0
/00009.0
/00096.0,
111
111
2
2
2
=+−+
=⇒
−==+
−
=
=
=<
∞
∞
Now from energy balance,
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 16/23
Solution
[ ]
( )
mU r
C m x
C T x xat
eT T T xT
dxC m
U r T xT T xT d
xT cmdx
d T xT U r x
ii
p
o
o
b
x
C m
U r
ob
p
o
ii
b
b
b p
o
oii
po
ii
45.272015206.5ln
2
6.5
)(
2)(
))((
)()(2
1
1
2
=++−=
⇒=⇒=
−=−⇒
−=−−⇒
∆−=−∆
−
∞∞
∞
∞
∞
π
π
π
π
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 17/23
e)
Solution
8/19/2019 Internal Flow BL
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Solution
f) Here Tm=0 oC and T b=3 Cq”Ai= heat flux based on inside area of cast iron
wall.
All resistances are also based on inside surface of
pipe wall.
8/19/2019 Internal Flow BL
http://slidepdf.com/reader/full/internal-flow-bl 19/23
Solution
)4(
1
2
)3()2(
4Pr 023.02
)2(
1
11.
2
)1(12
2ln
/00009.0
/0033.0
8.1
8.1
8.0
31
2
2
L
L
L
K
′+++
−=
′−
=″
⎟ ⎠
⎞⎜⎝
⎛ −
=⎟⎟ ⎠
⎞⎜⎜⎝
⎛
−=
⎟⎟ ⎠ ⎞⎜⎜
⎝ ⎛
−−=′
′⎟⎟
⎠
⎞⎜⎜⎝
⎛ −
=−
=′
−=⇒⎟
⎠ ⎞
⎜⎝ ⎛
−=
=
=
∞
−
iicewo
b
i
mb A
inin
i
i
i
o
i
in
ir
i
ii
ii
r
k R
iice
i
ii
ice
w
o
R R R R
T T
R
T T q
n
hh
d
d
d
m
d
k h
hr
r hd
d R
er K
d d
d
R
W K m R
W K m R
i
i
iceice
δ δ
δ π δ
δ δ
δ δ
8/19/2019 Internal Flow BL
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Solution
Which gives the necessary equations to solve δ.Unfortunately, it is a transcendental equations. We
can now linearize the equations assuring δ<<di and
then we solve it, or we can solve by iteration. Here
we take second path which is more straight forward.
8/19/2019 Internal Flow BL
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Solution
8/19/2019 Internal Flow BL
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Solution
g)
8/19/2019 Internal Flow BL
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h)
Solution