12. condensation
TRANSCRIPT
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12. CONDENSATION
Similar to natural (free) convection
But with phase change steam (vapor) to liquid
Liquid film or drops that form by condensation movesalways naturally downwards the role of gravitation
Takes place where the surface temperature is below
the saturation temperature Two modes film and dropwise condensation
Under what conditions
the dropwise condensation
can be reached and
maintained?
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Regardless of the condensation mode, the condensate
creates a resistance to heat transfer between the
vapor and the surface Resistance increases with condensate thickness
How to avoid an increase in condensate thickness?
Dropwise condensation is superior to film condensation
Various surfactants used to inhibit wetting (teflon,
silicon, waxes, fatty acids etc.) Coatings gradually lose their effectiveness fouling,
oxidation film condensation occurs
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Laminar film condensation on a vertical wall
2
2
f y
ugdx
dp
1
y
uvx
uu
++=
+
The same momentum PDE as for the natural convection
(gravitation in + x direction):
0p =No movement iny - direction
xp Same in the condensate filmand in the vapor
gp
v
=
Pressure difference results fromthe weight of the fluid column
2
2
f
v
y
ug
1
y
uv
dx
uu
+
=
+
x
y
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Laminar film condensation on a vertical wall
Assumption:
Momentum and energy transfer by advection in the film is
negligible what does it mean?
2
2
y
Ta
y
Tv
x
Tu
=
+
Energy PDE:
Momentum PDE:2
2
f
v
y
ug
1y
uvdx
uu+
=+
2
2
yTa
yTv
xTu
=
+
Energy PDE:
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x
y
Laminar film condensation on a vertical wall
Momentum equation:
0g
dy
ud vf2
2
=
+ y0,y
u == at
B.C.: u=0 at y=0
Solution:
21vf
CyC2
gu ++= ( )
=
22vf
y
y22
gu
Energy equation:
0dy
Td2
2
=B.C.: T=T
w
at y=0
T=Tsat at y=
21CyCT +=
( ) y
TTTT wsatw +=
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Laminar film condensation on a vertical wall
In both equations unknown condensate film thickness .
How to determine it? And how it develops along the surface?
Lets express the condensate mass flow in the film:
=)(
)(x
0fudyxm&
( )
=22
vf
y
y2
2
gu
3vf3vff
3
g
3
gm
=
=&
( )d
g
dx
d
mdx
x
mmd
2vf
=
=
=
&&&
Net inflow of condensate into the film:
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Laminar film condensation on a vertical wall
This inflow of condensate into the film is due to the release of
latent heat from the vapor:
mdlQd 23 && =
This heat transfer rate (tepeln tok) is transferred through the
condensate film by conduction:
mdldx.1
TTQd 23
wsat && =
=
( ) dxTTl
1
md wsat
23
=&
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Laminar film condensation on a vertical wall
( )d
gmd
2vf
=& from mass balance
( ) dxTTl1
md wsat
23=& from energy balance
( )( )dxTT
l
1
d
gwsat
23
2vf
=
Equation for the condensate film thickness:
B.C.: = 0 atx= 0
( )( )
41
vf23
wsat
gl
xTT4
=
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x
y
Heat transfer coefficient
( )wsatx0y TTdy
dTq == =&
Determine on the basis of the equality of latent heat transfer
by conduction and convection across the condensate film:
Linear temperature profile across the film
( )( )wsatx
wsat TT
TT=
( )( )
41
wsat
3vf23
xxTT4
gl
==
Try to analyze the influence of
major parameters on x
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Heat transfer coefficient
Local Nusselt number:
Average heat transfer coefficient on the wall
of the height L :
x
y
( )( )
41
wsat
3vf23x
x TT
xgl
0,707
x
Nu
=
L
L
0x
3
4dx
L
1 ==
Average Nusselt number:
( )
( )
41
wsat
3vf23
L
TT
Lgl0,9428
LNu
=
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Heat transfer coefficient
Or in a different form:
( )
41
III.
wsatp
23
II.
pf
I.
f
vf2
3
LTTc
l
c
gL0.9428Nu
=44 344 2132144 344 21
I Archimedes number
II Prandtl number
III - Jacob (or K) number
( ) 41L KPrAr0,9428Nu =
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Heat transfer coefficient
Experiments reveal that heat transfer more intensive than
predicted from the theory. Why?
Condensate film waves and the film isthus thinner - lower thermal resistance
higher heat transfer
( ) 41L KPrAr1,13Nu =
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Heat transfer coefficient on horizontal tubes
Height of the wall is replaced by the diameter of the
tube and the constant is lower:
( )( )
41
wsat
3
vf23d TT
dgl0,729Nu
=
An obvious question:
Whats better for the condensation?
Horizontal or vertical tube?