12. condensation

7/28/2019 12. Condensation

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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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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


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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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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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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( )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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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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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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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?

12. condensation

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