5. Condensation
5.1 Introduction
Condensation occurs when the temperature of a
vapor is reduced below its saturation temperature Tsat. This
is done by bringing the vapor into contact with a solid
surface whose temperature Ts is below the saturation
temperature of the vapor. Sometimes, Condensation occurs
when the vapor exposes a liquid or gas at a temperature
below its saturation temperature. For example, when the
temperature of moist air (dry air containing some water
vapor) is reduced to a temperature less than the water vapor,
droplet of water is condensed and suspended in air forming
fog.
Figure(1) Film and dropwise condensation
The condensation over cold surfaces can be
classified to two distinct forms, namely, film condensation and dropwise condensation. In film
condensation, the condensate wets the surface and forms a liquid film on the surface that slides
down under the influence of gravity. In dropwise condensation, the condensed vapor forms
droplets on the surface and the surface is covered by countless droplets of varying diameter as
shown in figure (1). In film condensation, the liquid film represents a thermal resistance of heat
liberated due to condensing process (hfg). On other hand in dropwise condensation, the cold
surface exposes, directly, the vapor. In accordance the heat transfer rate is more than 10 times
larger than that associated with film condensation.
5.2. Film Condensation
As shown in figure (2), the liquid film δ increases
in downward direction. Temperature profile and velocity
profile, also are shown, temperature varies from Ts at the
surface to Tsat at liquid-vapor interface, while the velocity
at the surface is zero and at maximum value at the film
edge. The condensate film flow can be laminar or turbulent
according to the value of Reynolds number (Re), which is
defined in this case as follows:
𝑅𝑒 =
𝐷ℎ 𝜌𝑙 𝑉𝑙
𝜇𝑙
=
4 𝐴𝑐 𝜌𝑙 𝑉𝑙
𝑝 𝜇𝑙
=
4 𝜌𝑙 𝑉𝑙 𝛿
𝜇𝑙
=
4 𝑚̇
(1)
𝑝 𝜇𝑙
Where
𝐷ℎ = 4 𝐴𝑐 /𝑝 = 4 𝛿 = hydraulic diameter of the
condensate flow, m
p = wetted perimeter of the condensate, m
𝐴𝑐 = p 𝛿 = wetted perimeter × film thickness, m2, crosssectional area of the condensate flow at the lowest Figure (2) Film condensation on vertical plate
part of the flow
𝜌𝑙 = density of the liquid, kg/m3
𝜇𝑙 = viscosity of the liquid, kg/m.s
𝑉𝑙 = average velocity of the condensate at the lowest part of the flow, m/s
1
𝑚̇ = 𝜌𝑙 𝑉𝑙 𝐴𝑐 = mass flow rate of the condensate at the lowest part, kg/s
The evaluation of the hydraulic diameter Dh for some common geometries is illustrated in figure
(3). That is, 𝐷ℎ = 4 𝐴𝑐 /𝑝 = 4(𝑝 × 𝛿)/𝑝 = 4 δ.
Figure (3) The wetted perimeter p, the condensate cross-sectional area Acr and the hydraulic diameter
Dh for some common geometries.
The heat released during condensation is latent heat of vaporization hfg J/kg. But since the
condensation temperature is below than saturation temperature and may be taken as the average
temperature of the film between Tsat and Ts, the heat transfer due to condensation is modified to
modified latent heat of vaporizationℎ𝑓𝑔 ∗ , defined as:
ℎ𝑓𝑔 ∗ = ℎ𝑓𝑔 + 0.68 𝐶𝑝𝑙 (𝑇𝑠𝑎𝑡 − 𝑇𝑠 )
(2)
where Cpl is the specific heat of the liquid at the average film temperature. When the vapor starts
condensation at a temperature higher than Tsat, modified latent heat of vaporization takes the
following definition:
ℎ𝑓𝑔 ∗ = ℎ𝑓𝑔 + 0.68 𝐶𝑝𝑙 (𝑇𝑠𝑎𝑡 − 𝑇𝑠 ) + 𝐶𝑝𝑣 (𝑇𝑣 − 𝑇𝑠𝑎𝑡 )
(3)
The rate of heat transfer can be expressed as:
𝑄̇𝑐𝑜𝑛𝑑𝑒𝑛 = ℎ 𝐴𝑠 (𝑇𝑠𝑎𝑡 − 𝑇𝑠 ) = 𝑚̇ ℎ𝑓𝑔 ∗
(4)
where As is the heat transfer area. According to equations (1) and (4), one can derive other
expressions of Reynolds number as:
4 𝑄̇
𝑅𝑒 = 𝑝 𝜇𝑐𝑜𝑛𝑑𝑒𝑛
=
ℎ ∗
𝑙
𝑓𝑔
4 ℎ 𝐴𝑠 (𝑇𝑠𝑎𝑡 −𝑇𝑠 )
𝑝 𝜇𝑙 ℎ𝑓𝑔 ∗
(5)
Equation (5) is used to determine Reynolds number when the condensation heat transfer
coefficient or the rate of heat transfer is known.
2
5.2.1 Flow Regimes
The Reynolds number for condensation on the outer
surfaces of vertical tubes or plates increases in the flow direction
due to increase of the liquid film thickness δ. The flow of liquid
film exhibits different regimes, depending on the value of
Reynolds number. It is observed that the outer surface of the
liquid film remains smooth and wave-free for about Re ≤ 30, as
shown in figure (4), and the flow is clearly laminar. Waves
appear on the free surface of condensate flow as Reynolds
number increases, and the condensate flow becomes fully
turbulent at about Re≈1800. The condensate flow is called
wavy-laminar in the range of 450<Re<1800 and turbulent for
Re>1800.
Figure (4) Flow regime during film
condensation on a vertical plate.
5.2.2 Heat Transfer Correlations for Film Condensation
(a) Vertical Plate
Referring to figure (5), and taking in account the forces
acting on the shown element, one can write and according
to the Newton’s second law of motion, the following
relations:
∑ 𝐹𝑥 = 𝑚 𝑎𝑥 = 0
𝐹𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑 = 𝐹𝑢𝑝𝑤𝑎𝑟𝑑
𝑊𝑒𝑖𝑔ℎ𝑡 = 𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝑠ℎ𝑒𝑎𝑟 𝑓𝑜𝑟𝑐𝑒 + 𝐵𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑟𝑐𝑒
𝜌𝑙 𝑔 (𝛿 − 𝑦) (𝑏𝑑𝑥) = 𝜇𝑙
𝑑𝑢
(𝑏𝑑𝑥) + 𝜌𝑣 𝑔 (𝛿 − 𝑦) (𝑏𝑑𝑥)
𝑑𝑦
And
𝑑𝑢 𝑔 (𝜌𝑙 − 𝜌𝑣 ) (𝛿 − 𝑦)
=
𝑑𝑦
𝜇𝑙
Figure (5) The volume element of condensate
on a vertical plate.
By integration
𝑔 (𝜌𝑙 − 𝜌𝑣 )
𝑦2
(𝑦𝛿 − )
𝜇𝑙
2
The mass flow rate of the condensate at a location x, can in accordance , determine as:
𝑢(𝑦) =
𝛿
𝑚̇(𝑥) = ∫ 𝜌𝑙 𝑢(𝑦)𝑑𝐴 = ∫ 𝜌𝑙 𝑢(𝑦) 𝑏 𝑑𝑦
𝐴
0
And
𝑚̇(𝑥) =
𝑔 𝑏 𝜌𝑙 (𝜌𝑙 − 𝜌𝑣 )𝛿 3
3 𝜇𝑙
3
𝑑𝑚̇
𝑑𝑥
𝑔 𝑏 𝜌𝑙 (𝜌𝑙 −𝜌𝑣 )𝛿 2 𝑑𝛿
=
𝜇𝑙
(6)
𝑑𝑥
Heat transfer through the condensate film is assumed to be by conduction only,
accordingly:
𝑑𝑄̇ = ℎ𝑓𝑔 𝑑𝑚̇ = 𝑘𝑙 (𝑏 𝑑𝑥)
&
𝑑𝑚̇
𝑑𝑥
=
𝑇𝑠𝑎𝑡 − 𝑇𝑠
𝛿
𝑘𝑙 𝑏 𝑇𝑠𝑎𝑡 −𝑇𝑠
ℎ𝑓𝑔
(7)
𝛿
From equation (6) and equation (7) and by integration:
4 𝜇𝑙 𝑘𝑙 (𝑇𝑠𝑎𝑡 −𝑇𝑠 ) 𝑥
𝛿(𝑥) = [
𝑔 𝜌𝑙 (𝜌𝑙 −𝜌𝑣 ) ℎ𝑓𝑔
4
]
(8)
The heat transfer rate from the vapor film to the plate at a location x can be
expressed as:
𝑞̇ 𝑥 = ℎ𝑥 (𝑇𝑠𝑎𝑡 − 𝑇𝑠 ) = 𝑘𝑙
𝑇𝑠𝑎𝑡 −𝑇𝑠
&
𝛿
𝑘
𝑙
ℎ𝑥 = 𝛿(𝑥)
(9)
From equations (8) and (9), one can express hx as:
3 1/4
ℎ𝑥 =
𝑔 𝜌 (𝜌 −𝜌 ) ℎ𝑓𝑔 𝑘𝑙
[ 4 𝑙𝜇 (𝑙𝑇 𝑣−𝑇 ) 𝑥 ]
𝑠
𝑙 𝑠𝑎𝑡
(10)
The average heat transfer coefficient over the entire plate is determined as:
ℎ = ℎ𝑎𝑣𝑒 =
𝐿
∫ ℎ
𝐿 0 𝑥
1
3 1/4
4
𝑑𝑥 = 3 ℎ𝑥=𝐿 =
𝑔 𝜌 (𝜌 −𝜌𝑣 ) ℎ𝑓𝑔 𝑘𝑙
0.943 [ 𝜇𝑙 (𝑇𝑙 −𝑇
]
𝑠) 𝐿
𝑙 𝑠𝑎𝑡
(11)
Taking in account the cooling of the liquid below the saturation temperature, the hfg is
replaced by hfg* in equation (11), one obtains an expression of heat transfer coefficient of
laminar film condensation over a vertical flat plate of height L:
∗
ℎ𝑣𝑒𝑟𝑡 =
3 1/4
𝑔 𝜌 (𝜌 −𝜌 ) ℎ𝑓𝑔 𝑘𝑙
0.943 [ 𝜇𝑙 (𝑇𝑙 𝑣−𝑇 ) 𝐿 ]
𝑠
𝑙 𝑠𝑎𝑡
(W/m2.K), 0<Re<30
4
(12)
Taking in account that ρv<< ρl
number can be written as:
(ρl - ρv) = ρl
&
and accordingly, the Reynolds
3
4𝑔𝜌𝑙 (𝜌𝑙 − 𝜌𝑣 )𝛿 3 4𝑔𝜌𝑙 2 𝑘𝑙 3
4𝑔
𝑘𝑙
𝑅𝑒 ≅
=
(
) =
(
)
3𝜇𝑙 2
3𝜇𝑙 2 ℎ𝑥=𝐿
3𝜈𝑙 2 3ℎ𝑣𝑒𝑟𝑡 /4
And the heat transfer coefficient hvert in terms of Re becomes:
1/3
𝑔
ℎ𝑣𝑒𝑟𝑡 ≅ 1.47 𝑘𝑙 𝑅𝑒 −1/3 (𝑣 2 )
0<Re<30
𝑙
&
𝜌𝑣 ≪ 𝜌𝑙
Wavy Laminar Flow on Vertical Plates
In wavy laminar condensate flow, the heat transfer rate is greater than that of laminar flow,
where 30<Re<1800. Heat transfer coefficient in this case takes the form:
𝑅𝑒 𝑘
𝑔
1/3
ℎ𝑣𝑒𝑟𝑡,𝑤𝑎𝑣𝑦 = 1.08 𝑅𝑒 1.22𝑙 −5.2 (𝑣 2 )
𝑙
30<Re<1800
&
𝜌𝑣 ≪ 𝜌𝑙
Another simpler relation for wavy laminar flow, is as follows:
ℎ𝑣𝑒𝑟𝑡,𝑤𝑎𝑣𝑦 = 0.8 𝑅𝑒 0.11 ℎ𝑣𝑒𝑟𝑡 (𝑠𝑚𝑜𝑜𝑡ℎ)
Knowing heat transfer coefficient, one can write an expression of Reynolds number as:
𝑅𝑒𝑣𝑒𝑟𝑡,𝑤𝑎𝑣𝑦 = [4.81 +
3.7𝐿𝑘𝑙 (𝑇𝑠𝑎𝑡 −𝑇𝑠 )
∗
𝜇𝑙 ℎ𝑓𝑔
𝑔
1/3 0.820
(𝑣2 )
]
,
𝑙
𝜌𝑣 ≪ 𝜌𝑙
Turbulent Flow on Vertical Plates
At Reynolds number of about 1800, the condensate flow becomes turbulent. For 𝜌𝑣 ≪ 𝜌𝑙 ,
the following relation for the heat transfer coefficient of turbulent flow can be written as:
𝑅𝑒 𝑘
𝑔
1/3
𝑙
ℎ𝑣𝑒𝑟𝑡,𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 = 8750+58 𝑃𝑟−0.5 (𝑅𝑒
0.75 −253) (𝑣 2 )
𝑙
, Re > 1800
&
𝜌𝑣 ≪ 𝜌𝑙
And
4/3
1/3
𝑅𝑒𝑣𝑒𝑟𝑡,𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡
0.0690 𝐿 𝑘𝑙 𝑃𝑟 0.5 (𝑇𝑠𝑎𝑡 − 𝑇𝑠 ) 𝑔
=[
( 2)
∗
𝜇𝑙 ℎ𝑓𝑔
𝑣𝑙
5
− 151 𝑃𝑟 0.5 + 253]
(b) Inclined Plates
The expression of vertical plate is used by replacing g by g cos θ,
as shown in figure (6).
The heat transfer coefficient for laminar flow and as an
approximation for wavy laminar, can be written as:
ℎ𝑖𝑛𝑐𝑙𝑖𝑛𝑒𝑑 = ℎ𝑣𝑒𝑟𝑡 (cos 𝜃)1/4
(c) Vertical Tubes
The expression of heat transfer coefficient of vertical plate;
equation (12) is valid in this case, since the film thickness is very
small compared with the diameter of the tube:
3 1/4
∗
ℎ𝑣𝑒𝑟𝑡 =
Figure (6) Film condensation on
an inclined plate.
𝑔 𝜌 (𝜌 −𝜌 ) ℎ𝑓𝑔 𝑘𝑙
0.943 [ 𝑙 ( 𝑙 𝑣 )
]
𝜇𝑙 𝑇𝑠𝑎𝑡 −𝑇𝑠 𝐿
(W/m2.K), 0<Re<30
(12)
(d) Horizontal Tubes and Spheres
In this case, there is an expression of heat transfer coefficient similar that of vertical tube
as:
∗
3
1/4
𝑔 𝜌𝑙 (𝜌𝑙 − 𝜌𝑣 ) ℎ𝑓𝑔 𝑘𝑙
ℎℎ𝑜𝑟𝑖𝑧 = 0.729 [
]
𝜇𝑙 (𝑇𝑠𝑎𝑡 − 𝑇𝑠 ) 𝐷
And for sphere the numerical constant is 0.81.
(e) Horizontal Tube Banks
As shown in figure (7), it is expected that heat transfer coefficient is
smaller than of that of single horizontal tube. The expression of heat
transfer coefficient, in this case, is as follows:
∗
3
1/4
𝑔 𝜌𝑙 (𝜌𝑙 − 𝜌𝑣 ) ℎ𝑓𝑔 𝑘𝑙
ℎℎ𝑜𝑟𝑖𝑧 = 0.729 [
]
𝜇𝑙 (𝑇𝑠𝑎𝑡 − 𝑇𝑠 ) 𝑁 𝐷
=
1
ℎ
𝑁1/4 ℎ𝑜𝑟𝑖𝑧,1 𝑡𝑢𝑏𝑒
Where N is the number of tubes per tier.
Figure (7) Film
condensation on
vertical tier of
horizontal tubes
5.3 Film Condensation inside Horizontal Tubes
In such a case, the vapor velocity has a considered effect on the film thickness and hence on
the coefficient of heat transfer coefficient. For low vapor velocity, the following relation is
used, see figure (8):
3
ℎ𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 =
𝑔 𝜌 (𝜌 −𝜌𝑣 ) 𝑘𝑙
0.555 [ 𝜇 𝑙 (𝑇 𝑙 −𝑇
𝑠)
𝑙 𝑠𝑎𝑡
1/4
3
(ℎ𝑓𝑔 + 8 𝐶𝑝 (𝑇𝑠𝑎𝑡 − 𝑇𝑠 ))]
6
For
𝜌𝑣 𝑉 𝐷
𝑅𝑒𝑣𝑎𝑝𝑜𝑟 = ( 𝑣 )
< 35,000
𝜇𝑣 𝑖𝑛𝑙𝑒𝑡
5.4. Dropwise Condensation
Figure (8) Condensate flow in a horizontal
tube.
In this case, the chance of vapor to be in contact with cold surface is greater than that of
film condensation and hence heat transfer coefficient is greater than ten times of that of
film condensation. The correlation for dropwise condensation of steam on copper surfaces:
51,104 + 2044 𝑇𝑠𝑎𝑡
22°𝐶 < 𝑇𝑠𝑎𝑡 < 100°𝐶
ℎ𝑑𝑟𝑜𝑝𝑤𝑖𝑠𝑒 = {
255,310
𝑇𝑠𝑎𝑡 > 100℃
7