Heat Transfer
Condensation
Dr. M. Subramanian
Department of Chemical Engineering
SSN College of Engineering
August 9, 2020
Dr. M. Subramanian Condensation
Introduction
Dr. M. Subramanian Condensation
Heat Transfer in Condensation
There are different resistances to heat transfer in condensation:
vapour phase resistance (convection and diffusion)
inter phase resistance
liquid phase resistance
The liquid phase resistance is controlling in most of the cases.
Dr. M. Subramanian Condensation
Types of Condensation
Film condensation This is the most common type where the
liquid film is formed which falls under gravity and always wets
the surface.
Drop wise condensation In this case wall is not wetted
completely hence it depends on the wetting behaviour of the
surface.
Dr. M. Subramanian Condensation
Nusselt Theory of Laminar Film Condensation
Nusselt made the following assumptions, in deriving the expression
for heat transfer coefficient for condensation:
The vapour is at rest and does not offer any shear at the
interface.
The flow of liquid is only by gravity and viscosity (no shear at
the interface).
The acceleration of liquid film (inertia) is negligible.
Heat transfer is by pure conduction across the film.
Properties are constant across the film and there is no
interface resistance.
The vapour is at saturation temperature.
Dr. M. Subramanian Condensation
Condensation on Vertical Surfaces
Dr. M. Subramanian Condensation
Condensation on Vertical Surfaces (contd..)
Dr. M. Subramanian Condensation
By force balance, shear force acting upward due to flow of fluid in
the downward direction is equal to the net gravitational force.
τ dx = (ρ − ρv )(δ − y )dx g
Using Newton’s law of viscosity,
du
µ dx = (ρ − ρv )(δ − y )g dx
dy
At the wall surface, the liquid velocity is zero:
u=0 at y = 0
Using this, and integrating the above, we get
g(ρ − ρv ) 1
u(y ) = δy − y 2
µ 2
where δ is the film thickness at any x .
Dr. M. Subramanian Condensation
By definition, mass flow rate = density × volumetric flow rate.
Therefore, for a unit width of plate, mass flow rate m(x ) through
the position x is
ˆ δ
m(x ) = ρ udy
0
We know,
g(ρ − ρv ) 1
u(y ) = δy − y 2
µ 2
Therefore,
ˆ δ
g(ρ − ρv ) 1
m(x ) = ρ δy − y 2 dy
0 µ 2
" #δ
ρg(ρ − ρv ) δy 2 1 3
= − y
µ 2 6 0
" # " #
ρg(ρ − ρv ) δ · δ2 1 ρg(ρ − ρv ) δ 3 δ 3
= − δ3 = −
µ 2 6 µ 2 6
ρg(ρ − ρv )δ 3
=⇒ m(x ) =
3µ
Dr. M. Subramanian Condensation
Differentiating the above, with respect to δ, we get
dm 3ρg(ρ − ρv )δ 2 ρg(ρ − ρv )δ 2
= = (1)
dδ 3µ µ
The rate of heat transfer (dQconden ) associated with rate of
condensation dm is given by
dQconden = λ dm
For a film of unit width, and thickness dx , the rate of transfer by
conduction(dQconduct ), through the film is given by
Tv − Tw
dQconduct = k dx
δ
Dr. M. Subramanian Condensation
Since, rate of condensation of vapor is equal to the rate of
conduction inside the condensate film, we get
dQ = dQconden = dQconduct
Tv − Tw
=⇒ λ dm = k dx
δ
dm Tv − Tw
=k (2)
dx δ
Eqn.(2) / Eqn.(1) gives
dδ µk(Tv − Tw ) 1
= (3)
dx gρ(ρ − ρv )λ δ 3
Integrating Eqn.(3), with the condition δ = 0 for x = 0 gives,
1/4
4µk(Tv − Tw )x
δ(x ) = (4)
gρ(ρ − ρv )λ
Dr. M. Subramanian Condensation
Since, we have established the relation for the thickness of the
condensate layer, the local heat transfer coefficient hx , for
condensation is determined from the definition:
Tv − Tw
hx (Tv − Tw ) = k
δ(x )
k
hx =
δ(x )
Using Eqn.(4) in above, we get
" #1/4
gρ(ρ − ρv )λk 3
hx = (5)
4µ(Tv − Tw )x
From the above expression, we could note that the local heat
transfer coefficient hx varies with the distance as x −1/4 . The
average heat transfer coefficient hm is given by
ˆ
1 L 4
hm = hx dx = hx (5)
L 0 3 L
Dr. M. Subramanian Condensation
Using Eqn.(4) in (5) we get,
" #1/4
gρ(ρ − ρv )λk 3
hm = 0.943
µ(Tv − Tw )L
In the above equation ρ, µ and k are properties of liquid.
For using the above equation for inclined plates, g is replaced by
g cos θ, where θ is the angle between the vertical and the surface.
However, it must be used with caution for larger values of θ and
does not apply if θ = 90◦ . The expression may be used for
condensation on the inner or outer surface a vertical tube of radius
R, if R δ.
The physical properties in the above equation including λ are
evaluated at the film temperature Tf
Tw + T v
Tf =
2
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Condensation on a Horizontal Tube
" #1/4
gρ(ρ − ρv )λk 3
hm = 0.729
µ(Tv − Tw )D
where D is the outside diameter of tube.
For condensation over a sphere of diameter D, the coefficient
0.729 is replaced with 0.826.
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Comparing Horizontal and Vertical Condensation
Comparing equations of horizontal and vertical condensation, we
get
1/4
hm,vert D
= 1.30
hm,horiz L
For L = 2.87 D,
hm,horiz
=1
hm,vert
For L = 100D,
hm,horiz
= 2.44
hm,vert
With this consideration, horizontal tube arrangements are generally
preferred to vertical tube arrangements in condenser design.
Dr. M. Subramanian Condensation
Condensation over Horizontal Tube Banks
For a vertical tier of N horizontal tubes, the relation between heat
transfer coefficient for the one tube (hD ) and the average heat
transfer coefficient over the N tubes (hD,N ) is given as
hD
hD,N =
N 1/4
Such an arrangement is often used in condenser design.
A reduction in the heat transfer coefficient with increasing N may
be attributed to an increase in the average film thickness for each
successive tube due to accumulation of drip from the upper tubes.
Obviously it is advantageous to stagger the tubes as the
accumulation of drip from the upper rows is at least partially offset
by the splashing effects, i.e., by the agitation caused by the drip as
it falls from one tube to another. That is, instead of square pitch
arrangement of tubes, triangular or rotated square pitch
arrangement would be better.
Dr. M. Subramanian Condensation
Tubes Arrangement
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Condensation Inside Horizontal Tube
" #1/4
gρ(ρ − ρv )λk 3
hm = 0.555
µ(Tv − Tw )D
where D is the inside diameter of tube.
Dr. M. Subramanian Condensation
Horizontal Condensers are Better—Why?
Regardless of whether it is in the form of film or droplets, the
condensate provides a resistance to heat transfer between the
vapor and cold surface. Because this resistance increase with
condensate thickness—which increases in the flow direction of
condensate (i.e., downwards by gravity), it is desirable to use short
vertical surfaces or horizontal cylinders.
Dr. M. Subramanian Condensation
Design Consideration of Condensers
Dropwise condensation is superior to filmwise condensation. In
dropwise condensation most of the heat transfer is through drops
of diameters of less than 100-µm and heat transfer rates are more
than an order of magnitude larger than those with filmwise
condensation.
To promote dropwise condensation, it is common practice to use
surface coatings to inhibit wetting. Slicones, Teflon, waxes and
fatty acids are often used for this purpose. However, such coatings
gradually lose their effectiveness due to oxidation, fouling, or
erosion, and filmwise condensation eventually occurs. For this
reason, condenser design calculations are often based on the
assumption of filmwise condensation.
Dr. M. Subramanian Condensation
