Condensation

CONDENSERS

Power plant – water is boiled in boiler and condensed in condenser

Oil refinery - oil is evaporated in distillation column and condensed into liquid fuels like gasoline and kerosene

Desalination plant – water vapor is produced by evaporation from brine and condensed as pure water

Condensation – enthalpy of phase change to be removed by a coolant

Enthalpy of phase change is relatively large, for water (2.5 106 J/kg) and associated heat transfer rates are also large

Heat transfer to phase interface – convective process – complicated by an irregular surface – bubbles and drops

CONDENSATION HEAT TRANSFER

• Film condensation

• Dropwise condensation

FILM CONDENSATIONCondensate wets the surface and forms a liquid film on the surface that slides down under the influence of gravity.

Surface is blanketed by a liquid film of increasing thickness, and this “liquid wall” between the solid surface and the vapor serves as a resistance to heat transfer

Liquid film

80° C

• Condensate film thickness are thin – heat transfer coefficients are large

• Example - steam at a saturation temperature of 305 K condenses on a 2 cm – O.D tube with a wall temperature of 300 K

• Average film thickness - 50m (0.05 mm) and the average heat transfer coefficient – 11,700 W/m2.K

• If the condensate flow rate is small, the surface of the film will be smooth and the flow laminar because

• Temperature difference is small

• Wall is short

• If the condensate flow rate is high, waves will form on the surface to give wavy laminar flow

• If the condensate flow rate is yet higher, the flow becomes turbulent

DROPWISE CONDENSATION

If the condensate does not wet the wall, because either it is dirty or it has been treated with a non-wetting agent, droplets of condensate nucleate at small pits and other imperfections on the surface, and they grow rapidly by direct vapor condensation upon them and by coalescence

When the droplets become sufficiently large, they flow down the surface under the action of gravity and expose bare metal in their tracks, where further droplet nucleation is initiated

THIS IS CALLED DROPWISE CONDENSATION

Droplets

80°C

Droplets slide down when they reach a certain size, clearing the surface and exposing it to vapor.

There is no liquid film in this case to resist heat transfer.

Heat transfer rates that are more than 10 times larger than those associated with film condensation can be achieved with dropwise condensation

Most of the heat transfer is through drops of less than 100m diameter

Thermal resistance of such drops is small; hence, heat transfer coefficients for dropwise condensation are large; values of upto 30000 W/m2.K have been measured.

Hence, dropwise condensation is preferred over filmwise condensation

Considerable efforts are put for non-wetting heat exchanger surfaces

If the surface is treated with non-wetting agent (stearic acid) to promote dropwise condensation, the effect lasts only few days, until the promoter is washed off or oxidised.

Continuous adding of the promoter to the vapour is expensive and contaminates the condensate.

Bonding a polymer such as teflon to the surface is expensive and adds additional thermal resistance

Gold plating is also expensive

Because of lack of sustainability of dropwise condensation, present day condensers are designed based on filmwise condensation

Filmwise condensation – conservative estimate

LAMINAR FLOW CONDENSATION ON A VERTICAL WALL

Temperature of the liquid-vapour interface is the saturation temperature that corresponds to Tsat

Vapour in the descending jet is colder than the vapour reservoir and warmer than the liquid in the film attached to the wall

Tsat

xx

Liquid Vapor

Velocity

0

Liquid

Vapor

Laminar

Turbulent

Wavy

Tw

T

Vapor reservoirCold wall

TsatTw

g

T

Tw

LAMINAR FLOW CONDENSATION ON A VERTICAL WALL

Consider a vertical wall exposed to a saturated vapour at pressure p and saturation temperature Tsat = Tsat(P).

The wall could be flat or could be the outside surface of a vertical tube

If the surface is maintained at a temperature Tw < Tsat, vapour will continuously condense on the wall, and if the liquid phase wets the surface well, will flow down the wall in a thin film

Provided the condensation rate is not too large, there will be no discernable waves on the film surface, and the flow in the film will be laminar

• Fluid dynamics of the flow of a thin liquid film

• Heat transfer during the flow of a thin liquid film

y + dy

d

gh d

satT T H + dH

Hy

u

v

Laminar film of condensate

Tsat

x

T

= Tsat

x

T

Tw

Tsat

x

v

0

Tw

0

x = δ(y)Interface

From reservoir of saturated vapor

0u

Zero shear ,y

ASSUMPTIONS

• Laminar flow and constant properties are assumed for the liquid film

• Gas is assumed to be pure vapour and at a uniform temperature equal to Tsat. The merit of this simplification is that it allows us to focus exclusively on the flow of the liquid film and to neglect the movement of the nearest layers

of vapour

• Shear stress at the liquid-vapour interface is assumed to be negligible

• With no temperature gradient in the vapour, heat transfer to the liquid-vapour interface can occur only by condensation at the interface and not by

conduction from the vapour

2

2

2

2

y

u

x

u

x

P

y

uv

x

uu LL

Steady state two dimensional incompressible flow

gy

v

x

v

y

P

y

vv

x

vu LLL

2

2

2

2

vanishesequationmomentumx,Hence,vu

L~y;~x

gy

v

x

v

dy

dP

y

vv

x

vu LLL

2

2

2

2

2

2

x

vg

y

vv

x

vu LvLL

pressurecHydrostatigpotioninviscidthefromimposedpressuredy

dPv

Neglected, y<<x

FRICTION

L

EFFECTSINKING

vL

INERTIA

L x

vg

y

vv

x

vu 2

2

Assuming inertia is negligible

02

2

gx

vvLL

Boundary conditions

0

00

x

vx

vx

1Cxgx

vvLL

Integrating

21

2

2CxC

xgv vLL

000 2 Cvx

110 CgCxgx

v

x

vx vLvLL

21

2

2CxC

xgv vLL

02 C

vLgC 1

2

2xx

gv

L

vL

xgx

gv vLvLL 2

2

2

2

2

1

xxg

y,xvL

vL

Film thickness is unknown function of (y)

Local mass flow rate per unit width (y)

0

dxvy L

0

2

2

2

1dx

xxgy

L

vLL

y

0

3

322

6

1

2

xxgy

L

vLL

622

L

vLL gy

3

3

L

vLL gy

3

2

L

vLL gy

3

3

L

vLL gy

3

3

L

vLL gbybm

B – width of the plate perpendicular to the plane of paper Flow rate is proportional to the sinking effect - g(L-v)Flow rate is inversely proportional to the liquid viscosity (Friction)

HEAT TRANSFER PROBLEMFilm velocity is low Temperature gradients in the y-direction are negligible since both wall and film surface are isothermal

02

2

dx

Td

211 CxCT;Cdx

dT

wsatwsatw

ww

sat

TTCTCTTxCT

TCTTx

TTx

CxCT

111

2

21

0

wwsat Tx

TTT

This is a linear temperature profile similar to the conduction in a plane wall

Heat flux into the wall = Heat flux across the film

wsatlwsat

wl

TTk

A

QTTh

dx

dTk

l

wsat

wsatl

wsat

wl

k

TT

TTk

TT

dxdT

k

h

lk

h

Determination of film thickness

;

gy

L

vLL

3

3

3

3

L

vLL gbybm

dy

dgbyb

dy

md

L

vLL

3

3 2

Rate of condensation of vapour over a vertical distance dy

Rate of heat transfer from the vapour = Heat releasead as vapour is condensed to the plate through the liquid film

wsatlfg

TTdybkhmdQd

wsat

fg

l TT

h

bk

dy

md

wsat

fg

l

L

vLL TT

h

bk

dy

dgbyb

dy

md

3

3 2

Cyhg

TTk

dyhg

TTkd

TT

h

k

dy

dg

fgvLL

wsatlL

fgvLL

wsatlL

wsat

fg

l

L

vLL

4

3

3

4

3

2

000 C,y

y

hg

TTk

fgvLL

wsatlL

4

4

4

144

yhg

TTky

fgvLL

wsatlL

4

14

4

yTTk

khgkh

wsatlL

lfgvLLl

L

wsatL

lfgvLLL

wsatL

lfgvLLL dyy

LTT

khgdy

yTT

khg

Lh

0

4

14

13

0

4

13

1

44

1

4

13

49430

LTT

khg.h

wsatL

lfgvLLL

3

3

L

vLL gbybm

4

144

yhg

TTky

fgvLL

wsatlL

4

344

3

yhg

TTkgbm

fgvLL

wsatlL

L

vLL

All liquid properties evaluated at

2wsat

fTT

T

Effect of subcoolingRohsenow refined• avoided linear temperature profile• Integral analysis of temperature distribution across the film Temperature profile whose curvature increases with the degree of subcooling Cp,L(Tsat-Tw)

Replace in previous equations

All liquid properties evaluated at

hfg and v are evaluated at the saturation temperature Tsat

wsatL,pfg'fg TTC.hh 680

'fgfg hbyh

2wsat

fTT

T

JAKOB NUMBERIs a measure of degree of subcooling experienced by the liquid film

fg

wsatL,p

h

TTCJa

wsatL,pfgfg TTC.hh 680

Ja.hh fgfg 6801

Reynolds Number

4

44

b

b

P

AD;u;

DuRe c

hL

mL

hmL

LLLLRe

44

LRe

4

3

3

L

vLLgy

LL

vLL

L

gRe

3

3

44

2

3

2

32

3

4

3

4

LL

LvL

ggRe

2

32

3

4

L

LvL

gRe

Lx

ll

h

kLx

h

k

avgLx h4

3h

3

avg

L2L

2L

3

Lx

L2L

2L

h43

k

3

g4

h

k

3

g4Re

3

1

23

1

471

llavg

gRek.h

Hydraulic diameter

44

c

c

P L

A L

ADh

P

δ

L

2

2

44

c

c

P L

A L

ADh

P

4

4

c

c

P D

A D

ADh

P

δ

D

D

δ

L

Wavy Laminar flow over vertical platesAt Reynolds number greater than about 30, it is observed that waves form at the liquid vapour interface although the flow in liquid film remains laminar. The flow in this case is Wavy Laminar Kutateladze (1963) recommended the following relation for wavy laminar condensation over vertical plates

3

1

2221 25081

l.

lwavy,vert

g

.Re.

kReh

lv,Re 180030

820

3

1

2

703814

.

lfgl

wsatlwavy,vert

g

h

TTLk..Re

Turbulent flow over vertical plates (Re > 1800)Labuntsov proposed the following relation

3

1

275050 253588750

l

..l

turbulent,vertg

RePr

kReh

Film condensation on an inclined Plates

Condensatecoshh vertinclined

2

1

3

18806440

3

12

10825

L

.L

.L

l

l

L PrRe.Regk

h

Non-dimensionalised heat transfer coefficients for the wave-free laminar and turbulent flow of condensate on vertical plates

Wave-free laminar

Wavy laminar

Turbulent

l

l

h( v g )

k

2 1 3

Pr = 10

5

3

2

1

10,0001800100010030100.1

1

Re

Problem: Saturated steam at atmospheric pressure condenses on a 2 m high and 3 m wide vertical plate that is maintained at 80C by circulating cooling water through the other side. Determine (a) the rate of heat transfer by condensation to the plate (b) the rate at which the condensate drips off the plate at the bottomSolution: saturated steam at 1 atm condenses on a vertical plate. The rats of heat transfer and condensation are to be determinedAssumptions: 1. steady operating conditions exist 2. The plate is isothermal. 3. The condensate flow is wavy laminar over the entire plate (will be verified). 4. The density of vapour is much smaller than the density of the liquid v<<l

Properties: The properties of water at the saturation temperature of 100C are hfg = 2257 × 103 J/g and v = 0.6 kg/m3. The properties of liquid water at the film temperature 90C are

96281

6750

4206

103260

103150

3965

902

80100

2

26

3

3

.Pr

K.m/W.k

K.kg/JC

s/m.

s.Pa.

m/kg.

TTT

l

pl

l

ll

l

l

wsatf

wsatL,pfgfg TTC.hh 680

801004206680102257 3 .h fg

kg/Jh fg3102314

4

1

3

34

13

4801001031504

675010002314396539658199430

49430

.

.....

LTT

khg.h

wsatL

lfgvLLL

Km

W.hL 222656

s/kg.mmhmQ

W.TTAhQ

sf

wsatsL

13290102314307464

307464801003222562

3

55623

13290

103150

4443

..

.b

mRe

LL

2

1

3

18806440

3

12

10825

L

.L

.L

l

l

L PrRe.Regk

h

2

1

3

18806440

3

126

962815562108255562819

103260

6750

.....

.

.

h ..L

Km

W.hL 247691

3

7691 4 2 3 100 80 2307420

2307420 2314 10 0 9972

L s sat w

sf

Q h A T T . W

Q mh m m . kg / s

3

4 4 4 0 99724221

0 315 10 3L L

m .Re

b .

This confirms that condensation is in turbulent region

Comments: This Reynolds number confirms that condensation is in Wavy laminar domain