8.3.1 Regimes of Filmwise Condensation - Thermal-Fluids ...

Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 1

Chapter 8: Condensation8.3 Filmwise Condensation

8.3 Filmwise Condensation8.3.1 Regimes of Filmwise Condensation

Figure 8.11 Flow regimes of film condensate on a vertical wall.


Chapter 8: Condensation

In order to use boundary layer theory to describe the condensation problem, the following assumptions must be made:

1. Tw, TI, Tv∞, ωlv∞, uvI, uI and uv∞ are independent of x.1. The condensate film and binary vapor boundary layers both develop

from the leading edge of the vertical surface, x=0.2. Condensation takes place only at the vapor-liquid interface. In other

words, no condensation takes place within the binary vapor boundary layer in the form of a mist or fog.

3. Both temperature and velocity are continuous at the vapor-liquid interface.

4. The condensate is miscible.5. The physical properties of the system are assumed to be constant with

respect to concentration and temperature except in the case of buoyancy terms.

6. The density of the condensate liquid is assumed to be much greater than that of the binary vapor.

7. The vapor mixture can be treated as an ideal gas.

8.3.2 Generalized Governing Equations for Laminar Film Condensation including Binary Vapor

8.3 Filmwise Condensation



Figure 8.12 Physical model and coordinate system for condensation of a binary vapor mixture.




The governing equations for the laminar film condensation of a binary vapor mixture can be given by taking the above assumptions into account and using boundary layer analysis, i.e.,For the condensate film:

(8.58)

(8.59)

(8.60)

0u vx y

∂ ∂+ =∂ ∂l l

2

2

1u u u dpu v gx y y dx

νρ

∂ ∂ ∂+ = + −∂ ∂ ∂l l l

l l ll

2

2

T T Tu vx y y

α∂ ∂ ∂+ =∂ ∂ ∂l l l

l l l




For the vapor boundary layer:(8.61)

(8.62)

(8.63)

(8.64)

Isobaric specific heat difference of the binary vapor

(8.65)

0=∂

∂+

∂∂

yv

xu vv

2

2 1v v v vv v v

v

u u uu v gx y y

ρνρ

∞ ∂ ∂ ∂+ = + − ∂ ∂ ∂ 2

1122

v v v v vv v v p

T T T Tu v Dcx y y y y

ωα∂ ∂ ∂ ∂ ∂+ = +∂ ∂ ∂ ∂ ∂

21 1 1

2v v v

v vu v Dx y y

ω ω ω∂ ∂ ∂+ =∂ ∂ ∂

1 2 1 212

1 1 2 2

p v p v p v p vp

p v v p v v pv

c c c cc

c c cω ω− −

= =+




Definitions of the terms in eqs. (8.58-8.64)(8.66)

(8.67) where

(8.68)

(8.69) Partial pressures of the system are determined by

(8.70)

(8.71)

11

vv

v

ρωρ

=

22

vv

v

ρωρ

=

vvv 21 ρρρ +=

1 2 1v vω ω+ =

1

1 21

2 1

1 v

v

Mpp M

ωω

−

= +

1

2 12

1 2

1 v

v

Mpp M

ωω

−

= +




Boundary conditions at the surface of the cold wall(8.72)

(8.73)

(8.74)

Boundary conditions at locations far from the cold wall(8.75)

(8.76)

(8.77)

0, 0u y= =l

0, 0v y= =l

, 0wT T y= =l

,v vu u y∞= → ∞

,v vT T y∞= → ∞

1 1 ,v v yω ω ∞= → ∞




Boundary conditions that exist at the liquid-vapor interface(8.78)

(8.79)

(8.80)

(8.81)

(8.82)

(8.83)

The mass fluxes of the vapor in the binary vapor system are (8.84)


vu u uδ δ δ= =l

vv

uuy yδ δ

µ µ ∂∂ = ∂ ∂ l

l

1 2v v vd du v u v m m mdx dxδ δ

δ δρ ρ ′′ ′′ ′′− = − = = + l l l & & &

vT T Tδ δ δ= =l

vv v

TTk h m ky yδ δ

∂∂ ′′= + ∂ ∂ l

l l &

1 1v vδω ω=

11 12 1( )1 2m D m + m

yωρ ω∂′′ ′′ ′′= − +

∂& & &



(8.85)the molar fluxes of the vapor in the binary vapor system are

(8.86)

(8.87) Equations (8.86) and (8.87) are often expressed in terms of partial

pressure, i.e., (8.88)

(8.89) The mass fraction of component 1 in the condensate film can be

found from the following expression:(8.90)

221 2 2( )2 1m D m + m

yωρ ω∂′′ ′′ ′′= − +

∂& & &

11 12 1 T

cn D x ny

∂′′ ′′= − +∂

& &

22 21 2 T

cn D x ny

∂′′ ′′= − +∂

& &

1 11 T

u

p pDn nR T y p

∂′′ ′′= − +∂

& &

2 22 T

u

p pDn nR T y p

∂′′ ′′= − +∂

& &

11

1 2

x

x x

mm m

ω′′

=′′ ′′+l&

& &



The flow is laminar. Constant fluid properties are assumed. Subcooling of the liquid is negligible in the energy balance, i.e., all

condensation occurs at the saturation temperature corresponding to the pressure in the liquid film near the wall.

Inertia and convection effects are negligible in the boundary layer momentum and energy equations, respectively.

The vapor is assumed stagnant and therefore shear stress is considered to be negligible at the liquid-vapor interface.

The liquid-vapor interface is smooth, i.e., condensate film is laminar and not in the wavy or turbulent stages.

8.3.3 Filmwise Condensation in a Stagnant Pure Vapor Reservoir8.3.3.1 Laminar Flow Regime




Figure 8.13 Overview of the control volume under

consideration in the Nusselt analysis.



Pressure in the liquid film(8.91)

Substituting eq. (8.91) into eq. (8.59)(8.92)

Neglecting the inertia term, eq. (8.92) becomes(8.93)

Integrating twice and applying boundary conditions(8.94)

Mass flow rate per unit width of surface(8.95)


vdp gdx

ρ=

2

2 ( )vu u uu v gx y y

ρ µ ρ ρ ∂ ∂ ∂+ = + − ∂ ∂ ∂ l l l

2

2 ( )vu g

yρ ρ

µ∂ = −∂ l

l

2( )( , )2

v g yu x y yρ ρ δµ

−= −

l

l

3

0

( )3

v gudyδ ρ ρ ρ δρ

µ−Γ = =∫ l l

ll



Heat flux across the film thickness(8.96)

Heat transfer rate per unit width for the control volume(8.97)

where . Latent heat effects of condensation dominate the process

(8.98) dΓ is found by differentiating the expression for mass flow rate per

unit surface eq. (8.95)

(8.99) Substituting into eq. (8.98)

(8.100)


( )sat wk T Tqδ

−′′ = l

k Tdq dxδ∆′ = l

sat wT T T∆ = −

vdq h d′ = Γl

2( )v gd dρ ρ ρ δ δµ−

Γ = l l

l

3

( )v v

k Tddx gh

µδδρ ρ ρ

∆=−l l

l l l



δ can be found be integrating eq. (8.100) and applying boundary conditions

(8.101) Local heat transfer coefficient

(8.102)

Local Nusselt number(8.103)

Mean heat transfer coefficient(8.104)

Substituting eq. (8.102) into eq. (8.104) and integrating(8.105)


1/ 44( )v v

k x Tgh

µδρ ρ ρ

∆= − l l

l l l1/ 4

3/ 4 ( )4

v vx

ghh kx T

ρ ρ ρµ

−= ∆ l l l

ll

1/ 43( )4

x v vx

h x gh xNuk k T

ρ ρ ρµ

−= = ∆ l l l

l l l

0

1 ( )L

L xh h x dxL

= ∫1/ 43( )0.943L v v

Lgh Lh LNu

k k Tρ ρ ρ

µ −= = ∆ l l l

l l l



Reynold’s number(8.106)

Substituting eq. (8.95) into eq. (8.107), Reynold’s number for laminar film condensation becomes

(8.107)

Substituting eqs. (8.106) and (8.101) into eqs. (8.103) and (8.105)

(8.108)

(8.109)

Energy balance at the interface that takes subcooling of the liquid into account

(8.110)


4Reδ µΓ=l

3

2

4 ( )Re3

v gδ

ρ ρ ρ δµ−= l l

l

1/ 321/ 31.1Re

( )x

v

hk g δ

µρ ρ ρ

− = −

l

l l l1/ 32

1/ 31.47Re( )v

hk g δ

µρ ρ ρ

− = −

l

l l l

0( )v p sat

k Tdq d dh c u T T dydx dx dx

δρ

δ′ ∆ Γ= = + −∫l

l l l



Substituting the velocity profile eq. (8.94) and using a linear temperature profile

(8.111)

To evaluate eq. (8.110) an energy balance (8.112)

where(8.113)

Rohsnow included convection and liquid subcooling effects to develop

(8.114)


1sat

sat w

T T yT T δ

− = −−

vk T dh

dxδ∆ Γ′=l l

l

( )318

p sat wv v

v

c T Th h

h − ′ = +

ll l

l

( )1 0.68 p sat w

v vv

c T Th h

h − ′ = +

ll l

l



Reynold’s number for onset of waves is related to Archimedes number by (8.115)

Archimedes number is defined as

(8.116)

Kutateladze gave the following correlation for the mean Nusselt number of film condensation on a vertical plate where wave effects are present

(8.117)


8.3.3.2 Wavy Condensate Regime

1/ 5Re 9.3Arδ > l

3/ 2

2 1/ 2 3/ 2( )v

Arg

ρ σµ ρ ρ

=−

ll

l l

2 1/3 1.22Re , 30 Re 1800

( / ) Re 5.2Lkhg

δδ

δν= ≤ ≤

−l

l



Mass flow rate of the condensate(8.118)


Rearranging eq. (8.119)(8.120)

Combining eq. (8.117) and (8.120) yields the Reynolds number for film condensation with waves

(8.121)


4 ( )Re L sat w

v

h L T Thδ ν

−=′l l

( )L sat w

v v

h L T Tqh h

′ −Γ = =′ ′l l

Re4 ( )

vL

sat w

hhL T T

δ ν′=

−l l

0.821/3

2

3.7 ( )Re 4.81 sat w

v

Lk T T ghδ µ ν

− = + ′

l

l l l



Labuntsov (1957) recommended the following empirical correlation:

(8.122)

Butterworth (1983) obtained the average heat transfer coefficient for film condensation that covers laminar, wavy laminar, and turbulent flow by combining eqs. (8.109), (8.117) and (8.122) as follows

(8.124)


8.3.3.3 Turbulent Film Regime

2 1/3 0.5 0.75Re , 1 Re 7200

( / ) 8750 58Pr (Re 253)Lkhg

δδ

δν −= < ≥+ −

l

l l

1/320.25 0.50.023Re Pr , Pr 10

( )x

v

hk g δ

µρ ρ ρ

− = ≥ −

ll l

l l l



The Reynolds number, , is needed in order to use eq. (8.124) to determine the heat transfer coefficient for turbulent film condensation.

Equation (8.120) was obtained by energy balance and it is valid for all film condensation regimes.

Combining eqs. (8.124) and (8.120), the Reynolds number for turbulent flow is obtained:

(8.125)


Reδ

4/ 31/30.50.5

20.069 Pr ( )Re 151Pr 253sat w

v

Lk T T ghδ µ ν

− = − + ′

l ll

l l l



Example 8.3 Saturated steam at 1 atm condenses on a vertical wall with

a height of L= 1 m and width of b= 1.5 m. The surface temperature of the vertical wall is 80 ˚C. What are the average heat transfer and condensation rates?



Solution: The saturation temperature of steam at 1 atm is Tsat =100

˚C. The vapor density at this temperature is ρv = 0.5974 kg/m3, and the latent heat of vaporization is = 2251.2 kJ/kg.

The liquid properties evaluated at Tf = (Tsat + Tw)= 90 ˚C are = 965.3 kg/m3, = 4.206 kJ/kg-K, = 0.315x10-3kg/m-s, = 0.675 W/m-K, and = 0.326x10-6 m2/s.

The revised latent heat of vaporization is

vhl

ρ l pc l µ lkl /ν µ ρ=l l l

' ( )1 0.68

4.206 (100 80) 2251.2 1 0.68 2308.4 kJ/kg2251.2

p sat wv v

v

c T Th h

h − = +

× − = × + =

ll l

l



Assuming the film condensation is laminar (as will be verified later), the heat transfer coefficient can be obtained from eq. (8.105), i.e.,

The heat transfer rate is then

The condensation rate is

1/ 43

1/ 43 3

3

2

( )0.943

965.3 (965.3 0.5974) 9.8 0.675 2308.4 100.9430.315 10 (100 80) 1

5340.2W/m -K

v vg k hhT L

ρ ρ ρµ

−

−= ∆

× − × × × ×= × × − × =

l l l l

l

5( ) 5340.2 1 1.5 (100 80) 1.602 10 Wsat wq hLb T T= − = × × × − = ×

5

3

1.602 10 0.0694 kg/s2308.4 10v

qmh

×= = =′ ×l

&



The assumption of laminar film condensation is now checked by obtaining the Reynolds number defined in eq. (8.106), i.e.,

which is greater than 30 and below 1800. This means that the assumption of laminar film condensation is invalid and it is necessary to consider the effect of waves on the film condensation.

It should be kept in mind the above Reynolds number of 588 is obtained by assuming laminar film condensation. For film condensation with wavy effects, the Reynolds number should be obtained from eq. (8.121), i.e.,

3

4 4 4 0.0694 5880.315 10 1.5

mRebδ µ µ −

Γ ×= = = =× ×l l

&

0.821/ 3

2

0.821/ 3

3 3 6 2

3.7 ( )Re 4.81

3.7 1 0.675 (100 80) 9.84.810.315 10 2308.4 10 (0.326 10 )

730.9

sat w

v

Lk T T ghδ µ ν

− −

− = + ′

× × × −= + × × × × =

l

l l l



The heat transfer coefficient is obtained from eq. (8.117), i.e.,

The heat transfer rate is then

The condensation rate is

which is much higher than the condensation rate obtained by assuming laminar film condensation.

2 1/3 1.22

26 2 1/ 3 1.22

Re( / ) Re 5.2

0.675 730.9 7160 W/m -K[(0.326 10 ) / 9.8] 730.9 5.2

khg

δ

δν

−

=−

= =× −

l

l

5( ) 7160 1 1.5 (100 80) 2.15 10 Wsat wq hLb T T= − = × × × − = ×

5

3

2.15 10 0.0931 kg/s2308.4 10v

qmh

×= = =′ ×l

&



8.3.4 Effects of Vapor Motion8.3.4.1 Laminar Condensate Flow The boundary layer momentum equation

(8.126)

Another pressure gradient exists along with the hydrostatic pressure gradient

(8.127)

The superimposed pressure gradient can be combined into a fictitious density

(8.128)

Substituting eq. (8.128) into eqs. (8.127) and (8.126)

(8.129)


2

2

u u dp uu v gx y dx y

ρ µ ρ ∂ ∂ ∂+ = − + + ∂ ∂ ∂ l l l

vm

dp dpgdx dx

ρ = + * 1v v

m

dpg dx

ρ ρ = + 2

*2 ( )v

u u uu v gx y y

ρ µ ρ ρ ∂ ∂ ∂+ = + − ∂ ∂ ∂ l l l



Neglecting inertia eq. (8.129) becomes(8.130)

Integrating eq. (8.130) twice and applying boundary conditions(8.131)

Mass flow rate per unit width of surface(8.132)

Heat flux across the film thickness can be obtained by Fourier’s Law(8.133)

Heat flow(8.134)

No subcooling, latent heat efffects of condensation dominate, thus(8.135)


2*

2 ( )vu g

yρ ρ

µ∂ = −∂ l

l* 2( )( , )

2v Ig yyu x y yρ ρ τδ

µ µ −= − +

l

l l

* 3 2

0

( )3 2

v Igudyδ ρ ρ ρ δ τ ρ δρ

µ µ−Γ = = +∫ l l l

ll l

( )sat wk T Tq

δ−

′′ = l

k Tdq dxδ∆′ = l

vdq h d′ = Γl



Differentiating eq. (8.135)(8.136)

Substituting into the mass and energy equation(8.137)

Integrating eq. (8.137) and applying boundary conditions(8.138)

Eq. (8.138) can be non-dimensionalized by(8.139)

(8.140)


* 2( )v Igd dρ ρ ρ δ τ ρ δ δµ µ

−Γ = +

l l l

l l

( )* 3 2v v I v

k Tddx gh h

µδρ ρ ρ δ τ ρ δ

∆=− +

l l

l l l l l

( ) ( )3

4* *

4 43

I

v v v

k x Tg gh

τ δ µδρ ρ ρ ρ ρ

∆ + =− −

l l

l l l l

FLδδ =*

* 4Pr

p

F v

c TxxL h

∆ =

l

l l



(8.141) where

(8.142) Eq. (8.138) can be rewritten as

(8.143) Nusselt and Reynolds number for laminar flow

with finite vapor shear(8.144)

(8.145)


( )*

*I

IF vL g

ττρ ρ

=−l

( )

1/ 32

*Fv

Lg

µρ ρ ρ

=

−

l

l l

( ) 3* * 4 * *4( )3 Ix δ δ τ= +

( ) ( )3 2* * *

* *

243

IL FF

h LNuk x x

δ δ τ= = +

l

( ) ( )3 2* * *4 4Re 23 Iδ δ δ τ

µΓ= = +l



For the case that gravitational force is negligible compared with the interfacial shear force imposed by the co-current vapor flow, Butterworth (1981) recommended the following correlation for local heat transfer coefficient:

(8.146)

(8.147)


1/ 2 1/ 21.41Re ( )x INu δ τ− +=

( )

1/32x

xv

hNuk g

µρ ρ ρ

= −

l

l l



(8.148)

(8.149)

where hgrav is heat transfer coefficient for gravity-dominated film condensation – determined with eqs. (8.103) or (8.108) – and hshear is heat transfer coefficient for shear-dominated film condensation, eq. (8.146).


( ) 2 / 3I

I

v gρ ττ

ρ ρ ρ µ+ =

−

l

l l l

2 2 1/ 2( )shear gravh h h= +



The boundary-layer equation for forced turbulent flow along a planar surface is

(8.153)

Eddy shear stress(8.154)

Apparent shear stress for turbulent flow(8.155)

Substituting eq. (8.154) into eq. (8.155)

(8.156)

(8.157)


1 1u u p uu v u vx y x y y

µ ρρ ρ

∂ ∂ ∂ ∂ ∂ ′ ′+ = − + − ∂ ∂ ∂ ∂ ∂

yuvu

∂∂=′′− ρ ερ

appu u vy

τ µ ρ∂ ′ ′= −∂

( )appu u uy y y

τ µ ρ ε ρ ν ε∂ ∂ ∂= + = +∂ ∂ ∂

1 εεν

+ = +

8.3.4.2 Turbulent Condensate Flow



Reδ

Figure 8.14 Variation of the mean film condensation heat transfer coefficient with Reynolds number and as predicted by Rohsenow et al. (1956).

Reδ,tr

Transition points



Eq. (8.156) simplifies to(8.158)

Reynolds number at the transition point in Fig. 8.14

(8.159)

Average heat transfer coefficient beyond the transition point into turbulent flow

(8.160)


( )1/ 3

1/ 21/ 2 *20.065PrL I

gh kv

τ

=

l ll

( )1/ 3

3* *,Re 1800 246 1 0.667 1v vtr I Iδ

ρ ρτ τρ ρ

= − − + +

l l

appuy

τ ρ ε ν+ ∂=∂



8.3.5 Turbulent Film Condensation in a Tube with Vapor Flow

Figure 8.15 Physical model of the condensation phenomena in contact with flowing vapor.



For countercurrent flow, a force balance results in(8.161)

Where the volume of vapor and liquid are(8.162)

(8.163) Substituting eq. (8.162) and (8.163) into eq. (8.161) and

dividing by Δx (8.164)

The shear stress at the liquid-vapor interface(8.165)


( ) ( )2 2 2v vdpp r gV gV p x r r xdx

π ρ ρ π τ π + + = + ∆ + ∆ l l

( ) 2vV R xδ π= − ∆

( ) 2vV R y x Vπ= − ∆ −l

( )( )

2

( )2 2v

RR y dpg gdx R y

δτ ρ ρ ρ

−− = − − − − l l

2I vR dpg

dxδτ ρ− = −




The film thickness is much smaller than the radius of the tube, eq (8.165) reduces to

(8.167)

If ρl >> ρv (8.168) Written in generalized form that includes concurrent flow

(8.169) The shear stress at the wall is

(8.170) Velocity profile when all axial terms and curvature are

neglected(8.171)


( )2 22 ( )

2( )I vR y R y ygR R y

δ δτ τ ρ ρδ

− − − += + − − − l

2I vR dpg

dxτ ρ = −

( )I g yτ τ ρ δ= + −l

( )I g yτ τ ρ δ= ± + −l

w I gτ τ ρ δ= ± + l

( ) 0md duv gdy dy

ε + + =

l



Boundary conditions(8.172)

(8.173) Velocity profile in the liquid film

(8.174)

The liquid Reynolds number(8.175)

Nondimensional variables(8.176)


0, 0u y= =

,Iu yy

µ τ δ∂− = ± =∂

[ ]0

( ) /y I

m

g yu dy

vδ τ ρ

ε− ±

=+∫ l

l

0

4Re 4Ludyδ

ρµ µ

Γ= = ∫ll l

fuδδ

ν+ =

l

, 1/32

*

gνδ δ

−

= l , fyu

yν

+ =l

, 2 /3

* ( )II

gτ ντρ

−

= l

l

f

uu u+ = , fxu

xv

+ =l

, 1 mm v

εε + = +l

, fDuD

ν+ =

l



Where uf is the fractional velocity defined as(8.177)

Applying these nondimensional variables eqs. (8.170), (8.174) and (8.175), their nondimensional forms are

(8.178)

(8.179)

(8.180) Energy balance for constant heat flux at the wall

(8.181)


1/ 2

wfu τ

ρ

= l

( ) ( )2/33 * 0f I fu u v g v gτ δ +− =l lm( )3

0

1 /y f

m

g y uu dy

νε

++

+ ++

−= ∫ l

0Re 4 u dy

δ + += ∫l

( / )Pr Pr

mp v

t

v dT dc q hdy dx

ε µρ µ Γ′′+ = = l l

l l l l



Eq. (8.181) can be nondimensionalized to(8.182)

where(8.183)

Local heat transfer coefficient found from eq. (8.174)(8.184)

Nondimensionalizing eq. (8.184) as a Nusselt number(8.185)

A modified Nusselt number

(8.186)


11

0

PrPr( / ) 1Pr Pr

tT m

t

d N dydx

δµ ε+

−−+ +

+

Γ = − + ∫l

( )Pr

sat w pT

v

T T cN

h−

= l

l

( )

1 1

0 Pr Prm

x psat w t

qh c dyT T

δ ενρ− −

′′ = = + − ∫ l

l l

11

0

PrPr 1Pr Pr

x tx m

t

hNu dyk

δδ δ ε+

−−+ + +

= = − + ∫

l

( )1/3

1/3fxx x

uh vNu Nu gk g

νδ

−++

= =

l

ll



The average modified Nusselt number is found from (8.187)

The dimensionless shear stress at the interface can be written as(8.188)

Fricition factor for vapor flow

(8.189)

where(8.190)


1

0

xNu Nu dL

++ +

+

=

∫

* 2 / 3 2, ,

( / )( ) ( ) ( )2E v

f v vf dg u u u u u

dxδ δ δρ µτ νρ

− + + + ++

Γ= + + +

ll l l

l

0.2

[1 0.045( 5.9)] Re 75[1 0.045Re ( 5.9)] Re 75E

v

f Mgf

f Mg

+

− +

+ − ≤= + − >

l

l

1/20.6

1/ 20.7

0.78Re Re 75

0.50 Re Re 75

I

v v c

I

v v c

Mg

ν ρ τν ρ τ

ν ρ τν ρ τ

+

≤ =

>

l ll l

l ll l



The characteristic stress(8.191)

(8.192)

where A+ = 25.1 and . This profile represents the eddy diffusivity in the inner layer closest

to the wall ( ), where the influence of the wall is important. In the outer layer ( ) the eddy viscosity is assumed to be constant, with a continuous transition to the inner layer.

(8.193)

(8.194)


0 0.6y δ+ +≤ ≤0.6 yδ δ+ + +≤ ≤

( )1 exp( / )Pr

1 exp Pr /t

y Ay B+

+ +

+−

− −= −

( )∑=

−+ =5

1

110 Prlog

i

iicB

2

221 1 1 0.64 1 exp exp 1.66 12 2m

w w w

yyA

τ τ τετ τ τ

++ +

+

= + + − − − −

11 23 3

wI

c I

τττ τ

−

= +



8.3.6 Other External Filmwise Condensation Configurations Nusselt analysis for a vertical plate

(8.195)

For laminar film condensation on a horizontal cylinder

(8.196)

For laminar film condensation on a sphere, the average heat transfer coefficient can be obtained by Nusselt analysis

(8.197)


( )( )

1/ 43cos0.943x v v

xg h xh xNu

k k Tρ ρ ρ θ

µ −

= = ∆ l l l

l l l

( ) 1/ 43

0.729D v vD

D h gh DNuk k T

ρ ρν

−= = ∆

l l

l l l

( ) 1/ 43

0.815D v vD

D h gh DNuk k T

ρ ρν

−= = ∆

l l

l l l



Nusselt analysis for a vertical array of horizontal tubes

(8.198)

To find the average heat transfer coefficient for a single tube array

(8.199)


( ) 1/ 43

0.729D v vD

D h gh DNuk nk T

ρ ρν

−= = ∆

l l

l l l

, 1/ 4D

D nhhn

=



For condensation on an upward-facing horizontal surface of a finite size, the condensate in the central region flows toward the edge where it is spilled. For condensation over a long horizontal strip with a width of L, the average heat transfer can be obtained by

(8.200) The heat transfer coefficient for film condensation over

an upward-facing horizontal circular disk with a diameter of D is

(8.201)

( )( )

1/ 53

1.079 v v

sat w

gh LhLNuk k T T

ρ ρ ρµ

′ −= = −

l l l

l l l

( )( )

1/ 53

1.368 v vD

sat w

gh DhDNuk k T T

ρ ρ ρµ

′ −= = −

l l l

l l l



Example 8.4 Saturated acetone at 60 ˚C condenses on the outside of a

copper tube with a diameter of D = 3.0 cm. The outer surface temperature of the copper tube is Tw = 40 ˚C. Find the heat transfer coefficient and the rate of condensation per unit length of the tube.



Solution: The vapor properties are evaluated at saturation

temperature Tsat = 60 ˚C. The vapor density at this temperature is ρv = 2.37 kg/m3, and the latent heat of vaporization is = 517 kJ/kg. The liquid properties evaluated at Tf = (Tsat + Tw)/2 = 50 ˚C are = 756.0 kg/m3, = 2255 J/kg-K, = 0.248x10-3 kg/m-s, = 0.172 W/m-K, and

The revised latent heat of vaporization is

vhlρ l

pc l µ l kl6 2/ 0.328 10 m /s.ν µ ρ −= = ×l l l

( )1 0.68

2.255 (60 40) 517 1 0.68 547.7 kJ/kg517

p sat wv v

v

c T Th h

h − ′ = +

× − = × + =

ll l

l



The heat transfer coefficient can be obtained from eq. (8.196), i.e.,

The heat transfer rate per unit width is then

The condensation rate per unit width is

1/ 43

1/ 43 3

6

2

( )0.729

(756.0 2.37) 9.8 0.172 547.7 100.7290.328 10 (60 40) 0.03

2331.3 W/m -K

v vg k hhTD

ρ ρν

−

′ −= ∆

− × × × ×= × × − × =

l l l

l

( ) 2331.3 0.03 (60 40) 4394.4 Wsat wq h D T Tπ π′ = − = × × × − =

3

4394.4 0.00802 kg/s-m547.7 10v

qmh

′′ = = =′ ×l

&



For condensation on the outside of a horizontal tube in crossflow, the heat transfer coefficient is affected by both free-steam velocity of vapor, u∞, and gravitational force, so that

(8.202)

For laminar film condensation on a horizontal flat plate in a parallel stream of saturated vapor, the average heat transfer coefficient is

(8.203)

( )

1/ 21/ 2

1/ 220.64Re 1 1 1.69 v

Dsat w

gh DhDNuk u k T T

µ

∞

′ = = + + −

l l

l l

1/ 31/ 21/ 2

3/ 2

Pr1.5080.872Re(1 Ja / Pr ) Ja

v vL

hLNuk

ρ µρ µ

= = + +

l

l l l l



Filmwise condensation for both stagnant vapor and forced convection has been analyzed using boundary layer treatment.

If film condensation takes place in an atmosphere where a noncondensable gas exists, the condensing vapor must diffuse through the noncondensable gas to the liquid-vapor interface (Stephan, 1992).

Therefore, a partial pressure gradient must exist in the vapor-gas atmosphere.

The partial pressure of the condensable gas, pcv, decreases from a constant value pcv,res in the vapor-gas reservoir to the value pcv,δ at the phase interface where the vapor is condensing to liquid.

The partial pressure of the noncondensable gas, pncg, on the other hand, increases from its reservoir value, pncg,res, to the value pncg,δ at the liquid-vapor interface.

At any point in space and time the summation of the partial pressures of this binary system must equal the constant total pressure.


8.3.7 Effects of Noncondensable Gas



Figure 8.16 Mass transfer in the

equivalent laminar film.



(8.204)

The partial pressure of the condensing gas decreases as it approaches the phase interface and its saturation temperature Tsat(pcv) also falls.

Depending on the noncondensable gas content, the temperature at the interface can be much lower than if no such gas were present.

The temperature difference across the interface would also be lower as a result, which would lead to a lower overall heat transfer coefficient.

This clearly demonstrates the benefit of removing as much noncondensable gas from the system as possible.

However, systematic purity cannot always be achieved and the noncondensable gas content must be taken into account.


v gp p p+ =


Chapter 8: Condensation8.3 Filmwise Condensation Molar flux of the condensable vapor at any point within the

equivalent laminar layer:

(8.205)

(8.206) Integrating eq. (8.206) over the equivalent laminar layer and

considering and cT are constants, one obtains

(8.207) Equation (8.207) can be rearranged to yield

(8.208)where

(8.209)

vv vg v v

cn D n xy

∂′′ ′′= − +∂

& &

1 /vg v

vv T

D cnc c y

∂′′ = −− ∂

&

vn′′&,

,

1v v

v

c

v vg T vcT v

n dy D c dcc cδ

δ δ

δ

∞+′′ = −

−∫ ∫&

, ,,

, ,

ln lnT vg T v T vv T m G

v T v T v

c D c c c cn c h

c c c cδ δ

δ ∞ ∞

− −′′ = = − − &

,vg

m Gv

Dh

δ=



Mass transfer coefficient related to the heat transfer coefficient through the Lewis equation

(8.210) Mass flow rate of condensable vapor to the liquid-vapor interface

(8.211) Energy balance across a differential control volume at the liquid-

vapor interface(8.212)

Substituting eq. (8.181) and (8.182) into eq. (8.180)(8.213)

Small inert gas content, eq. (8.183) reduces to(8.214)


,,

ln cvcv vg m G

cv res

p pm h Ap p

ρ −=−

&

,,

1 Gm G

vg p vg

hhcρ

=

( ),

, ,

ln cvG vI w vg I

p vg cv res

p ph hT T T Th c p p

δ ξ −

− = + − − l

l

,

, ,

ln cvG vI w

p vg cv res

p ph hT Th c p p

δ −− = −

l

l

( ) ( )w v v G vgh T T m h h T Tδ δ′′− = + −l l&



( )G vg Ih A T T− ( )wIh A T T−l

Figure 8.17 Energy balance at the liquid-vapor

interface for film condensation on a

vertical plate including the effects of non-condensable

gases.



No noncondensible gas in the vapor, the heat flux across the liquid film

(8.215) If noncondensible gas is present, the heat flux across the liquid film

(8.216)

Ratio of heat fluxes obtained by eqs. (8.216) and (8.215)

(8.217) Substituting eq. (8.214) into eq. (8.217)

(8.218)

or substituting from eq. (8.181)(8.219)


( )sat wkq T Tδ

′′ = −l

( )vg I wkq T Tδ

′′ = −l

1vg I w

sat w

q T Tq T T

′′ −= ≤′′ −

( ),

, ,

lnvg vcGO v

sat w vg res vc res

q p ph hq T T h c p p

δ′′ −=

′′ − −l

l

cvm&

( )vg v v

sat w

q m hq T T h

′′ ′′=

′′ −l

l

&



It can be seen from the above expression that for large (Tsat – Tw), the velocity or mass flow rate , must be made sufficiently large to acquire a large heat transfer coefficient for the heat transfer from the vapor-gas mixture to the liquid-vapor interface.

This must be done in order to make not too small and therefore removing the undesirable effects of the noncondensable gas as much as possible.


cvm&

vgq′′



Mass fraction of noncondensable gas

vgqq

′′′′

Figure 8.18 Falling film condensation of

steam with non-condensable gas

(Collier and Thome, 1994).



Example 8.5 A mixture of 20% steam and 80% air at 100 ˚C and 1 atm

flows across a horizontal cylinder. The diameter of the cylinder is 0.1 m and the velocity of the mixture is 30 m/s. The condensation rate on the cylinder is = 0.02 kg/m2-s. The properties of the mixture are ρvg = 0.944 kg/m3, μvg = 8.2x10-6 N-s/m2, Dvg = 3.64x10-5 m2/s, respectively. What is the temperature at liquid-vapor interface? If the temperature of the tube is 80 ˚C, what is the percentage of heat transfer reduction due to the existence of noncondensable gas?

vm′′&



Solution: The mass fraction of the steam is ωv,∞ = 0.9. The molecular mass of

water and air are Mv = 18.02 kg/kmol and Mg = 28.96 kg/kmol. The partial pressure of the steam can be obtained from eq. (8.70), i.e.,

The Reynolds number of the mixture is

The Schmidt number of the mixture is

1

,,

,

15 5

(1 )1

18.02 (1 0.9)1.013 10 1 0.9475 10 kPa28.96 0.9

v vv

g v

Mp p

Mω

ω

−

∞∞

∞

−

−= +

× − = × × + = × ×

6

0.944 30 0.1Re 345368.2 10

vgD

vg

u Dρµ

∞−

× ×= = =×

6

5

8.2 10Sc 0.2390.944 3.64 10

vg

vg vgDµ

ρ

−

−

×= = =× ×



The empirical correlation for forced convective heat transfer across a cylinder is

Analogy between mass and heat transfer gives us

The mass transfer coefficient is therefore

4/ 55/81/ 2 1/ 3

2 /3 1/ 4

0.62Re Pr ReNu 0.3 1[1 (0.4 / Pr) ] 282000

D DD

= + × + +

4/ 55/81/ 2 1/ 3

2 / 3 1/ 4

0.62Re Sc ReSh 0.3 1[1 (0.4 /Sc) ] 282000

D DD

= + + + 4 / 55/81/ 2 1/ 3

2 / 3 1/ 4

0.62 34536 0.239 345360.3 1 108.1[1 (0.4 / 0.239) ] 282000

× × = + + = +

5

,

Sh 108.1 3.64 10 0.03935 m/s0.1

D vgm G

Dh

D

−× ×= = =



The partial pressure of the vapor at the liquid-vapor interface, , can be obtained from eq. (8.211).

The interfacial temperature, Tδ, is the saturation temperature corresponding to the above partial pressure. It can be found from the Clapeyron-Clausis equation (2.168).

(8.220)

, ,,

5 5 5

5

( )exp

0.021.013 10 (1.013 10 0.9475 10 )exp0.944 0.03935

0.900 10 Pa

vv v

vg m G

mp p p phδ ρ∞

′′= − −

= × − × − × × = ×

&

, 1 1ln v v

g sat

p hp R T T

δ

δ

= − −

l


Chapter 8: Condensation8.3 Filmwise Condensation Equation (8.220) can be rearranged to obtain

The ratio of heat fluxes with and without noncondensable gas can be obtained from eq (8.217).

In other words, the heat transfer is decreased by 16.75% due to the presence of noncondensable gas.

1,

15

5

1 ln

1 0.4615 0.900 10ln 369.80K 96.65 C373.15 2251.2 1.013 10

g v

sat v

R pT

T h pδ

δ

−

−

= −

×= − = = ×

l

o

96.65 80.0 0.8325100.0 80.0

vg w

sat w

q T Tq T T

δ′′ − −= = =′′ − −



The Wallis correlation, in which experimental data were correlated for packed beds and countercurrent flow in tubes, is represented by the following empirical equation

(8.221)where

(8.222) For thermosyphons, as opposed to open systems, and are

related due to the fact that it is a closed system under steady state conditions.

(8.223)

(8.224)


8.3.8 Flooding or Entrainment Limit

( ) ( )1 2 1 2* *v wj m j C+ =l

( ) 1 2* 1 2i i i vj j gDρ ρ ρ

−= − l ( ), ,i v= l

*vj

*jl

vv

v

mjAρ

=&

mjAρ

= ll

l

&



Combining eqs. (8.221) – (8.224) and rearranging will result in an equation for the flooding limit based on an extension of the Wallis correlation for open systems to thermosyphons.

(8.225)

According to the Kutateladze correlation(8.226)

where and (8.227)

Comparing the Kutateladze correlation, eq. (8.227), with the Wallis correlation, eq. (8.221), the following relation is found assuming the two are identical.

(8.228)

( )( )

2

21 41

w v v v

v

C h gDqA

ρ ρ ρ

ρ ρ

−=

+

l l

l

( ) ( )1 2 1 2v kK K C+ =l

3.2kC =( ) 1 41 2

i i i vK j gρ σ ρ ρ−

= − l ( ), ,i v= l

( )

4

k

w v

CDC g

σρ ρ

= − l



The critical wavelength of the Taylor instability is:

(8.229) Setting eq. (8.229) (characteristic length) equal to eq. (8.228) and

choosing the upper limit in eq. (8.228) results in

(8.230) The Kutateldaze number decreases as the dimensionless diameter

decreases, which is called the Bond number. This trend requires that Ck is in terms of the Bond number

(8.231)or

(8.232)

( ) ( )crit 2 2 3vg

σλ π πρ ρ

=−l

:

3.2k

w

CC

=

( )

1 24

Bo k

w v

CDC g

σρ ρ

= = − l

1 4Bok

w

CC

=



With reference to the variation of the Kutateladze number versus the Bond number in the paper by Wallis and Makkenchery with , the function is introduced to account for the effect of the diameter on the flooding limit. If we let x = Bo1/4, eq. (8.232) results in

or(8.233)

Using experimental results for different working fluids, the following correlation is proposed

(8.234)or, for the maximum heat transfer rate,

(8.235)

* 1.0vj =tanhy x=

1 43.2 tanh Bo , with Cw = 1.0k

w

CC

=

( ) ( )1 2 1 2 1 43.2 tanh Bov kK K C+ = =l

0.142 2 1 4 2 1 4K tanh Bo tanh Bok

v

C Rρρ

′= = =

l

( ) 21 4 1 4 1 4max K v v vq h A gσ ρ ρ ρ ρ

−− − = − + l l l



Tien and Chung (1978)

R′ = 2.0, Faghri et al. (1989)

Cw = 0.7, Wallis (1969)

R′ = 3.2, Faghri et al. (1989)

Cw = 1.0, Wallis (1969)

0 5 10 15 20 25

0

1

2

3

4

5

K

Bond Number, Bo

Figure 8.19 Variation of the modified Kutataldze number versus the Bond number for a closed two-phase thermosyphon (Faghri et al., 1989).

8.3.1 Regimes of Filmwise Condensation - Thermal-Fluids ...

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