CHAPTER 6 Post Dryout Heat Transfer

Liquid

Vapor

q

Liquid droplets

Liquid deficient region

Vaporflow

Film boiling (inverted annular flow)•Topics to be discussed -Film boiling heat transfer (Inverted annular flow) -Heat transfer in the liquid deficient region (High void fraction film boiling) -Minimum film boiling temperature -Transition boiling heat transfer

6.1 Film Boiling with Inverted Annular Flow Pattern –Film boiling on vertical surface Brmoley’s model Ref.: Chemical Engineering Progress, vol.46, No.5, p.221-227 G.E. for vapor flow

Boundary conditions

)(2

2

vlv gdy

ud

)b(case0u

)a(case0dy

du,yat.2

0u,0yat.1

y

Liquid

u gVapor q

y

x

Solution for u(y)

-Heat transfer It is assumed that heat travels through vapor film by conduction. Thus

)b(case1

)a(case2C

]y2

1y

2

1C)[(

g)y(u 2

vlv

dxT

kdxdx

dwi vlv

)(

0

)(

0

)()()()(x

dxdx

dx

v dyyudyyuxwdxdx

dwxw

q

x

x+dx dxdx

dx

dxdx

dwxw

)(

)(

)(

)(

x

xw

T=Tsat

])dxdx

d2dx

dx

d3(

3

1

)dxdx

d2dx

dx

d(

2

C[

2

1)(

g

])dxdx

d(

3

1)dx

dx

d(

2

C[

2

1)(

gdy)y(u

]3

1

2

1C[

2

1)(

gdy)y(u

223

223vl

v

33vl

v

dxdx

d)x(

0

33vl

v

)x(

0

dx]

31

2C

[23

)(gi

Tkd

Tk

dx

d]

3

1

2

C[

2

3)(

gi

dx

d3]

3

1

2

C[

2

1)(

g

dx

dw

dxdx

d3]

3

1

2

C[

2

1)(

gdy)y(udy)y(u

vllvv

vsatv3

satv

2vl

vvlv

2vl

v

v

2vl

v

)x(

0

dxdx

d)x(

0

Boundary condition: δ=0 at x=0

Thus, the local heat transfer coefficient is:

4/1

)(]3

1

2[3

8)(

vlvlv

vsatv

giC

xTkx

4/13

4/1

)(

8

)3

1

2(3

)()(

xT

kgiC

x

kxh

vsat

vvlvlvv

The average heat transfer coefficient over the vertical plate is:

For case a C=2 C1=0.943 (Zero stress at film surface)

case b C=1 C1=0.666 (Zero velocity at film surface)

Considering the sensible heat of vapor:

vapor properties evaluated at film temperature(Tf=(Tw+Tsat)/2)

4/1

1

4/1

vsat

3vvlvlv

1

L

0 8

)31

2C

(3

3

4C

LT

k)(giCdx)x(h

L

1h

]34.01[lv

vlvlvlv i

TpCiibyreplacedisi

-Film boiling heat transfer from a horizontal surface Berenson’s model Ref.: J. Heat Transfer, vol.83, pp.351-358,1961

R H

Vv

1r

2r

Liquid

Solid

Vapor

r

z

0P

2P1P

Actual shape of liquid-vapor interface

Physical model of film boiling from a horizontal surface

4/1

3

4/1

)(

)(425.0

])()(

[35.2

vlsatvf

vlvflvvf

vlvlvflv

satvfvf

vfvf

gT

gik

ggi

Tk

kkh

Where the subscript vf stands for vapor properties evaluated at Tf=(Tw+Tsat)/2

•Analysis in Berenson’s model

rz

2P

1P

Vapor film

HgHl

2r

-Momentum equation in the vapor film

Boundary condition

2

2

dz

ud

dr

dPvf

0

000

dz

duu

z

uz

at

,0 exampleanasfirstdz

duConsider

z

212

11

2

2

2

1)(

1

1

CzCAzzu

CAzCzdr

dP

dz

du

Adr

dP

dz

ud

vf

vf

ACdz

du

Czu

z

1

2

0

00)0(

zAzA

zu 2

2)( 0

azdz

dufor

So the average velocity

3)1()(

1 2

0

AdzzuU

U33)

3

A(A)A(

dz

duvf

2

vfvfvf0z

vfw

The force balance requires that

2

3

),(1

U

onlycorrectelyapproximatisthisfactIndr

dP

vf

w

Similarly, for u(z=δ)=0,

From the energy balance

and assuming that heat transfer through the film is by conduction (i.e. the flow is laminar)

P drdr

dPP

r drr w

2

12

U

dr

dPvf

lvvf irUrrrq )(2)( 222

0)z(ufor12

0dz

dufor3

,r2

r2

i

Tk

dr

dP

2

1rwhere

r2

r2

i

Tk)r(U

Tkq

z

2

2lvvf

satvf

2vf

222

22

2lvvf

satvf

satvf

)(32

vlg

U(r)

r2r

q

2r

)(g35.2

2

Dr

dr)r2

r2(

i

Tkdr

dr

dP

22

2

vl

b1

r

r

r

r

22

2lvvf

satvf

2v

2

1

2

1

)(

8412

vllvvf

satvfvf

gi

TkPP

)2

1(

2)(

2

12

01

02

b

vl

v

l

DR

RgHPP

RgHPP

gHPP

Noting that

Assume bDH

)(2.3

vlgH

4/1

)()(

09.1

vllvvlvf

satvfvf

gig

Tk

4/1

)()(

09.14.1

vllvvlvf

satvfvf

gig

Tk

where“1.4”is to account for the ratio of total area /the area between bubble

ofvalueondepending

gig

Tkor

vllvvlvf

satvfvf

4/1

)()(9.1

6.2

The experimental data suggest that

4/1

vllvvlvf

satvfvf

)(gi)(g

Tk35.2

]50.01[

)(

)(45.0

4/1

3

lv

satvlvlv

vlsatvf

vlvflvvfvf

i

TCpii

gT

gikkh

-Film boiling on horizontal cylinders Following similar procedure to that for a vertical surface, Bromley obtained the following equation for film boiling heat transfer coefficient on horizontal cylinders.

Breen & Westwater correlation (1962)

interfaceliquidvaportheatvelocityzero

interfaceliquidvaportheatstressshearzeroC

DT

kgiCh

vfsat

vfvflvflv

512.0

724.0

])(

[

1

4/13

1

)(g2

]T

k)(gi][D/069.059.0[h

vlT

4/1

Tvfsat

3vfvflvflv

T

布林與韋斯特瓦特的膜沸騰經驗式 (Breen & Westwaater, 1962)

-Film boiling on spheres Dhir & Lienhard

-Film boiling heat transfer considering thermal radiation

4/13

])(

[67.0DT

kgih

vfsat

vfvflvflv

1/1/1

)TT)(TT(

TT

qh

h4

3hh

lw

satw2

sat2

wSB

satw

rr

rcond

•Application of boundary layer theory for inverted annular flow film boiling Ref: Sakurai et al., 1990, “A General Correlation for Pool Film Boiling Heat Transfer From a Horizontal cylinder to subcooled liquid Part I & Part II.” J. Heat Transfer, vol. 112 pp.430-450.

Governing Equations

0)()(

kkkk vy

ux

,vk

2

2

)(y

ug

y

uv

x

uu k

kkkxk

kkk

kk

,vk

2

2

)()(y

Tk

y

Tv

x

TuCp k

kk

kk

kk

,vk

， vertical surface ， horizontal cylinder ggx

)/sin( Rxggx

Boundary Conditions At y=0

wvvv TTvu ,0

At y=δ

uuv

y

u

y

uvv

virv

v iGy

Tkq

y

Tk

"

satv TTT

At y=∞

TTu ,0

Where Gi is the interfacial mass transfer rate,

v

dx

duv

dx

duG vvvi

/)()( "" rpr qq

31

31

/sin)(

43

31 ''sin4)(

od

)(1/1/1

44"satw

w

sbrp TTq

o

d 6122.0)(1

Similarity Transform Sakura et al. defined the following similarity variables.

y

oks

kkk vkdy

rN ,,

)(

vkM

fk

kkk ,,

)()(

TT

TT

TT

TT

satsatw

satvvv

)(,)(

vkx

vy

u k

k

ksk

k

k

ksk ,,;

vkvrgvM ksksksvsk ,,)/()( 41

23

vkvMN vskk ,,/

where

Thus,

vkffffksks

kkkkkk

ks

k ,,0/)(

/)(32

)(

)( "2'

'

"

vkfCp

Cp

k

kkk

ks

kksk

ks

k ,,0Pr3)(

)( '

'

'

0vAt

1)(,0)()( ' vvv ff

At

0'' f

and interfacial conditions. Numerical solutions is required.

次冷度對池膜沸騰的效應（ Sakurai et al., 1990 ）

Approximate solutions -Neglect convective terms in all the conservation equation-Vapor flow is driven by buoyancy force -Liquid flow is driven by vapor flow.

yg

yg

uvvv

v

v

vv

2

2

)(

2

)( 2

//

)/)(/(

2

)(

//

/

2

)( 2

v

vv

v

vv gz

gu

)()/( satwwv TTyTT

)()/( TTzTT satsat

Consider energy balance at the interface

)()('

TTkTTk

dyudx

di sat

o

satwvvvv

o

sat TTkdzTTu

dx

dCp

)()(

Thus,

213 )( vv

v

CCm

dx

d

C

mdx

d)( 3

Where

)/(),/()4( 2 ffmffmv

)](/[)(12 '1 vvvsatwvvv giTTkC

)](/[)(12 '2 vvvwsatvvv giTTkC

)](/[6 vvgCpkC

/,/ vf

-Integration of above two equations yields

oo

vovo

f

CCf

x )4(3

)()(4 214

3

4

3

)(4

o

o

f

fC

x

Combined above two equations and solve for 0

)3/(2 vvo CCVU

2121 , WWVWWU

)2/()3/(2)3( 12112

321 vvvvv CCfCCCCCW

)]27/(643/2)27/(4[ 221

222 fCCfCCW vvv

3121

21

31

2 )]3/()[/()]2/([)]/([ vvvvv CCCCCCfCC

Where

Thus,

41

33

)(4

o

o

f

xfC

The heat flux distribution is given by:

)(

)("

x

TTkq satw

v

Thus, the average heat flux can be obtained as

L

o

satwvxave L

TTkdxq

Lq

)(

)(

3

41 ""

The average Nusselt number for vertical surfaces can then be obtained as:

4

14

1

43

793.0)(4

3

3

4

)(3

4)( 3"

Lo

o

v

satwavevL M

fC

fL

L

L

k

LTTqNu

For a horizontal cylinder

41

43

41

612.0728.044

3M

L

DNu

k

DhNu vL

v

convD

Where

23' PrPr1Pr SpRSpEETTCpiGrM satwvvvvLL

23' PrPr1Pr SpRSpEETTCpiGrM satwvvvvD

ScBCABCAE 313

13

1

22223 Pr41Pr31271 SpRScSpRScA

22 Pr2732Pr32274 RSpScSpScB

2322 /272Pr41 RScSp

Pr21 2SpRC

vsatwvvv iTTCpii 5.01'

21

vvR

')( vsat iTTCpSc

)Pr'()( vvsatw iTTCpSp

)()( 23vvvvL vLgGr

)()( 23

vvvvD vDgGr

Note that

)(

')(

)(

'Pr 3

satwvv

vvv

satwv

vvvL

TTk

Lig

TTCp

iGr

)(

')(

)(

'Pr 3

satwvv

vvv

satwv

vvVD

TTk

Dig

TTCp

iGr

Thus, for a vertical surface

41

)(

')( 3

LTT

kigh

satwv

vvvvLcon

and for a horizontal cylinder

41

)(

')( 3

DTT

kigh

satwv

vvvvDcon

These expressions are same as those derived by thesimplied model of Bromley.

6.2 Heat Transfer in the Liquid Deficient Region

From: M.Andreani & G. Yadiaroglu “Prediction Methods for DispersedFlow Film Boiling” Int. J. Multiphase Flow, vol.20 suppl. Pp.1-51, 1994.

•Thermal nonequilibrium vapor is superheated

•Major heat transfer mechanisms Heat transfer from the surface to liquid droplet by either “wet” collisions or “dry” collision Convective heat transfer from the surface to the bulk vapor Convective heat transfer from the bulk vapor to suspended droplets in the vapor core

Liquid deficient region

qVaporflow

)( wdq )( wvq

)( vdq

•Bounding surface temperatures -Upper bound: complete departure from equilibrium all the heat added to fluid goes intosuperheating the vapor (toward this situation at low pressures and low velocity) -Lower bound: compete thermodynamic equilibrium all the heat added to fluid goes to evaporate the liquid droplets and Tg=Tsat. Surface

temperature decreases as a result of increasing the vapor flow due to evaporation. Toward this situation at high pressure approaching the critical condition and high flow rates (>3000 kg/m2s)

From: Collier, 1981

•Thermodynamic quality vs. actual quality (for uniform heat flux) -Thermodynamic quality (equilibrium quality) xe

-Actual quality (for vapor superheat of Tv-Tsat present)

-relation between xd0 and zd0

D

zz

Gi

qxx d

lvde

)(4 0

lv

satvvae

lv

satvvd

lvda

i

TTpCxx

i

TTpC

D

zz

Gi

qxx

)(

)()(4 0

dryout

zd0

xd0inlettubeatenthalpysubcoolingi

i

i

D

z

Gi

qx

isub

lv

isubd

lvd

,

,00

4

•Semi-theoretical model of Bennett et al (Ref. :Collier, 1981) -Assume -Assume the droplet number flow rate (N) is constant along the tube that is no drop breakup or coalescence occurs. But rd may

be changed due to evaporation.

-Mass balance gives

-Energy balance equation

0wdq

dz

drr

G

N

rdz

d

G

N

dz

dx

dd

l

dla

2

3

4

]3

4[)1(

ld

d

r

xGN

30

0

3

4)1(

rd : radius of the droplet

dz

dxAGiDq

dz

dTCpAGx a

lvv

va

-Momentum eq. For liquid droplets

-Evaporation of droplets (Ryley’s model)

84.1

16.084.0

16.1

)2(25.20

)]1()([

dl

vvl

l

vdvl

dd

rk

guukdz

duu

forcedraguuk dvl :)( 16.1

forcebouyancygl

v :)1(

uv: vapor velocity ;ud: droplet velocity

lvlsatvvd

dd

d

d

d

iTTFkdt

dr

dtudzNoteudt

dr

rdz

dr

/)()1(

)(1

2

1

2

2

3/1)()(

276.01vv

v

v

dvv

k

uu

factornventilatioF

RMk

vaportheincoefdiflusionselfk

vv

v

/)1(

.

M: molecular weight R: universal gas constant

statethesaturationat

dP

d vl )/()1(

-Using the Runge-Kutta method to solve this problem

P=69bar, vertical tube, L=5.8m, i.d=12.6mm 2rd=0.3mm(rd is an assumed

value) (From: Collier, 1981)

•Empirical correlations for liquid deficient film boiling heat transfer -Dougall & Rohsenow Correlation

see more correlations in Groeneveld & Snoek, 1986.6.3 The minimum film boiling temperature Ref. A. Sakurai, “Film Boiling Heat Transfer”, Proc. of 9th Int. Heat Transfer Conference, vol.1, pp.157-186.

)]1([Re

PrRe023.0

2

4.08.02

el

ve

v

vv

xxGD

whereD

kh

)(

)1(min

satII

satI

TTRaT

RaTTRaT

2/1

sss

lll

Cpk

CpkRa

)/17001(

)/(20exp26.0192.0

cr

crcrI P

PPTT TI: contact interface

temperature ;Pcr in kPa

It is expected that the minimum film boiling temperature on a solid surface with any thermal physical properties in ay liquid can be evaluated by above eq.(Sakurrai, 1990)

Minimum temperature vs. system pressure for cylinders of 1.2 and 3.0 mm diameters in Freon-11

Minimum temperature vs. system pressure for platinum and gold cylinders of 2.0 and 3.0 mm diameters in water.

6.4 Transition boiling

CHFq

q

satw TT

Transitionboiling

Nucleate boiling Film boiling

minq

Typical surface temperature history Typical local surface temperature history for a liquid-solid contact.

From: Lee, (J.C.) Chen and Nelson.

Model of Pan et al (1989)Ref.: Pan et al, “The Mechanism of Heat Transfer in Transition Boiling”, Int. J. Heat Mass Transfer, vol.32 No.7 pp.1337-1349, 1989.

(a) bubble departing (b) transient conduction (c) boiling incipience and heat transfer

(d) macrolayer evaporation (e) vapour covering

c

vUlU

V

Vapor bubble

Liquid film(macrolayer)

Vapor stemHeated surface

DDD

Vapor bubble

Liquid film

c

vapor structure near heated

surface at high heat fluxes Pool boiling at high fluxes on a

horizontal flat plate τ: vapor bubble hovering time.tme: time interval of macrolayer evaporation.

τ<tme: nucleate boiling.

τ=tme: critical heat flux.(Haramura & Katto’s model, 1983)

τ>tme: transition boiling / film boiling.)1( FqFqq rmetot

From M.Shoji, 1992

From: Katto, 1992

-Wall temperature during transient contact conduction

)()1)(1(

)1)(1()1(

)1)(1( 0 21

2112

21 t

nerfc

bb

bbbb

bb

TTTT

c

n

n

wI

0 21

2112

)(

)1(

)1)(1(

)1)(1()1(

nc

n

t

nerfc

bb

bbbb

21

)()(1 hc CpkCpkb 21

)()(2 CpkCpkb c

--Effective liquid thermal conductivity

tteff Cpkkkk )(

tconsempiricalC

T

TT

ggC

w

bw

vlt

tan00.1

)(

)()(

2/3

24/3

2/1

--Incipience criterion

v

vsat

csatvc i

vT

rTTtryT

2),(

0 21

2112

21 2)1)(1(

)1)(1()1(

)1)(1(),(

nc

n

w

t

n

t

xerfc

bb

bbbb

bb

TTTtyT

0 21

2112

)1(

2)1)(1(

)1)(1()1(

n c

n

t

n

t

xerfc

bb

bbbb

-Macrolayer evaporation heat flux

nbme qq

33.0"

7.1 )(Pr

)(

vv

nbsf

v

satcdI

gi

qC

i

TttTCp

-Macrolayer evaporation time "/ mevome qit δme: from Haramura & Katta’s theory

-Vapor hovering period

The average volumetric growth rate of the bubble in the period of macrolayer evaporation and vapour covering is given by

where λD is the most dangerous Taylor wavelength

In above equation, it is assumed that the unit heater area participating in the growth of one vapour bubble is

51

53

51

)(

)(4

4

3 1611

Vg v

v

)(/"2vvaveD iqV

21

21

)(23 vD g

2D

The average heat flux within these two periods is given as

-Liquid contact time fraction

-Transition boiling heat flux

""

" )( vmememeave

qttqq

mev tt

vmecd

mecd

vl

l

ttt

tt

tt

tF

vmecd

vvmemecdcd

ttt

tqtqtqq

From: Shoji, 1992

From: Shoji, 1992

From: Pan er al., 1989 From: Shoji, 1992

Empirical Correlation for Transition Boiling (for low quality) -Auracher’s correlation Ref.: Int. Froid 1988., vol.11, pp.329-335Nucleate boiling during liquid contact is the major mode of heat transfer.

3.1,)(

)/)((

)1/(1

,

3.10

3.0

nT

T

q

q

qqPCh

h

n

CHFsat

sat

CHF

h0.3: reference heat transfercoefficient at reduced pressureof 0.3

From: Auracher, 1988

From: Auracher, 1988