Boiling Heat Transfer and Two-Phase Flow.PDF

7/26/2019 Boiling Heat Transfer and Two-Phase Flow.PDF

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CHAPTER 2 FUNDAMENTALS OF BOILING HEAT

TRANSFER AND TWO-PHASE FLOW

2.1 Nucleation Theory

2.1.1 Homogeneous Nucleation

Thermodynamic Limit of Superheat

T = TC

T=T2>T1

T=T1

Critical point

Vapor spinodal

Vapor saturation

Mechanical

unstable region

Specific volume ()

Pressure

( p )

Liquid

saturation

Subcooledliquid

A

B C

E

F

G

"

"

"

"

""

"


2/440

#$%&'()*+,-./0123/0456

78/09:;?


3/440

Mechanical Stability Limit :

0


4/440

2.The minimum in the P-v isotherms , i.e., the minimum

pressure corresponding to a given T , is given by thefollowing two criteria :

Thus,

cc

n

c vbvRTa 3

1

8

9 1==

+

The state equation can be non-dimensionalized as

3

8

3

132

=

+

n

Where B=P/PC C=T/TC and D=v/vC

002

2

>

=


5/440

( )3

1

413

=+n

Applying these criteria one obtains the following equation along

the spinodal

Procedure to determine the thermodynamic limit

1.GiveCand solve for D2.Using the state equation to determine B

Eberhart & Schnyders J. physical Chemistry vol. 77 No.23, pp.2730-2736, 1973


6/440

Kinetic limit of superheat

At a given temperature liquid molecules have an energy distribution

such that there is a small but finite fraction having energies

considerably greater than the average, therefore is a small but finite

probability of a cluster of molecules with vapor like energies coming

together to form a vapor embryo of the size of the equilibrium nucleus.

Vapor phase is created and grown based on the surrounding

liquid superheat .Given a liquid superheat, an equilibrium radius of

embryo can be evaluated as follows

Refs. a. Collier & Tome, 1994, convective boiling and condensation, ch1.b. Hsu & Graham, 1989, Transport process in boiling and

two-phase system, ch1.

c. Van Stralen & Cole, 1979, Boiling phenomena, ch2.


7/440

Superheat needed to sustain a vapor nucleus of radius of r

Tsat = The saturation temperature corresponding to the liquid pressure, Pf

Tl

= The liquid temperature > Tsat (the liquid is superheated)

Question: Tl Tsat =? in order to sustain a vapor nucleus of radius of r ?

"

lPlT

r

vP

Tsat = The saturation temperature corresponding to the liquid pressure, Pf

Tl

= The liquid temperature > Tsat (the liquid is superheated)

Question: Tl Tsat =? in order to sustain a vapor nucleus of radius of r ?


8/440

The Clausius Clapeyron equation (for a flat vapor liquid interface)

The Clausius Clapeyron equation defines the slope of

saturation pressure vs. temperature.

P

Tsat

T

lvlv

lvlv

lvsat

lv

T

vvv

iii

T

i

dT

dP

sat

==

=

"

Vapor pressure curve


9/440

Laplace equation (mechanical force balance )

rPP lv 2= E : surface tension

C1

C2r1

r2

r2

r1C1r2C2

Vapor liquid interface

Pv

lP


10/440

Force balance requires that

( )

[ ]212121

1222111211

2

1sin2

2

1sin2

rr

rrrrPP lv

+

+

=

+=

21

11

rrPP lv

For a spherical bulle rPPrrr

lv

2

21 ===Form the Clausius-Clapeyron relation,

lv

lvsat

i

vTPT =

Neglecting the possible bubble curvature effect on the vapor pressure

( )

lv

lvsat

lv

lvsatvlsatl

i

vT

r

ivTPPTT

=

2

This is the superheat needed to sustain a vapor nucleus of radius r


11/440

Conversely the equilibrium radius for a nucleus in the liquid with a

superheat of Tl

-Tsat

is:

lv

lvsat

satl

ei

vT

TTr

=

2

Carey (1992) derived the following equation for rebased on the equilibrium of chemical potential:

( ) ( )[ ]{ } lllsatlllsate

PRTTPPvTPr

=

/exp

2


12/440

Now, the remaining question is that how many nuclei are

formed and grown per unit time per unit volume?

See the boiling process, the nucleation rate must be >109 to1013 m-3s-1.

The free energy to form a nucleus of radius r , FG(r) , is given by :

( )lv PPrrrG =32

3

44)(

Free energy Additional

surface energy

Work done by the vapor

to the surrounding liquid


13/440

At a given temperature (liquid superheat)

elv rPP

2

=

=

=

e

e

r

rr

rrrrG

3

214

2

3

44)(

2

32

1

FGmax

r/re


14/440

It can be shown that

[ ] erratdr

rGd == 0)(( )2

32

316

34)(

lv

eePP

rrG

==

Since the free energy is less, a nucleus smaller than re will collapse

and a nucleus larger than re will grow spontaneously.

If one more molecules collides with the equilibrium nucleus ,than the nucleus will grow and vice versa.

Rate of nucleation: J

J = the number of critical size nuclei per unit volume per unit timewhich grow to macroscopic size .


15/440

Boiling occurs if J > 109 to 1013 m-3s-1

)( erNJ=N(re) = number of equilibrium nuclei per unit volumeG = collision frequency

l

e

kT

rG

le NrN

)(

exp)(

=

N(re) is given by the Boltzmann equation

k = Boltzmann constant

Nl= Number density of liquid molecules

=Bernath

m

Westwaterh

kTl

2

1

2

h is Planks constant

m is the mass of a single liquid molecule


16/440

For water at 100H , G=1012 ~ 1013 s-1

Solution produce to determine the limit of superheat

1. At a give a superheat , evaluate re from the following eq.

lv

lvsat

satl

ei

vT

TTr

=

2

2. Evaluate 2

3

4)(

ee

rrG =

3. Evaluate le

kT

rG

le NrN

)(

exp)(

=

4. Evaluate le

kT

rG

lNJ

)(

exp

=

5. J > 109 to 1013 m-3s-1 , if yes, then Tl- Tsat is the superheat of

homogeneous nucleation , if not, repeat the procedure form 1 to 5

lT

J


17/440

Lienhards correlation:8

,,,, 095.0905.0 satrsatrsatrlr TTTT +=

Water Superheat for Homogeneous Nucleation

1 70 155

Van der Waals 175(174)* 11.2(14) * -5.75(-7.1) *

Berthelot 223(223) * 47.2(49.2) * 10.3(10.2) *

Kinetic 206 49.3 10.4

Lienhard 214 46.0 10.6

P(bar)T

l-Tsat

*IJKLMN)OP%.N


18/440

2.2 Heterogeneous Nucleation

Vapor formation with the presence of foreignbodies or container surface .

With the presence of non-condensible gases

a

e

lv

e

lav

Pr

PP

r

PPP

=

=+

2

2

The presence of dissolved gas reduces the

superheat requirement to maintain a nucleus of radius re

With the presence of a flat solid surface surface

wettability effect


19/440

Contact Angle

VaporLiquid

C

C: contact angle , measuring from liquid sideC= 0 Q perfect wetting ; eq. Freon on metal surface.

C= 180Qpoor wetting.Typically , C = 60Q~ 90 Q

Soild

"


20/440

Static advancing / receding contact angle

Cs,A

Cs,R

Liquid Supply Liquid Withdrawal

Static advancing Static receding

Cd,A

Cd,R

Vapor/air

Vapor/air

Dynamic advancing

Dynamic receding


21/440

Flat surface

B-C

C

re

vs ls

vV

lvA

vsA

Vapor

Force balance at the triple point

cos

)cos(

lvlsvs

lslvvs

==+

=lv

lP

vP

"


22/440

Free energy to form a vapor nucleous on a flat surface

)()(lvvlsvsvsvslvlve

PPVAAArG +=

[ ]

+=

+=

e

vvslv

e

lvvlsvsvslvlv

r

VAA

rVAA

2cos

2

lvlvA vsvsA +

lsvsA

)( lvv PPV

: Addition surface energy

: Liquid-solid interface replaced by vapor-solid interface

: Work done by the vapor on the liquid system

( )

( )

233

222

sincos3

cos13

2

sin;cos12

eev

evselv

rrV

rArA

++=

=+=


23/440

)(3

4

4

sincoscos22

3

4)(

2

22

e

ee

r

rrG

=

++=

2

3

4er

)(

)(

: free energy for forming an equilibrium nucleous in the

homogeneous liquid .

C= 0Q , i.e. in perfect wetting liquid.

C=180Q, i.e. poor wetting liquid.

0.0

0.5

1.0

0o

90o 180o

)(


24/440

))(

exp()( 32

g

ee

kT

rGNrN

=

At low contact angles, the superheat required for homogeneous

nucleation is lower than for heterogeneous nucleation with the

presence of a flat surface. At a contact angle of approximately 68Q

, the two modes are equally probable .For C= 90Q, the superheatis reduced by approximately 35%.

The reduction in superheat is insufficient to explain the very much

lower superheat found in practical situations, particularly with water,compared with those required for homogeneous nucleation.


25/440

Initiation heat-transfer and nucleation-site data(Form Van Stralen R Cole)Site q , kW /m 2 T w - T s, K R ,T m Type of s i te

1 27.9 6.8 5

2 28.1 7.2

4 29 .0 8 .0 6

8 29 .0 8 .3 4

18 29 .6 9 .0 5 .5

19 29 .6 9 .0 2 .5

20 29 .6 9 .0 1 .5

33 33 .9 9 .3 3

34 34 .3 9 .5 2

41 35 .1 9 .5 1 .5

42 35 .1 9 .5 1 .5

44 44 .6 10 .2 2 .5

45 44 .6 10 .5 1 .5

50 44 .6 10 .6 1

Pit , deep in par t

Largest cavi ty in c ra ter of 80T mdiameter

End o f g roove ,12 T m w ide

Pit , deep centra l par t of 4T m

diameter

Pit , e longated and d eep, 6 18 T m

Pit , deep in par ts

Si te a t end of groove, 20 T m

wide

Pit , deep in par ts

Pi t , deep in par ts

Elong ated pi t , deep a t one end

Pit, largest in crater of 90 T mdiameter

Pit , deep par t of 2 T m diamete r

Pit , deep par t of scra tch, 1T m w i d e

Pit , shal low cra ter 25T m w ide


26/440

Scanning-electron-mircoscope photographs of selected natural nucleation site.


27/440

Vapor Trapping Mechanisms Proposed by Bankoff

U

C

Liquid

For C>U, the flood of liquidwould tend to leave a vapor or

embryp in the cavity

U

C

BVC

LiquidGas(or vapor)

Gas(or vapor)

For C>BVU, gas or vapor woulddisplace the liquid in the cavity.

Advance of a liquid sheet

Retreating of a liquid sheat

(advancing of a gas-liquid interface)


28/440

Limiting cavity conditions

Case Type of cavity Condition Trapping Ability

1 steep (U small)

2 steep (U small)

3 shallow(U big)

4 shallow (U big)

Poorly wetted (C big,

C >U , C >B VU )

Well wetted(C smallC


29/440

C> 90

Surface tension tends to resist

further penetration of the liquid

Effective Nucleation Site

C< 90Surface tension at the interface

tends to lead liquid to penetrate

and cause the cavity to be ineffective


30/440

Re-entrant Cavity

C< 90 Natural re-enterant cavity

C

Artficially fabricated re-entrantcauity


31/440

Cavity Nucleation Process

ra

r2

r3

r1

Curvature of a liquid-vapor interface emerging form a conical cavity.

Form Van Stralen R Cole


32/440

U

C

rc

U

C

rc

Advancing liquid front

Vapor trapping model of Lorentz, Mikic, and Rohsenow


33/440

Form Tong et al. 1990 (Int. J. Heat Mass Transfer, vol.33, pp.91-103)

( )

3

1

2

3

2

3

3

2tan

2tan

1

1

2tan

2tan

2sin2sin32

2tan

2cos

2sin2sin

sin

+

+

=

cr

r CXU,in rad


34/440

2.3 Boiling Incipience and Size Range of Nucleation

Ref. Y.Y.Hsu, On the size range of active nucleation cavity on

a heating surface, J. Heat Transfer, pp.207-216 1962

Engineering surface is characterized by a distribution of cavities

with various size and geometries.

Experimental observations have indicated that for a given superheat

, there exists a size range of active cavities.

At higher heat fluxes or superheat, the range of active cavities isextended in both directions to both larger and smaller cavities.

Griffith and Wallis experiment:

For 25Tm diameter of artificial cavities, the wall superheat ofboiling incipience is 11k rather than 1.3k predicted by :

=

=

clv

lvsat

T

c

satri

vT

dT

dprTT

sat

212


35/440

Transient Heat Conduction in Thermal Boundary Layer

G.E.

t

T

y

T l

l

l

=

12

2

I.C. t =0, = TTl

B.C.

==

==

=

TTy

caseqq

caseTTy

o

wl

2

10

Zyb

y

rb

0qqorTT w ==

T[

Case 1

( )

+

=

=

=

2

22

1

expsincos2

tnyn

ny

TT

TT l

nw

l

yt

= ,


36/440

Bubble interior Temperature

lv

lvsat

b

satv

i

vT

rTT 2

=

Criterion for the bubble embryo to grow is the condition that the liquid

temperature at the bubble cap (y=yb)is equal to or greater than the

bubble interior temperature.

lv

lvsat

b

satbli

vTr

TyyT 2)( +=

2111

===== forrrycrcr cbbcb Rwhere


37/440

rc,maxrc,min

Tl-T

[

Tsat

-T[

t

Steady state temperature distribution

y or rc

lv

lvsat

b

satvi

vT

rTT 2

=


38/440

Size range of nucleation

max,min, cc rrr


39/440

Boiling incipience

No cavity will be effective if the discrininant in the above two

equation is negative

04

1

2


40/440

2.4 Bubble Dynamics

In homogeneous medium

In heterogeneous medium

More relevant to experimental boiling heat transferbut is more difficult to tackle

Extended Rayleigh Equation

"

R0

Rb

P

vP

l

Mechanical energy balance

drrtPdrrv b

b

R

RR l

222 4)(42

1

0

=

FP(t) = Excess pressure of bubble

Continuity equation

dt

dRRwhere

Rr

RrRRvr

b

b

b

b

bb

=

==

&

&&2

2

22)(44


41/440

Substituting the continuity equation into mechanical energy balance

equation results in

=

=

b

b

ob

R

Rbbl

R

RR

bbl

drrtPRR

drrtPdrrrRR

0

223

22

2

2

2

)()(2

1

4)(2

4

&

&

Operating on the above equation by

( )[ ]2223

)()(2

1

0 bb

R

Rb

b

bbl

b

b

RtPRdrrtPRdt

dR

RRdt

d

Rdt

dR

dt

d

b

=

=

=

&&

l

bbb

tPRRR

)()(

2

3 2 =+ &&&


42/440

( )Rrr

b

lvR

PPtP

+=2

)(

= Vapor pressure inside the bubble.vP Vapor pressure corresponding to the instantaneous liquidtemperature at the interface.

lP =Imposed static pressure well away form the bubble.

bR

2

=Capillary pressure due to the bubble interface.

= V

dr

dvrlRrr b

r

3

22)( =Normal stress due to the liquid motion

( ) bbRr Rdt

dRv &==

( )R

bb

b

rrr

r

RR

Rr

vv

Rvr

dr

d

rv

+=

+== &

&r 221 222


43/440

( )

= b

bR

blRrr

RRr

Rb

&& 1

3

4

3

4

( )

r

RrRR

r

R bbbr

R

b

b

+

=

&&&

0lim

b

b

R

b

b

bb

b

b

b

b

bbb

R

R

r

R

RR

rR

R

rR

R

rR

rR

RrRR

b

&&

&&&&&

)2(

211

1

1

)()(

2

22

2

=

=

+

=+

=+

b

bl

b

b

b

blrr

R

R

R

R

R

R &&& 4

3

4)2(

3

4=

=


44/440

Extended Rayleigh Equation

=+

b

b

l

b

lv

l

bbb

R

R

R

PPRRR&

&&&

421

)(

2

3 2

Pv and E are function of liquid temperature at the bubble interface.Therefore, the energy equation for the liquid region must be solved

simultaneously.

Liquid energy equation

r

Tr

rrT

t

Tv

t

Tlr

==

+ 2

2

2 1b

br R

r

Rv &

2

2

=

Initial conditions

l

b

b

TrT

R

RR

==

=

),0(

0)0(

)0(0

&


45/440

Boundary conditions

)(3

)3

4(

4

1

),(

3

2

3

2 vb

b

lv

bv

b

lv

R

l

l

Rdt

d

R

iR

dt

d

Ri

r

Tk

TtT

b

==

=

Inertial controlled bubble growth, valid for initial growth stage

Neglect the capillary pressure and viscous stress andPv-Pl = constant = FP Thus,

( )[ ]

( ) cRPRR

dRRP

dtdt

dRR

PdtRR

PRRd

PRR

dt

d

RR

PRRR

b

l

bb

bb

l

bb

l

bb

l

bb

l

bb

bb

l

bbb

+

=

=

=

=

=

=+

232

22232

32

2

2

3

12

222

)(2

1

)(2

3

&

&&

&

&

&&&


46/440

0)0(

0)0(

=2Uas Bankoff suggested and

rc,min


108/440

-Poisson distribution

n = local nucleation site density

nb= mean nucleation site density

a = the area of an area element

3.5 Boil ing Chaos

Nucleate boiling heat transfer is influenced by many variables such as

surface material, geometry orientation, working fluid and its

wettability, etc.Nucleate boiling heat transfer is highly nonlinear and boiling chaos is

thus of no surprise

)!(

)()( na

eannaP

anna

b

b

=


109/440

]^_3`ab;cCz{s+def


110/440

Ma & Pan, 1999


111/440

Ma & Pan, 1999a


112/440

Ma & Pan, 1990a


113/440

Ma & Pan, 1999b


114/440

Ma & Pan, 1999b


115/440

Kacamustafaogullari & Ishii, 1983


116/440

Point et al., 1996


117/440

Kenning and Yan, 1996


118/440

Kenning and Yan, 1996


119/440

Shoji, 1998


120/440

Nelson et al., 1996


121/440

Nelson et al., 1996


122/440

Sadasivan et al., 1995


123/440

Sadasivan et al., 1995

CHAPTER4 Nucleate Boil ing Heat Transfer


124/440

and Forced Convective Evaporation

Topics to be presented in this chapter

Onset of Nucleate Boiling

Subcooled Nucleate Boiling

Saturated Nucleate Boiling and Two-Phase Convective Heat Transfer


125/440

From: Collier, 1981

4.1 Onset of Nucleate Boil ing


126/440

Review of single-phase heat transfer

Energy balance

+=

=

=

=

=

=

z

l

lil

z

lilll

zdzqGDCp

TzT

w

smkgfluxmassG

skgGD

flowliguidofratemassm

TzTCpmDdzzq

0

2

2

0

)(4

)(

)/(

)/(41

))(()(

&

&dz

D

Z

Wf(kg/s)

Length of subcooled region, zscb h l h h b


127/440

Let zscbe the entrance length where , zsc can be

determined by the following equation.

subcoolinglocalzTTzTsubcoolinginletTTT

Tq

GDCpz

qzqge

zdzqGDCp

TTzT

lsatsat

ilsatisat

isub

l

sc

z

l

lisatscl

sc

====

=

=

+==

)()(

4

)(..

)(4

)(

,,

,

0

0

0

satl TT

Wall temperature distribution in single-phase region


128/440

tcoefficientransferheath

hzqzTzT

zTzThzq

lw

lw

=+=

=

/)()()(

)]()([)(

For turbulent flow in a pipe

Dittus-Boeiter correlation

10000Reand50z/DFor

023.0

PrRe023.0Nu

33.08.0

33.08.0

>>

=

=

lll

k

CpGD

k

hD

However, it is commonly used in entrance region.

Boiling incipience (Onset of subcooled nucleate boiling)

For boiling to occur T must be at least gT


129/440

For boiling to occur Tw must be at least gTsat

isublisat

l

sat

l

li

l

lil

z

l

lil

satlw

TTThGDCp

zqor

ThGDCp

zqT

zqGDCp

T(z)T

qzqif

dzzqGDCp

TzT

Th

zq

zTT

,

0

]14

[

]14

[

4

constand)(

)(4

)(

thatRecall

)(

)(

=+

++

+=

==

+=

+=

q

isubT ,

isub

l

ThGDCp

zq ,14 =

+

Non-boiling region

Recall that, assuming that a wide range of cavity size available,

ik1


130/440

for the onset of nucleate boiling.

For water, the correlation of Bergles and Rohsenows correlation

maybe used:

For a tube with a constant heat flux, , a mass flux, G, the location

of boiling incipience may be evaluated in the following way:

lvsat

lvl

ONBsatONBvT

ikTq

)2(4

1,

=

0234.0463.0

156.1, ]1082[556.0

PONB

ONBsat

q

T

=

q

hqzTzT lw /)()( +=

For the criterion for the onset of nucleate boiling,


131/440

Thus,

4)(

thatRecall

)(8

)()(

Thus,

)(8

,

5.0

5.0

5.0

5.0

l

ill

lvl

lvsatsatONBl

lvl

lvsatsatw

GDCp

zqTzT

qik

vTT

h

qzT

q

ik

vTTT

+=

=

+

=

5.0

5.0

)(84

qik

vTT

h

q

GDCp

zqT

lvl

lvsatsat

l

ONBli

=

+

+

+

= 5.0

5.0

)(8

4q

k

vT

h

qT

GDCpz lvsatisat

lONB


132/440

, )(4

qikhq lvl

isatONB

+

=

0234.0463.0

156.1,]

1082[556.0

4

PONBisat

lONB

P

q

h

qT

q

GDCpz

If water is used as the working fluid.

q Relation between heat flux and wall

h t t th iti i i i t


133/440

atw TTT =

ONBq superheat at the position incipientboiling (Form Hino & Ueda, 1985

Int. J. Multiphase Flow vol.11,No.3, pp.269-281 )

figuretheinEq

ri

TkTT

r

kq

figuretheinEq

vT

ikTTq

vlv

satlONBsatw

lONB

lvsat

lvlONBsatwONB

)4.(...........

..)(

2)(

)3.(...............

.)2(

)(4

1

2

maxmax

2

=

=

Boiling hysteresis


134/440

wT

q

ONBTsatT

Partial subcooled nucleate boiling

Fully developed subcooled nucleate boiling

Single-phase liquid

Non-boiling region

q KT


135/440

q KTw

satws TTT = mzDistanceBoiling curve hysteresis Wall temperature profiles

Form: Hino and Ueda, 1985, Studies on Heat Transfer and Flow

Characteristics in Subcooled Flow Boiling Part1, Boling

Characteristics, Int. J. Multiphase Flow, vol.11, pp.269-281


136/440

satEXP

lONB

ONBsat

TofvaluereportedfromX

q

T

X

= Pr)()(

5.0

05.0X

XEXP

Experimental data for the onset of boiling compared with eq.

(Forost and Dzakowic34) Form Collier, 1981

4.2 Heat Transfer in Subcooled Nucleate Boil ing


137/440

I. Partial subcooled boiling

i. Few nucleation sites

ii. Heat transferred by normal single-phase process

between patches of bubble + boiling

II. Fully developed subcooled boiling

i. Whole surface is covered by bubbles and their influence regions.

ii. Velocity and subcooling has little or no effect on

the surface temperature.

Partial subcooled boiling


138/440

Single-phase heat transferSubcooled boiling

heat transfer

Rohsenow correlation


139/440

=

=

+=

fluidsotherfor

waterfor

l

ll

vllvl

scBsf

lv

satl

scB

lwspL

scBspL

k

Cp

gi

qC

i

TCp

q

ncorrelatioBoelterDittush

TThq

qqq

7.1

0.133.0

)(

:

:

][

boilingnucleatesubcooled:scBliquidphasesingle:spL


140/440

Values of Csfin eq. Obtained in the reduction of the forced

(and natural) convection subcooled boiling data of various investigators

(Form Collier, 1981)

q Full developed


141/440

"

"

"

"

satT ONBwT )(

Cq

ONBq

wT Wall temp

Full developed

Partial

B

C

E

"

C

D

scBq

Bergles and Rohsenow correlation

50

[ ] 5.02)( ++= qqqq


142/440

Calculation procedure

1. Pick Tw

2. Evaluate and

3. Determine C, ie

form the incipience boiling model

4. Evaluate from fully-developed

equation but setting

5.02

11

+=

scB

C

spL

scB

spL

q

q

q

qqq

scBq )]([ satwspL TThq =

))(..( ONBWONB Tsvq

Cq ONBww TT )(=

[ ]5.0

2

11

)(

+=

++=

scB

c

spL

scBspL

CscBspL

qq

qqq

qqqq

Fully developed subcooled boiling


143/440

-Surface is covered by bubbles and their influence region

-Velocity and subcooling has little or no effect on the surface temperature


144/440

)ln( lw TT

)ln(q

)ln( lw TT

)ln(q

Fullydeveloped

boiling

H

L

H

L

Low subcooling

High subcooling

H Low G

L High G

Correlations for fully developed subcooled boiling

-Jens and Lottes correlation(1951)


145/440

-Ranges of data for water only

i.d = 3.63 to 5.74 mm , P = 7 to 172 bar , Tl = 115 to 340H

G = 11 to 1.0510-4

kg/m2

s , up to 12.5 MW/m2

-Thom et al (1965)

for water only

barin,Mw/mink,in

)(25

2

6225.0

PqT

eqT

sat

P

sat

=

q

barin,MW/mink,in

)(65.22

2

875.0

PqT

eqT

sat

P

sat

=

-Jens and Lottes correlation in terms of heat transfer coefficients

62250)(25

P

T


146/440

6225.0)(25sat eqT =

superheat.wallinincreasewithincreasesh

)(

25

1)(

25

1

flux.heatinincreasewithincreasesh

)](25

1[

)(25

3

4

62

4

4

62

75.062

6225.0

sat

P

sat

sat

P

sat

P

P

sat

Te

T

Te

T

qh

or

qe

eq

q

T

qh

=

=

=

=

=

=

??

15570,/10 26

==

hT

barandbarPmWq


147/440

?? == hTsat

At P=70bar

At P=150bar

kmWkmMWeh

keTsat

25275.062

70

62

70

25.0

/1024.1/124.01]25

1

[

1.8)1(25

===

==

kmWkmMWeh

keTsat

25275.062

155

62

155

25.0

/1087.4/487.01]25

1[

1.2)1(25

===

==

It is summarizes the present data

of heat transfer of subcooled flow

boiling of water in the swirl tube


148/440

boiling of water in the swirl tube

and hypervapotron under one side

heating conditions. The data areobtained by experiments in

regions fro non-boiling to highly

subcooled partial flow boiling

under conditions that surface heat

fluxes, flow velocities, and local

pressure range form 2 to 3

MW/m2, 4 to 16 m/s and 0.5, 1.9

and 1.5 MPa respectively. In the

figure, a new heat transfer

correlation for such subcooled

partial flow boiling under one

sides heating conditions on which

no literature exists is proposed.

Heat transfer of subcooled water flowingin a swirl tube(Form: S.Toda, Advanced

Researches of Thermal-Hydraulics under

High Heat Load in Fusion Reactor, Proc.

NURETH-8. pp.942-957,1997)

-Shah(1977), ASHRAE Trans. Vol.83, Part1, pp.202-217


149/440

satw

lsat

l

TP

satw

lsat

l

TP

satw

lsat

TT

TT

h

hthen

TT

TTif

h

hthen

TT

TTif

+=>

=10000

-Correlation of Bjorge, Hall and Rohsenow, 1982

Int. J. Heat Mass Transfer, vol.25, No.6 pp.753-757

For subcooled and low quality region


150/440

For subcooled and low quality region

f

fw

bl

blfl

flbl

FC

FC

lwFCsubsatFCFC

sat

ONBsatscbFC

Tbulkb

TTfilmf

k

CpGD

k

Dh

eqColburnbyecaluatedish

TThTThq

TTqqq

,

2023.0

.

)()(

)(1)()(

3/1

,

,,

8.0

,,

5.02

3

22

=

==

=

==

+=

.eqRohsenowbyevaluatedisqscB

3

8/18/58/98/7

8/18/198/172/15.0

)()( sat

satvllvl

vlllM

vllvl

scB TTi

CpkBgi

q

=

BM Depends upon boiling surface cavity size distribution

and fluids properties For water only, BM=1.8910-4 in SI units.

For rc,min>rmax (the radius of largest cavity, rmax)


151/440

[ ]

ONBsatlv

lvsatcritc

l

FC

FClvsat

lvl

subONBsat

critc

subONBsat

Ti

vTr

k

rhN

hvT

ik

TT

rrfor

TN

NN

T

)(

4

8

)41(12

1)(

)

4

1(

1

1)(

,

max

2/1

max,

=

=

=

++

=

=

=


172/440

CyT

dy

dT

+==

++

+

+

Pr

Pr

1

1

Applying the B.C. at y+ = Z+

++++

++++

++

+===

+==

Pr0Pr*0)0(

PrPr

Pr

sat

sat

sat

TyT

TyT

TC

5,Pr)( =

= ++

+

satw

w

sat TTq

CpuT

satT

+y

+

5


173/440

++

++

+

+++

++

+

++

+++==

++==

++=

+=

sat

sat

TyT

CTT

Cy

T

dydTy

)]1Pr

1

(5ln[5)]1Pr

1

(5

5

ln[5)5(

)]1Pr

1(5ln[5)(

)]1Pr

1(

5ln[5

]5

)1Pr1[(1

)(

1

1

+++

+ +++= satTy

T )]1Pr

1(

5ln[5)]1

Pr

1(

5ln[5

++y

sa

5 30

At y+=5, temperature evaluated from the buffer zone must be equal to

that from the viscous sublayer.

R ll th t f th i bl


174/440

Recall that for the viscous sublayer

Pr5)]1Pr

1(

5ln[5Prln50)0(

Pr5)]1Pr

1(

5ln[5Prln5

)]1Pr

1(

5ln[5)]1

Pr

1(1ln[55Pr)5(

Pr

2

2

2

2

++====

++=

+++=+=

+=

++

++

++

++

+

++

sat

sat

sat

TCyT

TC

TCT

CyT

Pr5)]1Pr

1(

5ln[5Prln5 +++=

++

satT

++=

+++=

+

+

)]15

Pr(1ln[Pr5

Pr5)]1Pr1(

5ln[5Prln5)(

satw

w

TTq

Cpu

or

30),3( >+Case

For the turbulent core region++ dTy

])11

[(1


175/440

Recall that

for the viscous sublayer,

for the butter zone

so, the continuity of temperature at y+ = 5 results in

+++

+

+

+++=

+=

satTy

T

dy

y

)]1Pr

1(

5.2ln[5.2)]1

Pr

1(

5.2ln[5.2

]5.2

)1Pr

[(1

++

= yT Pr

Cy

T ++=+

+ )]1Pr

1(

5ln[5

Pr5ln(Pr)5

ln(Pr)5)]1Pr

1(1ln[5Pr5

+=

+=++=

C

CC

So, for the buffer zone

Pr5ln(Pr)5)]1Pr

1(

5ln[5 +++=

++ yT


176/440

The continuity of temperature at y+ = 30 results in

Pr5ln(Pr)5]Pr

15ln[5

)]1Pr

1(

5.2

30ln[5.2)]1

Pr

1(

5.2ln[5.2

Pr5Prln5)]1Pr

1(6ln[5

)]1Pr

1(

5.2ln[5.2)]1

Pr

1(

5.2

30ln[5.2

++++

++=

+++=

+++

++

++

sat

sat

T

T

+

++++=

+

+ ]

)1Pr

1

5.2

30(

)1Pr

1

5.2(

ln[2

1)1Pr5ln(Pr5

satT

1Pr

)30

ln(2

1)1Pr5ln(Pr5

++++

+

if

Tsat

+

++

5Pr

)(

TTC


177/440

>

+++


178/440

U = average liquid flow rate assuming full pipe flow

+

+

+

+

+

+

=

====

=

=

=

sat

satsatsat

sat

sat

satw

TU

u

TU

u

v

DUv

T

Du

kT

CpDu

k

hDNu

T

Cpuh

h

CpuT

hq

TT

1RePr

1))()((

1

+

5Re

u


179/440

>

+++

+

=


190/440

TP

TP

TP

TP

TP

tttt

S

XX

eR70

0.70eR5.32

5.32eR

1.0

])e(R42.01[

])e(R12.01[

10.0)213.0(35.2

178.0

114.1


191/440

From: Collier, 1981

2


192/440

2

kWmq

CTsat Comparison of Chens correlation with Thoms and Jens-Lottess correlations.

P=10MPa, G=5.4Mgm-2s-1, De =0.3cm.

From: Hewitt, Delhaye & Zuber, 1986. Vol.2 ch3.

Modification Chens correlation

-Hahne, et al.s correlation, 1989.

Hahne Shen & Spindles 1989 Int J Heat Mass Transfer


193/440

Hahne, Shen & Spindles, 1989 Int. J. Heat Mass Transfer

, vol.32, No.10 1799-1808, 1989

[ ]

133.0

0

27.0

0

736.0

4.0

8.0

)())()1(

8.1

4.4()(1.2)(

1.0/1213.0)/1(35.2

1.0/11

Pr)1(

023.0

p

p

c

c

c

m

o

NCB

tttt

tt

ll

l

c

NCBcTPW

R

R

P

P

P

PP

P

q

q

hh

XXF

XF

D

kDxGh

ShFhh

++

=

>+=

=

=

+=

k

gh

kS

l

vl

c

l

5.0

5.0

])(

[041.0

)1(exp1

=


194/440

KmWh

Rfor

P

Pm

fluidsorganicfor

DINbydefinedroughnesssurfacetheisR

mRmWq

gh

c

p

p

l

vlc

23

0

3.0

0

6

0

24

0

/103.2

12

3.09.0

)4262(

101,/102

)(041.0

=

=

==

--Effect of flow direction is also studied in this paper. There is no clear

effect of flow direction

--Upwards or downwards with a minimum liquid velocity of 0.25 m/s

-Gungor-Winterton Correlation

Ref: Gungor. K.E and Winterton, R.H.S(1986),A general correlation

for flow Boiling in tubes and annuli, Int. J. Heat Mass Transfer.


195/440

Vol.29 pp.351-358

crrrNCB

lvtt

ll

l

c

NCBcTP

P

PPqMPPh

Gi

qBo

XBoF

D

kDxGh

ShFhh

==

=++=

=

+=

67.05.055.012.0

86.016.1

4.0

8.0

)())ln(4343.0(55

,)1

(37.1240001

Pr)1(

023.0

DxG

ES

weightMolecularM

l

117.126

)1(Re

]Re1015.11[

,

=

+=


196/440

KmWinhmWinqwhere

l

l

22 /;/

Re

=

)()(

&

05.0

)(.

)21.0(

2

2

satwNCBlwl

Fr

l

TTShTThqboilingflowsubcooledFor

FrSSEFrE

FrIf

gD

G

NoFroudeFr

boilingflowhorizontalFor

+=

==

00030,01)461(

0003.0,0.12305.0

5.0

BoifhBo

BoifhBo

l

l


205/440

=

>

+

=

0011.043.15

0011.070.14

1.0)74.2exp(

1.00.1)74.2exp(

0003.0,0.1)461(

15.05.0

1.05.0

Boif

BoifF

ifhFBo

ifhFBo

BoifhBo

h

l

l

l

NCB

lvGi

q

Bo

=

CHAPTER5 Critical Heat F luxIndividual

bubble

region

Vapor

mushroom

regionq

q


206/440

Topics to be discussed

Critical Heat Flux for Pool Boiling

Critical Heat Flux for Flow Boiling-Dryout: Dryout of the liquid film, theoretical modeling

-DNB: Departure from nucleate boiling, theoretical modeling

Empirical Correlation for Flow Boiling CHF

regionCHFq

trq

satw TT

5.1 Critical Heat Flux for Pool Boiling

Kutateladze (1951) used dimensional analysis to obtain that


207/440

Zubers modelRef.: N. Zuber, 1958, On the Stability of Boiling Heat Transfer,

Trans. ASME, vol.80, 711

Recall that for the Helmhotz instability to take place,

=

=

Lienhard

CollierC

gCiq vlvlvCHF

131.0

16.0

)]([ 4/12/1

2/12/1

2

2

)(

)(

2

)()(

=

==

++

vl

vl

lv

lv

lv

lv

g

gk

UU

k

From the Taylor instability and assume that the bubble is sphere,

Liquid


208/440

[ ]

[ ]

),)4

(3

4(

24)

2(

1)

4(

3

4

)()(

)()(

)()(

3

2

3

2/12

22

2/12

periodwavetheisvolumebubbletheis

iiq

gU

UUU

gUU

vlvvlvCHF

vl

vlvlv

vlv

vl

vl

vllv

==

+=

+=

Q

Vapor Vapor

At the critical heat flux point

vU

=


209/440

Haramura and Kattos model

Ref.: Haramura & Katto, 1983, Int. J. Heat and Mass Transfer,

vol.26, No.3, pp.389-399

4/12/12/1

4/12/12/1

4/12/1

)]([][131.0

)]([][24

)]([][24

vl

l

vlvlv

vl

l

vlvlv

vl

vl

vlvlvCHF

gi

gi

giq

+

=

+

=

+

=

Vapor bubble


210/440

22

2/1

2/1

)()(2

4

1

)(23

q

i

A

A

thicknessmacrolayer

g

yinstabilitTaylortheforlengthwavewavelengthdangerousmostThe

lvv

w

v

vl

vl

Hc

vl

D

+

=

=

=

D D D

c

At CHF

Hovering period for a bubble of volumetric growth ratevl

lvvwclwd iAAAq )( =

)11

(4 vl +


211/440

Haramura and Katto postulated that CHF appears when the liquid

film evaporates away at the end of the hovering period.

)()(),/(

])(

)

16

(

[)43(

23

2

5/15/35/1

DlvvDl

l

vl

vl

d

withinbubbleones

miqv

vg

=

=

modelsZubertheassame

theispredictedqAssume

A

A

A

A

A

A

g

i

q

CHF

v

l

v

l

w

v

v

l

v

l

w

v

w

v

v

vl

lvv

CHF

')1(

)116

11(

0654.0

)116

11(

)1(

132)(

2/1

5/3

16/5

5/3

16/58/516/1

211

4

4/1

2

+

+=

+

+

=


212/440

High heat flux boiling of water on

a horizontal 35-mm diameter

copper disk (Katto et al., 1970)

Behavior of vapor mass above a10-mm diameter disk observed by

Katto and Yokoya(1976) (Interval

between each frame: 11.3 ms for

No. 1-5 and 8.8 ms for No.5-10)


213/440

Void fraction at midplane of a

vertical rectangular plate of contact

angle of 27 deg measured by Liaw

and Dhir (1989)


214/440

Existing data and predictions for

initial macrolayer thickness in

water boiling at atm. Pressure.

Comparison of initial macrolayerthickness measured by Okuyama

et al. (1989) for R-113 boiling with

the prediction of Haramura and

Kattoss equation.

Parameter effect on pool boiling CHF


215/440

From El-Wakil, Nuclear Heat Transport, 1978 .


216/440

)/10( 26 mW

qCHF

Pressure (kPa)

Effect of system pressure on CHF (From Pan & Lin, 1990)

CHFT)(


217/440

From: J. H. Lienhard, Burnout on Cylinders, J. Heat Transport,

vol.110, pp.1271-1286,1988.

26

qCHF


218/440

)/10(26

mW

Tsub (K)

(From Pan & Lin, 1990)

]102.01[

1

)()(1.0

]1[

75.0

+

+=

=

+=

Ja

Jak

i

CpB

TBqq

sub

lv

l

v

l

subZuberCHF

lv

subl

v

l

vl

i

TCpJa

JaJa

k

=

=

=

75.0

75.0

)(

)/(

10


219/440

Horizontal ribbons oriented

vertically

horizontal ribbons, one side insulated

(From: Lienhard, 1981, A Heat Transfer Textbook)


220/440

The peak pool boiling heat flux on several heaters

(From: Lienhard, 1981, A Heat Transfer Textbook)


221/440

From: Lienhard, A Heat Transfer Textbook Prentice-Hall 1981

2/15/3


222/440

CHF

(MW/m2)

Contact Angle

Effect of surface wettability on the critical heat flux:

comparison between model prediction and experimental data

(From Pan & Lin, 1990)

2;00179.0

1

11611

5/3

==

+

+

=

nC

CA

A

v

l

v

l

n

w

v

5.2 Dryout Modeling (Dryout heat flux)

Following (1)Isbin and his co-workers (2)Whalley

Ref.: J.Weisman, 1985 Theoretically Based Predictions of Critical

Heat Flux in Rod Bundles


223/440

The liquid film flow (in a round tube) is determined by a balance

between entrainment,evaporation and deposition.

)(

)/(

][

mdiametertubeD

skgrateflowfilmliquidW

Ddzi

qDdzE

dzdz

dWW

dzDDW

l

lv

l

l

dl

=

=

+

++

=+

dz

Wf

D

Deposition

Entrainment

evaporation

DrateinflowW

smkgfilmthefromdropletsofratetentrainmenE

smkgfilmtheontodropletsofratedepositionD

l

d

=

=

=

)/(

)/(

2

2


224/440

][lv

d

l

i

qEDD

dz

dW =

dz

Wf

Deposition

Entrainment

evaporation

Similarly, the entrained liquid flow

rate satisfy the following eq.][ d

E DEDdz

dW=

ratenevaporatioDdzi

q

rateentraimentDdzE

rateoutflowdzdz

dWW

ratedepositiondzDD

lv

ll

d

=

=

=+

=

][

Deposition rate

=

=dsmtcoefficientransfermassordepositionThe

CD

)/(K

"


225/440

=

==

+=

=

==

z

zlv

v

vvlEE

vi

k

l

s

dzzqGDi

zx

Wzx

vaporofrateflowMassWmkgWWW

ionconcentratDroplet

HewittbyproposedequationempiricaluD

u

)(4

)(

)(

)/())/()//[(

C

)(/;][87

3

2/12

Entrainment rate

-Based on the observation that, under equilibrium conditions during

adiabatic operation, the entrainment and deposition rates equal.

Dd=mCE=E


226/440

Where CE is droplet concentration under equilibrium conditions

It is assumed that the same relationship holds away from equilibrium.

CE can be correlated as a function of iiZ/E

)/( 3mkgCE

/iCorrelation of equilibrium entrained droplet concentration

(Hutchinson and Whalley) Form Collier, 1981

To determine ii and Z, the triangular relationship may be used.[Relations between Wl, dP/dz, & Z]

Initial condition:


227/440

Given Wl , WE & WvSo that the above set of ODE can now be solved.

E.g. using an explicit scheme.

(1) Evaluate (Dp/dz) & _ at z = 0(2) Evaluate Z & ii(3) Evaluate C & CE(4) Evaluate Dd & DE

(5) Evaluate Wl , WE and Wv at z =z+dz(6) Repeat the process

CHF occurs while Wl=0

Summary

" Conservation of liquid film mass

][ dl qEDD

dW =


228/440

"Conservation of entrainment mass

"Deposition mass flux

C is droplet concentration

lvidz

][ dE DED

dz

dW=

v

v

l

E

E

l

l

v

i

d WW

W

DCD

+==

2

87k

"Entrainment flux

li

EEED

CCD

287k ==


229/440

"Equilibrium entrainment droplet concentrationlv

lv

G

GEln

vEln

vi

inE

i

Dq

dz

dW

D

WWWf

WWWfdz

dP

gdz

dPR

fC

=

=+=

+=

=

=

)1(2),,(

),(

][2

)(

)/(

, see, e.g., Martinelli correlation

, see, e.g., Martinelli

correlation

Ishii & Mishima model of Droplet Entrainment Correlation in

Annular Two-Phase Flow

Ref.: Ishii & Mishima, Int. J. Heat Mass Transfer, vol.32, No.10,

pp.1835-1846, 1989


230/440

-Inception criterion

(For Rel>2 for vertical downflow and Rel>160 for vertical up

or horizontal flow for Nuk1/15 )

l

lll

Dj

=Re

[ ]

2/12/1

8.0

3/18.02/1

))(/(/

1635Re

1635ReRe78.11)(

vlll

l

ll

l

vvl

gnumberViscosityN

forN

forNj

==


231/440

3/12

)(

)Re1025.7tanh(

v

vlvv

l

DjWe

We

=

=

Govan et al. (1988) model for droplet deposition and entrainment inannular two-phase flow (From: Collier & Thom, 1994)

2/1)(

Dk v

kC=D


232/440

65.02/1

2/1

)/(083.0)(

3.0

18.0)(

3.0

=

>

=

>>>+


356/440

>>>>>


357/440

>>>>


358/440

The characteristics of the set of equations can be evaluated by the

homogeneous part, i.e.,

0)()( =

+

z

uuB

t

uuA

vv )()(

tzieouuLet

=

0)()()()( =

tzieouuBuAi vvvThus,

The characteristic equation is hence given by:

0)()( = uBuA vv

The eigenvalue can then be determined. If is a complex number,

the system is unstable and the problem is ill-posed.

7.6 One-Dimensional Homogeneous F low Model

The homogeneous flow model assumes that both phase have same


359/440

velocity and temperature.

-Quality vs. void fraction

>>>>>>>


360/440

( ) zmvwvwwwmmmm

gD

Pz

wz

wt

++

=

+

ll

42

or

zmmm

e

mmmm gw

DfP

zw

zw

t

+

=

+ 2

2

2

2

11

Similarly the mixture energy equation can be obtained by combining

the phasic energy equation as:

mmw

H

mmmmm P

zwP

tq

Piw

zi

t+

+

+=

+ "

where

=mi mixture enthalpy[ ]>>


361/440

[ ] vvm

llll

( ) >>


362/440

Ref.: Zuber, N. and Findlay, J.A. , 1965, Average Volumetric

Concentration in Two-Phase Flow System, J. of Heat Transfer, pp.453-468.

-Define the vapor drift velocity as the difference between vapor velocity(in the axial direction) and mixture superficial velocity.

jww vvj =-Gas phase velocity:

>>=>=>>


364/440

The drift velocity is also a function of flow pattern:For bubbly and churn flow,

For slug flow

l /2.02.1 voC =

41

2

)(53.1

>>=>=


365/440

>>>=>>>=>=>>>=< vjom

vv

v wjCz

)1(

l


366/440

m

Similarly, the mixture momentum equation can be expressed as

zmmm

e

mmmm gwD

fz

Pw

zw

t +

=+

2

2

2

2

11

[ ]

>>


367/440

Based on the homogenous assumption, , thus,>>=>>>>>=


369/440

(2) Interactions between each phase and wall

(3) Turbulent model in two-phase flow

8.1 I nter facial Area Density

Recall that the time-averaged 3-D phasic equation includes the

following interfacial term

i

T kis

adensityarealInterfacia

volumemixture

arealInterfacia

LnVTs

====

11

vv

Ishii proposed that volumetric interfacial transfer rate=aidriving forceIshii and Mishima correlation

-For bubbly or dispersed droplet flow

sm

d

id

di

d

di

rAV

A

V

a 3

/3

3===

where Vd is the volume of a bubble or droplet,

Ai is the surface area of a bubble or droplet


370/440

i p

rsm is the Sauter mean radius . For a spherical bubble or droplet, rsm=r

Cross sectionally area averaged ai of bubbly, slug or churn-turbulent flow:

gs

gssm

gs

gs

gs

i forrD

a

>>==


373/440

--For subcooled boiling

Lahey and Moodys model

for water, H0=0.075H /s

-Local mass transfer rate

{ }])([

])([))((

)( 0lsatl

llvlsatlllvvldsatl

ldlvHv iTi

Cpv

H

iTiiiTi

ii

A

Pq

>=<

viv ma &=

for bubbly flow regime

where hli is the heat transfer coefficient between liquid and bubble.

Wolfert et al.s model

lv

illiv

i

TThm

)( =&

2/1]2

[t

llr

lllik

Cp

R

vCph =


374/440

For dispersed flow, i.e., liquid droplet flowing in superheated vapor

]1[l

ll

b

k

kk

R+

v

drv

dvd

d

vvd

lv

satvvd

Dv

D

kh

i

TThm

=+=

=

Re],PrRe74.02[

)(

3/12/1

&

8.3 Interfacial M omentum Transfer RateRecall that

w

k

L

k

v

k

D

k

d

k

d

kkkikikk

MMMMM

MPVM

+++=

++=v

=DkM=vkM

Interfacial drag force due to relative velocity

Virtual mass force due to relative acceleration


375/440

k

=LkM=wkM

Interfacial lift force due to velocity gradient

Wall force near the wall (still subject to discussion)-Interfacial drag force

Ishii shows that the interfacial drag force may be expressed as:

]

2

)4

(

4

[ rrc

i

dDi

D

k

VV

A

ACaM

=

CD is the drag coefficient.

Ad is the projected area in the moving direction of the particle.

A is the surface area of the article..

Table Local drag coefficients in multiparticle system(From: Ishii and mishima)


376/440

CdCdCL

D

k VVVCM = )( CL=lift coefficientThe theoretical value for CL from ideal fluid flow analysis is 0.25,

Wang et al. show that CL=0.02 for bubbly flow.

-Interfacial lift force


377/440

The present mixture viscosity model compared to existing models for solid particle system.(From: Zuber&Ishii)

-Virtual mass forceTheoretical result from ideal fluid flow analysis,

Ishii and Mishimas model gives:--For bubbly flow

dt

VdF rc

dv

d

v

2

=

][211

Cdrd

d

v

d VVVVV

Mvvvv

++

=


378/440

--For standard slug flow

][12

Crrdc

d

dd VVVVt

M +

][)](27.066.0[5 Crrdr

cbb

d

v

d VVVVt

V

L

DLM

vvvv+

+=

8.4 Interactions Energy Transfer RateRecall that

and have been discussed previously in the

models for mass transfer.

8.5 I nter facial between Each Phase and Wall

Heat transfer bet een all and each phase

)( kikikikiikikik qimaqai +=+ &

k kiq


379/440

-Heat transfer between wall and each phase

For single-phase liquid flow or for nucleate boiling region, i.e.,

the surface is all or mostly covered by liquid.Thus,

Conversely, for film boiling region the surface is mostly covered by

vapor. Thus,

CHFwCHF TTqq ,

0= lwH

lw qA

P

qPH

vwlw = ;0

For transition boiling, TCHF<


380/440

f

lv

v

vw

f

lv

l

lw

z

P

zM

z

P

z

zM

,22

,22

)(

)(

>==>>


381/440

where is the turbulent diffusivity. For two-phase bubbly flow,

it may be evaluated by a modified k-j model as following:

t

kv

tb

k

k

kt

k vk

Cv +=

2

where is the turbulent diffusivity due to bubble agitation,

and may be evaluated by the model of Sato et al.(1981)

tb

kv

rb

tb

l VRv 2.1=

kkand jkare the turbulent kinetic energy and dissipation rateand may be evaluated by a modified two-equation model for

two-phase flow with dilute concentration of dispersed phase.


382/440

Chapter 9 TWO-PHASE FLOW PRESSURE DROP9.1 Two-Phase Flow Pressure Drop Based on Homogeneous Flow Model

From the steady state momentum equation

zm

mm

gG

D

fG

dz

d

dz

dP

++

=

2

2

2

2

11

gradientpressureFrictionaldz

dP

F

=

( )vxvGfGf ll +==2

2

2

2

1111


383/440

( )vm

vxvGD

fD

f ll +2222

nture gradieonal pressAcceleratidz

dP

A =

( )

+=+=

=

dz

dP

dP

dvx

dz

dxvGvxv

dz

dG

G

dz

d v

vv

m

lll

22

2

Where the liquid phase is assumed to be incompressible.

tre gradiennal pressuGravitatiodz

dP

G =

v

zzm

vxv

gg

ll +==

The frictional pressure gradient can also be expressed as( ):

+

=

+=

l

l

ll

l

llv

vx

dz

dP

v

vxvG

Df

dz

dP v

oF

v

o 112

11

,

2

Let be the two-phase frictional multiplier defined as:

lv

xdz

dP

vF +=

12

+ 11v

x v

02 lff =


384/440

l

l

lv

x

dz

dP

oF

o +=

1

,

+= 11

lvx v

At low pressures vv >> vl , can be as high as several hundreds.

For example, for saturated water at 1 atm, vv/vl=1603, =17.0 for

x=0.01,for =61.2, for x=0.1 =802,for x=0.5, =802, for x=0.5

2

0l

2

0l

2

0l

2

0l 2

0l

Combining the three pressure gradients together yields,

dPdvxG

v

vxv

g

dz

dxvG

v

vxvG

D

f

dz

dP

v

v

zv

v

2

222

1

1

12

1

+

+

++

+

= ll

l

l

l

l

l

In general, 12


385/440

Considering a boiling channel with a uniform heat flux distribution.

Thus, the vapor quality is increased linearly from x1, to x2 . The qualitygradient in the channel can then be evaluated by the following equatio

12

12

zz

xx

dz

dx

= dxxx

zzdz

12

12

=or

Further assuming that , ,integration of

the pressure gradient equation gives:

12


386/440

9.2 Two-Phase F low Pressure Drop Based on Two-F luid Model

Recall the steady-state momentum equations for liquid and vapor

phase, respectively:

Liquid-phase momentum equation

zww gPdz

dw

dz

dlllllllll =

42

Vapor-phase momentum equation

Assume and combing above two equations yields.

zvvvwvwvvvvv gPdz

dw

dz

d =

42

PPPv==l

[ ] ( )vwvwwwvvvD

wwdz

d

dz

dP +++= lllll

422


387/440

( ) zvv g ++ llusing the following fundamental relationships with _v = _

=

1

)1( xvGw

l

l

xvGw

v

v=

The above equation can be expressed as:

2222

2

2

1

1

)1(o

ov vGD

fvxvx

dz

dG

dz

dPll

ll

+

+

=

[ ] zv g ++ l)1(

: Acceleratial pressure gradient F i i l di


388/440

: Frictional pressure gradient

: Gravitational pressure gradientThe accelerational pressure gradient can also be expressed as:

+

=

2

2

2

2

2

)1(

)1(

1

)1(22

v

pA

vx

x

vx

x

vxxv

dz

dxG

dz

dP lll

dz

dPvxvx

PdP

dvxG v

x

v

++2

2

2

222

)1(

)1(

l

+

+

+

+

=

2

2

2

222

2

2

2

2222

)1(

)1(1

)1()1(

1)1(22

211

v

x

v

v

p

voo

vxv

x

PdP

dvxG

vxvxx

vxxvdzdxGG

Df

dz

dP

l

l

lll

+

+

++

2

2

2

22

2

)1(

)1(1

])1[(

v

x

v

zv

vxv

x

PdP

dvxG

g

l

l

In general


389/440

In general,

Again, for a boiling channel with a constant heat flux distribution,

the pressure drop can be obtained by integrating above equation as:

1)1(

)1(2

2

2

222


390/440

Lockhart & Martinelli (1949) found that can be correlated as a

function of the Martinelli parameter, defined as:0l

vF

F

dz

dP

dz

dP

X

,

,2

= l

If both phases are flowing as turbulent flow

, it can be shown that:

channelsametheinalonephasevaporofdroppressurefrictional

channelsametheinalonephaseliquidofdroppressurefrictional=

2.08.1

2 1

=

v

vtt

x

xX

l

l

2

,

2 11

)(

)(

XX

c

zP

z

P

lF

F

++=

=l 2

,

2

1)(

)(

XcX

z

P

z

P

vF

F

v ++=

=

Liquid Vapor c


391/440

211

XcX

X

++=

where c is given by the following table

Turbulent Turbulent 20

Laminar Turbulent 12

Turbulent Laminar 10

Laminar Laminar 5


392/440

Lockhart & Martinelli, 1949

If both phase are turbulent, and is related by the following equation:20l

2l

8.122 )1( xo = ll 2.08.1

1.05.0

9.09.08.1 )1(20)1(

+

+=

l

l

l

l

v

v

v

v

xxxx


393/440

Martinelli & Nelson, 1949 Martinelli & Nelson, 1948


394/440

Martinelli & Nelson, 1948


395/440

Martinelli & Nelson, 1948

B is given in the following table

-Chisholms (1973) correlation for two-phase frictional multiplier

{ }nnno xxBxIx ++= 22/)2()2(22 )1()1(1)(l

=

=tuberough

ntubesmooth

I

v

n

v

v

,

2.0,

5.0

2/5.0

l

l

l

Where


396/440

I )/( 2smkgG

B

< 9 . 5

1900

1900500

500

I

IG

/21

)/(520 5.0

> 2 8 )/(1500 5.02 GI

-Friedels (1979) model for two-phase frictional multiplier

+=

vo

vo

f

fxxA

l

l22

1 )1(

2240780 )1(A

035.0

2

045.0

2

32

1

2 24.3

WeFr

AAAo +=l


397/440

224.078.0

2 )1( xxA =7.019.091.0

3

=l

l

l

l

vv

v

A

( )22

2 v

xvvgD

GFr

ll +=( )vxvv

DGWe ll +=

2

2

9.3 Secondary Two-Phase Pressure Drop-Pressure change through a sudden expansion

Considering force balance in the control volume shown in fig.:


398/440

Using the following basic equations:

)()( 12122221 lll wwWwwWAPAP vvv +=

1

1

11

111 vv

v

xG

A

xAGw ==

== 11112AxGxAG

wv

)1()1(

)1()1(

1

1

11

11

1

==

ll

lxG

AxAGw

=

=2

1

2

1

22

11

2)1(

)1(

)1(

)1(

A

AxG

A

xAGw

ll

l


399/440

2222

2AA

wvv

v

The pressure change through the sudden expansion can be expressed as:

+

+

=

2

2

2

2

2

1

1

2

1

2

2

12

112)1(

)1(

1

)1(

x

v

vx

A

Ax

v

vx

A

AvGPP vv

ll

l

If the void fraction is unchanged through the expansion,

+

=

22

2

1

2

12

1121

)1(1

x

v

vx

A

A

A

AvGPP v

l

l

-Pressure change through a sudden contraction

Considering force balance for the control volume shown in Fig.


400/440

21212222212

1

2

1

2

1

2

1vvvv wwwwPP += llll

2

11

22

2

11

22

2

2

2

2

2

2

2

1

)1(

)1(

2

1

2

1

)1(

)1(

2

1

+

=

v

v

v

vA

xGA

A

xGAxGxG

l

l

l

l

( ) ( )

+

=

2

1

2

12

2

112

2

2

2

2

2

22

2)1(

)1(

)1(

)1(

2

1

A

x

v

v

A

xx

v

vxvG vv

ll

l

If the void fraction is unchanged, the above equation can be simplied as:

( )

+

=

2

2

2

2

2

12

2

221)1(

)1(11

2

1

x

v

vx

AvGPP v

l

l

2112 /AAAwhere =


401/440

CHAPTER 10 Steady-State Two-Phase Pipe F lows

Topics to be discussed

l Two-phase flow characteristics in a boiling channel basedon the 1-D drift flux model

l Two-phase flow characteristics in a boiling channel basedon the 1-D two-fluid model


402/440

on the 1-D two-fluid model

l Two-phase flow characteristics in a boiling channel basedon the multi-dimensional two-fluid modell Adiabatic two-phase bubbly pipe flow

10.1 Two-phase flow characteristics in a boiling channel basedon the 1-D drift flux model

Zsc)( satlTii=

Saha & Zubers model for bubble departure point

l

l

k

DCpq"0022.0 000,70


403/440

ZONB

Zd

ldii=

G

Where X and is the liquidsubcooling at the bubble departure point. From the

energy balance the bubble departure point can

be determined as:

ll kGDCpPe /= dsubi ,

)("4"

)(,,

,,

dsubinsub

H

dsubinsub

d iiq

GD

Pq

iiGAz =

=

Conservation of mass for vapor phase

Conservation of mass for liquid phase

Assume both liquid and vapor densities are constant, the above

two equations can be expressed as:

vvvwdz

d=)(

[ ] vwdzd

= ll )1(

vwd

)( ( )d )1(


404/440

Combined the above two equations yields,

Thus,

vvv vwdz

=)( ( ) ll vwdz

v= )1(

[ ] vvvvv vvvwwdz

dlll ==+ )()1(

vvvjdz

dl=

For the saturated boiling region,

For the subcooled boiling region, the vapor generation rate maybe approximated by the following equation:

Di

q

v

satv

l

"4==

,satdsc

dv

zz

zz

= scd zzz

iGD


405/440

where

"4

,

q

iGDz

insub

sc

=

vsat

dsc

d vzz

zzl

scd zzz

=dz

dj

vsatvl sczz

Integration of the above equation gives:

The vapor velocity can then be determined based on the drift flux model:

Th h h l i b d

)(

)(

2

2

dsc

dvsatin

zz

zzvw

+ l scd zzz


406/440

Thus, the vapor phase velocity can be expressed as:

vj

dsc

dvsatino w

zz

zzvwC +

+

)(

)(

2

12

l scd zzz

Integration of mass conservation equation for the vapor phase gives,

)(

)(

2

12

dsc

d

vsatzz

zzv

scd zzz


407/440

distribution in the channel:

vjzz

zz

vsatino

zz

zz

vsat

wvwC

v

dsc

d

dsc

d

++

][)(

)(

21

)(

)(

21

2

2

l

scd zzz


408/440

+++

++

+ + vj

dsc

dpcho w

zzzzNC

)()(

211

2

+++


409/440

++

zzsc1

=+ )(zwl

++

++

+ )(

)(

12

11

)(1

12

dsc

d

v

pch

zz

zz

R

N

z l

+++

+ dscv

pch

zzzR

N

z 2

1

2

1

12

11

)(1

1

l

[ ]dsubsubpch

d NNN

z ,1 =+ subpch

sc NN

z 1=+

hspchRPeN0022.0 000,70Pe

numberchangephasevLPq

N vH

pch ==l

"


410/440

gpviGA v

pch

ll

numbersubcoolingv

v

i

iN v

v

insub

sub =

=l

l

l

,

numberPeclet

k

GDCpPe ==

l

l

LP

AR

H

hs =l

lv

vR vv =


411/440

NPCH=5.2 Nsub=2.3 Rvl=20 Pe=95600 Rhs=2.510-2

10.2 Two-phase flow characteristics in a boiling channel based on 1-D

two-fluid model

Ref.: Hu & Pan, 1995

Mass conservation equation for vapor phase:

Mass conservation equation for liquid phase:

vvv

wdz

d

=)(

[ ]wd

= )1(


412/440

Mixture momentum equation:

[ ] vwdz

= ll )1(

dz

dww

dz

dww vvv

l

ll )1( +

[ ] lll wvvv MgdzdPww += )1()(

Equation by eliminating pressure gradient in two momentum equations

[ ] Divivv

v Mwwwdz

dwA

dz

dwA =+ )1( l

ll

ll wv Mg ++ )()1(

[ ] vvv wA )1( = [ ] lll wA )1( =

Liquid phase energy equation

"Pd


413/440

[ ] vvh

i

"qP

iw)1(dz

d

= lll


414/440


415/440


416/440


417/440


418/440


419/440


420/440

10.3 Two-phase flow characteristics in a boiling channel based on

multidimensional two-phase flow model

Ref.: Kurul & Podowski, 1990, 1991; Lai & Farouk, 1993

Conservation of mass

Conservation of momentum

( ) ( ) kkkkkk Vt

=+

vk ,l=

( ) ( ) ( ) ( )[ ] gVPVVVt

kkk

t

kkkkkkkkkkk

v +++=+

+++ vD MMM vk l


421/440

Conservation of energy

+++ kv

k

D

k MMM vk ,l=

( ) ( )

( )[ ] vlkPVtP

AqiCpkk

Viit

kk

kk

kkwkwkikk

t

kkk

kkkkkkk

,/ =+

++++=

+

v

Some closure equations

( ) LvLviD

D

v

D VVVVaCMM ll == 8

1

( )

+

==

ll

l

ll

vvvvvvVVVV

t

V

t

VMM

vv

v

v

V

v

V 2

1

vki VVM lll ==

vblvvdbvw NfiDq 3

,6

1=

vwlw qqq =


422/440

10.4 Adiabatic two-phase bubbly two-phase flow

Ref.: Lahey and his coworkers


423/440

(b)

Steady-state mass conservation equation

0= kkk V vk ,l=

Steady-state momentum equation

( ) kkkT

kkkkkkkkk MgPVVvvvv ++

++= vk ,l=

Where

VVVstressyonldsRe tl'k

'kk

T

k === pWLVD MMMMMMllllll

++++=2


424/440

lMMv =

w

b

r

v

bW nRV

yRM 147.01.0

ll

+=

lll = ip PM

2

rvpi VCPP

lll =

lvb

l

lt

l VVD.kC vv +=

60

jland klare given by the k-j model:

( ) ( )llllllll += GkvkV t

+

=

l

l

l

ll

ll

l

llll

v

kC

k

GC

vV

t 2

21

Where Gl is the turbulence generation rate defined as:


425/440

Where Glis the turbulence generation rate defined as:

( lllll VVVvG Tt += :


426/440

CHAPTER 11 TWO-PHASE FLOW INSTABI LI TY

Ref.1 Lahey & Moody, 1977, The Thermal Hydraulics of a Boiling

Water Nuclear Reactor, ANS.

2. Lahey & Podowski, On the Analysis of Various Instability inTwo-Phase Flow, in Hewitt, Delhaye, Zuber, Multiphase

Science and Technology, Vol.4 Hemisphse publishing.

Two-phase flow instability


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Two-phase flow instability

-Interfacial instability: 1.Taylor instability2. Helmhotz instability

-Channel or system instability :1.Ledinegg instability

2.Density-wave oscillation

3.Nuclear-coupled Density wave

oscillation4.etc

11.1 Classification of instability

Static instabilities - can be explained in terms of steady-state laws.

Dynamic instabilities - require a consideration of the transient

conservation equations.

Examples of static instabilities:

1.Excursive (Ledinegg) instabilities

2.Flow regime relaxation instabilities

3.Nucleation instabilities


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Examples of Dynamic instabilities

1.Density-wave oscillations

2.Pressure-drop oscillations

3.Nuclear-coupled density wave instability.

11.2 Excursive

Boiling Heat Transfer and Two-Phase Flow.PDF

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