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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
"
"
"
"
""
"
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#$%&'()*+,-./0123/0456
78/09:;?
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Mechanical Stability Limit :
0
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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
>
=
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( )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
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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.
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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 ?
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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
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Laplace equation (mechanical force balance )
rPP lv 2= E : surface tension
C1
C2r1
r2
r2
r1C1r2C2
Vapor liquid interface
Pv
lP
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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
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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
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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
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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
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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 .
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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
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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
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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
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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
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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
"
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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
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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
"
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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
++=
=+=
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)(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
)(
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))(
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.
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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
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Scanning-electron-mircoscope photographs of selected natural nucleation site.
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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)
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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
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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
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Re-entrant Cavity
C< 90 Natural re-enterant cavity
C
Artficially fabricated re-entrantcauity
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Cavity Nucleation Process
ra
r2
r3
r1
Curvature of a liquid-vapor interface emerging form a conical cavity.
Form Van Stralen R Cole
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U
C
rc
U
C
rc
Advancing liquid front
Vapor trapping model of Lorentz, Mikic, and Rohsenow
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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
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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
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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
= ,
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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
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rc,maxrc,min
Tl-T
[
Tsat
-T[
t
Steady state temperature distribution
y or rc
lv
lvsat
b
satvi
vT
rTT 2
=
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Size range of nucleation
max,min, cc rrr
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Boiling incipience
No cavity will be effective if the discrininant in the above two
equation is negative
04
1
2
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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
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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 =+ &&&
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( )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
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( )
= 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=
=
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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
&
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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
&
&&
&
&
&&&
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0)0(
0)0(
=2Uas Bankoff suggested and
rc,min
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-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
=
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]^_3`ab;cCz{s+def
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Ma & Pan, 1999
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Ma & Pan, 1999a
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Ma & Pan, 1990a
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Ma & Pan, 1999b
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Ma & Pan, 1999b
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Kacamustafaogullari & Ishii, 1983
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Point et al., 1996
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Kenning and Yan, 1996
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Kenning and Yan, 1996
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Shoji, 1998
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Nelson et al., 1996
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Nelson et al., 1996
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Sadasivan et al., 1995
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Sadasivan et al., 1995
CHAPTER4 Nucleate Boil ing Heat Transfer
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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
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From: Collier, 1981
4.1 Onset of Nucleate Boil ing
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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
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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
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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
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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
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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,
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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
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, )(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
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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
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wT
q
ONBTsatT
Partial subcooled nucleate boiling
Fully developed subcooled nucleate boiling
Single-phase liquid
Non-boiling region
q KT
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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
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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
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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
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Single-phase heat transferSubcooled boiling
heat transfer
Rohsenow correlation
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=
=
+=
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
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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
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"
"
"
"
satT ONBwT )(
Cq
ONBq
wT Wall temp
Full developed
Partial
B
C
E
"
C
D
scBq
Bergles and Rohsenow correlation
50
[ ] 5.02)( ++= qqqq
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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
qqq
qqqq
Fully developed subcooled boiling
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-Surface is covered by bubbles and their influence region
-Velocity and subcooling has little or no effect on the surface temperature
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)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)
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-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
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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
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?? == 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
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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
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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
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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)
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[ ]
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,
=
=
=
++
=
=
=
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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
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++
++
+
+++
++
+
++
+++==
++==
++=
+=
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
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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
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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
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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
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>
+++
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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
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>
+++
+
=
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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
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From: Collier, 1981
2
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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
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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
=
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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.
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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[
,
=
+=
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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
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=
>
+
=
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
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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
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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
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[ ]
[ ]
),)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
=
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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
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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 +
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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
+
+=
+
+
=
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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)
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Void fraction at midplane of a
vertical rectangular plate of contact
angle of 27 deg measured by Liaw
and Dhir (1989)
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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
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From El-Wakil, Nuclear Heat Transport, 1978 .
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)/10( 26 mW
qCHF
Pressure (kPa)
Effect of system pressure on CHF (From Pan & Lin, 1990)
CHFT)(
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From: J. H. Lienhard, Burnout on Cylinders, J. Heat Transport,
vol.110, pp.1271-1286,1988.
26
qCHF
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)/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
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Horizontal ribbons oriented
vertically
horizontal ribbons, one side insulated
(From: Lienhard, 1981, A Heat Transfer Textbook)
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The peak pool boiling heat flux on several heaters
(From: Lienhard, 1981, A Heat Transfer Textbook)
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From: Lienhard, A Heat Transfer Textbook Prentice-Hall 1981
2/15/3
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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
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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
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][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
"
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=
==
+=
=
==
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
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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:
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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 =
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"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 ==
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"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
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-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
==
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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
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65.02/1
2/1
)/(083.0)(
3.0
18.0)(
3.0
=
>
=
>>>+
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>>>>>
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>>>>
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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
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velocity and temperature.
-Quality vs. void fraction
>>>>>>>
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( ) 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[ ]>>
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[ ] vvm
llll
( ) >>
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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:
>>=>=>>
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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
>>=>=
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>>>=>>>=>=>>>=< vjom
vv
v wjCz
)1(
l
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m
Similarly, the mixture momentum equation can be expressed as
zmmm
e
mmmm gwD
fz
Pw
zw
t +
=+
2
2
2
2
11
[ ]
>>
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Based on the homogenous assumption, , thus,>>=>>>>>=
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(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
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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
>>==
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--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 =
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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
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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)
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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
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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
++
=
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--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
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-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<
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f
lv
v
vw
f
lv
l
lw
z
P
zM
z
P
z
zM
,22
,22
)(
)(
>==>>
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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.
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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
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( )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 =
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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
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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
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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
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( ) 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
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: 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
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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
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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
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211
XcX
X
++=
where c is given by the following table
Turbulent Turbulent 20
Laminar Turbulent 12
Turbulent Laminar 10
Laminar Laminar 5
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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
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Martinelli & Nelson, 1949 Martinelli & Nelson, 1948
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Martinelli & Nelson, 1948
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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
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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
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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.:
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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
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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.
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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 =
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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
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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
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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(
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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
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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
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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
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distribution in the channel:
vjzz
zz
vsatino
zz
zz
vsat
wvwC
v
dsc
d
dsc
d
++
][)(
)(
21
)(
)(
21
2
2
l
scd zzz
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+++
++
+ + vj
dsc
dpcho w
zzzzNC
)()(
211
2
+++
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++
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
"
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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 =
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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(
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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
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[ ] vvh
i
"qP
iw)1(dz
d
= lll
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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
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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 =
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10.4 Adiabatic two-phase bubbly two-phase flow
Ref.: Lahey and his coworkers
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(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
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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:
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Where Glis the turbulence generation rate defined as:
( lllll VVVvG Tt += :
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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
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