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CHAPTER 6 Post Dryout Heat Transfer
Liquid
Vapor
q
Liquid droplets
Liquid deficient region
Vaporflow
Film boiling (inverted annular flow)•Topics to be discussed -Film boiling heat transfer (Inverted annular flow) -Heat transfer in the liquid deficient region (High void fraction film boiling) -Minimum film boiling temperature -Transition boiling heat transfer
6.1 Film Boiling with Inverted Annular Flow Pattern –Film boiling on vertical surface Brmoley’s model Ref.: Chemical Engineering Progress, vol.46, No.5, p.221-227 G.E. for vapor flow
Boundary conditions
)(2
2
vlv gdy
ud
)b(case0u
)a(case0dy
du,yat.2
0u,0yat.1
y
Liquid
u gVapor q
y
x
Solution for u(y)
-Heat transfer It is assumed that heat travels through vapor film by conduction. Thus
)b(case1
)a(case2C
]y2
1y
2
1C)[(
g)y(u 2
vlv
dxT
kdxdx
dwi vlv
)(
0
)(
0
)()()()(x
dxdx
dx
v dyyudyyuxwdxdx
dwxw
q
x
x+dx dxdx
dx
dxdx
dwxw
)(
)(
)(
)(
x
xw
T=Tsat
])dxdx
d2dx
dx
d3(
3
1
)dxdx
d2dx
dx
d(
2
C[
2
1)(
g
])dxdx
d(
3
1)dx
dx
d(
2
C[
2
1)(
gdy)y(u
]3
1
2
1C[
2
1)(
gdy)y(u
223
223vl
v
33vl
v
dxdx
d)x(
0
33vl
v
)x(
0
dx]
31
2C
[23
)(gi
Tkd
Tk
dx
d]
3
1
2
C[
2
3)(
gi
dx
d3]
3
1
2
C[
2
1)(
g
dx
dw
dxdx
d3]
3
1
2
C[
2
1)(
gdy)y(udy)y(u
vllvv
vsatv3
satv
2vl
vvlv
2vl
v
v
2vl
v
)x(
0
dxdx
d)x(
0
Boundary condition: δ=0 at x=0
Thus, the local heat transfer coefficient is:
4/1
)(]3
1
2[3
8)(
vlvlv
vsatv
giC
xTkx
4/13
4/1
)(
8
)3
1
2(3
)()(
xT
kgiC
x
kxh
vsat
vvlvlvv
The average heat transfer coefficient over the vertical plate is:
For case a C=2 C1=0.943 (Zero stress at film surface)
case b C=1 C1=0.666 (Zero velocity at film surface)
Considering the sensible heat of vapor:
vapor properties evaluated at film temperature(Tf=(Tw+Tsat)/2)
4/1
1
4/1
vsat
3vvlvlv
1
L
0 8
)31
2C
(3
3
4C
LT
k)(giCdx)x(h
L
1h
]34.01[lv
vlvlvlv i
TpCiibyreplacedisi
-Film boiling heat transfer from a horizontal surface Berenson’s model Ref.: J. Heat Transfer, vol.83, pp.351-358,1961
R H
Vv
1r
2r
Liquid
Solid
Vapor
r
z
0P
2P1P
Actual shape of liquid-vapor interface
Physical model of film boiling from a horizontal surface
4/1
3
4/1
)(
)(425.0
])()(
[35.2
vlsatvf
vlvflvvf
vlvlvflv
satvfvf
vfvf
gT
gik
ggi
Tk
kkh
Where the subscript vf stands for vapor properties evaluated at Tf=(Tw+Tsat)/2
•Analysis in Berenson’s model
rz
2P
1P
Vapor film
HgHl
2r
-Momentum equation in the vapor film
Boundary condition
2
2
dz
ud
dr
dPvf
0
000
dz
duu
z
uz
at
,0 exampleanasfirstdz
duConsider
z
212
11
2
2
2
1)(
1
1
CzCAzzu
CAzCzdr
dP
dz
du
Adr
dP
dz
ud
vf
vf
ACdz
du
Czu
z
1
2
0
00)0(
zAzA
zu 2
2)( 0
azdz
dufor
So the average velocity
3)1()(
1 2
0
AdzzuU
U33)
3
A(A)A(
dz
duvf
2
vfvfvf0z
vfw
The force balance requires that
2
3
),(1
U
onlycorrectelyapproximatisthisfactIndr
dP
vf
w
Similarly, for u(z=δ)=0,
From the energy balance
and assuming that heat transfer through the film is by conduction (i.e. the flow is laminar)
P drdr
dPP
r drr w
2
12
U
dr
dPvf
lvvf irUrrrq )(2)( 222
0)z(ufor12
0dz
dufor3
,r2
r2
i
Tk
dr
dP
2
1rwhere
r2
r2
i
Tk)r(U
Tkq
z
2
2lvvf
satvf
2vf
222
22
2lvvf
satvf
satvf
)(32
vlg
U(r)
r2r
q
2r
)(g35.2
2
Dr
dr)r2
r2(
i
Tkdr
dr
dP
22
2
vl
b1
r
r
r
r
22
2lvvf
satvf
2v
2
1
2
1
)(
8412
vllvvf
satvfvf
gi
TkPP
)2
1(
2)(
2
12
01
02
b
vl
v
l
DR
RgHPP
RgHPP
gHPP
Noting that
Assume bDH
)(2.3
vlgH
4/1
)()(
09.1
vllvvlvf
satvfvf
gig
Tk
4/1
)()(
09.14.1
vllvvlvf
satvfvf
gig
Tk
where“1.4”is to account for the ratio of total area /the area between bubble
ofvalueondepending
gig
Tkor
vllvvlvf
satvfvf
4/1
)()(9.1
6.2
The experimental data suggest that
4/1
vllvvlvf
satvfvf
)(gi)(g
Tk35.2
]50.01[
)(
)(45.0
4/1
3
lv
satvlvlv
vlsatvf
vlvflvvfvf
i
TCpii
gT
gikkh
-Film boiling on horizontal cylinders Following similar procedure to that for a vertical surface, Bromley obtained the following equation for film boiling heat transfer coefficient on horizontal cylinders.
Breen & Westwater correlation (1962)
interfaceliquidvaportheatvelocityzero
interfaceliquidvaportheatstressshearzeroC
DT
kgiCh
vfsat
vfvflvflv
512.0
724.0
])(
[
1
4/13
1
)(g2
]T
k)(gi][D/069.059.0[h
vlT
4/1
Tvfsat
3vfvflvflv
T
布林與韋斯特瓦特的膜沸騰經驗式 (Breen & Westwaater, 1962)
-Film boiling on spheres Dhir & Lienhard
-Film boiling heat transfer considering thermal radiation
4/13
])(
[67.0DT
kgih
vfsat
vfvflvflv
1/1/1
)TT)(TT(
TT
qh
h4
3hh
lw
satw2
sat2
wSB
satw
rr
rcond
•Application of boundary layer theory for inverted annular flow film boiling Ref: Sakurai et al., 1990, “A General Correlation for Pool Film Boiling Heat Transfer From a Horizontal cylinder to subcooled liquid Part I & Part II.” J. Heat Transfer, vol. 112 pp.430-450.
Governing Equations
0)()(
kkkk vy
ux
,vk
2
2
)(y
ug
y
uv
x
uu k
kkkxk
kkk
kk
,vk
2
2
)()(y
Tk
y
Tv
x
TuCp k
kk
kk
kk
,vk
, vertical surface , horizontal cylinder ggx
)/sin( Rxggx
Boundary Conditions At y=0
wvvv TTvu ,0
At y=δ
uuv
y
u
y
uvv
virv
v iGy
Tkq
y
Tk
"
satv TTT
At y=∞
TTu ,0
Where Gi is the interfacial mass transfer rate,
v
dx
duv
dx
duG vvvi
/)()( "" rpr qq
31
31
/sin)(
43
31 ''sin4)(
od
)(1/1/1
44"satw
w
sbrp TTq
o
d 6122.0)(1
Similarity Transform Sakura et al. defined the following similarity variables.
y
oks
kkk vkdy
rN ,,
)(
vkM
fk
kkk ,,
)()(
TT
TT
TT
TT
satsatw
satvvv
)(,)(
vkx
vy
u k
k
ksk
k
k
ksk ,,;
vkvrgvM ksksksvsk ,,)/()( 41
23
vkvMN vskk ,,/
where
Thus,
vkffffksks
kkkkkk
ks
k ,,0/)(
/)(32
)(
)( "2'
'
"
vkfCp
Cp
k
kkk
ks
kksk
ks
k ,,0Pr3)(
)( '
'
'
0vAt
1)(,0)()( ' vvv ff
At
0'' f
and interfacial conditions. Numerical solutions is required.
次冷度對池膜沸騰的效應( Sakurai et al., 1990 )
Approximate solutions -Neglect convective terms in all the conservation equation-Vapor flow is driven by buoyancy force -Liquid flow is driven by vapor flow.
yg
yg
uvvv
v
v
vv
2
2
)(
2
)( 2
//
)/)(/(
2
)(
//
/
2
)( 2
v
vv
v
vv gz
gu
)()/( satwwv TTyTT
)()/( TTzTT satsat
Consider energy balance at the interface
)()('
TTkTTk
dyudx
di sat
o
satwvvvv
o
sat TTkdzTTu
dx
dCp
)()(
Thus,
213 )( vv
v
CCm
dx
d
C
mdx
d)( 3
Where
)/(),/()4( 2 ffmffmv
)](/[)(12 '1 vvvsatwvvv giTTkC
)](/[)(12 '2 vvvwsatvvv giTTkC
)](/[6 vvgCpkC
/,/ vf
-Integration of above two equations yields
oo
vovo
f
CCf
x )4(3
)()(4 214
3
4
3
)(4
o
o
f
fC
x
Combined above two equations and solve for 0
)3/(2 vvo CCVU
2121 , WWVWWU
)2/()3/(2)3( 12112
321 vvvvv CCfCCCCCW
)]27/(643/2)27/(4[ 221
222 fCCfCCW vvv
3121
21
31
2 )]3/()[/()]2/([)]/([ vvvvv CCCCCCfCC
Where
Thus,
41
33
)(4
o
o
f
xfC
The heat flux distribution is given by:
)(
)("
x
TTkq satw
v
Thus, the average heat flux can be obtained as
L
o
satwvxave L
TTkdxq
Lq
)(
)(
3
41 ""
The average Nusselt number for vertical surfaces can then be obtained as:
4
14
1
43
793.0)(4
3
3
4
)(3
4)( 3"
Lo
o
v
satwavevL M
fC
fL
L
L
k
LTTqNu
For a horizontal cylinder
41
43
41
612.0728.044
3M
L
DNu
k
DhNu vL
v
convD
Where
23' PrPr1Pr SpRSpEETTCpiGrM satwvvvvLL
23' PrPr1Pr SpRSpEETTCpiGrM satwvvvvD
ScBCABCAE 313
13
1
22223 Pr41Pr31271 SpRScSpRScA
22 Pr2732Pr32274 RSpScSpScB
2322 /272Pr41 RScSp
Pr21 2SpRC
vsatwvvv iTTCpii 5.01'
21
vvR
')( vsat iTTCpSc
)Pr'()( vvsatw iTTCpSp
)()( 23vvvvL vLgGr
)()( 23
vvvvD vDgGr
Note that
)(
')(
)(
'Pr 3
satwvv
vvv
satwv
vvvL
TTk
Lig
TTCp
iGr
)(
')(
)(
'Pr 3
satwvv
vvv
satwv
vvVD
TTk
Dig
TTCp
iGr
Thus, for a vertical surface
41
)(
')( 3
LTT
kigh
satwv
vvvvLcon
and for a horizontal cylinder
41
)(
')( 3
DTT
kigh
satwv
vvvvDcon
These expressions are same as those derived by thesimplied model of Bromley.
6.2 Heat Transfer in the Liquid Deficient Region
From: M.Andreani & G. Yadiaroglu “Prediction Methods for DispersedFlow Film Boiling” Int. J. Multiphase Flow, vol.20 suppl. Pp.1-51, 1994.
•Thermal nonequilibrium vapor is superheated
•Major heat transfer mechanisms Heat transfer from the surface to liquid droplet by either “wet” collisions or “dry” collision Convective heat transfer from the surface to the bulk vapor Convective heat transfer from the bulk vapor to suspended droplets in the vapor core
Liquid deficient region
qVaporflow
)( wdq )( wvq
)( vdq
•Bounding surface temperatures -Upper bound: complete departure from equilibrium all the heat added to fluid goes intosuperheating the vapor (toward this situation at low pressures and low velocity) -Lower bound: compete thermodynamic equilibrium all the heat added to fluid goes to evaporate the liquid droplets and Tg=Tsat. Surface
temperature decreases as a result of increasing the vapor flow due to evaporation. Toward this situation at high pressure approaching the critical condition and high flow rates (>3000 kg/m2s)
From: Collier, 1981
•Thermodynamic quality vs. actual quality (for uniform heat flux) -Thermodynamic quality (equilibrium quality) xe
-Actual quality (for vapor superheat of Tv-Tsat present)
-relation between xd0 and zd0
D
zz
Gi
qxx d
lvde
)(4 0
lv
satvvae
lv
satvvd
lvda
i
TTpCxx
i
TTpC
D
zz
Gi
qxx
)(
)()(4 0
dryout
zd0
xd0inlettubeatenthalpysubcoolingi
i
i
D
z
Gi
qx
isub
lv
isubd
lvd
,
,00
4
•Semi-theoretical model of Bennett et al (Ref. :Collier, 1981) -Assume -Assume the droplet number flow rate (N) is constant along the tube that is no drop breakup or coalescence occurs. But rd may
be changed due to evaporation.
-Mass balance gives
-Energy balance equation
0wdq
dz
drr
G
N
rdz
d
G
N
dz
dx
dd
l
dla
2
3
4
]3
4[)1(
ld
d
r
xGN
30
0
3
4)1(
rd : radius of the droplet
dz
dxAGiDq
dz
dTCpAGx a
lvv
va
-Momentum eq. For liquid droplets
-Evaporation of droplets (Ryley’s model)
84.1
16.084.0
16.1
)2(25.20
)]1()([
dl
vvl
l
vdvl
dd
rk
guukdz
duu
forcedraguuk dvl :)( 16.1
forcebouyancygl
v :)1(
uv: vapor velocity ;ud: droplet velocity
lvlsatvvd
dd
d
d
d
iTTFkdt
dr
dtudzNoteudt
dr
rdz
dr
/)()1(
)(1
2
1
2
2
3/1)()(
276.01vv
v
v
dvv
k
uu
factornventilatioF
RMk
vaportheincoefdiflusionselfk
vv
v
/)1(
.
M: molecular weight R: universal gas constant
statethesaturationat
dP
d vl )/()1(
-Using the Runge-Kutta method to solve this problem
P=69bar, vertical tube, L=5.8m, i.d=12.6mm 2rd=0.3mm(rd is an assumed
value) (From: Collier, 1981)
•Empirical correlations for liquid deficient film boiling heat transfer -Dougall & Rohsenow Correlation
see more correlations in Groeneveld & Snoek, 1986.6.3 The minimum film boiling temperature Ref. A. Sakurai, “Film Boiling Heat Transfer”, Proc. of 9th Int. Heat Transfer Conference, vol.1, pp.157-186.
)]1([Re
PrRe023.0
2
4.08.02
el
ve
v
vv
xxGD
whereD
kh
)(
)1(min
satII
satI
TTRaT
RaTTRaT
2/1
sss
lll
Cpk
CpkRa
)/17001(
)/(20exp26.0192.0
cr
crcrI P
PPTT TI: contact interface
temperature ;Pcr in kPa
It is expected that the minimum film boiling temperature on a solid surface with any thermal physical properties in ay liquid can be evaluated by above eq.(Sakurrai, 1990)
Minimum temperature vs. system pressure for cylinders of 1.2 and 3.0 mm diameters in Freon-11
Minimum temperature vs. system pressure for platinum and gold cylinders of 2.0 and 3.0 mm diameters in water.
6.4 Transition boiling
CHFq
q
satw TT
Transitionboiling
Nucleate boiling Film boiling
minq
Typical surface temperature history Typical local surface temperature history for a liquid-solid contact.
From: Lee, (J.C.) Chen and Nelson.
Model of Pan et al (1989)Ref.: Pan et al, “The Mechanism of Heat Transfer in Transition Boiling”, Int. J. Heat Mass Transfer, vol.32 No.7 pp.1337-1349, 1989.
(a) bubble departing (b) transient conduction (c) boiling incipience and heat transfer
(d) macrolayer evaporation (e) vapour covering
c
vUlU
V
Vapor bubble
Liquid film(macrolayer)
Vapor stemHeated surface
DDD
Vapor bubble
Liquid film
c
vapor structure near heated
surface at high heat fluxes Pool boiling at high fluxes on a
horizontal flat plate τ: vapor bubble hovering time.tme: time interval of macrolayer evaporation.
τ<tme: nucleate boiling.
τ=tme: critical heat flux.(Haramura & Katto’s model, 1983)
τ>tme: transition boiling / film boiling.)1( FqFqq rmetot
From M.Shoji, 1992
From: Katto, 1992
-Wall temperature during transient contact conduction
)()1)(1(
)1)(1()1(
)1)(1( 0 21
2112
21 t
nerfc
bb
bbbb
bb
TTTT
c
n
n
wI
0 21
2112
)(
)1(
)1)(1(
)1)(1()1(
nc
n
t
nerfc
bb
bbbb
21
)()(1 hc CpkCpkb 21
)()(2 CpkCpkb c
--Effective liquid thermal conductivity
tteff Cpkkkk )(
tconsempiricalC
T
TT
ggC
w
bw
vlt
tan00.1
)(
)()(
2/3
24/3
2/1
--Incipience criterion
v
vsat
csatvc i
vT
rTTtryT
2),(
0 21
2112
21 2)1)(1(
)1)(1()1(
)1)(1(),(
nc
n
w
t
n
t
xerfc
bb
bbbb
bb
TTTtyT
0 21
2112
)1(
2)1)(1(
)1)(1()1(
n c
n
t
n
t
xerfc
bb
bbbb
-Macrolayer evaporation heat flux
nbme qq
33.0"
7.1 )(Pr
)(
vv
nbsf
v
satcdI
gi
qC
i
TttTCp
-Macrolayer evaporation time "/ mevome qit δme: from Haramura & Katta’s theory
-Vapor hovering period
The average volumetric growth rate of the bubble in the period of macrolayer evaporation and vapour covering is given by
where λD is the most dangerous Taylor wavelength
In above equation, it is assumed that the unit heater area participating in the growth of one vapour bubble is
51
53
51
)(
)(4
4
3 1611
Vg v
v
)(/"2vvaveD iqV
21
21
)(23 vD g
2D
The average heat flux within these two periods is given as
-Liquid contact time fraction
-Transition boiling heat flux
""
" )( vmememeave
qttqq
mev tt
vmecd
mecd
vl
l
ttt
tt
tt
tF
vmecd
vvmemecdcd
ttt
tqtqtqq
From: Shoji, 1992
From: Shoji, 1992
From: Pan er al., 1989 From: Shoji, 1992
Empirical Correlation for Transition Boiling (for low quality) -Auracher’s correlation Ref.: Int. Froid 1988., vol.11, pp.329-335Nucleate boiling during liquid contact is the major mode of heat transfer.
3.1,)(
)/)((
)1/(1
,
3.10
3.0
nT
T
q
q
qqPCh
h
n
CHFsat
sat
CHF
h0.3: reference heat transfercoefficient at reduced pressureof 0.3
From: Auracher, 1988
From: Auracher, 1988
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