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Akimoto1983-CondensationFlowingSteam.pdf
Journal
of NUCLEAR SCIENCE and TECHNOLOGY, 20[12],
Analysis
of Direct
Steam onto
Contact
Water
Yoshiyuki
with
of Two-Phase
Hajime
Japan
1983).
of Flowing
Multifluid
Flow
AKIMOTO,
Atomic Energy
KOZAWA,
(December
Condensation
Injected
Model
pp. 1006~4022
Akira
Research
INOUE
Institute*
and
Shigebumi
AOKI
Tokyo Institute of Technology**
Received
May
17,
1983
When subcooled water in accumulator
tanks is injected during a loss-of-coolant
accident
of a pressurized
water reactor, violent condensation
takes place in cold legs because of direct
contact of steam with water.
A flow model based on the multifluid model of the two-phase
flow has been developed to formulate the condensation and mixing processes in the injection
region.
The measured liquid film temperature
and pressure profiles in the injection
region
have been explained quantitatively
with the model which includes (a) drop generation arising
from atomization
of injected water, (b) condensation
of steam on the drops,
(c) flow contraction
resulting
from the formation of the dead water region, and (d) deposition of the
drops.
The calculated results show that the condensation rate depends mostly on the drop
diameter
generated near the water injection nozzle and the maximum drop mass fraction in
the injection region.
The present model can predict the effect of inlet thermal-hydraulic
conditions on the condensation rate qualitatively,
and then it has been confirmed that higher
condensation
rate with initial steam mass velocity is caused by the formation of finer drops
and the higher drop mass fraction in the injection region.
KEYWORDS:
reactor
vapor
condensation,
droplets,
liquids
safety,
direct
loss
contact,
I.
of
coolant,
two-phase
PWR
flow,
type
multifluid
reactors,
model,
accumulator,
steam,
water,
INTRODUCTION
If a loss-of-coolant accident (LOCA) should happen in a pressurized water reactor (PWR),
subcooled water in the accumulator tanks would be injected into steam flowing in the cold legs.
In this case, the steam comes into direct contact with the water and violent condensation takes
place because of the energy transferred from the steam to the water. It is generally agreed
that the direct-contact condensation rate should be evaluated based on both the steam-waterinterface area and the interfacial heat flux. However, the mixing process of injected water
with steam is so complicated that a certain condensation rate was assumed regardless of the
actual mixing process in the injection region in most of the previous studies.
Thermal equilibrium between steam and water was assumed in the so-called 1V1T
code such as the RELAP 4 code"" (2'. Only the integral mass, momentum and energy
balances were considered, and the details of the mixing of steam with water were not taken
into account at all in the 1V1T code. The assumption of thermal equilibrium caused the
condensation rate to overestimate and sometimes an instability to generate in the calcula*
Tokai-mura
** Ookayama
, Ibaraki-ken
, Meguro-ku,
319-11.
Tokyo
152.
34 —
Vol. 20, No. 12 (Dec. 1983)
ion triggered
The
1007
by the over-depressurization
condensation
rate
at the water
can be derived
injection
from the relaxation
(di dX)(rng)=(mg-m:)Ir
nozzle").
equation
:
,
(1)
where r indicates the relaxation length for the condensation
of steam and m; indicates the
steam mass velocity at the thermal equilibrium condition with complete mixing of steam
and subcooled
water.
Equation
( 1 ) gives
(n2,-41(7129,-4
Because
as
a
the
there
fitting
mixing
An
fluid
of steam
analysis
model,
stitutive
sents
is no physical
parameter
the
a typical
evaluated
with
based
for
determining
The
the
relaxation
parameter
should
length,
r
has
be determined
been
used
by
analyzing
In
the
water.
two-fluid
equations
both
analysis
for
analysis.
on the
energy
equations
basis
of the
(2)
=exp (-X/r),
the
with
model
for
steam
interfacial
the
has
area
two-fluid
been
performed
and
water
and
heat
model").
recently.
can be solved
flux.
The
The
interfacial
assuming
TRAC-PF1
area
and
two-
the
code
heat
conrepre-
flux are
by
Ai= f gV cell/1 ,
(3 )
hi=0.02
(4 )
Cpi p ai/Dn, ,
respectively.
Equations ( 3 ) and ( 4 ), however, were introduced without experimental basis
for the mixing process of steam with water.
The constitutive
equations should be assessed
and the physical meaning of the equations
mixing process in the injection region.
The authors
have
studied
should be clarified
the flow and heat-transfer
as to how they represent
mechanisms
of direct-contact
the
con-
densation phenomena in the injection region to develop a realistic model").
An experiment
was performed to observe flow behavior in the injection region.
Figure 1 shows a schematic
of the
observed
flow
behavior
in the injection
region.
Injected
water
comes
into
direct contact with flowing steam in the shape of a jet having almost the same width as
the water injection
nozzle.
The interface
between
the water jet and flowing steam is
disturbed, probably because the jet atomization occurs there.
The drops are dispersed in
flowing
steam
downstream
at 3,5
cm (x/Dh=1.0-1.8)
of the water
injection
nozzle.
In the vicinity of the water injection nozzle,
a dead water region is observed ; it disappears by 10,20 cm (X/Dh,--3.5-7.0)
downstream of the water injection nozzle.
It is
judged from the experimental
results that
the mixing process in the injection region
Fig.
1
was rapid and complicated.
Schematic of observed flow
behavior in injection region
In our previous paper, a simplified model was proposed to estimate
the effect
atomization
on the condensation,
assuming
that the injected water was atomized
taneously
into drops
having
the
same
size
at 3 cm
downstream
of the
water
of the
instaninjection
nozzle.
The rapid condensation of the steam in the injection region could be explained by
the enormous increase of the interfacial area resulting from the atomization of the injected
water.
However,
the calculation
the generated
in the previous
drop diameter
model.
The
was considered
to be a fitting
model could evaluate
35 —
neither
parameter
the effect
of
of the
1008
J. Nucl.
inlet thermal-hydraulic
conditions,
such as steam
and water
mass velocities,
Sci. Technol.,
nor the interac-
tion between the condensation
and the mixing of fluids in the injection region.
The objective of this paper is to present the feature of the revised model
which
has
been developed to describe the flow behavior in the injection region more reasonably.
multifluid model of the two-phase flow is applied in the revised model. The empirical
The
cor-
relations,
and the
concerning
deposition
evaluate
the generated drop diameter, the atomization
rate of the liquid film,
rate of the generated
drops onto the liquid film, are considered to
the interaction
between
the condensation
and the mixing
to explain the effect of the inlet thermal-hydraulic
steam in the injection region.
II.
PRELIMINARY
Ford
et al.")
was
in direct
with
the correlation
a
solid
studied
contact
sphere.
drop is limited
by
effect
mass
of
proximation
From
diameter
the
CONSIDERATIONS
the
growth
with
steam.
derived
from
Their
the
results
heat
drops
the
during
showed
solution
of the
unsteady
conduction
inside
the
drop.
due
to condensation
of steam
heat
for
agreed
equation
for
on a subcooled
also
suggest
a solid
be derived
water
rate
conduction
can be neglected
can
and
rate of
subcooled
growth
of steam
results
equation
on a drop
drop
rate
Their
of steam
of the condensation
rate.
unsteady
heat conduction
rate
when
the observed
condensation
with water
DEVELOPMENT
condensation
that
the
condensation
of steam
on the condensation
FOR MODEL
that
increase
the
of
They
indicate
in the evaluation
the solution
of the
of DI,
conditions
that
the
as a first
ap-
sphere
with
exactly
a
as
(5)
In the early
period of transient
(a1tk/Dk0.05),
Eq. ( 5 ) is approximated
by
(6)
The
total steam
mass condensed
( 5 ) for tk, given
on the drop mere can be derived
by the integration
of Eq.
by
(7)
where
in% =pDCprpr(Tsar-T
ko)/6Dh,
k indicates the total steam mass condensed on the drop by tk=oc.
The
comes
condensation
from
the
rate becomes
assumption
that
infinite
the
at tk=-Os according
thermal
resistance
to Eq. ( 5 ) or
( 6 ).
at the water-steam
This
interface
is
negligible.
Accurately, the condensation
rate should be limited by the thermal
resistance
at the water-steam
interface in the early transient.
The thermal resistance at the interface
was calculated
by Schrage(".
He derived the heat transfer
coefficient
resulting
from
thermal resistance at the interface and obtained the following equation for pure steam :
(8 )
Equation
( 8 ) gives
sure of steam
at tk=0s.
the heat
and water
transfer
with am=1.
coefficient
of 7.9 x 106 W/m2
This value gives a maximum
For a drop with 1.0 x 10-a m diameter,
( 8 ) is equal
to the
condensation
the condensation
K at atmospheric
heat transfer
rate calculated
rate at th-=1.5 x10-8 s based on Eq. ( 5 ).
36 —
pres-
coefficient
with Eq.
And at tk=
Vol. 20, No. 12 (Dec. 1983)
1.5x10-'s,
according
1009
the condensation
to Eq. (5).
rate for the drop becomes
The
1.5x10-8 and 1.5x10-6s,
the water-steam interface
total condensed
steam
only 1/10 of that at tk =1.5X 10-8 s
amounts
are 0.03 and 0.3% of nic*k by
respectively.
Therefore,
the effect of the thermal
resistance
at
is negligible and Eq. ( 5 ) can be used to calculate the condensation
rate of steam on the drop.
The drop diameter has never
measured
in the injection
region
of the condensing
two-
phase flow. But the drop diameter in the adiabatic annular two-phase flow was measured
by many researchersm">. These results were correlated
semi-empirically
by Tatterson
et al. with the following
equations
:
(Dk/Dh)({r6UlfsDh}/{20.1})612=1.6x10-2,
where
They
f,=0.046
derived
atomization
Rc0.2.
the non-dimensional
drops were generated
(9)
number
by the eruption
mechanism
in the left-hand
of wavelets
is consistent
side of Eq. ( 9 ) assuming
from the surface
with the observed
that
of the liquid film.
flow behavior
This
in our previous
ex-
periment.
It is expected that the atomization phenomena are determined by the local condition where the drops are generated.
Therefore,
Tatterson's
correlations are adopted to
analyze our experimental
results, based on the similarity of the atomization
mechanism
of
the injected
water.
Tatterson's
correlations
show that the drop diameter is proportional
to
In the
injection region of the condensing two-phase flow, steam mass velocity changes in the main
stream direction because of the steam condensation.
If a dead water region is established
in the injection
region, the flow area of the steam is changed by the flow contraction.
Therefore,
it can be considered that various sizes of drops are generated at different locations
from the water
injection
nozzle with the change
of steam
mixed up with flowing steam in the injection region.
steam on a single drop is a function the drop diameter,
condensation
rate of steam
on the drops with various
densation rate of each drop.
The condensation
rate of steam
the drop in steam, as described
evaluated by
velocity
sizes as the summation
on a drop is also a function
in Eq. ( 5 ).
The
and that they are
Because the condensation
it is necessary to evaluate
exposure
of the
rate of
the total
of the
exposure
con-
time
of
time of the drop at X can be
(10)
where Xk indicates the location where
exposure time of the drop is dependent
and X.
Let us consider
rate
is expressed
As previously
one depending
drops in order
the condensing
the condensation
rate at a certain
by the summation
mentioned,
the drops
the drop is generated.
Equation (10) shows that the
on the history of the drop velocity between
Xk
location
of the condensation
can have various
X.
The
total
condensation
rate on each drop passing
diameter
and
exposure
time
on the generated
location.
Therefore
it is necessary to trace
to calculate the total condensation
rate accurately in the injection
two-phase flow.
In the present
generated.
The
time if they have
analysis,
the
drops
are
grouped
by
the
location
where
each
there.
one
by
individual
region of
drop
is
drops generated at the same location Xi, should have the same exposure
the same diameter.
Even though various size of drops can be generated
37 —
1010
at
J. Nucl.
the
same
instead
we
are
steam
more
with
rate
location,
of considering
of
volumetric
the
size
interested
water
the
direction
the
such
liquid
of steam
in
film,
interaction
changes
and
diameter
of the
the
will
drops
between
of
the
deposition
be adopted
in the
the
generated
rate
of the
for
present
and
diameter,
generated
drop
diameter
analysis.
condensation
drop
the
Technol.,
drops
the
the
in the
Because
mixing
of
atomization
main
flow
flow.
III.
1.
the
as the
mean
distribution
Sci.
ASSUMPTIONS
AND FIELD
schematic
present
EQUATIONS
Assumptions
Figure
2 shows
the
of
the
model.
The
field
equations
describing
the interactions between the injected water
and the flowing steam are based on the
following assumptions :
(1)
The flow is steady
sional.
(2)
The multifluid model of the two-phase
flow is applied.
The field equations of
steam, liquid film and drops are solved.
The
drops are separated
and
one-dimen-
into subgroups
labeled by the location where
are generated.
(3)
Fig.
the drops
Schematic
of flow model
in injection region
The condensation
rate is calculated using the solution of the unsteady
heat conduction problem of a solid sphere with constant diameter for every drop groups.
The
volume mean diameter is used to evaluate
steam on the liquid film is negligible.
(4)
2
the condensation
rate.
The
condensation
of
The dead water region is assumed to be equivalent to the rigid flow blockage.
The
configuration
of the dead water region is given by the photographs
taken experimentally under
(5)
each flow condition.
Steam is in saturated
2.
Field
The
field
Continuity
condition.
Equations
equations
to be solved
are
:
(11)
(12)
(dIdX)mk=Efk—Ekf+Ggk
Momentum
(k=1, N),
(13)
:
(14)
(15)
Energy
(16)
:
(17)
38 —
Vol. 20, No. 12 (Dec. 1983)
1011
(d/dx)(mkho=Gkh9Gfhk+Efkhf
hg-=hsat
The
velocity
(18)
(k=1, N),
(19)
•
of each
fluid is given
mg=poUgf g
by the
mf= pit
following
ff
relations
:
mk=-piUkf k (k=1, N),
(20)
where
The foregoing equations are ordinary nonlinear differential equations with dependent
variables mg, mf, ink, Ug,U1,Uk, hf, hk, hp, f g, f f, f k and p. Preparatory to solving the
equations, relationships must be specified for the condensation rate rgo the entrainment
generation rate Efk, the deposition rate Ekf, the water injection rate 1"h,, and the shear
stresses Ff, Fgf and F, k.
3.
The
The
Constitutive
constitutive
condensation
Equations
equations
used
in the
Table
1
rate of each
present
Constitutive
drop
subgroup
analysis
are
summarized
in Table
1.
equations
Po,
is calculated
by
multiplying
the
number
density of drops nk by the condensation
rate of a single drop gg,
in Eq. ( 5 ).
The exposure time of each drop subgroup is calculated by solving the differential equation :
dtk/dX=
1 /Ilk
39 —
(k=1, N).
(39)
1012
J. Nucl.
Ishigai and Takagi developed a correlation
adiabatic
annular
two-phase
flow"2'.
From
liquid film, they derived
the frequency
liquid film and developed
for the atomization
rate of liquid film in the
the analysis of the interfacial
wave on the
of the eruption
a correlation
Sci. Technol.,
of the wavelet
for the atomization
at the surface
rate assuming
that
of the
the atomiza-
tion rate was proportional
to the frequency of the eruption of the wavelet.
Because their
atomization
mechanism
(that is, the drops are generated by the eruption of the wavelets at the
surface of the liquid film) is consistent
periment,
Ishigai
& Takagi's
with the observed
correlation
is applied
flow behavior
to calculate
present analysis, the atomization rate at the section between
to the k-th drop subgroup as described by Eq. (23) in Table
in our
previous
the atomization
rate.
Xk and X0-DXk
1.
exIn the
is assigned
McCoy et al."3' reviewed the previous measurements
and correlations for the deposition
rate in the adiabatic annular two-phase flow. They summarized that the deposition rate was
determined
by the inertia
of the drops and the radial fluctuating
they proposed their correlation as the combination of the
controlled
by the viscous diffusion of drops (7+<0.15),
component
of velocity
and
deposition rates for the region
the turbulent
diffusion of drops
(r+>22.9) and the transition region (0.15<r+<22.9).
The deposition rate of drops Ekf is
calculated
with the correlation
developed by McCoy et al., because the observed
flow
behavior in the injection region was similar to the annular mist flow of the adiabatic twophase flow. The deposition rate of each subgroup is calculated
velocity and the drop diameter of each drop subgroup.
The flow in the injection
region
of the condensing
separately
two-phase
using
the
mass
flow is not fully developed
and more disturbed than the adiabatic annular two-phase flow. Both Ishigai's and McCoy's
correlations
of E'fd and Eidf were developed based on the data in the fully developed region
of the adiabatic two-phase flow. Hence, both correlations
of Ekf and Efk are modified by
multiplying
the constants C. and Cd in order to take in account the effect of the developing
region
and the high disturbance
in the injection
region.
It can be expected
that
C, and
Cd are higher than unity because the high disturbance
seems to lead the high frequency of
the eruption of the wavelet at the surface of the liquid film and the high radial component
of velocity
4.
of drops entrained
Calculation
Procedure
The ordinary
Kutta-Gill method
1, and
with steam.
differential equations,
as an initial value
Eqs. (19) and
(20).
ing-=.171gt
U
The
771f=0,
pg ,
initial
The
initial
U f=0
velocity
values
where
In the
present
generation
the
and
,
study,
location,
are
the
that
1.
The
rate
Determination
present
of liquid
model
film and
of
by the following
relations :
=1, N),
tk=0
for the
drop
k-th
(40)
(k=1, N),
subgroup
p=pi
are
set
equal
to the liquid
generated.
drops
are
separated
m
two
deposition
into
(k=1,
RESULTS
Cd and
contains
the
set
N),
(k=1, N) ,
is, DzYk =0.001
IV.
(k =1,
were
k=U f Ia'=xk(k
enthalpy
drops
conditions
m k =0
hk—hrix=xk
film
Eqs. (11)~(18) and (39), were solved by the Rungeproblem with the constitutive
equations
in Table
subgroups
at
every
1 mm
of
the
N).
AND DISCUSSION
C,
unknown
rate
experimental
of drops
40 —
to liquid
constants
film,
i .e.
for
the
C, and
atomization
Cd.
Because
Vol. 20, No. 12 (Dec. 1983)
no experimental
result
rate in the injection
1013
has been reported
region
and Cd was performed
for either
of the condensing
to determine
the atomization
two-phase
these two constants.
rate and the deposition
flow , a parametric
study for Ce
This parametric study was made
for the test performed
with the conditions ; mg1=40 kg/m2.s,
m11=330 kg/m2-s,
323 K and 95=90.
Figure
3 shows the results of the parametric
study on the
film temperature
at 10 cm downstream
of the water
injection
nozzle.
T
liquid
The liquid film tem-
perature increases with both the atomization
and the deposition rates.
The steam transfers its latent heat to the drops due to the condensation,
and the heated drops transfer the
energy
hanced
to the liquid film due to the deposition.
The higher deposition rate results in enenergy transfer from drops to the liquid film , giving a higher liquid film temperature.
The higher atomization
rate results in greater
steam condensation
on the drops,
because the condensation rate is proportional
to the
number
higher
atomization
density
of the drops.
rate
also
The
results
in a
higher deposition rate of drops to the liquid
film, because the deposition rate is directly
proportional
to the drop mass velocity and
the inversely proportional to the steam mass
velocity.
From these reasons,
a higher
atomization
rate gives a higher
temperature.
that a certain
liquid
give the same liquid temperature
perimental
the water
film
The parametric
study shows
relation should be satisfied to
result at 10 cm
injection nozzle.
as the ex-
downstream
Fig.
3
Effect of atomization and deposition rates
on liquid film temperature
at 10 cm
downstream
of water injection nozzle
of
The calculated differential pressure between -5
and 10 cm from the water injection
nozzle was also compared with the experimental
result to determine the constants.
It was
found that the constant
set Ce=23 and Cd=6
of the liquid film temperature
differential pressure between -5
gave the same values simultaneously
in the experiment.
that the correlations
Even though it is not proved, it is assumed
for condensing two-phase flow can be expressed
as
for adiabatic
those
developed
used in all the following
for both
at 10 cm downstream of the water injection nozzle and the
and 10 cm from the water injection nozzle that observed
two-phase
flow.
The constant
in the present analysis
by the same functions
set C,=23
and Cd=6
was
calculations.
Figure 4 shows the comparison of the pressure
and liquid film temperature
profiles
between the calculated and measured results.
The calculated result shows good agreement
with the experimental
but
also
present
region.
2.
in the
result
whole
model reasonably
Drop Behavior
not only at 10 cm downstream
injection
region.
describes
in Injection
This result
the condensation
of the
also supports
and mixing
water
injection
nozzle
the conclusion
processes
in the
that the
injection
Region
Figure 5 shows the calculated atomization
and deposition rates, the drop mass fraction
in the liquid phase, and the steam mass velocity.
Near the injection nozzle (X< 1 cm) ,
the atomization
mass
transfer
velocity
from
rate is much
and
the
higher
than the deposition
low drop mass velocity.
rate
This results
because
of the
the liquid film to the drops and leads to the rapid increase
41 —
high
in an almost one-way
steam
mass
of the drop mass
J. Nucl.
1014
Fig.
4
fraction
rate
the
Comparisons
between experimental
and calculated results for differential
pressure and liquid film temperature
in injection region
in
this
of steam
steam
Sci. Technol.,
region.
The
corresponds
mass
velocity,
to
and
condensation
the
slope
Fig. 5
of
it is enhanced
Relations between overall drop
behavior
and condensation
of
steam in injection region
by the increasing of interfacial area between steam and water
hand, the atomization rate starts to decrease at 0.5 cm downstream
nozzle center.
This is caused by both the decreasing
in this region.
from the water
liquid film thickness
On one
injection
resulting
from
the atomization
and the decreasing
steam velocity arising from the condensation.
On the
other hand, the deposition rate of the drops increases because of both the decreasing steam
mass
velocity
and
the
increasing
drop
mass
velocity.
The
deposition
rate
equals
the
atomization
rate at 3.8 cm downstream
of the water injection nozzle ; the drop mass fraction
in the liquid phase has a maximum value of 91% at the same location.
This result shows
that almost all of the injected water is atomized into drops in this case.
The drop mass
fractions in the liquid phase are 74, 36, 18 and 13% at 10, 20, 30 and 50 cm downstream
of the water injection nozzle, respectively.
The rate of decrease in the downstream
region
is smaller than the rate of increase near the injection nozzle (X<1 cm), because the difference between
the deposition
the injection
nozzle.
rate and the atomization
The tendency
of the drop mass
rate is not so high as calculated
fraction
in this
calculation
near
is con-
sistent with the experimental
result in the downstream
region, because the wavy flow was
observed at 100 cm downstream
of the water injection nozzle in the experiment
under this
test condition.
These results show that the condensation
rate of steam is closely related
to the drop behavior
Figure 6 shows
subgroups
in the
in the injection region.
the calculated mass velocity
direction
of each drop subgroup
of steam
is proportional
main flow.
to the
and temperature
The
changes
initial increase
atomization
rate
of the
of certain
drop
of the mass velocity
liquid
film at
the
location where the drops are generated,
as described by Eqs. (13) and (23) in Table 1.
Once the drops are generated,
they are entrained with steam, contact with steam , condense
42 —
Vol. 20, No. 12 (Dec. 1983)
1015
steam and some of them deposit
The
drops are heated
of condensed
on the liquid film.
by absorbing
steam.
the latent
heat
Some of the drops deposit on
the liquid film and transfer
mass and energy to
the liquid film. The drops generated at Xk =0.5 cm
reach to 63% of the saturation
temperature
from
their
initial
temperature
at 1.1 cm from the water
injection nozzle. This shows that the relaxation
length for the condensation of steam onto the drops
is 0.6 cm under the calculated
condition.
While
63% of the drops generated at Xk =-0.5 cm deposit
on the liquid film by 13.6 cm from the water
injec-
tion nozzle. The relaxation length for the deposition
of the drops is 13.1 cm in this case.
The relaxation
length
for
the
condensation
is about 5% of
that for the deposition of the drops. These results
show that the deposition of the drops is of secondary
importance
for
the
condensation
of steam
in the
Fig.
injection
region, even though it deactivities
the
condensation
capability of the drops.
In the present model, the drops are classified
by
the
(k=1,
it is
where
N).
The
proportional
each
at
location
drop
the
location
steam
by
mass
summation
drop
that
than
mass
The
total
because
of the
the
that
generated
the
at every
1 mm,
steam
mass
that
length
on the
condensation
drops
of
densability
of each
subgroup,
mass
the
of the
on
each
for
is, GIXk =1 x 10-3 m
steam
drops,
This
subgroup
defined
of each
drop
on
means
drop
on
the
that
drop
and
is given
subgroup.
the
the
The
to the total
drops
effect
can be approximated
subgroup
X=00.
whole
of each
neglect
is generated
subgroup
the
condensation
we can
drop
X=Xk
of each
drops
separately
and
(Tsa,- T k,) of
the
drops
between
condensed
on the
of steam.
drop
rgk
contribution
steam
when
on the
rate
condensed
deposition
temperature
condensed
the
total
velocity
drop
mass
drop
condensation
condensed
the
the
to assess
the
relaxation
for
steam
of the
can be a measure
by
In case
are
T kt indicates
integration
condensation
shorter
where
Xk.
the
steam
the
drops
Mass velocity and temperature
change
of certain
subgroup
drops in injection region
condensation
rate of each drop subgroup
F5k is calculated
to the drop mass velocity
nzk and the initial subcooling
subgroup,
is given
the
6
drops
total
by
is
much
of the reducing
steam
the
initial
mass
con-
by
(41)
(k =1, N)
Figure
7 shows
the initial condensability
(41) and (42) for the results
shown
of each drop subgroup
in Fig. 6.
Because DXk
group, the calculated
condensability
can be a direct measure
Yk in order to assess the contribution
of each drop subgroup
sation.
The high initial condensability
water injection nozzle (X<=1 cm), because
(42)
calculated
is unique
for
all
using Eqs.
drop
sub-
without any modification of
to the total steam conden-D
can be realized for the drops generated near the
of the high atomization
rate and the high sub-
43 —
J. Nucl.
1016
cooling of the liquid film.
Sci. Technol.,
This result shows
that the condensation
of steam in the injection region depends mostly on the characteristics of the drops
injection
3.
generated
near the water
nozzle.
Effect
of Inlet
hydraulic
(1)
Thermal-
Conditions
Inlet Steam
Mass Velocity
Figure 8 shows the comparison of the
effectiveness of the condensation at 10 cm
downstream of the injection nozzle. The
effectiveness 72 is defined as the ratio of
the condensate mass velocity me to the condensability of the injected water mcmax, or
Fig.
(43)
7
Initial
condensability
of each
subgroup
drop
where
The
mum
condensability
mass
that
the
is a measure
injected
water
of the
maxi-
can condense
before reaching the thermal equilibrium
assuming
the complete mixing of water with steam.
The
effectiveness
is zero in the initial state (that is, in
the case when mc=0) and
thermal
equilibrium.
be considered
librium.
From
liquid
it is unity after
reaching
Hence, the effectiveness
to be a measure
of thermal
can
nonequi-
the energy balance relation among steam,
film and
the effectiveness
subgroups,
the equation
can be described
drop
as follows :
for
(44)
Fig. 8
where
(45)
Effect of
locity on
densation
of water
initial steam mass veeffectiveness
of conat 10 cm downstream
injection nozzle
(46)
(47)
(48)
(49)
44 —
Vol. 20, No. 12 (Dec. 1983)
1017
(50)
The
second
the mass
tion
terms
in the right-hand
velocity
resulting
side of Eqs. (49) and (50) account
from the condensation
of the liquid film and drop mass velocities
velocity
increase
due to the condensation.
of steam.
with the modification
If E ek=0
for the increase
of and ek indicate
concerning
(that is, no drop) or XF k k
the
of
frac-
the mass
-10
k=1
(that is, average drop temperature is equal to the liquid film temperature), ej.
Eq. (44). In our previous experiment,
only the liquid film temperature
is equal to 72 in
was measured and
72f was calculated using Eq. (45) to estimate the condensation
rate.
Experimental
results
are also shown in Fig. 8. From the calculated results, ef
was evaluated using the calculated liquid film temperature
to compare
with
experimental
results.
The
calculated e7
agrees with the experimental ef
qualitatively.
Both results show that higher steam mass
velocity results in higher 71.f. The calculated 72f is about 10% lower than the calculated
72 because the average
drop temperature
is higher than the liquid film temperature
as
shown in Fig. 6.
the same tendency
However,
both 7) and ef
show
with regard to the initial steam
mass velocity. Therefore, the liquid film temperature
can be considered to be another good measure for
the condensation
Figure
drop mass
maximum
rate in the injection
9 shows
fraction
region.
.
the calculated
results of the
and the drop diameter.
The
drop fractions
are 0.64, 0.91 and 0.97 at
4.5, 3.8 and 3.0 cm downstream
of the water injection nozzle for m0,=20,
40 and 60 kg/m2-s, respectively.
Higher
steam
mass
velocity
leads
to
quicker atomization
of the injected water near the
water injection nozzle.
The decreasing rate of the
drop
mass
fraction
from
smaller with steam mass
show that higher steam
its maximum
becomes
velocity.
These results
mass velocity results in
the more and quicker atomization
water and the slower deposition
of the injected
of the generated
Fig.
drops.
9
Effect of initial steam mass velocity on drop mass fraction
and generated
drop diameter
change in injection region
The drop diameter is determined mainly by the local steam velocity as described by
Eq. ( 9 ), and the higher steam mass velocity results in generating
the smaller drops. The
local steam mass velocity
is affected
by both the condensation
effect of steam
itself
and the
flow contraction
effect resulting from forming the dead water region (as mentioned in Chap.
II). The calculated results show that the drop diameter is nearly constant near the water
injection
nozzle
(Xsl
cm).
In this region,
the steam
velocity
is increased
by the flow con-
traction due to the formation of the dead water region,
whereas
it is decreased
by the
condensation.
Both effects compensate
each other near the water injection nozzle.
The
drop diameter
The
dead water
starts
to increase
region
at about 5 cm downstream
is decreasing
at this part.
of the
water
injection
Both the flow contraction
nozzle.
and the con-
densation effects act to reduce steam velocity and lead to the generation
of bigger drops.
The assumed
dead water region disappeared at 10 cm downstream
of the water injection
45 —
1018
J. Nucl.
Sci.
Technol.,
nozzle. In the downstream region (X 10 cm), the drop diameter is nearly constant because
the condensing two-phase flow is nearly thermal equilibrium.
Figure 10 shows the effect of the initial steam mass velocity on the maximum drop
mass fraction e, and on the effectiveness 72 at 10 cm downstream of the water injection
nozzle. The maximum drop mass fraction is a measure of the atomization of the injected
water because the total interfacial area is nearly proportional to the drop mass fraction.
In Fig. 10, 72,7,and Dm indicate the effectiveness and the diameter of the drops
generated at 0.5 cm downstream of the
water injection nozzle center. These drops
are representative of the drops generated
near the water injection nozzle (X<1 cm)
because the atomization rate is maximum
at 0.5 cm downstream of the water injection
nozzle as shown in Fig. 5. The maximum
drop mass fraction increases with the initial
steam mass velocity and it varies from 0.30
to 0.98 in the range of the initial steam
mass velocity of 10,-70 kg/m2-s.
Almost
all of the injected water is atomized into
drops over the range of 40 kg/m2- s. The
drop diameter Dm decreases with the initial
steam mass velocity.
overall effectiveness
while
it changes
Fig.
along
the en,
line in the higher
rate in the injection
steam
region
drop mass fraction and the condensation
effectiveness
nozzle. One concludes that the enhancement
of the
(1)
(2)
Effect of initial steam mass velocity
on maximum drop mass fraction and
effectiveness of condensation at 10cm
downstream of water injection nozzle
This results in increasing the effectiveness of the drops
72 changes along the e, line in the lower steam mass velocity
that the total condensation
mass velocity
10
mass
velocity
is determined
range.
This
The
range
means
by both the maximum
of drops generated
condensation
with
near the injection
the initial steam
is caused by
the increase of the maximum drop mass fraction,
the increase of the condensation
effectiveness of the drops
injection nozzle because the finer drops are formed.
generated
near the water
The calculated
results
range (mg,520 kg/m2-s).
show that factor (1) is dominant in the low steam mass velocity
Whereas factor (2) is dominant in the high steam mass velocity
range (mg, >40 kg/m2.^)
in this range.
because
(2 )
Inlet Water
almost all of the injected
water
is atomized
into the drops
Mass Velocity
Figure 11 shows the effect of the inlet water mass velocity on the condensation in the
injection region. The calculated e f is consistent with the experimental ef,
and is almost
constant regardless of the inlet water mass velocity. Figure 11 also shows the calculated
drop diameter Dm that is generated at 0.5 cm downstream of the water injection nozzle and
the calculated maximum drop mass fraction em. Both the drop diameter and the maximum
drop mass fraction are almost constant. This indicates that the atomization rate of the
injected water is nearly proportional to the inlet water mass velocity near the water
injection nozzle and that the feedback effect of the condensation of steam on the drop
diameter is not so significant near the water injection nozzle under these conditions.
46 —
Vol. 20, No. 12 (Dec. 1983)
1019
However, there might be a feedback effect in
the higher range of water mass velocity because the steam
very rapidly,
mass velocity would reduce
resulting
in an increase
generated
drop diameter.
rate would increase
with
mass velocity
because
is nearly proportional
in the
The deposition
the inlet water
the condensation
rate
to the inlet water mass
velocity, resulting
in a shorter relaxation
length for the deposition of drops.
Even
though the relaxation length for the deposition
is longer than that for the condensation
in
the
present
analysis
conditions,
ation length for the deposition
the relax-
might become
Fig.
the same order as that for the condensation
in the high water mass velocity range because of the
increasing
drops.
These
near the water
deposition
situations
injection
of
would be realized
nozzle of the oscil-
latory flow of the condensing
two-phase
( 3 ) Water Injection Angle
In the present model, the
difference
rate
in the configuration
flow.
effect
of the
water
of the dead water
and 90.
The water injection at 45- results
because of the initial velocity component in
main
stream
direction.
This
Effect of inlet water mass velocity on effectiveness
of condensation at 10 cm downstream
of water
injection
nozzle,
maximum drop mass fraction
and drop diameter
generated
at 0.5 cm downstream of water
injection nozzle
injection
region
of the injected
water in the main stream direction.
the observed configurations
of the dead water region
the
11
angle
is evaluated
and the initial velocity
by the
component
Figure 12 shows the comparison of
for the water injection angle of 45
in the smaller
dead water
region
than at 90
should
result in the weaker effect of the flow contraction
due to the dead water formation.
The
configulation
was
assumed
of the dead water
in the
calculations
region
based on
the observed results as shown in Fig. 12.
Figure 13 shows the effect of the water
injection
angle
on
the
condensation
phe-
Fig.
Fig.
12 Effect of water injection angle
on shape of dead water region
47 —
13
Effect of water injection angle on
effectiveness
of condensation
at
10 cm downstream
of water injection nozzle, maximum drop mass
fraction and drop diameter
generated at 0.5 cm downstream of water
injection nozzle
J. Nucl.
1020
nomena in the injection region.
the water injection at 45- results
results
in the lower
45 and
90-.
The
)ef.
Sci.
Technol.,
Both the calculated
and experimental
results show that
in a lower 71f than at 90-. The smaller flow contraction
The
present
calculation
shows
model predicts
almost
that the maximum
the
same
drop mass
drop
diameter
fraction
than that for 90-. The formation of the dead water region enhances
the injection region resulting from the higher drop mass fraction.
the
for
for 45- is less
condensation
in
The calculation results show good qualitative agreement
with the experimental
results
for the parametric
study of the inlet steam and water mass velocities and the water injection angle.
Therefore,
one can conclude that the present model is reasonable
the condensation
phenomena in the injection region of the condensing two-phase
However,
checked
different
the correlation
for the atomization
for a wider range of experimental
pressure ranges.
The atomization
rate
and
the
deposition
should
be
conditions, especially for different facilities and
rate should have a strong
dependency
on the
water injection method and on the geometries near the water injection
think that it is necessary to review the correlations
of the atomization
plication of the present model to other facilities.
deposition in the injection region of the condensing
general
rate
to predict
flow.
nozzle. The authors
rate before the ap-
More studies of the atomization and the
two-phase flow are required to get more
correlations.
V.
The
direct-contact
to formulate
condensation
the condensation
CONCLUSIONS
of flowing
and mixing
steam
processes
on injected
water
in the injection
has
been
region.
studied
A flow model
based on the multifluid model of the two-phase
flow has been developed
from the flow
observation results in the injection region of the condensing two-phase flow. In the model,
the drops are grouped by the location where each drop is generated
in order
the condensation
rate accurately.
The conclusions
from the present study
marized
(1)
to evaluate
can be sum-
as follows :
The liquid film temperature
and the pressure
explained quantitatively
with the model which
arising
profiles in the injection
includes
(a)
drop generation
from atomization
of the injected
(b)
(c)
condensation
of flowing steam on the drops,
flow contraction
resulting from the formation
(d)
deposition
region
can be
water,
of the dead water
region,
of the drops on the liquid film.
(2)
The present model can reasonably
predict the effect of the inlet steam and water
mass velocities and the water injection angle on the condensation
rate in the injection
region.
(3)
The present model shows that the condensation
rate of steam depends mostly on both
the drop diameter generated near the water injection nozzle (X<1 cm) and the maximum drop mass fraction in the injection region.
Finer drops and a higher drop mass
fraction
(4)
result
in a higher
It is confirmed
region enhances
drop mass fraction
(5)
condensation
rate of steam
in the injection
region.
from the calculation results that the formation
of the dead water
the condensation
in the injection region resulting from the higher
arising
from the higher
steam
velocity
caused
by flow contraction.
The condensation
capability of the drops are deactivated
due to the deposition on the
liquid film. However, its effect is weak because the relaxation
length for the condensation is shorter than that for the deposition of the drops in the present study.
48 —
1021
Vol. 20, No. 12 (Dec. 1983)
(6)
It is clarified from the calculation results that higher condensation
rates with initial
steam mass velocity is caused by the formation of finer drops and the higher drop
mass fraction in the injection region.
[NOMENCLATURE]
a : Thermal diffusivity(mz/s)
Sc : Schmidt number
T: Temperature(K)
b: Width of water injection nozzle
(m)
t: Time(s)
Ce : Constant for atomization
rate
U: Velocity(m/s)
Ca: Constant for deposition rate
Ugo: Superficial steam velocity(m/s)
CD: Drag coefficient for drop
U10: Superficial water velocity(m/s)
CDF:
Drag coefficient for liquid film
Veen Cell volume(m3)
Dh:
Hydraulic diameter(m)
X : Distance from water injection
D: Drop diameter(m)
nozzle center(m)
Eaf:
Deposition rate of drops onto liquid
XI,: Location where drops in k-th
subgroup are generated(m)
k=1
Xk:
Section
length D
where drops in k-th
Ed f : Deposition rate of drops on liquid
subgroup
are
generated(m)
film from correlation developed
Xtt: Lockhart-Martinelli parameter
by McCoy et a/.(kg/m3-s)
{=,(nil/moo-87s oo-87s
(0/01)0.5(tit/ti g)0.1•z51
Efa: Atomization rate of liquid film
(fm: Accommodationcoefficient
P: Mass transfer rate(kg/m3-s)
(kg/m3-S)
5=1
rgk: Condensation rate onto a drop
(kg/s)
E'fa: Atomization rate of liquid film
Of: Liquid 5Im thickness(m)
from correlation developed
vLf: Effectiveness defined by Eq. (45)
by Ishigai & Takagi
(kg/m3-s)
: Effectiveness
Efk: Atomization rate of liquid film (kg/m3-s)
: Fraction of mass velocity l
Ehf: Deposition rate of k-th subgroup
t : Friction factor for water flow
drop onto liquid film(kg/m3-s)
{= 0.3164(plUDD,,/
0.25}
F1 : Shear stress for two-phase mixture (Pa/m)
g Friction factor for steaml flow
F01: Interfacial shear stress between
{=0.3164(peUgoph/pg)-O.25} m
: Viscosity(kg/
m s)
steam and liquid film(Pa/m)
: Density(kg/m3) s
r
F0k: Interfacial shear stress between
steam and k-th subgroup drop
(Pa/m)
: Surface tension(N/m) t
: Relaxation length(m)
f: Hold up
iy : Wall shear stress
t
(=DI,Ff/4)
(Pa)
g : Acceleration of gravity(m/s2)
+ : Stopping length I=
(18A)) t
h: Specific enthalpy (J/kg), or height
of test channel(m)
9: Lockhart-Martinelli parameter p
h,: Heat transfer coefficient
(W/m2•K)
(=1+21X12+Xt)
7: Lockhart-Martinelli
p
parameter
g: Latent heat of condensation
D (J/kg)
L: Width of test channel(m)
(-1+21/Xit+1/.100
m: Mass velocity(kg/m2-s)
(Suffix)
rne: Condensate mass velocity
(kg/m2. s)
f: Liquid film,
9: Steam,
I: Water
d: Total or average of whole drop subgroups
mcmax: Condensability(kg/m2-s)
k: k-th drop subgroup
N: Number of drop subgroup
i: Inlet condition
nk: Number density of drops(m-3)
in : Injection condition
p: Pressure(Pa)
m: Maximum or at location of maximum
dp1dX: Pressure gradient(Pa/m)
atomization rate
(dp/dX)":
Frictional pressure gradient
o Initial state without condensationof steam
for single-phase water flow
(Pa/m)
sat: Saturation
R: Gas constant of steam(m2/K
s)
* : Thermal equilibrium
Re: Reynolds number
film(EE51)(kg/m's)
(=ENEf
k)
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50 —
Fly UP