7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
MODELING AND ANALYSIS OF DIRECT STEAM CONDENSATION IN A PASSIVE
SAFETY SYSTEM OF ADVANCED PWR
D. Shaver"*, S. Antal1, M. Podowski1, D. Kim2
1 Rensselaer Polytechnic Institute, Mechanical Aerospace, and Nuclear Engineering
110 8th St., Troy, New York, 12180 USA
2 Korean Atomic Energy Research Institute
Daejeon, Korea
(*) corresponding author
shaved@irpi.edu, antals@irpi.edu, podowm(@rpi.edu, dhkim8(gkaeri.re.kr
Keywords: Directcontact condensation, diabetic twophase flow, CMFD, Lagrangian perspective
Abstract
A safety system based on a steam jet injection into a pool of cold water is considered in the design of a next
generation 1400 MWe pressurized water reactor (APR1400) in Korea. Because of the complexity of the physical
phenomena governing the interaction between a highspeed impinging jet and liquid water, extensive combined
experimental, theoretical and computational studies are necessary to fully understand the performance of such
systems. The purpose of this paper is to present the results of model development and computer simulations aimed
at capturing the fundamental physical phenomena governing steamjet/water interaction. Two models have been
developed, crosscompared, and validated against the experimental data of Kim et. al. (2001). One is a simplified,
theoretical, 2D model using the combined Lagrangian and Eulerian frames of reference. The other is a
multidimensional computational multiphase fluid dynamics (CMFD) model based on a multifield modeling concept
which has been implemented in the state of the art CMFD code, NPHASECMFD (Tiwari, Antal, & Podowski
2006). It has been shown that the predicted droplet concentrations, temperatures and steam condensation rate are
consistent with the experimental observations.
Introduction
Due to the high effectiveness of direct steam
condensation, it is being considered as a passive
safety system for a next generation nuclear reactor in
Korea. The proposed design for the 1400MWe
pressurized reactor (APR1400) involves injecting
steam into the incontainment refueling water storage
tank (IRWST). This will quickly condense the steam
and dissipate the excess heat into the subcooled water
contained in the tank. Significant theoretical and
computational studies are needed to understand the
governing physical phenomena associated with the
steam jet. The objective of this work is to develop a
consistent model which captures the effects needed to
predict the behavior of these phenomena.
Because of the complexity of the physical
phenomena governing directcontact condensation of
highspeed steam injected into a pool of cold water,
the modeling of the governing combined twophase
flow and heat transfer phenomena is a challenging
task. Two independent models have been developed
for this purpose. One is a simplified, theoretical
model using the combined Lagrangian and Eulerian
frames of reference in two dimensions. The other is a
multidimensional computational multiphase fluid
dynamics (CMFD) model which has been
implemented in the stateoftheart computer code,
NPHASECMFD.
Consistent with experimental observations, it is
assumed that a highspeed steam jet injection breaks
up the interface between the steam and the
surrounding water into microscopicsize liquid
droplets which travel through and along the jet.
These droplets interact with the steam via two major
mechanisms: drag force and condensation heat
transfer.
Condensation on a subcooled liquid droplet can be
best examined using the Lagrangian frame of
reference. The heat transfer to the droplet is a
transient effect which depends more strongly on the
time the droplet is exposed to steam than the specific
path of the droplet through the flow. As such, the
simplified, theoretical model solves the momentum
and energy balance equations for individual droplets
in the Lagrangian system of reference. This model
calculates droplet size, velocity, and temperature as
functions of time and initial conditions. Heat transfer
to each droplet is dominated by latent heat from
condensation. A transient solution for the
temperature profile inside the droplet gives the rate of
condensation on the droplet surface, the rate at which
the droplet heats up, and the rate of growth of the
droplet. From these parameters, thermodynamic and
hydrodynamic conditions for the droplets can be
determined as functions of position. This provides a
description of the flow in the Eulerian perspective,
which can then be compared to a detailed CMFD
model.
The theoretical model is used to parametrically study
the effects of different droplet sizes and the velocity
at which droplets enter the jet. This model has been
calibrated against the experimental data in order to
provide reasonable input data for the CMFD
simulation. Since the simplified, theoretical model
does not predict the effect of the droplets on steam
flow conditions, a complete CMFD model is
necessary to capture the underlying local phenomena.
Such a model has been formulated and implemented
in the stateoftheart code, NPHASECMFD
(Tiwari, Antal, & Podowski 2006). The NPHASE
CMFD code solves the individual transport equations
for mass, momentum, energy, and turbulence
quantities for the continuous steam and dispersed
droplet fields. The formulation of the governing
equations is based on the ensembleaveraging
concept (Podowski 2009). Phenomena modeled by
the CMFD model include the entrainment of droplets
by the jet, interfacial forces between the droplets and
the steam jet, condensation heat transfer from the
steam to the droplets, as well as the effects of local
velocity fields and droplet concentrations. The
results obtained from the CMFD simulations have
been compared to those obtained from the theoretical
model, and both have been validated against the
experimental data of Kim et. al. (2001).
The objective of this experimental study was to take
measurements of direct contact condensation of a
stable steam jet discharging into a quenching tank
with cold water. The test facility used to observe
steam jet condensation in a liquid pool consisted of a
steam generator, a quench tank, drain line, coolant
supply line, steam supply line, preheat line, valves
and the necessary instruments. The steam
generator's maximum operating pressure was 1.03
MPa, and the maximum steam flow rate was 1000
kg/hr. The system produced a steady flow of steam
at a quality higher than 99 %. The horizontal
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
quenching tank was open to the atmosphere, and its
diameter and length were 1 m and 1.5 m,
respectively.
TwoDimensional (2D) Theoretical Modeling
Formulation
The rationale behind the development of a two
dimensional model was to gain insight into the
physical phenomena governing the interaction
between liquid droplets and condensing steam based
on purely theoretical considerations. The model
follows the progression of individual droplets
through the steam jet. The domain used to model the
steam jet is taken as a 2 cm wide, 10 cm long
channel.
Due to the high speed of the steam jet, the dominant
force acting on the droplets is drag. The model
assumes that the droplets are formed due to liquid
atomization at the jet/liquidpool interface and then
penetrate the steam volume and cause steam
condensation. Since this process produces droplets
with injection velocities covering a large spectrum of
values, the total population of the newly born
droplets has been dividend into several groups, each
corresponds to a different velocity at injection. Since
the effect of the droplets on steam flow is not
accounted for, this model is valid only when droplet
volume fraction is low. The cumulative local droplet
concentration can be determined by adding together
the contributions from each individual group of
droplets.
The equations of motion for each group of droplets in
the axial (x, u) and lateral (y, v) directions have been
obtained from the corresponding force balance.
du 3 CdP ag ) (1)
dt 4 pD
dv 3 Cdg Ug (2)
dt 4 pD
Here, u(0)=u0 and v()= vo are the initial droplet
velocities at injection and un varies with different
droplet groups.
The drag coefficient has been determined from a
standard model for small spherical particles, as a
function of the relative Reynolds number. In general,
these equations must be solved numerically in order
to account for the effects of increasing droplet
diameter due to condensation, as well as of changing
drag coefficient. However, based on test calculations
it has been found that due to a short distance covered
by the droplets and high liquidtosteam density ratio,
the increase in droplet diameter is minimal and the
drag coefficient can be approximated by the
following expression
C K (3)
Red
where Kd is a coefficient.
This, in turn, allows for obtaining an analytic solution
for the velocity components in the Lagrangian frame
of reference
0(t) Ujet
no UJet
4 D2 f
exDI t
*[ 4D^p1 )
= exp (5)
vo 4 D D2p, f
By converting Eqs.(4) and (5) into the stationary
(Eulerian) frame of reference, both velocity
components can be expressed as functions of position
across the steam jet.
u(y) U,0
UH/ UJet
no U.1et
3 KdUPg (y o)
41 2 V
4 D2p, vo
v(y) 3 Kdl g (y yo
vo 4 D2pf vo
Because all droplets start from the same boundary,
these equations are therefore valid for all droplets of
a given field.
Interestingly, both velocity components vary linearly
with the transverse coordinate. Due to the small
diameter of the steam jet and the high velocities
associated with it, it is expected that the droplets will
be unable to slow down significantly within the jet
domain and the paths traced by the droplets will be
close to linear.
With the velocity components and boundary
conditions independent of the axial coordinate, it
follows that the concentration of droplets in each
group should also be independent. The superficial
velocity of the droplets in the transverse coordinate
must then remain constant if mass transfer due to
condensation is neglected. Using the expression for
the velocity distribution in the lateral direction, the
volume fraction across the jet can be found
a =J (8)
3 Kdpg
4 D2pf
It is important to notice that the results shown above
are based on several simplifying assumptions.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
However, several assumptions can be readily
removed at the expense of losing the clarity of an
analytic solution but with no significant impact on
the accuracy of solutions. Thus, the results of
predictions shown in the last section of the paper
have been obtained by numerically solving Eqs. (1),
(2) and (8) for a variable droplet diameter and a
rigorous model of drag coefficient. Then, to
determine the rate of droplet size change along the
flow, the kinematic part of the model has been
coupled with the steam/droplet interfacial heat
transfer model.
The directcontact energy transport between
condensing steam and subcooled liquid droplets is
due to two mechanisms: latent heat released due to
condensation and heat conduction inside the droplets.
A single water droplet can be modeled as a sphere
surrounded by a thin interface at the saturation
temperature. The energy equation for the droplet
including the interface is
dmd dm
d (mh,)= shdin (9)
dt gdt
(6) where q is the convective heat transfer rate from the
steam to the interface. If only the droplet without the
interface is considered, the energy equation becomes
(10)
(mdhd)= q + h, U
dt dt
where q, is the conductive heat transfer rate from the
interface into the droplet. Combining Eq.(9) and
Eq.(10), the heat transfer equation at the interface
between the condensing steam and a liquid droplet
becomes
dm, dIm
q, hg = qd + h dt
where
dmd_ dm_
dt dt s"
(11)
(12)
is the steam condensation rate. Since in the present
case, steam superheat can be ignored, Eq.(ll)
simplifies to
dmid
h d qd A q "
Sdt
(13)
where Ad = rD2 is the droplet surface area and q"d
is the heat flux due to conduction inside the droplet
volume.
From the Fourier law, the heat flux into the droplet
can be modeled
q k T(r,t)
q f or R
For spherical droplets, the radiallydependent droplet
temperature is given by the heat conduction equation
in spherical coordinates
1 OT 1 C98 OT) (15)
!2! rr2 (15)
af dt r2 9r 9r
Where a, is the thermal diffusivity of the droplets.
Defining the dimensionless variables as
0 =T (16)
T, T,
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
very small, it is still applicable to the current
situation.
Since the interface between a droplet and the steam is
assumed to have a constant temperature
Bi =c
and the roots of Eq.(25) become
n = nz
(26)
(27)
Consequently, Eq.(24) simplifies to
0(r ,Fo)= 2cos()exp( 2Fo)S1n(L
(28)
Eq.(15) can be nondimensionalized to obtain
S 2+ (18)
dFo r*2 2r* dr*
The corresponding boundary and initial conditions
are
aO(0,Fo) =0 (19)
9r*
dOFo) BiO(1,Fo) (20)
Dr*
(r*,0)= 1 (21)
Where the Biot and Fourier numbers, respectively,
are
HeqDd
aft
Fo 
R2
The solution to the boundary value problem given by
Eqs.(18)(21) can be written as
0(r ,Fo)=
4(sin(2,) cos(2,,)) p ( 2o sin(2r) (24)
n= 2), sin(22,) V 2r
Where the coefficients are given as the roots of the
following equation
12,, cot(,)= Bi (25)
Rigorously speaking, Eq.(24) has been obtained for
constantdiameter spheres. However, because the
change in droplet diameter due to condensation is
Differentiating Eq.(28) with respect to r
0Or*,Fo)
Or
(29)
Z2cos(A,)exp (_2Fo siAr cos( r
n=l Lr* r
yields the following expression for the temperature
gradient at the surface of the droplet
d0r *,Fo
Or
2 exp (n22Fo)
n=l
(30)
Converting Eq.(30) into a dimensional form and
substituting into Eq.(14), the surface heat flux
becomes
q11 d = k h, f h, exp
D c n1
(31)
/2 4aft
D2 2
Fortunately, the series in Eq.(31) converges very fast,
so that a relatively small number of terms is sufficient
to obtain an accurate estimate of the temperature
distribution inside the droplet.
For a given heat transfer rate, the timedependent
mass of a droplet can be determined from the rate of
condensation
md (t)= D3
6
t s
jfr (t')dt'+m,
0
(32)
The timedependent average enthalpy of a droplet
was then found from Eq.(10). Then, since the
hydrodynamic model predicts the location of droplets
at any given time, the timedependent temperature of
individual droplets in the Lagrangian system can be
readily converted into a positiondependent
temperature of the groups of droplets in the Eulerian
system of reference.
Results of Calculations
The theoretical model described above has been used
to perform calculations for the experimental
conditions reported by Kim at al. (2001). In parallel,
a complete multifield model of gas/droplet flow has
also been implemented in the NPHASECMFD code.
Details about the formulation of the NPHASE/CMFD
model can be found in (Podowski, 2009).
In both models the total population of droplets has
been divided into five groups, Based on extensive
parametric testing on the effect of individual
assumptions, in the final comparisons against data
droplets 300 ptm in diameter have been used. The
initial droplet temperature has been prescribed equal
to the water pool temperature (4'i (). Also,
according to the photographic evidence from the
experiments, the droplet injection velocity (due to
atomization in contact with steam flowing at an
average velocity of over 400 m/s) has been assumed
to be 70 m/s.
The remaining initial conditions for the droplets in
each group are shown in Table 1.
Table 1: The assumed boundary conditions for steam
jet simulations, used for the 2D model.
Group uo [m/s] jo [m/s]
1 100 0.00449
2 170 0.1326
3 240 0.3438
4 310 0.8112
5 380 1.373
In the theoretical model, the steam velocity does not
change and the initial droplet velocity in the axial
direction remains constant. At the same time, the
CMFD model is capable of capturing the effect of
droplets on condensing steam. Thus, the
corresponding boundary conditions have been
modified, and the droplet axial velocities have been
defined as fractions of the average steam velocity
along the flow. These boundary conditions are listed
in Table 2. For a constant steam velocity of 412 m/s,
these boundary conditions would exactly match those
used in the theoretical model.
Table 2: The assumed boundary conditions for steam
jet simulations, used for the CMFD model.
Group u.. I\ Uetave(x) jo [m/s]
1 0.243 0.00449
2 0.413 0.1326
3 0.583 0.3438
4 0.752 0.8112
5 0.922 1.373
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Figure 1 shows the positiondependent droplet
velocities of droplets in groups 2 and 4. Naturally,
since the motion of droplets is axisymmetric, upon
reaching the opposite wall, they deposit on the wavy
surface of the liquid and/or get redirected back into
the steam region. As can be seen, due to a short
distance traveled across the jet and a high injection
velocity, the relative decrease in droplet velocity is
only about 5%. The results obtained using the
theoretical model compare quite well against the
NPHASECMFD predictions.
72odel G
72D Model Group 2
71 2MD ModelGroup 4
CMFD Group 2
70 CMFD Group 4
70 Eq (7)
E 69
S68
> 67
66
65
001 0005 0 0005 001
y [m]
Figure 1: Predicted transverse velocity of droplets at
x/L = 0.4. Results are shown using an analytic
solution, Eq.(7); the theoretical model; and the
complete multifield CMFD model.
The predicted lateral profiles of droplet concentration
at an axial location equal to the jet diameter for the
individual groups of droplets are shown in Figure 2.
Again, the agreement between both models is quite
good.
0 02
Group 5
0016
.
0 00812
0 004 /
Group
001 0005 0 0005 001
y [m]
Figure 2: Predicted volume fraction profiles for
droplet groups at x/L = 0.2. The solid lines show the
theoretical model results, the dashed lines show the
CMFD model results.
Summing up the concentration for the individual
groups yields the local droplet volume fraction as a
function of both the distance from the center of the
jet as well as from the steam injection zone.
The lateral distributions of the cumulative droplet
volume fraction at different axial locations along the
steam flow are shown in Figure 3. The observed
trend shows an evolution of those profiles from wall
peaked to center peaked. The theoretical model
clearly underestimates the droplet concentration at
the centerline.
012
0 08
006
0 04
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The corresponding plots are shown in Figure 5. As it
can be noticed, the results of the theoretical model
agree well against the predictions of the NPHASE
CMFD code as long as x/L < 0.7. This is due to the
fact that the theoretical model does not account for
decreasing steam velocity and concentration due to
condensation. In fact the predicted distance needed
to condense the entire amount of steam in the jet is
approximately equal to x/Diet = 3.75 for the
theoretical model and x/Djet.= 5 (or x/L = 1)
according to the CMFD model. Interestingly, the
latter results agree quite well with the experimental
observations.
x/L =08
x/L =0 4
'x/L=02
001 0005 0 0005 001
y [m]
Figure 3: Predicted volume fraction of the droplets
across the steam jet, taken at various axial locations.
The solid lines show the theoretical model results, the
dashed lines show the CMFD model results.
Since in reality, droplets create a visible cloud inside
the steam jet, these profiles can be readily compared
to the experimental data shown in Figure 4.
2D model
CMFD
0 02 04 06 08 1
x/L
014
2D model
CMFD
0 08
0 02
(a) Conical shape (nozzle ID: 20
mm, mass flux: 280 kg/m's)
Figure 4: Image of the steam jet from Kim, et. al.
(2001). The marked distances correspond to 20mm
intervals.
The profiles shown in Figure 3 can be used to deduce
the axial distributions of droplet volume fraction
along the centerline, as well as the crosssectional
area averaged volume fraction along the channel.
0 02 04 06 08
x/L
Figure 5: Volume fraction of droplets along the
centerline of the jet (a) and the cross sectional area
averaged volume fraction of the droplets (b).
One of the measured parameters in the experiments
of Kim et al. (2001) was the average temperature of
the steam/water mixture along the centerline of the
jet. The measurements were performed using a
thermocouple. This, in turn, indicates that for
modeling purposes, the averaging concept should
reflect the fractional average contact area ratio
between the microscale droplets and the
thermocouple. Consequently, the following
definition of the weightedaverage temperature has
been used in the present work
kT1a. 3+kl,T(a)3
kg(1 a3 + k,( an 3
The definition given by Eq.(33) accounts for the
effect of both thermal conductivities of each phase,
and the volume fraction occupied by each phase. The
two thirds exponent is used based on a weighting of
the volumetoarea ratio, the former being a measure
of the droplet volume, the latter a measure of the
dropletthermocouple contact area. This is consistent
with the averaging used by Kim, Podowski and Antal
(2008).
In the current model, another level of averaging has
been used to account for the different temperatures of
droplets belonging to different groups.
Combing the calculated droplet concentrations at the
center line, shown in Figure 2, with Eq.(33), the
average steam/droplet temperature at the centerline
has been evaluated for both the simplified theoretical
model and the NPHASECMFD model The results
are shown in Figure 6.
120
110
100 1T 
o Kim et al (2001)
o 90 2D Model
S8CMFD Model
800 CMFD Model (Shifted)
70
I 60
50
40
0 025 05 075
x/L
Figure 6: Comparison of centerline two phase
mixture temperature predicted by the theoretical
model, the CMFD model, and that observed
experimentally.
As it can be seen, a good agreement between the
predictions and the data has been obtained for both
models. In particular, the trend in temperature
change has been matched quite well. Some of the
observed differences are due to the fact that in the
experiments the steam was slightly superheated at the
inlet. Also, due to the nature of the computational
multifield model, a very small (and uniformly
distributed) droplet concentration is normally
assumed to exist everywhere inside the steam
volume. This, in turn, has caused the observed
temperature drop near the inlet. However, when the
plot is shifted to account for the actual position when
droplets reach the centerline for the first time, the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
predicted distribution is very close to the
experimental data.
The advantages of CMFD simulations can be clearly
seen in Figure 7, where complete contour plots of the
steam velocity and droplet volume fraction are
shown. It can be seen in Figure 7(a) that the steam
jet slows down from the initial velocity of about 412
m/s and reverses near the location where complete
condensation occurs. Such a dramatic change in
steam velocity affects the motion of droplets and
droplet/steam interaction. Figure 7(b) shows the
cumulative droplet concentration throughout the
steam jet. The region near the end of the jet domain
exhibits the greatest droplet concentration. Naturally,
this region coincides with the region of a dramatic
slowdown of steam and a decrease in the flow rate
of uncondensed steam.
Steam U Velocity (m/s)
400 200 0 200 400
425, 11 I 1415
425.934 414.159
(a)
Droplet Volume Fraction
0.04 0.08 0.12 0.16
I4.300I I 0.1
4.3e005 0.16816
(b)
Figure 7: CMFD predictions of local parameter
evolution inside the steam volume: (a) steam
velocity, (b) droplet volume fraction.
Conclusions
Two different models for investigating fluid flow and
heat transfer in a high speed steam jet have been
presented. It has been shown that a simplified 2D
model can predict the hydrodynamic behavior of
groups of droplets where condensation is negligible.
However, it cannot predict the distribution of droplets
beyond x/L = 0.7. The CMFD model has been used
to predict the droplet distribution and centerline
temperature throughout the domain, and the overall
condensation rate. The results from the CMFD
model have been compared against the experimental
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
data of Kim et al. (2001). The consistency of the
proposed approach has been demonstrated and a good
agreement has been shown for all the parameters
mentioned above.
References
Kim, D., Podowski, M. Z. & Antal S. P., "Modeling
and Computer Simulation of Circulate Steam Jet
Injection into a Subcooled Liquid". 7th International
Topical Meeting on Nuclear Reactor Thermal
Hydraulics, Operation and Safety, Paper 125 (2008)
Kim, H. Y., Bae, Y. Y., Song, C. H., Park, J. K. &
Choi, S. M., "Experimental Study on the Stable
Steam Condensation in a Quenching Tank".
International Journal of Energy Research, Vol. 25,
pp. 239252 (2001)
Tiwari, P., Antal, S. P., & Podowski, M. Z., "Three
Dimensional Fluid Mechanics of Particulate Two
Phase Flows in UBend and Helical Conduits".
Physics ofFluids, Vol. 18 (4), pp. 118 (2006).
Podowski M.Z., "On the consistency of mechanistic
multidimensional modeling of gas/liquid twophase
flows", Nuclear Engineering and Design, 239, pp.
933940 (2009).
