Evaluation of slip ratio correlations in two-phase flow
Nuclear Science and Technology, Vol.10, No. 1 (2020), pp. 21-30
Evaluation of slip ratio correlations in two-phase flow
Thanh Tram Tran1,2,3, Hyun-Sik Park2,3
1Nuclear Training Center, 140 Nguyen Tuan str. Thanh Xuan, Ha Noi, Viet Nam
2University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon, 34113, Korea
3Korea Atomic Energy Research Institute,
111 Daedeok-daero 989, Yuseong-gu, Daejeon 34057, Republic of Korea
Email: [email protected]
(Received 19 November 2019, accepted 25 February 2020)
Abstract: Critical flow is one of the essential parameters in LOCA accident analysis in which
pressure difference is very high. Void fraction (α), in another term, slip ratio, s, is the key parameter
that could affect critical flow prediction. Henry-Fauske (HF) model is the model for critical flow
calculation existing in current computer codes such as MARS, RELAP, TRACE. However, the
limitation of this model is slip ratio s=1. By modified the slip ratio correlation, the paper focuses on
evaluating the HF model. Among the chosen correlations for slip ratio, Smith correlation is the best
option for this purpose. The results in our paper showed that while the original Smith correlation with
k=0.4 is suggested for horizontal tests, the modified one with k=0.2 could be applied for vertical tests.
Keywords: Void fraction, slip ratio, critical flow.
one
component
steam-water
system.
I. INTRODUCTION
Furthermore, based on the assumption of the
HF model, it could be applied to a well-mixed
condition, thermal equilibrium. TR model,
however, is applied to the two-component
system, air-water. Together with this, to predict
the critical mass flow, the HF model uses the
upstream conditions, while the TR model bases
a void fraction to predict the critical flow. Void
fraction, defining as the fractional occupied
area of the gas phase, is one important
parameter for two-phase flow, based on which
the component pressures, flow rate, and heat
transfer are determined. Because the gas phase
normally moves faster than that of the liquid
phase, the void fraction could not be directly
calculated from the mass flow rates of each
phase separately. Therefore, void fraction
depends on the phase velocity ratio, so-called
slip ratio. Slip ratio can be influenced by many
variables, such as mixture quality, temperature
The critical flow phenomenon takes
place when liquid, gas, or mixture leaks from a
system at high pressure to the ambient at lower
pressure through a break such as in a break of
safety valves or safety injection lines during a
loss
of
coolant
accident
(LOCA).
Understanding of break flow and its modeling
are important things in a LOCA scenario.
When the system approaches the critical flow
condition, the discharge flow through the
broken exit could reach a maximum value, and
the flow rate becomes independent from the
downstream pressure. Henry-Fauske (HF) [1]
applicable models for critical flow calculation
in the current safety analysis codes among
application of these models is different, they
should be used carefully while doing the
calculation. HF model is only applicable for
©2020 Vietnam Atomic Energy Society and Vietnam Atomic Energy Institute
EVALUATION OF SLIP RATIO CORRELATIONS IN TWO-PHASE FLOW
and pressure, the direction of flow, circulation
- No mass transfer in the expansion
- Being in thermal equilibrium
- Isentropic expansion (sg=sl)
- The same liquid temperature.
- Polytropic expansion of vapor at the
effect of diameter was the most important one
influencing the critical flow. His explanation
for this effect was related to the slip ratio.
exit (as the ideal gas).
- The critical mass flow rate reaches a
maximum value concerning the throat pressure,
dG/dpt =0.
This paper first reproduced the HF
numerically. Our reproduced model for HF
showed a good agreement in comparison
with the selected database. Furthermore,
based on Kim’s suggestion, various slip ratio
correlations were taken into account for
evaluating the critical flow rate. However,
the model of HF has only considered the slip
ratio is unity. Therefore, our focus is on
evaluated the HF model with different slip
ratio correlations. Our result showed that
for predicting the critical flow. Based on his
original k =0.4 value, the author modified
this value. It could conclude that among
chosen slip correlations, Smith correlation
gave a good agreement while varying the k
parameter for both horizontal and vertical
tests. While the original Smith correlation
could be used for the horizontal test, the
modified k=0.2 for this correlation could be
applied for a vertical one.
Based on these approximations, the flow
rate can be determined as follows:
[
{
( )
}]
Integrating the momentum equation
from the stagnant to the throat locations:
( )
[ ]
II. HENRY-FAUSKE CORRELATION
A. Henry-Fauske correlation
Substitution eq. (1) into the eq. (2) and
then rearrange this equation, the compact form
can be obtained:
for a non-equilibrium model, using two
continuity and one momentum equations of the
single component flow (water–vapor system)
without considering the wall shear stress and
heat exchange. This model also developed
using the following approximations:
[
]
Where:
- Same phase velocity (k=1)
22
THANH TRAM TRAN, HYUN-SIK PARK
data at low pressure. It means that this model
in the current MARS code could be useful for
high-pressure systems.
( ) (
)
( )
( )
( )
and
,
a
Fig. 1. Comparison of reproducing HF model with
For the given stagnant conditions of P0
experimental data at 0.12MPa (17.6 psi) [1].
and x0, by iteration until the η values in two
Eq. (3) and Eq. (4) are converged, the critical
pressure, , can be obtained. The critical mass
flow rate can finally be calculated.
B. Comparison of HF model
The main author’s difficulty is that the
MARS code is not commercial software. It is
only allowable for Korean students to use its
source code. Therefore, to modify the
correlations in the HF model, the author has
numerically reproduced this model based on
the main equations given in the original papers
[1] and compared the results with original ones.
This work can be seen in detail in our paper [5].
Comparisons have been made for one
component, steam-water. The critical mass flux
data versus stagnation quality are taken from
Fig. 2. Comparison of reproducing HF model with
experimental data at 1.38MPa (200 psi) [1].
HF [1] at different pressures.
Figs.
1, 2 and 3 show the comparisons between
reproducing the HF model and the
reproducing model gives similar results to the
original ones.
Fig. 3. Comparison of reproducing HF model with
experimental data at 2.76 MPa (400 psi) [1].
III. SLIP RATIO CORRELATIONS
From Figs. 1, 2, and 3, one can see that
the HF model tends to be better at high
pressure, and its result is under-predicted the
The
estimation
of
hydrostatic,
acceleration and friction pressure drops, and
23
EVALUATION OF SLIP RATIO CORRELATIONS IN TWO-PHASE FLOW
critical mass flow rate in the two-phase
mixture is based on the knowledge of the
void fraction, the relative volumes of gas and
liquid phases at a given location. A huge
number of empirical correlations and
theoretical models for void fraction have
theoretical ones are mainly based on
the flow regime, and therefore the results
cannot be of general applicability to two-
phase phenomena such as boiling process
including several different flow regimes,
from bubbly to separate flow. Other models
are developed based on a limited range of
experimental conditions such as pressure,
fraction depends on the phase velocity ratio,
function of quality (x), density (ρ), and
viscosity (µ):
or
)
(
) (
(
) ( )
Where the slip ratio, s, is determined
as follows:
(
)
(
)
Select the void fraction correlations
having the form in Eq. (7), the author has
compared them with the available experimental
data (Table I).
Void fraction correlations related to slip
ratio are listed in Table II.
Where: Re and We are Reynolds and
Weber numbers, D is the equivalent diameter.
APRM, APA and M parameters are determined as
followings:
{
},
)
(
) ( ) (
)
(
,
In reality, only several types of
reactors for gas/liquid exist. Moreover, one
could be noted that the contribution of the
, (
)
( ) ,
viscosity component as shown in Table I,
,
, a
( ) , is less dominant. Therefore, this
1+log( ) log( ) ,
parameter could be negligible in the void
fraction form. Void fraction correlations then
could be reduced as follows:
, σ is surface
tension.
Table I. References related void fraction data measurement.
Reference
Cook [12]
Diameter[mm] Working fluid
Geometry
Pressure [psi]
114 - 600
127
127
Steam-Water
Steam-Water
Steam-Water
Rectangular, vertical
Rectangular, vertical
Pipe, horizontal
114 - 600
Haywood [13]
12.7-38.1
250 - 2100
24
THANH TRAM TRAN, HYUN-SIK PARK
Table II. Void fraction correlations.
Correlation
A
1
1
b
1
1
1
c
1
0.5
0.67
d
0
0
0
1
Smith [7]
1
1
0
√
(
)
Chisholm [17]
1
1
0
√
(
⁄
)
Spendding & Chen [10]
Tuner & Wallis [19]
Thom [21]
Baroczy [22]
Madsen [24]
Chen [25]
2.22
0.26
1
0.28
1
0.65
0.67
0.72
0.64
1
0.74
1
M
0.65
0.33
0.4
0.36
0.89
0.65
1
-0.5
0.33
-0.126
0
0
0.08
0.07
0.18
0.13
0
0
0.07
0
1
APRM
1
0.18
APA
0.6
-0.2
Table III. Slip ratio correlations.
Correlation Slip ratio, s
HEM [14]
Fauske [15]
1
(
)
)
Zivi [16]
Smith [7]
(
√
(
)
Spedding & Chen [10]
Hamersma & Hart [18]
2.22(
)
( )
0.26(
)
( )
slip ratio is the main parameter that could affect
the diameter effect. In the past, some experimental
work showed that critical mass flow rate increased
while reducing the diameter of the throat (Sozzi
could be explained mainly based on the diameter
effect. The raise of vaporization may be higher at
the choking place with a decrease in sub-cooling
upstream conditions at a very low sub-cooling
temperature nozzle. It means that the slip ratio
may increase for low sub-cooling upstream
conditions. Several correlations of slip ratio were
reviewed before taking them into account their
critical flow predictions.
25
EVALUATION OF SLIP RATIO CORRELATIONS IN TWO-PHASE FLOW
By comparing with the experimental
data, void fraction correlations listed in Table I
are evaluated. The six selected correlations are
rewritten in the slip ratio form, as shown in
Table III. Those slip ratio correlations will be
evaluated in more detail using different
A. Evaluations using horizontal test data
Fig. 6. Void fraction vs. quality in comparing with
the Haywood data at 14.48 MPa (2100 psi) [13].
Chosen correlations are compared with
Haywood data for a horizontal test in a high-
pressure range from 1.72 to 14.5 MPa, as
shown in Figs. 4, 5, and 6.
B. Evaluations using vertical test data
performed experiments using the same vertical
test facility, but at different pressures varied
from 0.69 to 4.14 MPa. The predictions of
chosen correlations are compared with the
vertical test data, as shown in Figs. 7, 8, and 9:
Even data changes in a high-pressure
range, the original Smith’s correlation remains
the best candidate. We can conclude that
Smith’s correlation with his recommended k =
0.4 gives the best predictions.
Fig. 7. Void fraction vs. quality in comparing with
the experimental data at 0.79 MPa (114.5 psi) [11].
Fig. 4. Void fraction vs. quality in comparing with
the Haywood data at 4.14 MPa (600 psi) [13].
Fig. 5. Void fraction vs. quality in comparing with
Fig. 8. Void fraction vs. quality in comparing with
the Haywood data at 8.62 MPa (1250 psi) [13].
the Experimental data at 1.89 MPa (274.3 psi) [11].
26
THANH TRAM TRAN, HYUN-SIK PARK
The comparison of predictions between
the chosen correlations and the experimental
show that the modified Smith’s correlation
with k = 0.2, as well as Speeding and Chen
[10] one, are the best predictions.
parameter k, which is the ratio between the
mass liquid in the homogenous mixture and
the total liquid mass, varies from 0 to 1. If k =
0, then slip ratio, s= ( )
, Smith
correlation becomes Fauske one, in this case,
the velocity heads of both liquid and gas are
equal, . If k = 1, then s = 1,
Smith correlation becomes HEM one, .
However, Smith's correlation was developed
for the stratified flow with a homogeneous
mixture phase and a liquid phase. In the
original HF model, the slip ratio is a constant, s
=1, which means that the mixture is well
mixing in a homogeneous equilibrium state. By
changing the chosen slip ratio correlations in
the HF model, the predictions of mass flux at
high quality are quite the same, while they
become different at low quality.
Fig. 9. Void fraction vs. quality in comparing with
the experimental data at 4.24 MPa (614.4 psi) [12].
From these above comparisons, we
could conclude that while the original Smith
correlation with k=0.4 seems to be the best
correlation for the horizontal test, its
modification with k=0.2 could be the best
correlation for the vertical test.
IV. EVALUATION OF SMITH’S
CORRELATION
Fig. 10. Evaluation of critical mass flux for the HF
model using the experimental data at 0.12 MPa [1].
their correlation in assuming that slip ratio
equals 1. This slip ratio in the original HF
model is modified by using the chosen slip
correlations to predict the critical mass flux.
The results are listed in Figs 10, 11, and 12.
at 0.12, 1.38, and 2.76 MPa.
The critical mass flux result of original
HF is plotted in the continuous black line, and
its modified one using Smith correlation with
k = 0.4 is in the red long dash line. It should
be noted that in Smith's correlation, the
Fig. 11. Evaluation of critical mass flux for the HF
model using the experimental data at 1.38 MPa [1].
27
EVALUATION OF SLIP RATIO CORRELATIONS IN TWO-PHASE FLOW
model showed similar results with that
calculated by using the original HF model. Slip
ratio correlations, which correlate with critical
mass flux predictions, were chosen and
evaluated using both horizontal and vertical
tests. From this work, we could conclude that
the original Smith correlation with k=0.4 is the
best choice for horizontal tests, while the
modified one with k=0.2 is applicable for the
vertical test. HF model was developed based
on the slip ratio of unity. Therefore, we suggest
modifying this ratio to get a better result for the
two-phase critical flow simulation. Further data
evaluation is needed for a wider range of
pressure for both horizontal and vertical tests
to get a clear picture of the best option for
critical flow rate prediction.
Fig. 12. Evaluation of critical mass flux for the HF
model using the experimental data at 2.76 MPa [1].
At a quality higher than 0.1, the results
at low pressure (Fig. 10) showed a similar
result for all correlations. However, they
become different while reducing qualities
lower than 0.1. At hight pressures, Figs 11 and
12, the results using Spedding & Chen give a
bad prediction at very low quality (less than 10-
3), while this correlation gave as good
prediction as Smith one for vertical test data as
can be seen in Figs. 7, 8, and 9.
NOTATION
c
Sound velocity ( m/s)
C Vitural mass (kg)
D Diameter (m)
G Mass flux (kg/m2/s )
Base on the evaluation work for the
chosen slip correlations, we could conclude
that the results given by Smith correlation with
k=0.4 show the best critical mass flux
prediction. This slip ratio correlation is based
on simple assumptions for stratified flow. In
the current computer codes for critical flow
calculation, HF and TR models are still the
most popular tools for critical flow predictions.
HF model, however, keeps using the slip ratio
of unity. Based on the result of our paper in
predicting the critical flow rate, the modified
slip ratio should be considered in the HF to get
a better prediction.
h
k
Enthalpy (J/kg)
liquid mass in the homogeneous mixture
over total liquid mass
L Length (m)
N the partial phase change at the throat
s
S
x
P
uf Liquid velocity (m/s)
ug Gas velocity (m/s)
vf Specific volume of liquid (m3/kg)
vg Specific volume of gas (m3/kg)
We Weber number
Slip ratio, (ug/uf)
Entropy (J/K)
Quality
Presssure (MPa)
Re Reynolds number
V. CONCLUSIONS AND FUTURE WORK
Superscripts
Due to the limitation in software
handling, the reproducing model of HF has
been successfully evaluated. The reproducing
0
t
Stagnant location
Throat location
28
THANH TRAM TRAN, HYUN-SIK PARK
[9]. M.A. Woldesemayat, A.J. Ghajar, “Comparison
g
f
Gas component
Fluid component
of void fraction correlations for different flow
patterns in horizontal and upward vertical
pipes”, International Journal of Multiphase
Flow, 33, 347–370, 2007.
Greeks
α
γ
λ
η
Void fraction
[10].P. L. Spedding, J. J. J. Chen, “Holdup in two-
phase flow”, Int. J. Multiphase Flow, 10, 307-
339, 1984.
Isentropic exponent
The root of characteristic equation
Critical pressure ratio
ρf Liquid density (kg/m3)
ρg Gas density (kg/m3)
µf Liquid viscosity (Ns/m2)
µg Gas viscosity (Ns/m2)
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