Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Twophase flow pressure change across sudden expansion in small channels
Ing Youn Chena, ChihYung Tsengb, ChiChuan Wange
aMechanical Engineering Department, National Yunlin University of Science and Technology, Yunlin, Taiwan 640
bEnergy & Environment Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan 310
'Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, 300
Keywords: Twophase flow, Pressure change, Sudden expansion
Abstract
In this study, the authors first give a short overview on the singlephase flow across sudden expansion, followed by a
thorough review of the relevant literature for twophase flow across sudden expansion, and examine the applicability of the
existing correlations. Also, 282 data available from 5 publications are collected. None of the existing correlations can
accurately predict the entire database. Most of the correlations highly over predict the data with a mini test section which has
a Bond number being less than 0.1 in which the effect of surface tension dominates. Also, some of the correlations
significantly under predict the data for very large test sections. Among the models/correlations being examined, the
homogeneous model shows a poor predictive ability (a standard deviation of 143%) than the others, but it is handy for the
engineering application. Hence by taking account the influences of Bond number, Weber number, Froude number, liquid
Reynolds number, gas quality and area ratio into account, a modified homogeneous model is proposed that considerably
improves the predictive ability over existing correlations with a mean deviation of 23% and a standard deviation of 29% to all
the data with wider ranges for application.
Introduction
The flow of twophase mixtures across sudden expansions
and contractions is relevant in many applications such as
chemical reactors, power generation units, oil wells and
petrochemical plants. It is well known that the gasliquid
interactions in sudden flow area changes such as pipeline
connections and heat exchangers are a complex function of
the flow rates of the two phases, their physical properties
and pipe geometry. As the twophase mixture flows
through the sudden area changes, the flow might form a
separation region at the sharp corner leads to an
appreciable pressure loss due to irreversibility. Twophase
flow studies in constant area pipes have been widely
studied in the literature, however, frictional performance
arisen from singularities such as expansion and contraction
are among the least studies of twophase system (Chalfi, et
al., 2008). Recently, the authors had investigated the
twophase flow pressure change across sudden
contractions with small channels. A correlation was
proposed that considerably improves the predictive ability
(with a mean deviation of 30%) over existing correlations
with their 156 data and 357 available literature data (Chen
et al., 2009).
Though, there are several correlations for the twophase
flow across sudden expansions available in the literature.
Most of the correlations can only predict their own
database but failed to predict the data from the others.
Therefore, a short overview on the singlephase flow
across sudden expansion is firstly given, followed by a
thorough review of the relevant literature for twophase
flow across sudden expansions. From the available data
collected in the literature, the correlations/models are
tested to see their applicability. Finally, a correlation with
all the significant parameters for the twophase flow across
sudden expansions is proposed with an acceptable
accuracy for the engineering application.
Review of Literature
Singlephase pressure change across sudden expansion
For singlephase flow, this pressure change across the
sudden expansion was estimated by Delhaye (1981) as a
simple correlation from a simplified momentum balance
equation.
SG20,1 A A
eP = 1
Where p is the fluid density, and mass flux, G, is calculated
based on the smaller cross sectional area of the inlet tube,
and oA is the passage cross section area ratio (oA < 1).
Figure 1 shows a typical change of static pressure along
the axis for flow across the expansion. Due to the
deceleration of the flow in the transitional region, the static
pressure initially increases at the expansion area. After the
pressure reaches the maximum, the pressure gradient
merges with the downstream pressure gradient line. The
Paper No
pressure change at the sudden expansion is defined as the
pressure difference for upstream and downstream fully
developed pressure gradient lines extended to the
expansion position, i.e., APEX, aS shown in Figure 1.
Flow path
Figure 1: Idealized pressure variations across sudden
expansion
Twophase pressure change across sudden expansion
Romie (1958), Richardson (1958), Lottes (1961), Mendler
(1963), and McGee (1966) were firstly to investigate
twophase flow through sudden expansions. This study was
continuously investigated by Chisholm and Sutherland
(1969), Delhave (1981), Wadle (1980), Schmidt and
Friedel (1996), Attou et al. (1997), Attou and Bolle (1997),
Abdelall et al. (2005), and Chalfi et al. (2008). In the
majority of the previous studies, the void fraction (u), area
ratio (GA) and gas quality (x), as well as the densities of gas
and liquid (pL PG) WeTC USed to estimate the pressure
recovery across the sudden area expansion. The pressure
change equations were obtained from the mass and
momentum balances without considering the structure of
the flow and the frictional effect on the pipe wall. Ahmed
et al. (2007) had shown that the pressure recovery is
dependent on both the wall shear stress in the developing
region immediately downstream of the expansion and the
wall pressure on the downstream face of the expansion in
the flow developing region. The upstream and downstream
void fractions were estimated using the appropriate
correlations on the local flow pattern. Recently, Ahmed et
al. (2008) also correlated the length of the developing
regime to the area ratio and the upstream liquid Reynolds
number. However, Ahmed et al. model (2007) is
unpractical for the engineering application.
For allowing a relative velocity between the phases, Romie
(1958) derived an expression for sudden enlargement from
the momentum balance.
AP= (1 xy x
G'P (1 a, (1a ) P
Where the subscript 1 denotes upstream of expansion, and
2 represents downstream of expansion. If the void fraction
remains unchanged, Eq. (2) is simplified by Delhave (1981)
as:
GCr4a(1cA)r (1x)2 (pL /G X2 (3)
McGee (1966) measured pressure drops for steamwater
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
mixtures flowing through sudden expansions. His test
conditions are summarized in Table 1. The predicted
pressure change by Romie's Eq. (2) using the measured
upstream and downstream void fractions compared fairly
well with the data. The agreement with the predictions and
his 64 point data was in the order of + 40%. Neglecting
the density change and replacing a; and at by their average
value, the predictions of Eq. (3) did not affect the results
significantly with a 39% standard deviation as shown in
Table 2. Considering a homogeneous flow from
momentum balance, Eq. (3) is reduced to:
AP =Gaq(7 )(1 x)P Px(4
The data of McGee (1966) predicted by the
homogeneous model were significantly higher than the
data with a standard deviation of 138% as shown in Table
2. Attou et al. (1997) and Abdelall et al. (2005) had
pointed out that the homogeneous model tends to
significantly overestimate the experimental results due to
the assumption of no slip between the phases.
Richardson (1958) simplified the energy balance model
and assumed that the pressure recovery is proportional to
the kinetic energies of the phases:
G2(10 ~A) 0 (1 x)2(
hP,= 2 PL 1 I
Lottes (1961) ignored the gas mass flow rate and assumed
that all losses of dynamic pressure head occur in the liquid
phase only to derive the twophase pressure change.
Assuming a remains unchanged, his equation is simplified
to:
AP = Gcr (1 ) (6)
Mendler (1963) measured pressure drops for steamwater
mixtures flowing through sudden expansions. His test
conditions are listed in Table 1. His dada predicted by
homogeneous model and the simplified Romie's equation,
Eq. (3) were fairly as 65% and 36%, respectively, shown
in Table 2. Chisholm and Sutherland (1969) developed a
heterogeneous model based on the momentum balance.
Their model was compared with the airwater bubbly flow
data (a < 0.35). Their predictions remained reasonably
good agreement with data, but were slightly under
estimated by Attou et al. (1997). The underestimation of
the Chisholm model was also observed by Wadle (1980).
Wadle assumed the liquid is decelerated much less than the
gas when it passes through the expansion due to its higher
inertia. The pressure recovery at the sudden expansion is
caused by the bulk deceleration effect and a formula was
proposed from his test data to describe the pressure
recovery in an abrupt diffuser. The model includes an
artificial constant K in connection with different working
fluids (K = 0.667 for steamwater, K = 0.83 for airwater).
G K(1o, ) (1X2 7
AP (1 ) x (7)
2 p,
Attou et al. (1997) had mentioned that Wadle's model
a (2
Greek letters
D correction factors given in Eq. (12)
a mean void fraction
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
flow area expansion and contraction using air and water.
Their new data failed within an even lower all liquid
Reynolds number (Reno < 500) than that of Abdelall et al.'s
data (2578 < Reno < 3530). The Zivi slip flow model along
with the Armandtype void fraction, a = 0.5JG JL + G)
fairly under predicted their 24 data up to about 30%, where
Jis the superficial velocity.
From the foregoing review of the twophase pressure
change across the sudden contraction, the void fraction is
utilized for predictions except the homogeneous model,
Wadle (1980) and Chisholm & Sutherland (1980)
correlations. The void fraction may vary in the short length
of the sudden expansion due to the flow separation,
velocity and geometry changes.
MOst of the pressure change correlations based on the inlet
conditions to give a constant void fraction from
measurements or predictions from conventional
correlations, or by the individually developed empirical
correlations. Some of the investigations included the
upstream and downstream void fractions from the
measurements by Schmidt and Friedel (1996), Abdelall et
al. (2005) and Ahmed et al. (2007). Ahmed et al. (2007)
also predict the flow patterns at the upstream and
downstream, then using the appropriate correlations to
predict the corresponding void fractions. Recently, Dalkilic,
et al. (2008) had surveyed the void fraction correlations
and a summary of 35 correlations was given in a table. The
predictions of correlations had compared with data, and
Thom's void fraction correlation (1964) was found as one
of the bests in their study. Hence the void fraction
correlation proposed by Thom (1964) is implemented for
those correlations in need of void fraction.
Most of the proposed correlations/models are only
applicable to their own database. Also, some of the
correlations are not handy for the engineering application.
The objective of this study is therefore to provide a
simplified and reliable correlation with meaningful
physical parameters from the large literature database to
predict the twophase pressure change at sudden
expansions.
Nomenclature
Paper No
widely overestimates their airwater bubbly flow data (a <
0.35). Owen et al. (1992) had also observed that, with K =
0.22 (very different from 0.667), Wadle's model agreed
quite well with their measurements. The difference of the
K values was speculated by the difference of the expansion
geometries and flow conditions with a lower mass flow
rate. The parameter, K, is an empirical constant, an
adequate modeling is needed. Attou and Bolle (1997)
simplified the jet line emerging from a sudden expansion
as a straight line, and applied the momentum balance
within the boundary of the conical jet, they obtained a
correlation for the twophase flow pressure recovery from
a sudden expansion:
AP, = G a,(1 OA)[
Where 0 = 2/(apG) + 1X2(1 aL], a = 3/[1 + 04 +
04] and r is a correction factor related to the plwsical
properties of the mixture. For a gas quality x = 0, Eq. (8)
can be reduced to Eq. (1). The best fitting to the correction
factor is r = 1 for steamwater mixture and r = 1.4 for
airwater mixture. The mean quadratic errors for Eq. (8)
are about 23.4%. The correlation is particularly good for
small mass velocities, but it is inapplicable to high quality
flows (x > 0.2).
Based on the momentum and mass transfer balance,
Schmidt and Friedel (1996) developed a new pressure drop
model for sudden expansion which incorporates all of the
relevant boundary conditions. Assuming constant
properties, their developed model predicts several
experimental data sets from 8 test sections with several
conventional inlet (17.2~44.2 mm) and outlet (44.2~72.2
mm) diameters. The average of the logarithmic ratios of
measured and predicted values is less than 3%, the scatter
equals to 61%. The waterair data for the test sections with
area ratio of 0.0568 and 0.0937 conducted at 250C and 5
bar shown in Fig. 4.1 of Schmidt and Friedel (1996) are
listed in Table 1. Though, small error may be included in
the data reading, the data sets are still applicable in this
study.
Recently, Abdelall et al. (2005) investigated airwater
pressure drops caused by abrupt flow area expansions in
two small tubes. The tube diameters were 1.6 and 0.84 mm
with a sudden area change ratio of 0.276. Their
measurements indicated the occurrence of significant
velocity slip. With the assumption of an ideal annular flow
regime in association with the velocity slip ratio given by
Zivi (1964):
(9)
Bo
d
g
Fr
G
K
P
dPe
r
Re
S
uG
us
x
Fe
Bond number defined Eq. (10)
internal diameter of circular tube, m
gravity, ms2
Froude number defined in Eq. (10)
total mass flux, kgm s
loss coefficient
pressure, Pa
preSsure change across the sudden expansion, Pa
correction factor given in Eq. (8)
Reynolds number, pud p
ship ratio,
actual gas velocity, ms'
actual liquid velocity, ms'
gas quality
Weber number defined in Eq. (10)
21G _1 L _p pL
S
21L Gl G x~,P
where uG and uL are the actual gas and liquid velocities of
gas and liquid phases, respectively. The predictions of
Abdelall et al.'s slip flow model are slightly higher than the
experimental data, but the homogeneous model, Eq. (4)
significantly over predicts their data. The error margins for
the predictions of the homogeneous model and Abdelall et
al.'s slip flow model to their 14 data were not reported in
their study.
Based on the same test facility of Abdelall et al. (2005),
Chalfi et al. (2008) recently reported more data for
singlephase and twophase flow pressure drops caused by
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
for the models/correlations had being examined; the results
are shown in Table 2. None of the existing correlations can
accurately predict the entire database. Most of the
correlations highly over predict the data with a mini test
section which has a Bond number being less than 0.1 in
which the effect of surface tension dominates. Also, some
Of the correlations significantly under predict the Schmidt
& Friedel data (1996) with very large test sections.
Delhaye (1981) correlation seems to have the best
predictions to the data with a standard deviation of 54%,
but it over predicts the Abdelall et al. data (2005) (139%
standard deviation) and under predicts the Schmidt &
Friedel data (1996) (59% standard deviation), followed by
Richardson (1958) whose standard deviation is 63%, but it
significantly under predicts the Schmidt & Friedel (1996)
data (82% standard deviation). Attou and Bolle (1997)
correlation is with a standard deviation of 66%, but it also
significantly under predicts the Schmidt & Friedel data
(1996) (83% standard deviation). The above four
correlations all utilized the void fraction for the prediction.
Currently, there are so many correlations available in the
literature, which one shows the best predictive ability still
requires further examination. Yet the void fraction is not a
constant in the short path of the sudden expansion
Therefore, exploitation of the void fraction raises
additional uncertainty and inconvenience from the
engineer's perspective. In addition, the correlations of
Attou and Bolle (1997) and Wadle (1980) include an
artificial correction factor which is varied for different
twophase flow mixtures.
The highly over predictions by the existing correlations to
the Abdelall et al. data (2005) could be attributed to the
very small outlet section (d,,= 0.84 mm, dout = 1.6 mm). In
addition, the significantly under predictions for most of the
correlations to the Schmidt & Friedel (1996) data could be
due to the very large test sections (d, = 17.2 mm, dout =
72.2 mm and d,, = 19.0 mm, dout = 56.0 mm). For
obtaining a better predictive ability for mini test sections,
one should also take into account the influence of surface
tension force (Tripplet et al., 1999). The balance of
buoyancy force and surface tension force can be
represented by Bond number (Bo). Considering the effects
of total mass flux and gas quality to the surface tension,
Schmidt and Friedel (1996) had proposed a Weber number
(We) to correlate the twophase pressure change across
sudden expansions. The Froude number (Fr) which has the
ratio between the mixture inertia and the buoyancy force
was utilized by Friedel (1979) for the twophase frictional
pressure drop correlation in conventional straight tubes. In
addition, the liquid Reynold number, Reco = Gd/pt, was
USed as a significant parameter for correlating the
developing length downstream of a sudden expansion by
Ahmed et al. (2008).
Paper No
p density, kg m3
Ap density difference between liquid and gas, kg m3
0 surface tension, Nm1
oA CTOsssectional area expansion ratio, O < aA< A
6 function of the area ratio defined in Eq. (8)
pu viscosity, Nsm
Subsripts
G gas phase
in sudden expansion inlet
L, liquid phase
LO all hiquid flow
m twophase mixture
out sudden expansion outlet
Results and Discussion
To test the validity of the foregoing described
models/correlations from the existing literatures, 90 data
from Mendler (1963), 64 data from McGee (1966), 90 data
from Schmidt & Friedel (1996), 14 data from Abdellal et al.
(2005) and 24 data from Chalfi et al. (2008) are collected
and their test conditions, as well as the ranges of the
significant parameters are also listed in Table 1. The data
are compared with the previously described
correlations/models of Delhaye (1981) for Eq. (3),
Homogeneous model for Eq. (4), Richardson (1958) for Eq.
(5), Chisholm and Sutherland (1969), Wadle (1980) for Eq.
(7), Schmidt and Friedel (1996), Attou and Bolle (1997)
for Eq. (8) and Abdelall et al. (2005). Table 2 shows the
standard deviations of the comparison results for each data
set and the total 282 data. The standard deviations of the
relevant predictions to the total data are 54%, 143%, 63%,
74%, 92%, 69%, 66%, 230%, respectively. None of them
can accurately predict the entire database.
10 .
t Abddellal (25 Ot Dat~ao4 **
O Chalfi (2008, 24 Data)
:0 Standard Deviation= 657%
V Scmidt r(19961,90 Dat )
Standard Deviation= 31%
S101 Totaldtandardeviation=143%
Total Data= 282
lo '" n l 0. 1 .. .. ,.
Apgd 2
Bo =, We
G
lo~ lo lo~ loo
ape (Experiment Data,kPa)
lo~ 10
G 2d
Fr
op ,,
G 2
p gd
Figure 2: Comparison of Homogeneous predictions, Eq.
(3), with data.
Figure 2 is the comparison of Homogeneous predictions
with data which has a poor agreement with a standard
deviation of 143% to all the data. The predictive capability
Though, homogeneous model shows a little poor than the
others, but it is handy to use. Hence by taking account the
where pm =[ (1 x) _Px
Paper No
forgoing parameters in Eq. (10), into the original
homogeneous model, Eq. (4), the proposed modified
Homogeneous model takes the form as:
Modzfy iromogen~eous 1 23
WeBo 1x03
st = ( >2 x "
Reo x
2? = 0.2 x ( ~r)"
1
FrOX
02 = 0.4x( x 03'+0.3xel6/Resl '
3 x
0.4x(P L 02
G
This proposed modified homogeneous correlation
considerably improves the predictive ability over existing
correlations with a mean deviation of 23% and a standard
deviation of 29% to all the data as shown in Figure 3. The
standard deviations for the predictions of the
Homogeneous model to the data sets of Abdellal et al.
(2005), Chalfi et al. (2008), Mendler (1963), McGee (1966)
and Schmidt & Friedel (1996) are greatly improved by the
modified Homogeneous model, from 468%, 207%, 65%,
138% and 31% to 26%, 32%, 31%, 34% and 24%,
respectively. In summary, the proposed correlation shows a
very good accuracy against the existing data and is capable
of handling the effects of gas quality, area ratio, mixture
inertia, surface tension and buoyancy force, and is valid for
much wider ranges of: 506 < G < 562 gm2S 002
< 0.99, 0.057 < GA < 0.607, 0.84 < d,,, < 19 mm, 0.095 <
Bo < 92, 1.03E+1 < Fr < 9.19E+5, 1.0E+2 < We < 8.3E+4,
and 4.35E+2 < ReLo < 4.95E+05. '
los
a Abdellal (2005, 14 Data) +50% .
: tandar d(D~e~v t nna26%
102 Standard Deviation= 32%0%
y ta~ndr rD vato a)
33 Mcgee (1966, 64 Data)
: Standard Deviation= 24%
S10a _
S10 
10 2Total Standard Deviation = 29% .
Total Data= 282
103
103 102 101 100 101 102 10"
aPe (Experiment Data,kPa)
Figure 3: Comparison of the data with modified
Homogeneous model, Eq.(12).
Conclusions
In this study, the authors present a thorough review of the
relevant literature for twophase flow across sudden
expansions, and examine the applicability of the existing
correlations. Also, 282 data available from 5 publications
are collected. None of the existing eight correlations can
accurately predict the entire database. Most of the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
correlations highly over predict the data with a mini test
section which has a Bond number being less than 0.1 in
which the effect of surface tension dominates. Some of the
correlations significantly under predict the data with very
large test sections. The significant parameters of Bond
number, Weber number, Froude number, and liquid
Reynolds number to the frictional twophase pressure
change across sudden area expansions are discussed.
Among the models/correlations being examined, the
homogeneous model though shows a poor predictive
ability than the others, but it is handy for the engineering
application. Hence by taking account the influences of
Bond number, Weber number, Froude number, liquid
Reynolds number, gas quality and area ratio into the
original homogeneous model for correlating with the data,
a modified homogeneous correlation is proposed that
considerably improves the predictive ability over existing
correlations with a mean deviation of 23% and a standard
deviation of 29% to all the data with wider ranges for
application.
Acknowledgements
The supports from the Energy Bureau and Department of
Industrial Technology of the Ministry of Economic Affairs
and National Science Committee (NSC
972221E224054) of Taiwan are greatly appreciated.
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Table 1. Available data for twophase pressure change across sudden expansions
Schmidt &
Abdellal et al. Chalfi et al. Mendler (1963) McGee (1966)
Resarcers (2()(5) 1 tube (2()(8) 1 tube 3 tubes 3 tubes Friedel (1996)
2 tubes
G (kg/n~s) 3()74 4212 5()6 664 692 5642 544 248() 1()(() 3()()
1 ().()(2 ).()13 ).()285().182 ).()32 ().469 ().()(1 .3()4 ().)1 .99
~AirWater AirWater SteanlWater Stean1Water AirWater
Working fluid
25 oC 25 oC 194252 oC 141195oC 25 oC
din (nm) ().84 ().84 9.55, 12.9, 17.63 8.64, 11.68 17.2, 19
().145,().264 ().332, ().546,
CT ().276 ().276 '(.()568, ).()937
().493 ().6()7
Bo =~ gd (.()95 ).()95 19,8 92.1 13.53().7 4() 48.8
G2
Fr = 935() 167()()( 264()(91 9()() 25.7 376()( 1().3957()( 39.5 172()()(
G~d
We = 312 133() 1()) 594 1()2() 455()( 185155()( 716 836()
Gd
ReLo 258() 453() 435 571 9)2()() 495()()( 336()(1 37()() 19()(() 63()()(
AP, (kPa) ).726 4.62 ().8()1 3.71 ().575 15.6 ).()69 14.4 ().2 36.()
Data Points 14 24 9() 64 9()
Table 2. The standard deviations for the pndctions of existing com relations to the data
Abdellal Schmidt
Chalfi et Mendler McGee Total
Data et al. & Friedel
al., 1 tube 3 tubes 3 tubes 282
Correlations 1 tube 2 tube
(2008) (1961) (1966) Points
(2005) (1996)
Delhaye (1981) 139% 28% 36% 39% 59% 54%
Homognous 468% 207% 65% 138% 31% 143%
Richardson (1958) 57% 51% 54% 44% 82% 63%
Chisholm & Sutherland (1969) 250% 77% 37% 56% 49% 74%
Wadle (1989) 178% 46% 52% 66% 123% 92%
Schmit & Friedel (1996) 259% 28% 43% 47% 41% 69%
Attou & Bolle (1997) 93% 53% 54% 52% 83% 66%
Abdelall et al. (2005) 133% 46% 169% 80% 360% 230%
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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