Void fraction prediction for separated flows in the nearly horizontal tubes

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Void fraction prediction for separated flows in the nearly horizontal tubes. The void fraction is calculated without any discontinuity at flow regime transitions by considering continuous changes of the interfacial geometric characteristics and interfacial friction factors across three typical separated flows, namely stratifiedesmooth, stratifiedewavy, and annular flows. An evaluation of the proposed model against available experimental data covering various types of fluids and flow regimes showed a satisfactory agreement.
Available online at www.sciencedirect.com
ScienceDirect
Original Article
VOID FRACTION PREDICTION FOR SEPARATED FLOWS IN THE
NEARLY HORIZONTAL TUBES
TAE-HWAN AHN, BYONG-JO YUN*, and JAE-JUN JEONG
School of Mechanical Engineering, Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan 46241, South Korea
a r t i c l e
i n f o
a b s t r a c t
Article history:
A mechanistic model for void fraction prediction with improved interfacial friction factor
Received 13 April 2015
in nearly horizontal tubes has been proposed in connection with the development of a
Received in revised form
condensation model package for the passive auxiliary feedwater system of the Korean
27 May 2015
Advanced Power Reactor Plus. The model is based on two-phase momentum balance
Accepted 2 June 2015
equations to cover various types of fluids, flow conditions, and inclination angles of the
Available online 11 August 2015
flow channel in a separated flow. The void fraction is calculated without any discontinuity
at flow regime transitions by considering continuous changes of the interfacial geometric
Keywords:
characteristics and interfacial friction factors across three typical separated flows, namely
Concave interface
stratifiedesmooth, stratifiedewavy, and annular flows. An evaluation of the proposed
Interfacial friction factor
model against available experimental data covering various types of fluids and flow re-
Passive auxiliary feedwater
gimes showed a satisfactory agreement.
system
Copyright © 2015, Published by Elsevier Korea LLC on behalf of Korean Nuclear Society.
Separated flow
Stratified flow
Void fraction
1.
Introduction
condensation heat transfer coefficient in the horizontal
tubes of a condensing heat exchanger similar to that of PAFS
The Korean Advanced Nuclear Power Plant Plus (APRþ) is
[1e3]. This is because most condensation heat transfer
expected to adopt a passive auxiliary feedwater system
models of the horizontal tubes are based on empirical cor-
(PAFS) consisting of a condensation heat exchanger having
relations that are not applicable to a variety of conditions
nearly horizontal tubes (3 downward) as one of the passive
including the types of flowing fluids and inclination angle of
safety systems. Recently, many experimental studies and
the heat exchanger tube. As an alternative approach to
analyses have been conducted to verify the cooling perfor-
achieve
better
predictions,
a
mechanistic
condensation
mance of PAFS. These comprehensive evaluations revealed
model is considered applicable to the nearly horizontal
that
most
of
the
existing
models
underestimate
the
tubes
by
distinguishing
two
different
heat
transfer
* Corresponding author.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://
creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any me-
dium, provided the original work is properly cited.
1738-5733/Copyright © 2015, Published by Elsevier Korea LLC on behalf of Korean Nuclear Society.
dP
dP
S
t S t S
wg
wl i i
1
2
1
2
 
1
670
mechanisms
in
the
separated
flow
regimes
typically
2.
Void fraction prediction model
observed in the PAFS heat exchanger tubes [4]. For such an
approach, estimation of void fractions is of crucial impor-
The proposed model is based on the concept of equilibrium-
tance in considering different heat transfer mechanisms in
separated flow, proposed by Taitel and Dukler [8]. The
nearly horizontal condensing tubes.
configuration of the ideal separated flow, which is typically
To predict the void fraction in a two-phase flow, empir-
generated in the nearly horizontal tube, is schematically
ical correlations that consider slip ratio parameters have
depicted in Fig. 1. The separated flow model used in this study
typically been used [5,6]. These methods are applicable to
focused on the simple adiabatic conditions of negligible phase
some dispersed flows such as bubbly flow and intermittent
change and droplet entrainment. Therefore, the momentum
flow but not to separated flows whose slip ratios are usually
balance equations for the two phases with the assumption of
large or non-negligible [7]. By contrast, a mechanistic model
fully developed flow in the steady state condition are as
proposed by Taitel and Dukler [8] is widely used to predict
follows:
the void fraction by iterative schemes for separated flow in
horizontal tubes. However, this model is only applicable to
fully stratified flow with the assumption that the gaseliquid
 
Aa dz g twgSg tiSi rgAag sin q ¼ 0
(1)
interface is flat. Barnea et al [9] predicted the void fraction by
a separated flow model that used geometric parameters
applicable to annular flow. Ullmann and Brauner [10] and
 
Að1 aÞ dz l twlSl þtiSi  rlAð1 aÞg sin q ¼ 0
(2)
Chen et al [11] proposed improved geometric models for a
curved interface caused by large interfacial shear stress in a
stratifiedewavy flow. The interfacial friction factor, another
important parameter for the mechanistic prediction of the
void fraction, has been widely studied for the strat-
ifiedesmooth, stratifiedewavy, and annular flows [8,12,13].
Assuming equal pressure difference between the two
phases, the combined momentum equation for the separated
flow is finally obtained as follows:
tAag  Að1 laÞþ Aað1 aÞrl  rgg sin q ¼ 0 (3)
Ottens et al [14] reviewed such correlations and reported
that some had large errors when compared with experi-
mental data owing to limitations in their applicability
depending on the flow conditions. Moreover, the use of
different models for the friction factor and geometric char-
acteristics according to flow regime may also lead to dis-
continuities at flow regime transitions in the void fraction
calculations.
To determine the void fraction by using Eq. (3), it is
necessary to define the constitutive equations for the shear
stresses twg, twl for each phase at the wall and ti (positive
when ug > ul) at the phase interface, as well as for the contact
perimeters Sg, Sl, and Si over which the shear stresses act.
The shear stress terms for the wall and interface are
calculated by applying single-phase expressions as follows:
As mentioned above, accurate prediction of the void
fraction in such separated flows requires sophisticated
twg ¼ 2fgrgug
(4)
constitutive models on the interfacial characteristics.
Therefore, the objective of this study is to develop a new
mechanistic model for better prediction of the void fraction
twl ¼ 2flrlul
(5)
under separated flows in a nearly horizontal tube, especially
focusing on the continuous changes in geometric shape and
friction factor at the phase interface. That is, the geometric
relations that assume an ideal arc for the curved interface
were used to define the continuous flow regime transition
from stratified to annular flows with the improved interfacial
friction factors.
ti ¼ 2firgug ulug ul (6)
where the actual velocities ug and ul are expressed by the su-
perficial velocities jg and jl of gas and liquid phases, respec-
tively, as well as functions of the mass flux G, flow quality x,
and void fraction a as follows:
Fig. 1 e Schematic depiction of the configuration and coordinates of the separated flow. (A) Flow parameters. (B) Geometric
parameters.
j
x
g
r
g
¼
¼
u
¼
S
n
g
c
l
¼
D
S
þ
a ¼ 1 
(13)
671
ug ¼ a ¼ Ga
(7)
Sl ¼ g1ðD=2Þ
(15)
jl Gð1 xÞ
l ð1  aÞ rlð1 aÞ
(8)
g2D sinðg1=2Þ
i 2 sinðg2=2Þ
(16)
Additionally, the wall friction factors fg and fl are based on
the Blasius friction factor [15] as follows:
2.1.
Flow regime transition model
fg ¼  cg. 
rgugDhg mg
(9)
To calculate the void fraction, the flow regime must be iden-
tifiedbecauseit affectsthegeometricparameters expressed in
fl ¼ ðrlulDhl=mlÞnl (10)
For both friction factors, the constants c and n were defined
as 16 and 1.0 for laminar flow; the corresponding values for
turbulent flow were set to 0.046 and 0.2, respectively. The
symbols mg and ml are the dynamic viscosities of the gas and
liquid phases, respectively, and the hydraulic equivalent di-
ameters in the corresponding phases are given by:
Eqs. (13)e(16) and the interfacial friction factor in Eq. (6). The
separated flow regimes expected in the horizontal condensing
tubes are typically classified into stratifiedesmooth, strat-
ifiedewavy, and annular flows. Fig. 2 depicts the ideal inter-
facial shapes for such flow regimes. The stratifiedesmooth
flow, whose interface is flat, occurs under low superficial ve-
locities for gas and liquid phases. As the gas superficial ve-
locity increases at a given liquid flow condition, the flow
regime develops into a stratifiedewavy flow in which liquid
4Ag
hg Sg þSi
(11)
tends to climb up the tube wall by the pumping action of the
disturbance waves and the interface becomes far from flat,
owing to the increased interfacial shear stress [17]. As the
Dhl ¼ 4Al (12)
l
The hydraulic diameters for the friction factors were
defined differently according to the phases, as in the equa-
tions above; the interface is generally considered as stationary
(wetted) with respect to the flow of the faster phase and as free
with respect to the flow of the slower phase [16].
The phase interface shown in Fig. 1B is assumed to have an
ideal arc shape that changes continuously according to the
flow condition. The geometric relations of the flow cross-
section are defined by trigonometric relationships under the
condition of downward concave curvature (g1 > g2). From
these relationships, the void fraction and contact perimeters
are obtained as follows:
g1  sin g1 g2 sin g2 1 cos g1
2p 2p 1 cos g2
superficial gas velocity increases further, the liquid film
eventually covers the entire tube wall and the flow regime
transforms into an annular flow.
For geometric models of flow regime, the conditions for
transitions between the stratifiedesmooth, stratifiedewavy,
and the annular flows were derived from the change in the
interfacial shape. Here, the interface shape of stratifiedewavy
flow was assumed to be an ideal arc as a time-averaged shape
formed by wave spreading toward the top. Based on this
geometric model, the transition between stratifiedesmooth
and stratifiedewavy flows is assumed to occur when the
wetted angle g1 in Fig. 2 is equal to the stratification angle gss
defined by the flat interfacial configuration. In contrast, the
transition between stratifiedewavy and annular flows was
defined as the condition in which the wetted angle becomes
2p. In general, the region classified as the annular flow also
includes the intermittent and dispersed flow regimes under
high liquid flow conditions in the nearly horizontally arranged
heat exchanger tubes. Such flow regimes occur when the void
Sg ¼ pD g1ðD=2Þ
(14)
fraction calculated in the annular configuration is < 0.76,
which is the spontaneous blockage proposed by Barnea [18]. It
Fig. 2 e Ideal interfacial shapes for separated flow regimes in the horizontal tube. (A) Stratifiedesmooth. (B) Stratifiedewavy.
(C) Annular.
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Void fraction prediction for separated flows in the nearly horizontal tubes. The void fraction is calculated without any discontinuity at flow regime transitions by considering continuous changes of the interfacial geometric characteristics and interfacial friction factors across three typical separated flows, namely stratifiedesmooth, stratifiedewavy, and annular flows. An evaluation of the proposed model against available experimental data covering various types of fluids and flow regimes showed a satisfactory agreement..

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Nucl Eng Technol 47 (2015) 669e677 Available online at www.sciencedirect.com ScienceDirect journal homepage: http://www.journals.elsevier.com/nuclear-engineering-and-technology/ Original Article VOID FRACTION PREDICTION FOR SEPARATED FLOWS IN THE NEARLY HORIZONTAL TUBES TAE-HWAN AHN, BYONG-JO YUN*, and JAE-JUN JEONG School of Mechanical Engineering, Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan 46241, South Korea a r t i c l e i n f o Article history: Received 13 April 2015 Received in revised form 27 May 2015 Accepted 2 June 2015 Available online 11 August 2015 Keywords: Concave interface Interfacial friction factor Passive auxiliary feedwater system Separated flow Stratified flow Void fraction a b s t r a c t A mechanistic model for void fraction prediction with improved interfacial friction factor in nearly horizontal tubes has been proposed in connection with the development of a condensation model package for the passive auxiliary feedwater system of the Korean Advanced Power Reactor Plus. The model is based on two-phase momentum balance equations to cover various types of fluids, flow conditions, and inclination angles of the flow channel in a separated flow. The void fraction is calculated without any discontinuity at flow regime transitions by considering continuous changes of the interfacial geometric characteristics and interfacial friction factors across three typical separated flows, namely stratifiedesmooth, stratifiedewavy, and annular flows. An evaluation of the proposed model against available experimental data covering various types of fluids and flow re-gimes showed a satisfactory agreement. Copyright © 2015, Published by Elsevier Korea LLC on behalf of Korean Nuclear Society. 1. Introduction condensation heat transfer coefficient in the horizontal tubes of a condensing heat exchanger similar to that of PAFS The Korean Advanced Nuclear Power Plant Plus (APRþ) is expected to adopt a passive auxiliary feedwater system (PAFS) consisting of a condensation heat exchanger having nearly horizontal tubes (3 downward) as one of the passive safety systems. Recently, many experimental studies and analyses have been conducted to verify the cooling perfor- mance of PAFS. These comprehensive evaluations revealed [1e3]. This is because most condensation heat transfer models of the horizontal tubes are based on empirical cor-relations that are not applicable to a variety of conditions including the types of flowing fluids and inclination angle of the heat exchanger tube. As an alternative approach to achieve better predictions, a mechanistic condensation model is considered applicable to the nearly horizontal that most of the existing models underestimate the tubes by distinguishing two different heat transfer * Corresponding author. E-mail address: bjyun@pusan.ac.kr (B.-J. Yun). This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http:// creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any me- dium, provided the original work is properly cited. http://dx.doi.org/10.1016/j.net.2015.06.005 1738-5733/Copyright © 2015, Published by Elsevier Korea LLC on behalf of Korean Nuclear Society. 670 Nucl Eng Technol 47 (2015) 669e677 mechanisms in the separated flow regimes typically 2. Void fraction prediction model observed in the PAFS heat exchanger tubes [4]. For such an approach, estimation of void fractions is of crucial impor-tance in considering different heat transfer mechanisms in nearly horizontal condensing tubes. To predict the void fraction in a two-phase flow, empir-ical correlations that consider slip ratio parameters have typically been used [5,6]. These methods are applicable to some dispersed flows such as bubbly flow and intermittent flow but not to separated flows whose slip ratios are usually large or non-negligible [7]. By contrast, a mechanistic model proposed by Taitel and Dukler [8] is widely used to predict The proposed model is based on the concept of equilibrium-separated flow, proposed by Taitel and Dukler [8]. The configuration of the ideal separated flow, which is typically generated in the nearly horizontal tube, is schematically depicted in Fig. 1. The separated flow model used in this study focused on the simple adiabatic conditions of negligible phase change and droplet entrainment. Therefore, the momentum balance equations for the two phases with the assumption of fully developed flow in the steady state condition are as follows: the void fraction by iterative schemes for separated flow in horizontal tubes. However, this model is only applicable to fully stratified flow with the assumption that the gaseliquid interface is flat. Barnea et al [9] predicted the void fraction by a separated flow model that used geometric parameters applicable to annular flow. Ullmann and Brauner [10] and Aa dz g twgSg tiSi rgAag sin q ¼ 0 (1) Að1 aÞ dz l twlSl þtiSi rlAð1 aÞg sin q ¼ 0 (2) Chen et al [11] proposed improved geometric models for a curved interface caused by large interfacial shear stress in a stratifiedewavy flow. The interfacial friction factor, another important parameter for the mechanistic prediction of the void fraction, has been widely studied for the strat-ifiedesmooth, stratifiedewavy, and annular flows [8,12,13]. Ottens et al [14] reviewed such correlations and reported that some had large errors when compared with experi-mental data owing to limitations in their applicability depending on the flow conditions. Moreover, the use of different models for the friction factor and geometric char-acteristics according to flow regime may also lead to dis-continuities at flow regime transitions in the void fraction calculations. Assuming equal pressure difference between the two phases, the combined momentum equation for the separated flow is finally obtained as follows: tAag Að1 laÞþ Aað1 aÞrl rgg sin q ¼ 0 (3) To determine the void fraction by using Eq. (3), it is necessary to define the constitutive equations for the shear stresses twg, twl for each phase at the wall and ti (positive when ug > ul) at the phase interface, as well as for the contact perimeters Sg, Sl, and Si over which the shear stresses act. The shear stress terms for the wall and interface are calculated by applying single-phase expressions as follows: As mentioned above, accurate prediction of the void fraction in such separated flows requires sophisticated constitutive models on the interfacial characteristics. Therefore, the objective of this study is to develop a new mechanistic model for better prediction of the void fraction twg ¼ 2fgrgug (4) twl ¼ 2flrlul (5) under separated flows in a nearly horizontal tube, especially focusing on the continuous changes in geometric shape and friction factor at the phase interface. That is, the geometric relations that assume an ideal arc for the curved interface were used to define the continuous flow regime transition from stratified to annular flows with the improved interfacial friction factors. ti ¼ 2firgug ulug ul (6) where the actual velocities ug and ul are expressed by the su-perficial velocities jg and jl of gas and liquid phases, respec-tively, as well as functions of the mass flux G, flow quality x, and void fraction a as follows: Fig. 1 e Schematic depiction of the configuration and coordinates of the separated flow. (A) Flow parameters. (B) Geometric parameters. Nucl Eng Technol 47 (2015) 669e677 671 ug ¼ a ¼ Ga jl Gð1 xÞ l ð1 aÞ rlð1 aÞ (7) Sl ¼ g1ðD=2Þ g2D sinðg1=2Þ (8) i 2 sinðg2=2Þ (15) (16) Additionally, the wall friction factors fg and fl are based on the Blasius friction factor [15] as follows: 2.1. Flow regime transition model fg ¼ cg. rgugDhg mg (9) To calculate the void fraction, the flow regime must be iden- tifiedbecauseit affectsthegeometricparameters expressed in fl ¼ ðrlulDhl=mlÞnl (10) For both friction factors, the constants c and n were defined as 16 and 1.0 for laminar flow; the corresponding values for turbulent flow were set to 0.046 and 0.2, respectively. The symbols mg and ml are the dynamic viscosities of the gas and liquid phases, respectively, and the hydraulic equivalent di- ameters in the corresponding phases are given by: Eqs. (13)e(16) and the interfacial friction factor in Eq. (6). The separated flow regimes expected in the horizontal condensing tubes are typically classified into stratifiedesmooth, strat-ifiedewavy, and annular flows. Fig. 2 depicts the ideal inter-facial shapes for such flow regimes. The stratifiedesmooth flow, whose interface is flat, occurs under low superficial ve-locities for gas and liquid phases. As the gas superficial ve-locity increases at a given liquid flow condition, the flow regime develops into a stratifiedewavy flow in which liquid 4Ag hg Sg þSi tends to climb up the tube wall by the pumping action of the (11) disturbance waves and the interface becomes far from flat, owing to the increased interfacial shear stress [17]. As the Dhl ¼ 4Al (12) l The hydraulic diameters for the friction factors were defined differently according to the phases, as in the equa-tions above; the interface is generally considered as stationary (wetted) with respect to the flow of the faster phase and as free with respect to the flow of the slower phase [16]. The phase interface shown in Fig. 1B is assumed to have an ideal arc shape that changes continuously according to the flow condition. The geometric relations of the flow cross-section are defined by trigonometric relationships under the condition of downward concave curvature (g1 > g2). From these relationships, the void fraction and contact perimeters are obtained as follows: g1 sin g1 g2 sin g2 1 cos g1 2p 2p 1 cos g2 superficial gas velocity increases further, the liquid film eventually covers the entire tube wall and the flow regime transforms into an annular flow. For geometric models of flow regime, the conditions for transitions between the stratifiedesmooth, stratifiedewavy, and the annular flows were derived from the change in the interfacial shape. Here, the interface shape of stratifiedewavy flow was assumed to be an ideal arc as a time-averaged shape formed by wave spreading toward the top. Based on this geometric model, the transition between stratifiedesmooth and stratifiedewavy flows is assumed to occur when the wetted angle g1 in Fig. 2 is equal to the stratification angle gss defined by the flat interfacial configuration. In contrast, the transition between stratifiedewavy and annular flows was defined as the condition in which the wetted angle becomes 2p. In general, the region classified as the annular flow also includes the intermittent and dispersed flow regimes under high liquid flow conditions in the nearly horizontally arranged heat exchanger tubes. Such flow regimes occur when the void Sg ¼ pD g1ðD=2Þ (14) fraction calculated in the annular configuration is < 0.76, which is the spontaneous blockage proposed by Barnea [18]. It Fig. 2 e Ideal interfacial shapes for separated flow regimes in the horizontal tube. (A) Stratifiedesmooth. (B) Stratifiedewavy. (C) Annular. 672 Nucl Eng Technol 47 (2015) 669e677 1 0.1 0.01 Stratified–smooth flow France & Lahey (1992) Abdul-Majeed (1996) Chen et al. (1997) Badie et al. (2000) Ottens et al. (2001) Stratified–wavy flow France & Lahey (1992) Paras et al. (1994) Abdul-Majeed (1996) Chen et al. (1997) i g Badie et al. (2000) Ottens et al. (2001) Annular flow France & Lahey (1992) Abdul-Majeed (1996) Srisomba et al. (2014) 1E–3 0.002 fi=fg 0.004 0.006 0.008 0.01 Wall friction factor of gas phase, fg Fig. 3 e Relationship between fi and fg in the separated flow. should be noted that the intermittent and dispersed flow re-gimes were not considered in the present study because our study is limited to separated flows. 2.2. Constitutive relations according to the flow regime 2.2.1. Geometrical parameters 2.2.1.1. Stratifiedesmooth flow. The geometrical relation for the stratifiedesmooth flow regime is based on the flat inter-face, which is same as the Taitel and Dukler model [8]. In the present model, stratifiedesmooth flow represented by a flat interface occurs as the central angle g2 of the eccentric arc reaches zero (see Fig. 2A). Substituting the condition into Eqs. (13) and (16), the void fraction and interfacial contact perim-eter are simply derived by L'Hopital's rule as follows: a ¼ 1 g1 sin g1 (17) Si ¼ D sinðg1=2Þ (18) & (adjusted R2 Fig. 4 e Ratio of fi to fg with g* for the stratifiedewavy flow. Nucl Eng Technol 47 (2015) 669e677 673 The wetted angle in Eq. (17) can be calculated implicitly by an iterative procedure or explicitly by the Biberg [19] approx- imation as follows: 1=3h i gss ¼2pð1 aÞþ 2 2 2a1 þð1 aÞ1=3 a1=3 h n oi 100að1 aÞð2a1Þ 1 þ4 ð1 aÞ2 þa2 (19) 2.2.1.2. Stratifiedewavy flow. Stratifiedewavy flow was – assumed to have the concave interface shape of an ideal arc. – Fig. 2B shows the geometrical configuration for this flow regime and the relevant geometric parameters defined in Eqs. (13)e(16). The wetted angle, which is a requisite parameter for calculating the wall contact perimeter, is calculated from the following correlation proposed by Hart et al [20]. – g1 ¼ 2p 0:52ð1 aÞ0:374 þ0:26Fr0:58 (20) 2 Fr ¼ l (21) rl rg Dg cos q Here, the inclination angle q in the Froude number Fr is added to the original model in order to take into account the angle of inclination of the flow channel from the horizontal pipe. 2.2.1.3. Annular flow. Annular flow is a limiting condition for both the wetted angle g1 and central angle g2, whose value is 2p (see Fig. 2C). Assuming symmetric shapes of the interface in the annular flow, the void fraction and interface perimeter are simply expressed as a function of annular film thickness as follows: a ¼ ð1 2d=DÞ2 (22) Fig. 5 e Procedure for the calculation of void fraction. Si ¼ pðD 2dÞ (23) For the annular flow, the void fraction calculated by the constitutive models above needs to satisfy a value of > 0.76 on the basis of the spontaneous blockage criterion [18]. 2.2.2. Interfacial friction factor 10 Dispersed or intermittent 1 (Inapplicable data) The interfacial friction factor in Eq. (6) has a marked effect on calculation of the void fraction from the momentum balance equation as in Eq. (3) and may vary according to the flow regime. In the present study, the interfacial friction factors fi were investigated with the present void fraction prediction model against the available experimental data tabulated in Table 1, which are for separated flows regimes under the 0.1 Annular France & Lahey (1992) ( Air–Water, 1.0 bar) Data observed by the author 0.01 Stratified flow Stratified–wavy Intermittent flow (Plug) Intermittent flow (Slug) Stratified–smooth The present model 1E–3 0.01 0.1 1 10 100 Superficial gas velocity, jg (m/s) Fig. 6 e Comparison of flow regime identification with data by Franc¸a and Lahey [21]. horizontal and nearly horizontal pipes. The interfacial friction factor fi deducted from the data analysis is expressed with the gas phase wall friction factor fg as shown in Fig. 3. The results, althoughdespitesomescatter, explicitly show that theratio of fi to fg is around unity in the stratifiedesmooth flow, ranges from 1 to 10 for the stratifiedewavy flow, and is > 10 for the annular flow. The results confirm the previous investigations showing that the interfacial friction factor is assumed to be fi ¼ fg for stratifiedesmooth flow and fi ¼ 10fg for annular flow [8,26]. However, fi should be changed continuously from strat- ifiedesmooth flow to annular flow, unlike the case in previous 674 Nucl Eng Technol 47 (2015) 669e677 models, in order to ensure continuity of interfacial friction factor as well as better prediction of the void fraction. In this study, fi for the stratifiedewavy flow is expressed by an angle g* normalized ranging from the stratifiedesmooth to annular flow regimes, as follows:

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