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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: