Nuclear Engineering and Design 243 (2012) 200–213

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Nuclear Engineering and Design

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Entrained liquid fraction prediction in adiabatic and evaporating annular

two-phase flow
Andrea Cioncolini 1 , John R. Thome ∗
Heat and Mass Transfer Laboratory, Swiss Federal Institute of Technology-EPFL, EPFL-STI-IGM-LTCM, Station 9, 1015 Lausanne, Switzerland

a r t i c l e i n f o a b s t r a c t

Article history: A new method to predict the entrained liquid fraction in annular two-phase flow is presented. The under-
Received 1 March 2011 lying experimental database contains 2460 data points collected from 38 different literature studies for
Received in revised form 8 November 2011 8 different gas–liquid or vapor–liquid combinations (R12, R113, water–steam, water–air, genklene–air,
Accepted 9 November 2011
ethanol–air, water–helium, silicon–air), tube diameters from 5.0 mm to 95.3 mm, pressures from 0.1 to
20.0 MPa and covers both adiabatic and evaporating flow conditions, circular and non-circular channels
Keywords:
and vertical upflow, vertical downflow and horizontal flow conditions. Annular flows are regarded here
Annular two-phase flow
as a special form of a liquid atomization process, where a high velocity confined spray, composed by the
Entrained liquid fraction
Liquid film atomization
gas phase and entrained liquid droplets, flows in the center of the channel dragging and atomizing the
Shear-driven liquid film annular liquid film that streams along the channel wall. Correspondingly, the liquid film flow is assumed
Entrainment to be shear-driven and the energy required to drive the liquid atomization is assumed to be provided in
Deposition the form of kinetic energy of the droplet-laden gas core flow, so that the liquid film–gas core aerody-
namic interaction is ultimately assumed to control the liquid disintegration process. As such, the new
prediction method is based on the core flow Weber number, representing the ratio of the disrupting
aerodynamic force to the surface tension retaining force, a single and physically plausible dimensionless
group. The new prediction method is explicit, fully stand-alone and reproduces the available data better
than existing empirical correlations, including in particular measurements carried out in evaporating
flow conditions of relevance for boiling water nuclear reactor cooling.
© 2011 Elsevier B.V. All rights reserved.

1. Introduction micro evaporators and micro heat sinks for the thermal man-
agement of microelectronic components, computer chips, laser
Annular two-phase flow is one of the most frequently observed diodes and high energy physics particle detectors, while also for
flow regimes in practical applications involving gas–liquid and refrigeration, air-conditioning and petrochemical piping and pro-
vapor–liquid two-phase flows, such as steam generators, refrigera- cesses.
tion and air conditioning systems, nuclear reactors and chemical A crucial parameter for predicting and modeling annular flows is
processing plants. In annular flows, a part of the liquid phase the entrained liquid fraction e, defined as the ratio of the entrained
flows as a continuous film that streams along the channel wall, liquid droplets mass flow rate to the total liquid mass flow rate. The
while the rest of the liquid phase is transported in the gas core as entrained liquid fraction is a flow parameter bounded between 0
entrained droplets. Annular flows have been extensively investi- and 1, with values close to 0 being characteristic of annular flows
gated in the last decades, particularly in connection with nuclear with an almost perfect segregation between liquid and vapor and
reactor cooling applications. Nonetheless, this topic is currently most of the liquid concentrated in the film, while values close to 1
experiencing a renewed interest, driven in particular by nuclear are typical of annular flows close to the transition to dispersed mist
reactor power uprates and nuclear reactor fuel optimization, appli- flow where most of the liquid is in the form of entrained droplets.
cations where more accurate and reliable prediction methods The accurate prediction of the entrained liquid droplets mass flow
for system computer codes are required. Sound prediction meth- rate is of paramount importance since this is liquid not flowing in
ods for annular flows are as well required for the design of the annular film, and hence has an important influence both on
the gas/vapor core and annular film flows. As a matter of fact, an
accurate knowledge of the entrained liquid fraction is required in
most thermal–hydraulics predictions, such as the onset of dryout
∗ Corresponding author. Tel.: +41 21 693 5981; fax: +41 21 693 5960.
in boiling channels, post-dryout heat transfer and the effectiveness
E-mail addresses: andrea.cioncolini@epfl.ch (A. Cioncolini), john.thome@epfl.ch
(J.R. Thome). of nuclear reactor core cooling, particularly during transient and
1
Tel.: +41 21 693 5984; fax: +41 21 693 5960. accident scenarios.

0029-5493/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.nucengdes.2011.11.014
 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213 201

at the same speed, then the droplet-laden gas core density c , the
Nomenclature core flow velocity Vc and equivalent diameter dc are as follows:

ag gas sonic velocity (m s−1 ) x + e(1 − x) xG

c = ; Vc = ;
atpf two-phase flow sonic velocity (m s−1 ) (x/g ) + (e(1 − x)/l ) εg
Aflow channel cross section flow area (m2 ) 
d tube diameter (m) xl + e(1 − x)g
dc = d ε (2)
dc core flow diameter (m) xl
e entrained liquid fraction (–)
where x is vapor quality, l and g are the liquid and vapor densi-
g acceleration of gravity (m s−2 )
ties, G is the mass flux, ε is the cross-sectional void fraction of the
G mass flux (kg m−2 s−1 )
channel and d the tube diameter. The measured entrained liquid
M Mach number (–)
fraction values from Cioncolini and Thome (2010) are displayed
Pwet channel wetted perimeter (m)
in Fig. 1 versus the core flow Weber number defined in Eq. (1),
Jg superficial gas velocity (m s−1 )
together with the proposed correlating equation:
Jl superficial liquid velocity (m s−1 )
Vc core flow velocity (m s−1 ) e = (1 + 13.18 Wec−0.655 )
−10.77
(3)
Vg gas velocity (m s−1 )
Vtpf average two-phase flow velocity (m s−1 ) Besides outperforming existing prediction methods, as dis-
Wec core flow Weber number (–) cussed by the authors, a significant advantage of Eq. (3) with
x vapor quality (–) respect to other empirical correlations is that it is based on a sin-
ε cross sectional void fraction (–) gle and physically plausible dimensionless number, which is also a
c core density (kg m−3 ) controlling group in determining the wall shear stress and associ-
g vapor density (kg m−3 ) ated frictional pressure gradient of annular flows (Cioncolini et al.,
l liquid density (kg m−3 ) 2009b). The experimental database used to derive Eq. (3) con-
 surface tension (kg s−2 ) tained 1504 measurements of the entrained liquid fraction and
covered 8 gas–liquid or vapor–liquid combinations (R12, R113,
water–steam, water–air, genklene–air, ethanol–air, water–helium,
Recently, Cioncolini and Thome (2010) proposed a prediction silicon–air) and 19 different values of the tube diameter from
method for the entrained liquid fraction based on the assumption 5.0 mm to 57.1 mm. Although this experimental database was quite
that annular flows can be regarded as a special form of a liquid large, it is essentially limited to adiabatic annular flows, as it con-
atomization process, where a high velocity confined spray, com- tains only 16 points obtained in diabatic flow conditions. As such,
posed by the gas phase and entrained liquid droplets, flows in the the application of Eq. (3) to evaporation in channels is not straight-
center of the channel dragging and atomizing the annular liquid forward and requires some extrapolation. Besides, as can be seen
film that streams along the channel wall. In particular, the liquid in Eq. (2), the core flow density c depends on the entrained liquid
film flow was assumed to be shear-driven and the energy required fraction e, so that Eq. (3) has to be used iteratively, a complica-
to drive the liquid atomization was assumed to be provided in the tion that might limit its applicability into existing simulation tools.
form of kinetic energy of the droplet-laden gas core flow, so that Finally, this prediction method is not completely stand-alone, since
the liquid film–gas core aerodynamic interaction was ultimately the void fraction ε is required as input in Eq. (2) to calculate the
assumed to control the liquid disintegration process. Accordingly, core flow velocity Vc and equivalent diameter dc . As such, an addi-
the dimensionless group used by the authors to fit entrained liq- tional empirical correlation is actually required to provide the void
uid fraction data is a core flow Weber number Wec , representing fraction.
the ratio of the disrupting aerodynamic force to the surface tension The purpose of the present study is to improve the prediction
retaining force and defined as: method for the entrained liquid fraction proposed in Cioncolini
and Thome (2010). In particular, the method is here extended
c Vc2 dc to cover evaporating flow conditions and non-circular channels.
Wec = (1)
 Moreover, the method is also simplified and made explicit and
If the slip between the carrier gas phase and the entrained liquid fully stand-alone. The underlying experimental databank has been
droplets is neglected so that gas and droplets are assumed to travel significantly expanded from the 1504 data points initially used by

1
Entrained Liquid Fraction

0.8

0.6 Eq. (3)

0.4

0.2

0 1 2 3 4 5
10 10 10 10 10
Core Flow Weber Number

Fig. 1. Entrained liquid fraction vs. core flow Weber number as defined in Eq. (1), from Cioncolini and Thome (2010).
202 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213

Table 1
Experimental annular flow data bank for vertical flow in circular tubes.

Reference Fluids d (mm) P (MPa) G (kg m−2 s−1 ) x e (1) (2) (3) No. points

Lopez de Bertodano et al. (2001) R113 10.0 0.32–0.53 205–1089 0.35–0.90 0.18–0.89 350 ↑ a 48
Han et al. (2007) H2 O–Air 9.52 0.10 181–344 0.10–0.27 0.02–0.07 166 ↑ a 30
Okawa et al. (2005) H2 O–Air 5.00 0.14–0.76 265–1939 0.05–0.78 0.05–0.69 320 ↑ a 170
Sawant et al. (2008) H2 O–Air 9.40 0.12–0.40 115–890 0.06–0.91 0.02–0.75 210 ↑ a 66
Jepson et al. (1989) H2 O–Air H2 O–He 10.3 0.15 55–219 0.07–0.75 0.01–0.26 292 ↑ a 49
Nakazatomi and Sekoguchi (1996a) H2 O–Air 19.2 0.30–20.0 87–3350 0.06–0.99 0.01–0.99 474 ↑ a 203
Azzopardi and Zaidi (2000) H2 O–Air 38.0 0.15 47–154 0.21–0.73 0.13–0.50 118 ↑ a 28
Schadel 25.4 0.15 51–312 0.35–0.91 0.10–0.76 150 ↑ a 64
et al. H2 O–Air 42.0
(1990) 57.1
Jagota et al. (1973) H2 O–Air 25.4 0.28–0.42 135–591 0.12–0.69 0.06–0.56 150–261 ↑ a 103
Mayinger and Langner (1976) R12 14.0 1.10 300–1000 0.30–0.90 0.06–0.94 na ↑ d 17
Hinkle (1967) H2 O–Air 12.6 0.28–0.62 155–733 0.15–0.71 0.01–0.45 119–262 ↑ a 129
Andreussi (1983) H2 O–Air 24.0 0.15 76–523 0.09–0.84 0.13–0.62 208 ↓ a 66
Steen and Wallis (1964) H2 O–Air Silicon–air 15.9 0.10–0.41 48–462 0.04–0.73 0.01–0.98 na ↓ a 251
Adamsson and Anglart (2006) H2 O–Steam 14.0 7.0 750–1750 0.29–0.76 0.55–0.96 na ↑ d 153
Ueda and Kim (1982) R113 10.0 0.33 486–1155 0.26–0.71 0.29–0.95 na ↑ d 27
Al-Yarubi and Lucas (2008) H2 O–Air 50.0 0.12 67–158 0.15–0.46 0.07–0.14 na ↑ a 15
Würtz (1978) H2 O–Steam 10.0 3.0–9.0 500–3000 0.08–0.80 0.11–0.92 450–900 ↑ ad 180
20.0
Assad et al. (1998) H2 O–Air 9.50 0.37 128–757 0.22–0.85 0.04–0.80 440 ↑ a 31
Whalley et al. (1974) H2 O–Air Genklene–Air 31.8 0.12–0.35 78–789 0.10–0.90 0.15–0.97 590 ↑ a 158
Gill et al. (1964, 1969) H2 O–Air 31.8 0.10–0.34 20–245 0.14–0.88 0.03–0.70 167–520 ↑ a 48
Brown (1978) H2 O–Air 31.8 0.17–0.31 158–316 0.33–0.66 0.42–0.86 420 ↑ a 30
Cousins et al. (1965) and Cousins 9.53 0.14–0.41 106–475 0.15–0.81 0.03–0.65 230–480 ↑ a 123
H2 O–Air
and Hewitt (1968a,b) 31.8
H2 O–Air 6.00 0.23–0.86 111–1290 0.03–0.90 0.02–0.87 na ↑ a 102
Minh and Huyghe (1965)
Ethanol–Air 12.0
Hewitt and Pulling (1969) H2 O–Steam 9.30 0.24–0.45 295–299 0.14–0.75 0.01–0.68 390 ↑ a 72
Keeys et al. (1970) H2 O–Steam 12.6 3.5–6.9 1308–2765 0.25–0.68 0.65–0.86 290 ↑ a 18
Singh et al. (1969) H2 O–Steam 12.5 6.9–8.3 517–4242 0.11–0.93 0.13–0.82 180 ↑ a 39
Nigmatulin et al. (1977) H2 O–Steam 13.3 1.0–10.0 500–4000 0.10–0.90 0.07–0.98 300 ↑ a 45
Milashenko et al. (1989) H2 O–Steam 13.1 7.0 1000–3000 0.06–0.42 0.35–0.93 na ↑ d 28

(1) – Dimensionless distance L/d of test section inlet from mixer (2 component flows) or preheater (saturated flows) (for adiabatic tests only).
(2) – Flow direction: ↑, vertical upflow; ↓, vertical downflow.
(3) – Type of test: a, adiabatic; d, diabatic.

Cioncolini and Thome (2010) and now contains 2293 data points, expanded and now includes 2293 measurements of the entrained
taken in vertical circular tubes under both adiabatic and evaporat- liquid fraction collected from 31 different literature studies.
ing flow conditions, including now measurements carried out in The collected data cover 8 different gas–liquid or vapor–liquid
evaporative channels of relevance for boiling water nuclear reactor combinations (both single-component saturated fluids such as
applications. Additionally, 71 data points for non-circular chan- water–steam and refrigerants R12 and R113 and two-component
nels (annulus and rod bundle) and 96 data points for horizontal fluids, such as water–air, genklene–air, ethanol–air, water–helium,
and inclined circular tubes are also included in the databank. The silicon–air) and 24 different values of the tube diameter in the
new, improved prediction method proposed here is still based on range of 5.0 mm to 57.1 mm. The experimental database now
the core flow Weber number, a single and physically plausible includes additional measurements carried out in evaporating flow
dimensionless group, and reproduces the available data better than conditions with both water–steam (Würtz, 1978; Milashenko
existing empirical correlations. Besides, being explicit and fully et al., 1989; Adamsson and Anglart, 2006) and refrigerants R12
stand-alone, this new method is much simpler to use than the orig- (Mayinger and Langner, 1976) and R113 (Ueda and Kim, 1982).
inal one proposed by the authors and can be easily implemented In particular, Adamsson and Anglart (2006) performed their
into existing thermal–hydraulic system codes and simulation tools. measurements using different axial power distributions (uniform,
Such an explicit method is particularly useful for simplifying cal- inlet-peaked, middle-peaked and outlet-peaked) at conditions
culations during the optimization of thermal systems and eventual typical of boiling water nuclear reactors (pressure of 7.0 MPa, mass
flow stability analysis. This new method is part of a unified annu- flux of 750–1750 kg m−2 s−1 and tube diameter of 14.0 mm). As
lar flow modeling suite that is currently being developed by the can be seen in Table 1, most of the data refer to vertical upflow, but
authors that also includes methods to predict the axial frictional some data taken in vertical downflow are also included. Adiabatic,
and total pressure gradient, the convective boiling heat transfer two-component annular flows can be quite slow in approaching
coefficient, the annular liquid film thickness and the liquid film and fully developed flow conditions and losing any memory effect of the
gas core velocity profiles (Cioncolini et al., 2009a,b; Cioncolini and mixing device (Wolf et al., 2001). As can be seen in Table 1, however,
Thome, 2010, 2011). most of the adiabatic test rigs have been designed with calming
sections long enough to significantly damp out any dependence on
inlet conditions. As such, inlet effects in adiabatic two-component
2. Experimental database description flows are not taken into account in the present study.
As can be seen in Table 1, 73.0% of the collected data are for adi-
The main details regarding the extended experimental annular abatic upflow, while 13.8% are for adiabatic downflow and 13.2%
flow databank for vertical flow in circular tubes are summarized in cover evaporating upflow conditions, so that the databank is biased
Table 1, while a selection of histograms that further describes the towards adiabatic upflow conditions. Besides, as can be seen in
collected data is shown in Fig. 2. The database has been significantly Fig. 2, most of the data were taken at operating pressures close to
 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213 203

600 2000

Number of Data Points

Number of Data Points

500
1500
400

300 1000

200
500
100

0 0
0 10 20 30 40 50 60 0 2 4 6 8 10 12 14 16 18 20
Tube Diameter [mm] Pressure [MPa]

700 200
Number of Data Points

Number of Data Points

600
500 150

400
100
300
200
50
100
0 0
0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2
Mass Flux [kg/m s] Vapor Quality

500 400
Number of Data Points
Number of Data Points

400
300

300
200
200

100
100

0 0
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Void Fraction (W&G) Void Fraction (DIX)

Upflow Data from Table 1

150 250
Number of Data Points
Number of Data Points

200

100
150

100
50
50

0 0
0 1 2 3 4 5 6 7 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Eq.(4) Left-Hand Mach Number

Fig. 2. Selected histograms describing the experimental database in Table 1.

atmospheric and at low mass fluxes. As such, additional entrained Fig. 2 are predicted according to Woldesemayat and Ghajar (2007)
liquid fraction measurements appear more than welcome, partic- and to the Dix model (Chexal et al., 1986), labeled W&G and DIX in
ularly at high operating pressures, high mass fluxes and under Fig. 2, respectively.
evaporating flow conditions. Typically, the predominant mode of liquid entrainment in ver-
As noted by Levy (1999), the transition from intermittent to tical annular flow is the shearing off of the crests of the disturbance
annular flow typically corresponds to a cross sectional void frac- waves that slide on top of the liquid film. If the hydrodynamic con-
tion between 0.7 and 0.8. As can be seen in Fig. 2, the vast majority ditions are appropriate, however, other entrainment mechanisms
of the data collected in Table 1 correspond to a local void fraction may come into play (Hewitt and Hall-Taylor, 1970), such as wave
above 0.8. As such, the contamination of the data from intermittent undercutting. In this latter case, a disturbance wave is deformed
flow can be expected to be minimal, and is therefore neglected in into a liquid ligament by the pulling action of gravity and then the
the present study. In particular, the void fraction values included in ligament protrudes into the gas core where it is finally atomized.
204 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213

Table 2
Additional annular flow data for non-circular channels and non-vertical flows.

Reference Fluids d (mm) P (MPa) G (kg m−2 s−1 ) x e (1) (2) (3) No. points

Würtz (1978) H2 O–Steam 9.0a

3.0–9.0 400–2000 0.15–0.61 0.06–0.88 889 ↑ ad 52
Feldhaus et al. (2002) H2 O–Air 11.8a 0.15 300–450 0.25 0.11–0.42 171 ↑ a 19
Williams et al. (1996) H2 O–Air 95.3 0.11 123–215 0.44–0.80 0.64–0.82 273 → a 7
Dallman et al. (1984) H2 O–Air 25.4 0.11–0.23 64–665 0.12–0.91 0.06–0.95 500 → a 44
Paras and Karabelas (1991) H2 O–Air 50.8 0.11–0.20 65–308 0.21–0.87 0.10–0.75 300 → a 17
Ousaka and Kariyasaki (1992) H2 O–Air 26.0 0.11 80–254 0.09–0.47 0.05–0.47 142 → a 12
Geraci et al. (2007) H2 O–Air 38.0 0.14 32–47 0.69–0.84 0.07–0.09 132 → a 16

(1) – Dimensionless distance L/d of test section inlet from mixer (2 component flows) or preheater (saturated flows) (for adiabatic tests only).
(2) – Flow direction: ↑, vertical upflow; →, horizontal.
(3) – Type of test: a, adiabatic; d, diabatic.
a −1
Hydraulic diameter (4 Aflow Pwet ).

As such, wave undercutting can be expected to play a role in the et al. (2002) used water–air at low pressure in a vertical channel
entrainment process whenever gravity affects the flow. A prelimi- designed to mimic the four subchannels among ten fuel rods in a
nary and qualitative check of the influence that gravity may exert light water nuclear reactor bundle, with and without spacer grids.
on the flow can be obtained by extrapolating a criterion proposed The other data collected in Table 2 (Williams et al., 1996; Dallman
by Wallis (1961) for predicting flow reversal, the condition at which et al., 1984; Paras and Karabelas, 1991; Ousaka and Kariyasaki,
in an initially cocurrent annular upflow some of the liquid in the 1992; Geraci et al., 2007) cover low pressure air–water flows in
film starts flowing downward under the pull of gravity. This flow horizontal circular tubes, covering diameter values from 25.4 mm
reversal condition reads as follows: to 95.3 mm. Geraci et al. (2007), in particular, tested channel incli-
∗ ∗ nations of 0◦ , 20◦ , 45◦ , 70◦ and 85◦ with respect to the horizontal.
Jl + Jg < 1 ⇒ Flow reversal (4)
It is worth highlighting that the new prediction method that
where the non-dimensional average volumetric fluxes of liquid Jl∗ will be described in Section 3 is based on the vertical tubular data
and vapor Jg∗ are: of Table 1. The additional data collected in Table 2 will then be
  used to extend the applicability of the new prediction method to
l g non-circular and non-vertical channels.
Jl∗ = Jl ; J ∗ = Jg (5)
gd(l − g ) g gd(l − g )

where g is the acceleration of gravity and the superficial velocities 3. New prediction method
of liquid Jl and vapor Jg are:
In the new prediction method, the core flow Weber number
(1 − x)G xG is simplified with respect to the original formulation in Eq. (1) as
Jl = ; Jg = (6)
l g follows:
As shown in Fig. 2, the influence of gravity on the flow should be c Jg2 d
negligible for the vast majority of the data, so that shear-induced Wec = (8)

liquid atomization can be regarded as the dominant entrainment
mechanism. It is worth noting that the histogram in Fig. 2 showing where the core flow density c is calculated as indicated in Eq. (2),
the left-hand of Eq. (4) is limited to upflow data from Table 1. while the core flow velocity Vc and core equivalent diameter dc are
In order to check the relevance of compressibility effects in the approximated with the superficial gas velocity Jg and tube diame-
flow the following form of the Mach number M is used: ter d, respectively. Within the limits of the present study, both the
Vtpf Vg superficial gas velocity and the tube diameter provide reasonably
xG
M= ≈ = (7) good approximations to the core flow velocity and core equiva-
atpf ag g εag
lent diameter, so that the simplified form of the core flow Weber
where Vtpf and atpf are the average velocity and the sonic velocity for number in Eq. (8) is on average within about 15% of the original
the two-phase flow. As can be seen in Eq. (7), the average two-phase formulation in Eq. (1) and can therefore be expected to still cap-
velocity is approximated with the gas velocity Vg calculated in a ture the essence of the liquid film atomization process. The great
segregated-phase approximation, which provides a slight upper- advantage of this new core flow Weber number, with respect to the
bound to the average core velocity for a cocurrent annular flow. original formulation, is that the dependence on the void fraction ε
The sonic two-phase velocity, on the other hand, is approximated is removed, so that a completely stand-alone prediction method for
in Eq. (7) with the gas sonic velocity ag , which provides a lower- the entrained liquid fraction can now be designed.
bound to the sonic velocity of a high void fraction two-phase flow The measured entrained liquid fraction values from Table 1 are
(Tong and Weisman, 1996). As such, Mach number values predicted displayed in Fig. 3 versus the core flow Weber number as defined
with Eq. (7) can be expected to be slightly conservative, i.e. approx- in Eq. (8). Notwithstanding a significant scatter, the majority of the
imate the ‘true’ Mach number from above. As can be seen in Fig. 2, data points can be seen to cluster reasonably well on a sigmoid
almost all data points in Table 1 have a predicted local Mach num- trend. It is worth emphasizing that measuring the entrained liq-
ber value below 0.3. Assuming that this same threshold normally uid fraction is very challenging, and all the measuring techniques
used in single-phase incompressible flow applications is applica- proposed and used so far are quite invasive and may significantly
ble for annular flows as well, compressibility effects can then be perturb the annular flow that is being investigated. As such, some
neglected. scatter in the measurements is to a good extent unavoidable, par-
The main details regarding the additional annular flow data for ticularly when merging data from different studies. Moreover, it
non-circular cross sections and channel inclination other that verti- can be noticed in Fig. 3 that the scatter tends to be higher for
cal are summarized in Table 2. In particular, Würtz (1978) measured lower entrained liquid fraction values. This can be justified con-
the entrained liquid fraction in a vertical annulus with water–steam sidering that the two most frequently used measuring techniques
in both adiabatic and evaporating flow conditions, while Feldhaus for the entrained liquid fraction, namely the liquid film suction
 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213 205

0.8

Entrained Liquid Fraction

0.6

0.4

0.2

0 1 2 3 4 5
10 10 10 10 10
Core Flow Weber Number

Fig. 3. Entrained liquid fraction data of Table 1 vs. core flow Weber number as defined in Eq. (8).

and the core flow sampling, both become progressively less accu- data. Here, in particular, a robust fitting technique and a simple
rate in the limit e → 0+ . From a fluid dynamics point of view, this model-based outlier detection algorithm (Pearson, 2005) are used
limit corresponds to the so called ideal annular flows, character- to provide a correlation between e and Wec . First, a preliminary
ized by an almost perfect segregation between the phases. Since fitting equation is derived using the entire databank (i.e. outliers
these conditions are rarely encountered in practice, the high scat- included). Since the bulk of the data in Fig. 3 is clustered on a
ter in the data in the limit e → 0+ can be considered not critical sigmoid trend, a generalized logistic function (Jukić and Scitovski,
for practical applications. Experimental observations generated in 1996) is selected for use here among the several ‘S’ shaped fitting
fields as diverse as biology, demography, economics, chemistry and functions available, which for a target variable bounded between 0
medicine frequently collapse on sigmoid trends. These trends are and 1 as e in the present case reads as:
typical of dynamic systems characterized by a positive feedback in −1/g
their early evolution, which gives rise to an exponential growth, e = (1 + be−cgx ) ; b, c, g > 0 (9)
followed by a negative feedback that dampens the initially expo- where c is the growth rate, b is related to the abscissa of maximum
nential growth and brings the system to saturation. In the present growth and g is an asymmetry coefficient. Noting that the x-axis
context, the initially exponential growth is limited to low values in Fig. 3 is reported in logarithmic scale, the use of Eq. (9) in the
of the core flow Weber number Wec and entrained liquid fraction present context yields the following correlating equation for the
e, corresponding to annular flows where most of the liquid flows entrained liquid fraction:
in the film. An increase in the entrained liquid fraction e yields an
−1/g cg
increase of the core flow density and kinetic energy that enhance e = (1 + b Wec−d ) ; d= (10)
ln(10)
the liquid film atomization, thus triggering a further increase in e
that gives rise to a positive feedback. The final damping and sat- where the parameters b, c and g have to be determined from the
uration is reached for high values of the core flow Weber number experimental data solving a nonlinear regression problem. Since
Wec and entrained liquid fraction e, corresponding to annular flows the regression is done using the entire databank, it is wise to use
where most of the liquid is entrained and the liquid film is corre- outlier-resistant or so called robust procedures for estimating the
spondingly very thin. As e further increases, less and less liquid regression model coefficients b, c and g, in order for the prelimi-
remains available for further atomization, giving rise to a negative nary fit to properly capture the trend of the bulk of the data and not
feedback. As can be noticed in Fig. 3, although the majority of the being excessively distorted by the outliers. In particular, the robust
data points cluster, some anomalous records that appear to deviate method of least-absolute-residuals is used here. This method min-
from the behavior seen in the bulk of the data are as well recog- imizes the sum of the absolute values of the residuals, instead of
nizable. These outliers, which are almost always present in large minimizing the sum of the squares of the residuals as happens with
datasets, can be normally traced back to measuring errors, sen- least-squares methods. This guarantees that extreme points in the
sor malfunctions, measurements carried out outside the range of databank have a milder influence on the fit that, therefore, better
optimal operation of the experimental stand or errors occurred in captures the trend of the bulk of the data. Once the preliminary
the recording, post-processing or reporting of the data. In particu- fitting equation is available, every data point in the databank is
lar, these outlying observations can be seen in Fig. 3 to be mostly tested to determine whether it is an outlier according to the fol-
concentrated above the trend of the bulk of the data. This asym- lowing outlier detection rule, called standard symmetric boxplot
metry in the outliers distribution is believed to be the consequence (Pearson, 2005):
of the slight contamination of the databank already discussed in  
if R − median (R) > 2 · IQD ⇒ Current record is outlier (11)
Section 2. In particular, the present dataset contains a residual tail
of points taken close to the intermittent to annular flow transi- where the residual R is the difference between the measured
tion, where more liquid is typically present in the core flow in the entrained liquid fraction value and the prediction of the prelimi-
form of ligaments and bridges, and a residual tail of points taken nary fit generated in the first step of the analysis, the median of
close to the flow reversal limit, where gravity assists the liquid film the residuals provides an outlier-resistant estimate of the nomi-
atomization. nal residual value, while the interquartile distance IQD provides an
It is well known that even a few of these anomalous records in outlier-resistant estimate of the scale of natural variation of the
a large dataset can have a disproportionate influence on the ana- residuals about the nominal value. According to this decision rule,
lytical results derived from analyzing the data, so that in general a data point is identified as an outlier if its residual deviates from
outliers have to be handled properly during the modeling of the the median of the residuals more than two times the interquartile
206 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213

1
All Data
Clean Data
0.8

Entrained Liquid Fraction

Eq. (12)

0.6

0.4

0.2

0 1 2 3 4 5
10 10 10 10 10
Core Flow Weber Number

Fig. 4. Entrained liquid fraction data of Table 1 vs. core flow Weber number as defined in Eq. (8).

distance of the residuals. Once outliers have been identified, they without success, however. Probably, the significant scatter in the
are removed from the databank and the preliminary fitting equa- available data is large enough to hide any fine details, so that only
tion derived in the first step of the analysis is refined using the clean the predominant influence of the core flow Weber number clearly
data only. The correlating equation that is finally obtained is: emerges, while any second order effect is lost. As such, future
−2.209 research should concentrate on developing new and less invasive
e = (1 + 279.6Wec−0.8395 ) ; 101 < Wec < 105 (12) measuring techniques for the entrained liquid fraction.
where the core flow Weber number is as defined in Eq. (8). The Since the core flow density c depends on the entrained liquid
correlating Eq. (12) is displayed in Fig. 4 together with the entire fraction e, as can be seen in Eq. (2), in principle the proposed cor-
databank in Table 1 (black circles) and the cleaned databank that is relating equation, Eq. (12), should be used iteratively. In order to
obtained from Table 1 after the outliers have been removed (red simplify its use for engineering applications, however, the imple-
dots). (For interpretation of the references to color in this text, mentation of Eq. (12) that is proposed here follows an explicit
the reader is referred to the web version of the article.) As can be predictor–corrector scheme. Predictor–corrector methods, in gen-
seen, Eq. (12) provides a reasonably good fit of the data and nicely eral, are explicit algorithms that proceed in two steps. First, in the
follows the trend of the bulk of the measurements. It is worth high- predictor step a rough approximation of the desired quantity is cal-
lighting that outliers are removed from the databank of Table 1 culated. Then, the corrector step refines the initial approximation.
only to derive Eq. (12), but in all the comparisons that will be In the present context, in particular, in the predictor step the core
presented the entire databank of Table 1 is always used, outliers flow density c is approximated with the gas density g . An esti-
p
included. It is worth remembering that Eq. (12) is appropriate for mate of the core flow Weber number Wec is then evaluated and
shear-driven annular flows only, where the entrainment process is used with Eq. (12) to get a preliminary estimate of the entrained
essentially due to the shearing off of the disturbance wave crests liquid fraction ep :
by the droplet-laden gas core flow. As can be seen in Fig. 4, the
majority of the records that appear to deviate markedly from the g Jg2 d
p p−0.8395 −2.209
trend of the bulk of the data are actually identified as outliers by the c ≈ g ⇒ Wec = ; ep = (1 + 279.6Wec ) (13)

procedure described above. In particular, the procedure identifies
190 data points as outliers, corresponding to a dataset contami-
nation of 8.3%. Considering that contamination levels between 1% In the corrector step, the preliminary estimate of the entrained
and 10% are quite common (Pearson, 2005) and remembering that liquid fraction ep is used to calculate the core flow density c , as
the entrained liquid fraction is a particularly difficult parameter indicated in Eq. (14). The core flow Weber number is then updated
to measure, the 8.3% contamination level predicted here appears and used with Eq. (12) to get the final prediction for the entrained
reasonable. Further inspection of Fig. 4 shows that the outlier detec- liquid fraction:
tion procedure used here seems to be less effective at low core flow
Weber number values, since a tail of entrained liquid fraction values x + ep (1 − x) c Jg2 d
above the trend of the bulk of the data is not identified as anoma- c = ; Wec = ;
(x/g ) + ((ep (1 − x)/l ) 
lous. As already noticed, however, the region e → 0+ is not critical
−2.209
for the applications. Besides its efficiency (Pearson, 2005), the out- e = (1 + 279.6 Wec−0.8395 ) (14)
lier detection procedure used here was also selected because it can
be easily coded into most computing software tools using built-in
features only. It is worth highlighting, however, that many other
outlier detection algorithms are available in the robust statistics The above predictor–corrector scheme is simple, explicit and
literature and many more are likely to become available in the can be easily coded into existing simulation tools. In what fol-
near future, since automatic outlier detection is currently being lows, the predictions from Eq. (12) are always obtained using the
extensively investigated for numerous data-mining applications, above predictor–corrector scheme. With respect to the old predic-
such as credit card fraud detection, clinical trials, voting irregular- tion method proposed in Cioncolini and Thome (2010), therefore,
ity analysis, network intrusion detection and athlete performance the updated method proposed here is much simpler to use, being
analysis. explicit and fully stand-alone, and is based on a much wider
The inclusion into the proposed correlation, Eq. (12), of fur- experimental databank that besides adiabatic flows now covers
ther dimensionless groups to improve its accuracy was attempted evaporating flow conditions and non-circular geometries as well.
 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213 207

0
10

Entrained Liquid Fraction:

Experimental
-1
10

+ 50 %

-2
10
- 50 %

-3
10 -2 -1 0
10 10 10
Entrained Liquid Fraction: Predicted

0
10
Entrained Liquid Fraction:
Experimental

-1
10

+ 50 %

-2
10
- 50 %

-3
10 -2 -1 0
10 10 10
Entrained Liquid Fraction: Predicted

Fig. 5. Entrained liquid fraction: experimental data from Table 1 vs. predictions of Eq. (12) [top] and predictions of Ishii and Mishima (1989) correlation [bottom].

4. Results and discussion (1996b), Utsuno and Kaminaga (1998), Pan and Hanratty (2002),
Sawant et al. (2008, 2009) and Cioncolini and Thome (2010). Fur-
The comparison of the measured data from Table 1 with the ther details regarding these correlations and their implementation
predictions of Eq. (12) is presented in Fig. 5 (top). The statistical can be found in the Electronic Annex, available in the online ver-
comparison between measured data from Table 1 and predic- sion of this article. As can be seen in Table 3, the new prediction
tions is reported in Table 3, which includes both the predictor and method proposed here fits the available data better than the other
the corrector steps and also the results of the correlations pro- methods considered, with a mean absolute percentage error of
posed by Paleev and Filippovich (1966), Wallis (1968), Oliemans 34.1% and about 6 points out of 10 captured to within ±30% (in
et al. (1986), Ishii and Mishima (1989), Nakazatomi and Sekoguchi the present study, the empirical correlations are ranked on the

Table 3
Statistical comparison between experimental data of Table 1 and correlations.

(1) (2) (3) (4) (5)

Paleev and Filippovich (1966) 185.6 −59.5 10.5 20.4 33.5

Wallis (1968) 85.5 −61.6 18.4 36.9 55.2
Oliemans et al. (1986) 68.4 −41.2 36.5 52.1 67.0
Ishii and Mishima (1989) 39.9 4.7 27.7 53.0 72.7
Nakazatomi and Sekoguchi (1996) 155.3 −123.8 11.4 25.2 42.5
Utsuno and Kaminaga (1998) 242.6 237.0 15.3 18.5 23.0
Pan and Hanratty (2002) 48.3 40.3 20.5 42.9 66.5
Sawant et al. (2008) 54.4 0.4 29.7 50.3 69.7
Sawant et al. (2009) 43.2 33.5 24.4 43.2 63.4
Cioncolini and Thome (2010) 34.9 0.7 39.2 61.4 79.0
Present study Eq. (12): predictor step 37.3 18.1 27.8 55.2 75.0
Present study Eq. (12): corrector step 34.1 1.9 38.7 61.0 78.9

n

100 |eexp −ecal |

(1) – Mean absolute percentage error (%) n eexp
.
1

n

100 e exp −ecal

(2) – Mean percentage error (%) n eexp
.
1
(3) – Percentage of experimental data captured within ±15%.
(4) – Percentage of experimental data captured within ±30%.
(5) – Percentage of experimental data captured within ±50%.
208 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213

1
Gill et al. (1964,1969) + 30 % Eq. (12)
Cousins et al. (1965,1968a,b)

Entrained Liquid Fraction

0.8 Andreussi (1983)
Steen&Wallis (1964)
Assad et al. (1998) - 30 %
0.6

0.4

0.2

0 1 2 3 4 5
10 10 10 10 10
Core Flow Weber Number

1
+ 30 %
Hewitt&Pulling (1969)
Keeys at al. (1970)
Entrained Liquid Fraction

0.8 Singh at al. (1969) Eq. (12)

Nigmatulin at al. (1977)
Wurtz (1978) - 30 %
0.6

0.4

0.2

0 1 2 3 4 5
10 10 10 10 10
Core Flow Weber Number

1
Adamsson&Anglart (2006) + 30 % Eq. (12)
Wurtz (1978)
Entrained Liquid Fraction

0.8 Milashenko et al. (1989)

Mayinger&Langner (1976)
Ueda&Kim (1982) - 30 %
0.6

0.4

0.2

01 2 3 4 5
10 10 10 10 10
Core Flow Weber Number

Fig. 6. Entrained liquid fraction data of Table 1: selected water–air low pressure data [top], adiabatic water–steam data [middle] and data taken under evaporating flow
conditions [bottom].

basis of their mean absolute percentage error; if two correla- the proposed method has been designed with the present data-
tions yield comparable values of the mean absolute percentage bank while the other correlations are based on different data sets.
error, i.e. difference within ±1%, then the ranking is done accord- Nonetheless, the good correlating capability of Eq. (12) suggests
ing to the percentage of data captured within ±15%, ±30% and that considering annular flows as a form of a liquid atomization
±50%). Even the preliminary estimate provided by the predictor process is a promising assumption in their modeling, at least as
step is quite accurate and slightly outperforms existing predic- long as the entrainment process is mostly due to the shearing
tion methods, although the corrector step definitely improves the off of the disturbance wave crests by the droplet-laden gas core
prediction accuracy. The best predictions by an existing method flow.
are given by the correlation of Ishii and Mishima (1989), often Selected low pressure air–water data are presented in Fig. 6
employed in thermal–hydraulic simulation codes. The compari- (top), together with the proposed correlation Eq. (12). The avail-
son of the measured data with the predictions of the correlation able water–steam adiabatic data are presented in Fig. 6 (middle),
of Ishii and Mishima (1989) is presented in Fig. 5 (bottom). Anal- while Fig. 6 (bottom) shows the available data taken under evapo-
ogous figures for the other correlations considered are included in rating flow conditions with water–steam and refrigerants R12 and
the Electronic Annex. Direct comparison of the prediction method R113. As can be seen, the agreement of the data with Eq. (12) is for
proposed here with other correlations is somewhat unfair, as the most part quite satisfactory.
 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213 209

1
Adiabatic + 30 % Eq. (12)
Evaporating
0.8

Entrained Liquid Fraction

- 30 %
0.6

0.4

0.2

0 1 2 3 4 5
10 10 10 10 10
Core Flow Weber Number

0.6
No Grid Spacers + 30 % Eq. (12)
EggCrate-Upstream
0.5
EggCrate-Downstream
Entrained Liquid Fraction

- 30 %
ULTRAFLOW-Upstream
0.4 ULTRAFLOW-Downstream

0.3

0.2

0.1

0 2 3 4
10 10 10
Core Flow Weber Number

Fig. 7. Entrained liquid fraction data of Table 2: annulus data of Würtz (1978) [top] and rod bundle data of Feldhaus et al. (2002) [bottom].

The proposed correlation Eq. (12) is based on the databank of is extrapolated to the non-circular geometries in Fig. 7 using the
−1
Table 1, which is limited to vertical flow through circular tubes hydraulic diameter (4 Aflow Pwet ) in place of the tube diameter. As
and is biased towards adiabatic, fully developed, low pressure and can be seen, the agreement is for the most part satisfactory. As can
low mass flux flow conditions, as already discussed. As such, the be noticed in Fig. 7 (bottom), the measurements carried out down-
application of Eq. (12) to nuclear reactor operating conditions, char- stream of the ULTRAFLOW grid spacer are consistently below the
acterized by non-circular channel geometry, annular flows entered other data, as expected since this grid spacer is provided with flow
from intermittent flow without any flow developing length and vanes designed to force the entrained liquid droplets to deposit
evaporating flow conditions with non-uniform heat flux, is not onto the liquid film. The egg-crate grid spacer tested, on the other
straightforward. In what follows, therefore, the application of Eq. hand, does not appear to affect the entrained liquid fraction, since
(12) to nuclear reactor operating conditions is addressed, using the measurements carried out upstream and downstream of this
both the tubular data in Table 1 and the additional data collected grid are consistent and compare favorably with the data measured
in Table 2. without grid spacer.
Measurements in Table 2 carried out by Würtz (1978) in a ver- The available tubular data taken in evaporating flow conditions
tical annulus (water–steam, pressure of 3.0–9.0 MPa, mass flux of in Table 1 are presented in Fig. 8 (top) as the ratio of the mea-
500–2000 kg m−2 s−1 , tube diameter of 26.0 mm, rod diameter of sured entrained liquid fraction to the prediction of the proposed
17.0 mm, hydraulic diameter of 9.0 mm) under both adiabatic and correlation Eq. (12), plotted versus the dimensionless distance L/d
evaporating flow conditions (electrical heating, uniform heat flux) between the measurement point and the location upstream in the
are presented in Fig. 7 (top), while adiabatic measurements car- channel where annular flow is entered from intermittent flow. As
ried out by Feldhaus et al. (2002) in a vertical channel designed to already noted, the transition from intermittent to annular flow
mimic a light water nuclear reactor fuel bundle (water–air, pressure typically corresponds to a local void fraction between 0.7 and 0.8
of 0.1–0.2 MPa, mass flux of 300–450 kg m−2 s−1 , hydraulic diam- (Levy, 1999). Here, in particular, this transition is assumed to occur
eter of 11.8 mm) with and without grid spacers are presented in at a local void fraction of 0.75, and the local void fraction is pre-
Fig. 7 (bottom). Although the local pressure at the measuring posi- dicted according to Woldesemayat and Ghajar (2007). It is well
tion is not specified, the information provided by Feldhaus et al. known that this transition, in reality, is not localized and static
(2002) allows bounding the operating pressure between 0.1 MPa but smeared along a certain tube length, and any void fraction
and 0.2 MPa. Accordingly, the results included in Fig. 7 (bottom) correlation is at most accurate to within a few percent. Nonethe-
are for an operating pressure of 0.15 MPa, and the horizontal error less, the above outlined simplified procedure provides a useful, first
bars account for a ±50 kPa variation in the operating pressure. The approximate estimate of the annular flow boiling length that is not
corresponding variation in the core flow Weber number is signifi- experimentally available. Notwithstanding a few outliers, the pro-
cant, as can be seen in Fig. 7 (bottom), as a result of the strong air posed correlation Eq. (12) seems to perform well in the region of
density dependence on pressure. The proposed correlation Eq. (12) L/d > 50, as can be seen in Fig. 8 (top). In the region of 0 < L/d < 50, on
210 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213

1
10
Adamsson&Anglart (2006)
Wurtz (1978)

Entrained Liquid Fraction:

Milashenko et al. (1989)

Measured vs. Predicted

Mayinger&Langner (1976)
Ueda&Kim (1982)

0
10

-1
10
0 50 100 150 200 250 300 350 400
Annular Flow Boiling Length L/d
1.25
Uniform
1.2 Inlet peaked
Entrained Liquid Fraction:

Middle peaked
Measured vs. Predicted

1.15
Outlet peaked
1.1

1.05

0.95

0.9

0.85

0.8
60 80 100 120 140 160 180 200
Annular Flow Boiling Length L/d

Fig. 8. Ratio of the measured entrained liquid fraction to the prediction of Eq. (12) plotted vs. annular flow boiling length L/d: all evaporating flow conditions data from
Table 1 [top] and Adamsson and Anglart data (2006) with different heat flux profiles [bottom].

the other hand, the data of Milashenko et al. (1989) are correctly length estimated as described above is L/d > 50. Application of Eq.
reproduced, while the data of Würtz (1978) are partly captured (12) for L/d < 50 may underpredict the entrained liquid fraction,
and partly underpredicted. Although the majority of the data in although not enough data in this region seem available to draw
this region is correctly reproduced, the application of the proposed any definite conclusion. The axial power profile is not found to
correlation Eq. (12) in this range should be done with care, as only appreciably affect the entrained liquid fraction measurements, so
few data from two independent studies are available. The data in that only the total power supplied seems relevant but not its actual
Fig. 8 (top) characterized by L/d = 0, that correspond to void frac- shape profile. A developing length of L/d ∼ 50 for evaporating annu-
tion estimates at the measurement point below 0.75, tend to be lar flows entered from intermittent flow is much smaller than the
underpredicted by the proposed correlation Eq. (12). This seems developing lengths typically found in adiabatic, two-component
plausible, as in intermittent flows the phases are less separated annular flow, that can reach as high as L/d ∼ 200–300 (Wolf et al.,
than in annular flows, and more liquid is present in the central 2001). Adiabatic, two-component annular flows are typically gen-
part of the channel in the form of ligaments and bridges. The mea- erated from a single-phase gas flow by injecting the liquid phase
surements of Adamsson and Anglart (2006), obtained at conditions either as a film at the channel wall or as a jet at the channel axis. In
typical of boiling water nuclear reactors (water–steam, pressure the former case a perfectly segregated, ideal annular flow with no
of 7.0 MPa, mass flux of 750–1750 kg m−2 s−1 , tube diameter of entrained droplets is generated (e ∼ 0+ ), while in the latter case the
14.0 mm, electrical heating) are displayed in Fig. 8 (bottom) as the liquid jet gets atomized and a perfectly mixed mist flow is gener-
ratio of the measured entrained liquid fraction to the prediction of ated (e ∼ 1− ). Both flow conditions are significantly different from
the proposed correlation Eq. (12), plotted versus the dimension- a real annular flow, and this explains why a very long developing
less annular flow boiling length L/d, predicted as discussed above. length is required to damp out inlet effects. In boiling channels,
Besides using a uniform axial power distribution, Adamsson and on the other hand, annular flow is entered from intermittent flow,
Anglart (2006) made also tests with inlet-peaked, middle-peaked which from a fluid-dynamics point of view is not that different
and outlet-peaked power profiles, in order to better approximate from an annular flow, having only less phase segregation, lower slip
the actual thermal–hydraulics of boiling water nuclear reactors. As and more liquid in the center of the channel. As such, the shorter
can be seen, the predicted annular flow boiling length L/d is in the developing length for evaporating annular flows appears physically
range of 70–190, the data are nicely reproduced by the proposed plausible.
correlation Eq. (12) and no effect of the non-uniform heat flux can Horizontal and inclined channel data for low pressure air–water
be noticed. flows from Table 2 are presented in Fig. 9, together with the pro-
In conclusion, and within the limits of the present study, the posed correlation Eq. (12). Although the scatter is significant and
proposed correlation Eq. (12) seems suitable for nuclear reactor the data quite limited, the trend seems correctly captured by Eq.
operating conditions provided that the hydraulic diameter is used (12). This is noteworthy, since these data refer to strongly asym-
in place of the tube diameter and that the annular flow boiling metric annular flows, where the liquid film flowing on the bottom
 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213 211

1
Williams et al. (1996) + 30 % Eq. (12)
Dallman et al. (1984)

Entrained Liquid Fraction

0.8 Paras&Karabelas (1991)
Ousaka&Kariyasaki (1992)
Geraci et al. (2007) - 30 %
0.6

0.4

0.2

0 1 2 3 4 5
10 10 10 10 10
Core Flow Weber Number

Fig. 9. Entrained liquid fraction: horizontal flow data from Table 2.

of the channel is much thicker than the liquid film flowing on the is more intense. According to Geraci et al. (2007), this compensa-
top. This is consistent with the findings of Geraci et al. (2007), who tion is responsible for the mild dependence of the entrained liquid
tested pipe inclinations of 0◦ , 20◦ , 45◦ , 70◦ and 85◦ from the hor- fraction on pipe inclination.
izontal and found that there is not a strong dependence of the Finally, selected simulations showing the sensitivity of the pre-
entrained liquid fraction on pipe inclination. The interpretation dictions of the proposed correlation Eq. (12) to the vapor quality,
provided by Geraci et al. (2007) is related to the effect of gravity on operating pressure, mass flux and tube diameter are included in
the morphology of the disturbance waves. In vertical annular flows, Fig. 10. As can be seen, the entrained liquid fraction is predicted to
disturbance waves are present over the entire tube periphery, so increase with an increase of vapor quality, an increase of the mass
that the entire tube perimeter is participating in the entrainment flux, a decrease of the operating pressure and an increase of the tube
process. In horizontal or inclined annular flows, however, distur- diameter. All these predicted trends appear physically consistent.
bance waves are present only at the bottom of the tube, since the An increase of the vapor quality or mass flux triggers an increase
liquid film streaming on the tube top is too thin for disturbance of the kinetic energy of the core flow, which provides the energy
waves to develop and only small ripples are present, not partic- that drives the liquid film atomization and thus yields an increase
ipating in the entrainment process. As such, only the lower part in the entrained liquid fraction. A reduction in the operating pres-
of the tube periphery is participating in the entrainment process sure is followed by a higher liquid to vapor density ratio that yields
in horizontal or inclined channels. Disturbance waves are however a higher slip between the phases and consequently a higher inter-
thicker than those appearing in vertical flows, so that the entrain- facial shear, which increases the liquid film atomization and the
ment process, although limited to a fraction of the tube perimeter, entrained liquid fraction. Finally, a tube diameter reduction yields

-2 -1 -2 -1
P=7.0 MPa; G=1000 kgm s ; d=10 mm G=1000 kgm s ; d=10 mm
1 1
Entrained Liquid Fraction

Entrained Liquid Fraction

0.8 0.8

0.6 0.6

0.4 0.4
P=0.1 MPa
P=1.0 MPa
0.2 0.2
P=5.0 MPa
P=10.0 MPa
0 0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Vapor Quality Vapor Quality

-2 -1
P=7.0 MPa; d=10 mm P=7.0 MPa; G=1000 kgm s
1 1
Entrained Liquid Fraction

Entrained Liquid Fraction

0.8 0.8

0.6 0.6

0.4 -2 -1
G=500 kgm s 0.4
G=1000 kgm s
-2 -1 d=5 mm
0.2 -2 -1 d=10 mm
G=1500 kgm s 0.2
d=15 mm
-2 -1
G=2000 kgm s d=20 mm
0 0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Vapor Quality Vapor Quality

Fig. 10. water–steam simulations for a pressure of 7.0 MPa, a mass flux of 1000 kg m−2 s−1 and tube diameter of 10 mm. Entrained liquid fraction predicted with Eq. (12) vs.
vapor quality [top-left] and sensitivity to pressure [top-right], mass flux [bottom-left] and tube diameter [bottom-right].
212 A. Cioncolini, J.R. Thome / Nuclear Engineering and Design 243 (2012) 200–213

an increase in the stability of the liquid film, as the surface tension Dallman, J.C., Laurinat, J.E., Hanratty, T.J., 1984. Entrainment for horizontal annular
force scales as d−1 , so that the smaller the tube diameter the better gas–liquid flow. Int. J. Multiphase Flow 10, 677–690.
Feldhaus, G., Azzopardi, B.J., Zeggel, W., 2002. Annular flow experiments in rod
surface tension resists the liquid film atomization. bundles with spacers. Nucl. Eng. Des. 213, 199–207.
Additional figures that show the ratios of the entrained liquid Geraci, G., Azzopardi, B.J., van Maanen, H.R.E., 2007. Inclination effects on circumfer-
fraction values predicted with the new method to the measured ential film flow distribution in annular gas/liquid flows. AIChE J. 53, 1144–1150.
Gill, L.E., Hewitt, G.F., Lacey, P.M.C., 1964. Sampling probe studies of the gas core in
data displayed versus the tube diameter, operating pressure, vapor annular two-phase flow-II: studies of the effect of phase flow rates on phase and
quality and mass flux are included in the Electronic Annex. velocity distribution. Chem. Eng. Sci. 19, 665–682.
Gill, L.E., Hewitt, G.F., Roberts, D.N., 1969. Study of the behavior of disturbance waves
in annular flow in a long vertical tube. Report AERE-R 6012, UKAEA, Harwell,
5. Conclusions Oxon.
Han, H., Gabriel, K.S., Wang, Z., 2007. A new method of entrainment fraction mea-
surement in annular gas–liquid flow in a small diameter vertical tube. Flow
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