7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Improved Modeling of TwoComponent Annular Flow
DuWayne Schubring*, Timothy A. Sheddt, and Evan T. Hurlburtt
Department of Nuclear and Radiological Engineering, University of Florida, Gainesville, FL 32611, USA
t Department of Mechanical Engineering, University of WisconsinMadison, Madison, WI, 54476, USA
SBettis Laboratory, West Mifflin, PA, 15122
dlschubring@ufl.edu, shedd@engr.wisc.edu, hurlburt@bettis.gov
Keywords: gasliquid flow, annular flow, correlations, modeling
Abstract
The model for base film thickness by Schubring and Shedd (2009) and the twozone interfacial shear model
by Hurlburt et al. (2006) have been adjusted and integrated to produce a more comprehensive model for airwater
annular flow at low pressure. Only flow rates, absolute pressure, temperature, and tube diameter are used as inputs.
Base film and wave zones are modeled separately. The shear is both zones is converted to pressure gradients and
weighted by wave intermittency, for which a correlation is used, to compute average pressure gradient, dP/ldz. The
film flow rate is estimated using the UVP in the waves and a piecewise linear profile in the base film. Wave velocity is
taken as that in the wave zone at the gasliquid interface. For vertical flow (22.423.4 mm ID), mean absolute errors
of 8% (Sbase), 9% (6wave), 9% waveve, and 19% (ddP/dz) are found. For horizontal flow (8.826.3 mm ID), MAE's
of 17% (Sbase), 14% (vwave) and 10% (dP/ldz) are obtained.
Introduction
The understanding of twophase gasliquid flow is of
ten tied to the flow regime (the general distribution of
the two phases within the flow area). One of the most
frequently seen flow regimes in industrial heat trans
fer equipment is annular flow. Devices such as steam
generators often experience annular flow as the domi
nant flow regime due to the wide range of flow quali
ties at which it is seen. Annular flow is characterized
by a thin film along the walls (periphery of the tube)
with a fastmoving gas core in the center of the flow.
The thin film has been observed to include disturbance
waves (Azzopardi (1986); Jayanti et al. (1990); Sawant
et al. (2008)), structures in which a segment of the film
is thicker and travels at a higher velocity, as well as base
film, which is relatively smooth and slowly moving (He
witt et al. (1990)).
Most frequently, annular flow is understood through
the triangular relationship, advanced by Hewitt and
Hall Taylor (1970) among others. This asserts a close re
lationship among liquid film flow rate, liquid film thick
ness, and gasliquid interfacial shear. The liquid film
flow rate is computed by integrating a velocity profile
across the liquid film thickness, with the velocity profile
nondimenionalized using shear. The shear at the inter
face or the wall can be used for this purpose; they are
linked through a simple force balance. Most frequently,
the Universal Velocity Profile (below, as reported by
Whalley (1987)) is selected:
y+ if y < 5
S 3.0 + 5.01n(y+) if 5 < y+ <30 (1)
5.5 + 2.51n(y+) if 30 < y+
Interfacial shear and film thickness are most often
linked through the film roughness concept. A wide va
riety of correlations have been developed for the in
terfacial fanning friction factor, Cf,i, as a function of
gas Reynolds number, Re,, average film thickness 6,
and other parameters. One early example is the Wal
lis (1969) correlation, which asserted an offset linear re
lationship between interfacial friction and the average
film thickness. Other correlations have been more com
plex, such as those of Fore et al. (2000) and Hurlburt and
Newell (2000), although a linear relationship between
average film thickness and effective roughness is usu
ally maintained. An exception is the work of Kishore
and Jayanti (2004), which asserted the proportionality of
roughness with film thickness to the 1.25 power (along
with explicit inclusion of both gas and liquid flow rates).
In most models, a measurement of the liquid film
thickness is required. A review of the experimental
techniques for this is supplied by Clark (2002). Tradi
tionally, this has been accomplished using conductance
probes, which use the difference in electrical conductiv
ity between the phases and an assumed qualitative ge
ometry (liquid film around gas core) to estimate film
thickness. The details of the construction, implemen
tation, and calibration of these devices varies among re
searchers (see, for example, Brown et al. (1978), Fossa
(1998), Fore et al. (2000), and additional references from
the review of Clark (2002)). More recently, the present
authors have improved a planar laserinduced fluores
cence (PLIF) system for film thickness measurements
Schubring et al. (2010a,b) based on the work of Ro
driguez (2004).
Some models, such as that of Owen and Hewitt
(1987), have required an estimate of entrained fraction
as an input. By liquid mass conservation, an estimate
of entrained fraction is tantamount to one of total liquid
film flow rate if total liquid flow is known. The model
of Hurlburt and Newell (2000), although focused on the
circumferential distribution of film thickness in horizon
tal flow, also required an estimate of entrained fraction to
compute local interfacial shear. Such a measurement is
arguably even more complex than a film thickness mea
surement; work on entrainment in annular flow is dis
cussed in the review of Azzopardi (1997).
This traditional modeling paradigm is problematic
even for analysis of existing twophase systems, as it
frequently replaces a simple measurement with a more
complex one (e.g., pressure gradient with film thickness,
film thickness with entrained fraction). For design pur
poses, a model that requires only flow rates and ther
modynamic states) is preferable, as device performance
could be optimized without expensive prototyping. It is
particularly important to know the film thickness distri
bution (due to its relationship with heat transfer) and the
pressure gradient (relating to required pumping power).
The present model has been developed to predict film
thickness and pressure gradient for twocomponent (air
water) annular flow data. In addition, estimates of wave
velocity and entrained fraction are obtained, which can
be compared to experimental data and expected physics
to gain additional insight into the flow.
Using only flow rates and thermodynamic states as
inputs, the model predicts base film thickness, wave
height, base film shear, wave shear, disturbance wave ve
locity, entrained fraction, and pressure gradient. These
are estimated through the following steps:
Film thickness (base and wave) is estimated us
ing a revised version of the critical fiction factor
model by some of the present authors (Schubring
and Shedd 1,21 1 ,
The shear of the base film is estimated using this
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
modeled film thickness and the friction factor ad
vocated by Hurlburt et al. (2006). A similar friction
factor is used to model wave roughness.
Velocity profiles within the film for both waves and
base film are used to model the liquid film flow rate;
entrained fraction is estimated from this through a
mass balance.
A second shear term in the wave zone, related to
sharp transitions between base film and disturbance
waves, is computed using knowledge of the core
flow rate, including entrained droplets. This shear
is solved iteratively along with the other wave zone
term.
The wave velocity is estimated through the wave
zone velocity profile.
Within both wave and base film zones, the shear es
timates are expressed as a pressure gradient. Wave
intermittency (INT,, the fraction of time waves
are present at a given axial location), is used to es
timate average pressure gradient.
Description of Model Predicting Film
Thickness
The prediction of the film thicknesses centers around
two correlations for (1 .inirii' i friction factor. The first is
the Blasius relation, increased by a constant factor ORR.
The second is the friction factor suggested by Hurlburt
et al. (2006), with CB,base set to 0.8. The overall results
of the model are not strongly sensitive to the details of
these friction factors use of a McAdams smooth tube
relation and the Haaland or Colebrook rough tube corre
lations produces results that are not significantly differ
ent than those piLcnill reported.
Cf,i,ba
[0base
se
 0.07.., Re 0 25
S base
f,i,base
0.58 2
,CB, a + 1.05+ Q 11
The Reynolds number for the gas core in the base film
zone (Rec,base) and the relative roughness bases) are
evaluated considering geometry, flow rates, and base
film height.
The roughness is evaluated using:
^base 2 (1 LFbase) base
26base
base
D base
Here, D is the inside diameter of the pipe, bbase is the
average base film thickness, and LFbase is the fraction of
the base film modeled as experiencing a linear velocity
profile. Based on the interfacial shear predicted by this
model or from direct wall shear measurements in similar
flows (e.g. Govan et al. (1989); Vlachos et al. (1997)), it
appears unlikely that turbulence is sustained in the base
film. As a result, a linear (viscous) velocity profile is
used.
Based on experimental data (Schubring et al.
(2010a,b)), the standard deviation of base film measure
ments is modeled as 30% of the average. As a result,
LFbase is set to 0.7, so that Ebase (roughness) is set equal
to 60% of 8bas.
The Reynolds number of the gas core over the base
film is computed using:
Rec,base
Dc,base
Ac,base
Pg Uc,baseDc,base
Py
D 26base
7D2
c,base
4
Uc,base gU,base l,i,base
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
erties, v, and pi, are the kinematic viscosity and density
of the liquid, respectively. A superscript + refers to di
mensionless wallcoordinates: s, U1 b
base' 1,i,base
Other authors (e.g., Owen and Hewitt (1987)) have
suggested that wall shear or some combination of wall
(Tm) and interfacial shear characteristic of the film
should be used instead of Ti. For the flow of interest,
the difference between Ti,base and Tw,base is small. Con
version between Ti,base and Tw,base requires an estimate
of pressure gradient; this estimate is not achieved until
the second stage of the model. As a result, using Tw,base
to nondimensionalize film velocities would produce a
fullycoupled, singlestage model, greatly increasing the
difficulty of evaluating the equations and not represent
ing any Nigl'iliik.im improvement in the accuracy or un
derlying physics of the model.
It is useful to compute the mass flow rate of the base
film zone, r film,base, by integrating the velocity profile.
pi is the dynamic viscosity of the liquid.
film,base
Ug,base sg Ac,
Acbase
[ LF a
L + LFbse (1
S(bse )
(11) nfilm,base = film,baseDpi
LFbase)]
(17)
(18)
In the computation of the Reynolds number, the ge
ometry of the core is also computed. A refers to flow
area of the tube and Ac,base to the flow area of the gas
core over the base film. Dc,base is the diameter of the
gas core. Ug,base is the average physical velocity of the
gas over the base film. Us, is the superficial velocity
of the gas. Gas properties are required to evaluate these
equations: p, is the gas density and t, is the dynamic
gas viscosity.
The core Reynolds number is computed using Uc,base,
which includes the effect of the moving base film
through the velocity of the liquid at the gasliquid inter
face in the base film, Ul,i,base. This velocity is estimated
through a computation of shear:
Based on observations discussed in Schubring et al.
(2010a,b), the average wave height, Swave, is modeled
as twice the base film height:
8wave
26base
The base film model is closed by using experimental
data to correlate ORR:
1;, +m1I
SP9Ucbase
f,i,base
2
T=,ba se
P1
ba se (14)
U+ 6+ LFb e (15)
1,i,base base ase (15)
Ulb,i,base Ui,base ase (16)
Ti,base is the interfacial shear for the base film; .. is
the liquid friction velocity for this zone. The liquid prop
The flow quality, x, is defined using the gas and liquid
mass flow rates, ; and rz.
Given only external geometry (tube diameter), fluid
properties thermodynamicc state), and flow rates, the
yields predictions for base film and wave heights, as well
as an interfacial shear in the base film zone.
Ti,base
Base
Description of Model Predicting Wave
Behavior, Entrainment, and Pressure Gradient
The geometry and velocity of the core are handled anal
ogously to that for the base film:
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
zone roughness is set to twice this value:
wave 0.4 .ave
t6wave
D wave
Dcwave
Ac,wave
Uc,wave
D 26Oave
SD c,wave
4
U ,wav Ui,i,wave
A
UY wave U9 A (25)
A6ave wCwave
For the wave zone, the UVP (Equation 1) is assumed.
For the flows of interest, the wave zone gasliquid in
terface is %.i'nsisicil\ within the log layer (y+ > 30) of
the film, based on a nondimensionalization using wave
zone shear.
The following calculations for the interfacial velocity
of the waves, U, iwave, and wave zone liquid film flow
rate, ;, are performed:
U*A
wave
6+
wave
U+
1,i,wave
Ul,i,wave
film ,wave
wavee
P1
5.5 + 2.51n (6Wav,)
SU+
l,i,wave wave
64 + 36+,av
+ 2.56,ave In (6wav,)
m film,waveDt1
The shear from wave roughness is then computed with:
Ti,wave,rough
u2
PY9 Ucwave
fJiwa e 2~
The second part of wave zone shear is due to
the sudden transitions between base film and waves.
This drag force is hypothesized to be proportional
to the surface area on which the gas impacts waves,
7D (Swave Sbase); the density of core including en
trained droplets, pc; and the square of a velocity charac
terizing the local gas flow, Ugtan,. This force is bal
anced by the shear at the bottom surface of the waves,
Ti,wave,drag, acting over an area estimated as 7fDLwave.
An empirical correlation (from Schubring et al.
(2010c)) is used to estimate the length of the disturbance
waves, Lwave:
Lwave = 0.53x 06D (36)
It deserves to be emphasized that this length is the size
of the disturbance wave, rather than a wavelength in the
) sense of a spacing between waves.
(28) The characteristic gas velocity, Uytrans is found us
g trans
6wave base Ti,base
VY V Ps
Ug,trans (Ul,i,base U,i,wave)
Wave zone shear is computed as the sum of two terms.
The first, Ti,wave,ough, relates to the roughness of waves
and is computed in an analogous manner as the base film
roughness. The second, Ti,wave,drag, relates to the sud
den transitions from flow over base film to flow over
waves.
To compute T,wave,rough, a friction factor must be
estimated. This is done using the form recommended in
the Hurlburt et al. (2006) model, which employs a value
Of CB, wave of 2.4:
Sf,i,wave
0.58
(wauee 1)2 1n cBuaue+l 0
In Schubring et al. (2010a,b), the standard deviation
of wave height measurements was found to be approx
imately constant at 20% of the mean wave height. In
analogy with the modeled base film roughness, the wave
S Ti, base 9,an j6I
\ pqY trans 
V PS \ bg,trans 0
[+ (y+)]2 dy+ (38)
The nondimensional distance 6grans represents the
penetration of the wave into the boundary layer in the
gas core over the base film. The characteristic velocity
is estimated as the sum of two terms. The first, in paren
theses in Equation 38, adjusts for the change in interfa
cial velocity between the wave and base film zones. The
second, larger, term computes the root mean square ve
locity within 6,trans from the base film interface. That
is, the characteristic gas velocity encountering a wave is
selected to produce a good estimate of the core kinetic
energy relative to the wave. The UVP (Equation 1) was
assumed in the boundary layer above the base film.
An empirical factor (analogous to a drag coefficient)
was fit to the present vertical FEP tube data as a value of
2.
2
P Uian2 (eTac base)
i,wave,dra trans (6wave bae(39)
Lwave
The density of the core (gas and entrained droplets),
pc appears in this equation. This is estimated by mass
conservation in the liquid phase and an assumed homo
geneous flow in the core:
ml,Ent
E
mi ;' film,base (1
'; film,wave Tw
mlEnt
INTw)
Snl,Ent + ';'
pc A(Us, + UE)
The entrained fraction, E; droplet mass flux rate,
T4l,Ent; and liquid superficial velocity, Us8, are also
computed through these equations.
The wave intermittency, INT,, is estimated by
an empirical correlation, also from Schubring et al.
(2010c), based on the liquid superficial Reynolds num
ber, Rel:
INT, 0.1 + R (44)
40000
plUs/D
Re =l p (45)
P[i
The droplet deposition flux, RD, is required in the
evaluation of pressure drop. The correlation of Ishii and
Mishima (1981) is used to compute this, based on the
entrained fraction, E, computed above with a mass bal
ance and a superficial gas Reynolds number, Re,:
\RDP Pg ) )
Re, = D (47)
The average interfacial shear in the wave zone,
Ti,wave, is computed as the sum of two parts:
Ti,wave
Ti,wave,rough + Ti,wave,drag (48)
Estimation of wave zone pressure loss requires solv
ing the following equation for dP/dxwave:
7i,wave
D ,wave
4
PcU g,wave dP
Pabs dx wave
pc cwav RD (Uc,wave Ui,,wave) (49)
4
A similar expression is employed to estimate pressure
gradient in the base film:
Ti,base
Dc,base
4
PcUg,base dP
Pabs dx base
 Dc,base RD (Uc,base Ul,i,wave) (50)
4
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The timeaveraged pressure gradient can then be
found:
dP dP dP
= (1INT) + INT, (51)
dx dx base dx wave
The timeaveraged film thickness is computed by:
6 (1 INTw) base + INTw6.wav (52)
The model has therefore produced zonal estimates of
film height, interfacial velocity (wave velocity in the
wave zone), pressure gradient, and film flow rate. Global
estimates of these parameters and entrained fraction are
also produced.
Vertical Quartz Tube
Data for pressure gradient and wave velocity were ac
quired in a 23.4 mm (I.D.) quartz tube. A development
length of 150 times the tube diameter was provided. Ac
cording to Wolf et al. (2001), entrainment rate (or liq
uid film flow rate) is the last flow behavior to develop;
a criterion for fully developed entrainment is provided
by Ishii and Mishima (1981), also reported by Kataoka
et al. (2000). Based on this criterion, the development
length is sufficient for the flows of interest. Further de
tails regarding the measurements and the flow loop are
available from Schubring et al. (2010c). Flows with gas
superficial velocities from 32 to 91 m s1 and liquid su
perficial velocities from 0.04 to 0.39 m s1 were studied
at pressures between 101 and 116 kPa.
A useful way to compare the agreement of a cor
relation or model is with mean absolute error (MAE),
defined as below for a number of flow conditions,
nFc, modeled result XXmod, and experimental result
XXexp.
MA 1 E XX .. XX p ,
MAE Xcorr XX 1,. (53)
nec C XXexp
The MAE's for pressure gradient and disturbance
wave velocity for these data were found to be 19% and
9%, on par with purely empirical, singlebehavior esti
mators. If flows with Ugs above 75 m s 1 are excluded,
the MAE for pressure gradient improves to 14%. At
these high gas flow rates, a wispyannular flow regime
is often observed, in which the entrained droplets form
clouds, lines, or some other sort of structure. Further,
the wave frequency is observed to increase (with a cor
repsonding decrease in wave spacing); see Schubring
et al. (2010c). The twozone modeling approach with
sharp base/wave transitions may not be appropriate for
this flow condition that more closely resembles homo
geneous roughness.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The modeled and experimental results are compared
for the quartz tube in Figure 1. The series labeled with
constant Us, are based on an average Us, for the data
plotted in that series; the series are based on constant air
meter readings. For pressure loss, the high mass flow
rate predictions deviate strongly from the data. How
ever, at lower mass flow rates particularly lower gas
flows good agreement is obtained. The wave velocity
is wellpredicted in the quartz tube.
30
25
20
E 15
I
3 4 5 6
v waveQuartz [ms 1]
wave,Quartz
Figure 1: Performance of model in vertical quartz tube.
(Top) dP/dx. (Bottom) vwave.
The model predicts three sources of interfacial shear:
roughness in the base film zone, roughness in the wave
zone, and the drag force linked to the sudden transition
from base film to waves. Figure 2 shows an estimate of
each of these, as functions of gas and liquid flow rates.
Each source of shear is shown, as weighted by the appro
priate function of wave intermittency: INT, for wave
zone effects, (1 INT, ) for the base film effect. While
these three components do not sum to a onezone interfa
cial shear (due to nonlinear effects such as acceleration
pressure loss), comparing the figures allows for a better
understanding of the effect of each of the three modeled
sources of shear.
35
30
S25
20
15
10
5
* 34  U [ms 1]
843  sg
T

<1
A
53
65
76
T   
!_ i i W 1
5 10 15 20 25
U [cm s 1]
30 35 40
S34 U [ms 1]
35 43  g
30 < 65 
S25 
20  
> 15 5   
0  I  I
0 5 10 15 20 25 30 35 40
Us [cm s 1]
40 1 I
34 U [ms ]
35 43  g
v53
H <76
. 25 g7 
i i i i
S20
15 
0 10
5 
0
0 5
10 15 20 25 30 35 40
Us [cm s 1]
Figure 2: Components of T, from model for vertical
quartz tube, by U,,. (Top) Base film rough
ness. (Middle) Wave roughness. (Bottom)
Wave drag.
The modeled effect of base film roughness is primar
ily a function of Us,, increasing approximately as the
square of gas flow rate, typical of flow over a rough sur
face. In contrast, the wave drag component is dominated
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
by liquid flow rate, with a weaker trend with gas flow.
The final term, wave zone roughness, is most often the
smallest of the three and is a strong function of both gas
flow (linked to increased KES,) and liquid flow (linked
to increased INT,).
The estimate of entrained fraction, Emod, from the
model cannot be directly compared with experimental
data, but can be qualitatively evaluated with respect to
its similarity to expected trends. This estimate is shown
in Figure 3. The entrained fraction increases with both
gas and liquid flow rates. The increase with gas flow
rate matches trends generally seen in entrainment corre
lations and what is expected based on visual observation
of wave videos. The increase with liquid flow rate and
the sharp drop towards an entrained fraction of 0 at low
Us8 are in agreement with the excess liquid concept, in
which there is some critical Us1 below which no entrain
ment is present.
1.0
0.8
S0.6
S0.4
U [ms 1f>
sg
5 10 15 20 25 30
U [cm s1]
* 34
R43
v53
< 65
A76
35 40
Figure 3: Modeled entrained fraction, Emod in quartz
tube.
Vertical FEP Tube
The planar laserinduced fluorescence (PLIF) measure
ments were taken in a fluorinated ethylene propylene
(FEP) tube with an inside diameter of 22.4 mm. An
array of five gas meter readings and five liquid meter
readings were explored, along with several intermediate
flow conditions. Further details on these experiments
are available from Schubring (2009) and Schubring et al.
(2010a,b).
Base film thickness and wave height are directly avail
able from the FEP tube data; the mean absolute errors
(MAE's) are found to be 8% and 9%, respectively. Fig
ure 4 indicates the predicted base and Swave with series
of liquid flow rate, along with the performance of the
model for film thickness.
200  a A 
r
150 ^    
S100  
50 
O0 50 100 150 200 250 300 350
8 [i m]
base,exp m
36 U [ms ]
1 46 sg
400 7 57      s e *
0A 80
e 00 mod e
0 100 200 300 400 500
6 [p m]
The model's performance for 6 shows no strong trends
with gas flow and is comparable to the experimental un
certainties for PLIF data. For both wave and base film
heights, the range in 6 with liquid flow and constant gas
flow is underpredicted.
Horizontal Tubes
Three horizontal tube loops were constructed with di
ameters of 8.8, 15.1, and 26.3 mm (I.D.). Development
lengths of 400, 330, and 210 (respectively) times the
tube diameter were provided. Gas superficial velocities
of 30 to 90 m s1 and liquid superficial velocities of
0.04 to 0.31 m s1 were studied at pressures of 101 to
121 kPa. A total of 185 flow conditions are available,
with estimates of base film thickness, wave velocity, and
pressure gradient. Base film thickness was studied us
ing the total internal reflection technique developed by
Shedd (1998). Waves were investigated using two in
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
frared light emitting diodes and two phototransistors and
signal analysis (crosscorrelation) in an apparatus simi
lar to Hawkes et al. (2000). Additional details regard
ing the experiments are available in the open literature
(Schubring and Shedd (2008a,b, 2009)) or from the dis
sertation of the first author (Schubring (2009)).
In fully annular horizontal flow in these small tubes,
the circumferential asymmetry in base film thickness is
relatively small (factor of 23, often less). As a result,
the model will be taken as an estimate of circumferential
averages for interfacial shear and film thickness. The
estimates of interfacial shear need to be adjusted to the
following, as there is no axial gravitational effect:
Ti,base
wi,wave
Dcbase
4
PcU, base dP
Pabs d base
RD (Uc,base Ul,i,wave) (54)
D ,wave PUg,wave dP
4 Pabs dx wave
RD (Ucwave Uli,,wave) (55)
Mean absolute errors for pressure gradient, base film
thickness, and disturbance wave velocity were found to
be 10%, 17%, and 14%, respectively, on par with com
putations from purely empirical correlations and single
behavior models. The results are shown by gas flow rate
in Figure 5 and by tube diameter in Figure 6. Estimates
of entrained fraction are shown in Figure 7.
Conclusions
A twozone model of interfacial shear has been pro
duced that requires only thermodynamic states, ge
ometry, and flow rates as inputs.
Film thickness is estimated using a critical friction
factor model.
The model explicitly considers the sharp transitions
between base film and waves, rather than merely
summing two homogeneous roughness terms.
Good agreement (to within 20% MAE) is obtained
on pressure gradient and film thickness measure
ments; agreement to within 20% is also obtained
on wave velocity data.
The trends in estimated entrained fraction agree
with those anticipated by the excess liquid concept.
Future work with this model includes applying it to
other databanks as well as application and adaptation to
flows other than airwater at low pressure.
200 75 I  
150 
100  
50
0 50 100 150 200 250 300
8 [i m]
base,exp m]
30
30   7 
36 < U [m s ]
n 43 sg
25 5 1   
8 51
< 64
20 A 75 
S+20%
S15i
S10  I/ i'S9 ^  
0 2 46
0O 5 10 15 20 25 30
dP/dx [kPa m1]
exp
10 
*10 36 us U [m s
1 43 s
8 v 51
A 75
6  20%  
4    
2          
0 2 4 6 8 10
v [m s ]
wave
Figure 5: Performance of model in horizontal tubes, by
Us,. (Top) .base. (Middle) dP/dx. (Bottom)
Vwave
Acknowledgements
The financial support of Bettis Laboratory is gratefully
acknowledged.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
1.0
8.8
a 15.1
0.8 V26.3
S0.6
M 0.4
Vmm
mm
mm  
 "~ Q
.
20 40 60 8(
U [ms 1]
sg
0 100
0 100
20 30 40 50 60 70
dP/dx [kPa m1]
exp
1.0
36 U [m s
,143 1 U [m s
0.8 V5
6 6
'7:
0.6
S0.4
5 10 15
Usi [cm s1]
20 25 30
S_20%
6
4   '
n i
4       
S 2 4 6 8 10
v [m s1]
wave
Figure 6: Performance of model in horizontal tubes, by
D. (Top) base. (Middle) dP/dx. (Bottom)
Wave.
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