Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 3.7.3 - Characteristics of Separated Laminar Flows in Inclined Pipes
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00096
 Material Information
Title: 3.7.3 - Characteristics of Separated Laminar Flows in Inclined Pipes Interfacial Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Goldstein, A.
Ullmann, A.
Brauner, N.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: stratified flow
core-annular flow
inclined pipes
interfacial shear
surface tension
 Notes
Abstract: A complete set of exact solutions for laminar separated flows in inclined pipes is presented. These include all possible configurations of stratified flows (STF) with concave and convex interfaces, and core-annular flows (CAF) with various core eccentricities. The exact solutions are used to analyze the local and averaged wall and interfacial shear stresses for these various flow geometries. In particular, the possibility of reducing the pressure gradient of a viscous fluid in upward inclined flows by adding a lubricating phase is investigated. It is shown that independently of the properties of the second phase introduced to the flow, the effect of hydrostatic pressure gradient always adversely affects the possibility to reduce the pumping requirement for the flow of the viscous phase. The limiting behavior of the local shear stresses at the triple-point, where the interface meets the pipe wall, is shown to be determined by the fluids viscosity ratio and the contact angle. Closed-form analytical expressions for the average interfacial and wall shear stresses were obtained for all cases of STF. These are useful for testing and improving closure relationships for approximate two-fluid models.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Characteristics of Separated Laminar Flows in Inclined pipes


Ayelet Goldstein, Amos Ullmann and Neima Brauner

Tel-Aviv University, Faculty of Engineering, School of Mechanical Engineering
Ramat Aviv, Tel-Aviv, 69978, Israel
brauner(eng.tau.ac.il, ullmann @eng.tau.ac.il


Keywords: stratified flow, core-annular flow, inclined pipes, interfacial shear, surface tension

Abstract

A complete set of exact solutions for laminar separated flows in inclined pipes is presented. These include all possible
configurations of stratified flows (STF) with concave and convex interfaces, and core-annular flows (CAF) with various core
eccentricities. The exact solutions are used to analyze the local and averaged wall and interfacial shear stresses for these
various flow geometries. In particular, the possibility of reducing the pressure gradient of a viscous fluid in upward inclined
flows by adding a lubricating phase is investigated. It is shown that independently of the properties of the second phase
introduced to the flow, the effect of hydrostatic pressure gradient always adversely affects the possibility to reduce the pumping
requirement for the flow of the viscous phase.
The limiting behavior of the local shear stresses at the triple-point, where the interface meets the pipe wall, is shown to be
determined by the fluids viscosity ratio and the contact angle. Closed-form analytical expressions for the average interfacial
and wall shear stresses were obtained for all cases of STF. These are useful for testing and improving closure relationships
for approximate two-fluid models.


Introduction

Laminar two-phase flows are frequently encountered in
small diameter pipes and in liquid-liquid flows. In many
cases the flow regime can be characterized as separated or
pseudo-separated flows. These include all possible
configurations of stratified flows (STF) with concave and
convex interfaces, and core-annular flows (CAF) with
various core eccentricities (Figure 1). Each of these flow
regimes may occupy the entire pipe, or parts of its length
(e.g., long Taylor bubbles). In principle, exact analytical
solution for the two-dimensional velocity profiles and shear
stress profiles can be obtained for fully developed laminar
flows. Such solutions are also needed as benchmark
problems for testing the validity of the numerical methods
for solving general two-phase separated flows (e.g., Ng et
al., 2002, Berthelsen and Ytrehus, 2004) and for testing
closure relations for simplified one-dimensional two fluid
models (Ullmann et al, 2004).
Analytical solutions available in the literature for STF with
curved interfaces and for eccentric CAF are restricted to
horizontal flows (Bentwich, 1964, Bentwich et al., 1970,
Brauner et al., 1996a, Rovinky et al., 1997). These are not
applicable to inclined systems in case of different densities
of the two phases. Compared to horizontal flows, the
solution for inclined systems is more complicated in case
the fluids differ in their density, as the axial driving force in
the two phases is not the same. Counter-current flow of the
two phases, which is the commonly used flow configuration
in various direct contact heat transfer and mass transfer
operations (e.g. the settling section in the phase transition
extraction column), is feasible only in inclined systems
(Ullmann et al, 2003a).
For inclined pipes analytical solutions are available in the
literature only for laminar fully-developed STF with a plane


interface (Biberg and Halvorsen, 2000, Goldstein, 2002,
Ullmann et al., 2004) and for concentric CAF (Bai et al.,
1992, Ullmann and Brauner, 2004). However, stratified flow
with a plane interface is typical to gravity-dominated
systems. In surface tension dominated systems (e.g.,
microgravity, capillary systems and in liquid-liquid systems),
the Eotvos number, EoD =ApgD2/o and the fluids/wall
contact angle are important parameters for determining the
interface shape in smooth-stratified flow (Brauner et al.,
1996b, Gorelik and Brauner, 1999). The wetting liquid tends
to spread over the tube wall resulting in a curved (convex or
concave) interface (see Figure 1). In fact, the possible
stratified flow configurations extend from fully eccentric
core of the upper phase to fully eccentric core of the lower
phase.
In this paper a complete set of analytical solutions is
presented, which covers all the above separated flow regimes
in inclined pipes. Such a complete set of solutions is not yet
available in the literature. The solutions obtained can be used
to study the effects of the pipe inclination, the interface
curvature and the core eccentricity on the characteristics of
counter current and co-current upward and downward flows.
The pronounced effects of core eccentricity in CAF, or
interfacial curvature in STF are demonstrated and discussed.
The effects of the physical properties (e.g., viscosities,
densities, surface tension and wall wetting) on the holdup,
frictional and total pressure gradients are examined. In
particular, the possibility of reducing the pressure gradient of
a viscous fluid in inclined flows by adding a lubricating
phase is investigated. The local wall and interfacial shear
stresses in the various STF geometries are analyzed,
including their limiting behavior at the Triple Points (TPs),
where the interface meets the pipe wall. The averaged wall
and interfacial shear stresses are used to test the validity of
closure relations for Two-Fluid (TF) models.






Paper No


Nomenclature


D
E
e
EoD
F
f
g
P,p
Po
Q
q
R
Re
S
U
u
X2
Y
x,y
Greek


a


Ii
B
U


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

stratified flows in the case of constant interface curvature
(represented by * in Figure 1).
Stratified flow- In stratified flow, the assumption of
constant curvature of the interface is consistent with the
exact solution for the interface shape in the both extremes
cases of gravity dominated systems, EoD >1, or surface
tension dominated systems, Eo < 1. In-between these two
extremes, using the model of Brauner et al. (1996b) for the
characteristic interface curvature was shown to yield a fair
approximation to the exact interface shape for Eo, = O {1,
(Gorelik and Brauner, 1999).


ApgD2/


Pipe diameter (m)
Core eccentricity,, Eq.(ll)
Core displacement (m)
System Eotvos number, Eo,
Interaction factor, Eq.(32)
Friction factor
Gravitational constant (ms-2)
Pressure (Nm 2)
Power dimensionlesss)
Volumetric flow rate (m3s-')
Flow rates ratio
Pipe radius (m)
Reynolds number
Wetted perimeter (m)
Average axial velocity (ms-1)
Local axial velocity (ms-1)
Martinelli parameter
Inclination parameter, Eq.(6)
Cartesian coordinates, (m)
letters
Contact angle
Pipe inclination
Holdup of the heavy phase
Lower phase wetted perimeter


Convex Interface
a >o *' < X


Q, U.S


interface curvature
ipolar coordinates
ni-polar coordinates


Concave Interface
a -




a


S1, 2

17
p

S




H,L
T

Subsrip
a,c
f
g
H,L
s


Viscosity (m2s-')
Pressure gradient dimensionlesss)
Density (Kgm3)
Shear stress (Nm 2)
frequency
ts
annular, core phase
frictional
gravitational
Heavy, Light phase
superficial
dimensionless


Exact solutions for laminar separated flows

Given the location of the fluids interface, the 2-D velocity
profiles in a steady and fully developed axial laminar pipe
flow (LPF) of two separated phases are derived from the
solution of the Navier-Stokes equations (in the z direction,
see Figure 1):

SV2U, =-- p gsin, j =H,L (1)
In the case of CAF the annular phase can be either the light
(L) or the heavy (H) phase. Obviously, in the case of
stratified flow the heavy phase forms the lower layer. The
required boundary conditions follow from the no-slip
condition at the pipe wall and continuity of the velocities
and tangential shear stresses across the fluids' interface.
Exact analytical solutions for Eq.(1) can be obtained in the
bipolar coordinate system for Eccentric CAF, and for


Q1_Q Vs

Y Zu~z


Figure 1: Schematic description of the STF and CAF
configurations and parameters

In the bipolar coordinate system (, J), the pipe perimeter
and the interface between the fluids are iso-lines of
coordinate (, so that the upper section of the tube wall
bounding the lighter phase is represented by 0, while the
bottom of the tube, bounding the denser phase, is
represented by 0 + T The interface coincides with the
curve of = *. Thus, the two-phase domains map into two
infinite strips in the domain defined by:
-o < < ,* > ( > for the upper phase, and
-o0< 0> for the lower phase. The
relations between * and the geometrical variables (e.g.,
flow areas, wetted perimeters) are given elsewhere (Brauner
et al., 1998). A plane interface corresponds to a constant
curvature arc, (* = /r. In this case the flow geometry can
also be described in terms of the thickness of the lower fluid
layer, h =0.5(1-cos (0). In the case the pipe center is at


Plane Interface,
a = o0,'=





Paper No


x,y =(0,0), point (0,f) corresponds to :
S=_+ sinhr sin -cosh4coshc +cos ( ) (2)
cosh cos Y cosh cos (
The solution obtained for the velocity profiles in the two
phases is in the form of the following Fourier integrals:
4sinssin(-)P r rH ,
uH = sin+ sHin(- )- + f sinh[ m( r- )] cos cdc
cosh -cos f io
4 sin 0 sin (0 ) sinhos
cosh -cos 0
PL = (dp/dz- pHLgsing)/(- dpf/dz) (3)
UH, L = H,L/UL are velocities normalized with respect to
the lighter phase superficial velocity, fi = u /1 is
the viscosity ratio, (dp /dz) is the superficial pressure
gradient of the light phase (R is the pipe radius),
UL, = R2 (-dp/dz)L /(8PL), while H and iP represent
the dimensionless driving force in the heavy and light
phase, respectively.
In the case of a plane interface, = z :


8sino {Y tanh[o (r-o)]coso( + (PL-PH/i)csino (4.1)


sinh (co) cosh (c6of) {i tanh [c (/ )] + tanh (c (qL)/=, = (4.2)
-8 sin Y tanh (o) coso ( P P -P /) sin
sinh (cW ) cosh c[ ( -)] i tanh c ( -e ) + tanh (co)
/-=/ tanh I (tr- o)) + tanh (cOo) (4.3)
In the case of a curved interface, 0* # :;


(q.. .


8 sin (A


cosh ['(q )]


H 0*,, u)Sics3i3 *ysinh(o )cosh [O(0 i o)]
Y t anh[co(O* 0,)][(- ,+ COSnO+ Cos(0* 0))


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

positive. In co-current flow, UH,UL, are both positive in
case of downward flow and negative for the case of upward
flow. Integration of the velocity profiles over the
corresponding flow cross-sectional area yields the flow rates
of the two fluids:


ULs

f, f fuLJ(1)djdo =1


(7.1)

(7.2)


where J(, )= sin2 0O/(cosh-_cos)2 is the transforming
Jacobean from bipolar to Cartesian coordinates. For
specified 0* Eqs.(7) provide the phase flow rates
corresponding to a given pressure drop and holdup (or 0),
since geometrical relations yield:
110*- =
{ -0.5 sin(2) (sin /sin*)2 [ -- 0.5 sin(20*)]}/V
s,= [0 0.5 sin(20)]/T (8)
Equations (7.1 and 7.2) can be manipulated to obtain
explicit expressions for the two unknowns, P, ,. When
these are introduced in Eq.(6), an implicit equation for the
unknown 0 is obtained, once is specified. It is worth
noting that although Eqs. (7) involve triple integrals (over
a for obtaining the local velocity and then over the flow
area (,, 0) ), the integration of the various terms included in
MI,2 with respect to and ( can be carried out
analytically, whereby the effort involved in the numerical
computations is reduced drastically (Goldstein, 2010).
The dimensionless frictional pressure gradient and the
additional hydrostatic pressure gradient (compared to single
phase flow of the light phase) are given by:


(-dpf /dz)
(-dp,/dz) ,

(5.1) (dgL (/dZ) / dzL
-dp / dz)L


(9.1)


(9.2)


tanh[to' (q5 ,)]Cos( -,) qosin2 *]
+(P, PH/o)sin (0* -,)tanh[a)o (0 -.)] sin2 *


8sin o


(c \ .


il'L 0*# cosi v/sih( co~r)U oh [C" ( 0')

-(Y cosh [' (* /)l tanh [' (0* 0)]-
.[(- Cos 0 + Cos0* Cos 0* 0))

(co sin 0* cos 0* tanh [' jP* '9
tanh ['(0*- T lCo (0* -0,) in 20 +


(P PH)/f )csin2 sin(* 0) sinh l[ (*- -r)]}

= itanh o) )] tanh ) ) (5.3)
The dimensionless inclination parameter, Y is given by:
Y = PL -PH = (P -p )gsinf /(-dp ldz)s (6)
Note that the inclination angle, 6 is always taken as


For a specified interface curvature, the solutions so-obtained
for the holdup and dimensionless frictional and gravitational
pressure gradients are dependent on three dimensionless
parameters f, q and Y. Note that in laminar flows, the
Martinelli parameter, which represents the ratio of the
superficial frictional pressure drop in the two phases,
is X2 =(dp/dz)H (dpf/dz)L =uq and can replace
either q or ft. As the characteristic interface curvature is
dependent on the system EoD and the fluid/wall contact
angle, a (Brauner et al., 1996b), the complete solution for
the general case of laminar stratified flows with smooth
interface is dependent on five dimensionless
parameters: q, f, Y, EoD, a .
From the view point of possible pressure drop reduction in
inclined systems by introducing a second phase, the ratio
between the total pressure gradient in the two-phase system
and that obtained in single phase under the same inclination
should be examined. In case the reference system is single
phase flow of the light phase, the following dimensionless


V


v





7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


pressure gradient (i.e., pressure factor) is used:
S (-dp/dz) H -Ye-r/(p-1)
(-dp/dz), 1-Y/(p-1)
The corresponding dimensionless pumping power (p
factor) needed is given by:
Po_ (-dp/dz) (U,,+U,,)_ -Ye-/(-1)[
(-dp/dz) UL, 1-Y/(p-1) [1+q
Note that the term Y / ( 1) represents the ratio bet
the gravitational pressure and frictional pressure gradie
single phase flow of the light phase. The density
= PH PL is an additional dimensionless paran
which has to be prescribed in order to obtain the
pressure and power factors.
In case the reference system is single phase flow o
heavy phase, the following definition of the dimensio
pressure gradients and power are used:
(-dp / dz) -P,+YE x2
-H= x2 ; = \q
S /dp / dz)
(dp / dz) (dp / dz), y( )
(-dp, / dz)H X
_,_-sY /X2 (Y /X2) (p- 1)


S 1-(Y/X2)p (p-l)

(Y/X2)pI/ (p5 1)


8p fi sinh2 sinh[m(f
(9.3) [tanh[m(yc -7)] +] tanhy cosh[m(

power + 8Pa sinh27 sinh[m( -
[tanh[m(7 -7)]+ i] tanhy cosh[[m(y-
(9.4) +sinhycoshy, e- sinh[m( -7y)]
cosh[m(7c -w)] e cosh[m(7c -7)]
ween
wnt in [tanh[m(c -7 )] +i]cosh[m( -7)]]}+
nt in
ratio, Ye sinh[m(f-7y )] sinh2
neter, cosh[m(7c -7)]
total 1 + 1 [ tanh[m(7c -y)]

f the tanhy/c msinh2 ,) [tanh[m(7y )]+P]


(10.4)


Eccentric Core Annular Flow- The bi-polar coordinates
fits also the geometry of eccentric CAF. For this geometry
the tube wall is represented by = w,, while the two fluids'
interface coincides with J = y Hence the eccentric CAF
configuration in the (x-y) domain maps into a semi infinite
strip in the ( ~) domain. For the annular
phase y < yc < oo; 0< I 2f where:

7 =cosh-1 I1- -ecl R= =
_2Ro J' = -- -; R


7y =cosh1 -I-2 + ,
S2e ,


E=-
1 -R


R, denote the core radius, and e =e IR is the cor
dimensionlesss) eccentricity (e in Figure 1). In the case th
pipe center is at x,y =(0,0), point (0,4) corresponds to
= sinhy sin. cosh(r- ;) cosh ; cos (12
cosh cos cosh cos 0
The solution obtained for the velocity profiles in the tw
phases is in the form of the following Fourier series:
iua = 4P. sinh2 cosh 4P sinh y cosh y (13.1
cosh -cosy (13.1
sinh2 n
+ 4Y(7, ) + (I cosm
S sinh2 c m=1
S i2 coshq 1 ]
i = 4P sinh2 7 ah- -
cosh q cos 1 tanh Cj
4% sinhW, sinh(7C w) 4(_ cw) sinh2 iy
-s4Y + -D_ -hcosm
A sinh A sinh2 yc m=1
(13.2
where


e
e


-7w)] .
-r)]


7w)]








1]


_________ sinh2
(ch = -8e-' I- +
e [[tanh [m (y w)] + c i] tanh 7y

+ sinhycoshy e'"- sinh2r
cosh [m (y w)] e-"Y tanhT )

+Y tanh[m (C T)] sinh2 7. msinh- +
Lfi t ;talnh1 smihs =2 /c
S. dp/dz- pg sin /
Pc ( dp/dz)c


(14.1)













(14.2)





(14.3)


) y (Pa c-)gsingP
a (- dp/dz)c,

Integration of the velocity profiles over the corresponding
flow cross-sectional area yields the flow rates:


1 2ff
-f dJf if (& ,)J(O,)do =q
Tw 0
1 2f
1 df i, (0, ) J (0, ) do = I
d) d 1
if J (^ ./( ) =


(15.1)


(15.2)


where J(0,;)=sinh2 y/(cosh -cos)2 is the transforming
Jacobean. Fortunately, the integration of the velocity terms,
which correspond to the homogenous solution u~h,Uch
(for m > 1) can be carried out analytically (Goldstein, 2010).
Given q, Y E and ii, Eqs. (15) can be solved for the
unknown P4,P, which are then substituted in Y= -Pp
to yield an explicit equation for q(Y,/i,E,se), whereby
s (q,i, Y,E) can be obtained.


-) Fully Eccentric Core Annular Flow- A configuration of
fully eccentric core represents the boundary between
o stratified flow and eccentric CAF configurations. Although
the bipolar coordinate system is also appropriate for solving
stratified flows and eccentric core-annular flows, it fails
) when in the extreme of fully eccentric core. When the limit
of the fully eccentric core-annular configuration is
approached, calculations become tedious and time
consuming. This is due to the dramatic increase of the
frequency range over which the Fourier integral/series must
be numerically computed in order to obtain convergent
solution for the velocity profiles.
To handle the geometry of a fully eccentric core, a special
) ('unipolar') coordinate system was introduced by Rovinsky
et al. (1997). That solution is limited to horizontal systems


Paper No






Paper No


and is herein extended to include the effect of the pipe
inclination. In the uni-polar coordinate system:
R 2Ry R 2Rx
I1 ---~ _2 2 2 (16)
r' X 2+y 2 +y2
The solution obtained for the dimensionless velocity profiles
of the annular and core phase iic = ua Uas is in the form
of the following Fourier integrals:
8P
i= = -;2 + (a), )cos()W2)do) (17.1)


f = i zp + (Jo, (o) cos(wof2)dco (17.2)
2 2 0


S--8 (e e1-) 1- (l+()I
T y "** *'


+e 2 ) (l(l i )] ( e (1


ft 1P- 2o e +e2- 1 e ) 204

--
Y L L JJ



U= jI J l+e/k
+Ye l+ 1+A 1+ey(1-(^))1

TY=e 2' (1- ;)-( 1+I); =1/Rc

ppp/;P dp / dz pg sin/
(-dp, /dz)_
(p, )g sin -
Y (-dp i _d)


,) + (-2))


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

dependent on the following dimensionless parameters: the
viscosity ratio, the inclination parameter, Y, and the flow
rates ratio, q (or the Martinelli parameter, X2). In order to
use a unify notation for all three models, the results below
will be presented by referring to the heavy phase subscriptt
H) and light phase subscriptt L), and the dimensionless
parameters indicated on the figures refer to the definition
used in the stratified flow (STF) model. Note however, that
the solution for (eccentric and FE) CAF refers to the core
and annular phases. Therefore, when considering all
possible separated flow configurations for a particular
two-phase system, care must be taken to use consistent
dimensionless parameters in each of these models for
representing the system under consideration (Goldstein,
2010).


08
07
S06
" 05-
" 04
03
02
01
0


()]




(18)



(19)


Integration of the velocity profiles (Eqs. (17.1) and (17.2))
over the corresponding flow cross-sectional area of the
annular and core phases, with J (,2)=4/(2 +22)
yields the flow rates of the two fluids:

If d2 f (1, 42 )J (1, )d = 1
0 1 (20)
2 1 Uas
J ( ile (, 2 )J (, ) = =d - -
o7 0 i q UVs'

Eqs. (20) can be solved for P,Pe which are then
substituted in r= p_ to yield an explicit equation for
q(Y, /,ec) whereby s =1/ 2 =F(q, Y) can be
obtained.

Separated Flow Characteristics Discussion

The set of exact solutions obtained for laminar two-phase
separated flows enables the investigation of the effect of the
interface curvature on the various flow characteristics for all
possible flow configurations. These extend from fully
eccentric core of the heavier phase, through convex, flat, or
concave interfaces, up to the other extreme of fully eccentric
core of the lighter phase (see Figure 1). Additionally, the
exact solutions for annular flows, extending from fully
eccentric to concentric core-annular flows, enable studying
the effect of the core eccentricity on the CAF characteristics.
In all these separated-flow configurations, the solution is


0


EoEoO, 2
a E0o =2- *=



C Dominated ~
G Dommated -
YPH/M -0 1
Y=-5


20.


-04 -03 -02 -01 0 01 02 03 04 05
Counter-current X Z=HQH/tLQL Co-currentUp


Figure 2: The effect of the interface curvature on the
holdup for 0* = (Eo,, a)

For stratified flows, a closure for the interface curvature is
also required. Based on the balance between, gravity and
surface tension forces, a closure for the interfacial curvature,
has been obtained by Brauner et al (1996b ) and Gorelik
and Brauner (1999). The results obtained by applying that
closure for the interface curvature are demonstrated in
Figure 2. In this figure the solution obtained for holdup vs.
X2 (which for constant p represents variation in the flow
rates ratio, q) is shown for specified system
parameters /, Y,Eo, a In the extreme of gravity
dominated systems (EoD>>I), the interface is practically
plane (q* ->* z) independently of the holdup. In the other
extreme of a surface tension dominated system (EoD< the interface curvature is given by = o a + ,
(0o < 0* < o + 7r). With this closure, ideal wetting by the
heavy phase (a= 0) results in a configuration of a fully
eccentric core of the light phase (q* = 2z ); whereas in the
opposite case of ideal wetting by the light phase, a =7
the solution corresponds to fully eccentric core of the heavy
phase, (* = 0. The solution for the case of equal wetting of
the wall by both phases, a = 7/2, is also shown in Figure
2. In this case, the curvature varies in the range
ofz-/2 < 0* < 3 -/2, where the lower limit is approached for
low holdups (4o 0) and the interface is thus convex; the
upper limit is approached for high holdups ( o ->* ),
whereby the interface is concave. For 0 = -/2


I


1






Paper No


(holdup=0.5) the interface is plane even though the system
is surface tension dominated. To conclude, the model
predicts that in EOD<<1 and ideal wetting systems, the basic
flow configuration is core-annular flow with the wetting
phase flowing in the annulus. However stratified flow, even
with a plane interface, is still feasible in cases of similar
wall wetting by the two-phases.
For core annular flows, a closure is needed for the core
eccentricity. The two main modeling approach to determine
the core eccentricity suggest it is determined by a balance
between gravity and inertial forces (e.g., Bai et al, 1996), or
by a balance between gravity and lubrication forces (e.g.
Omms et al, 2007). Both mechanisms for the core
stabilization assume a wavy core. Each of these models
introduces additional dimensionless parameters, which may
not be readily available. Therefore, for the sake of
generality, the separated flow characteristics are studied
here by considering the interfacial curvature as a parameter
of STF and the core eccentricity as a parameter of the CAF
models. To elucidate the effect of pipe inclination, results
obtained in horizontal flows are presented first.
The flow configuration has a pronounced effect on the
separated flow characteristics. Figures (3a,b) shows results
obtained for the holdup in a horizontal pipe. The light phase
(e.g., oil) is 10 times more viscous then the heavy phase
(e.g., water).








09


0 05 1 15 2 25 3 35 4
q=QH/QL
Figure 3: Effect of the separated flow configuration on the
holdup in a horizontal pipe a. various flow configuration
corresponding to e = 0.2, b. holdup vs. q.

To demonstrate of the effects of the curvature and core
eccentricity, sketches showing the flow configuration for a
constant water holdup, E = 0.2, are included in Figure 3.
These different configurations cover a wide range of flow
rate ratios. On the other hand, maintaining a constant flow
rate ratio, the water holdup monotonically increases as it
becomes more affected by the wall and slowed down. The
holdup increases when the interface become less convex,
more concave, approaching a configuration of CAF with the
water in the annulus. Obviously the variation of the holdup
with the interface curvature in the stratified flow
configuration is much more pronounced compared to its
variation with the eccentricity in the CAF configuration. The
larger effect of the core eccentricity is observed in the case


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

where the viscous core approaches the pipe wall
(E 1 with the heavy less viscous phase in the annulus).


0 05 1 15 (a) 2
q=QH/QL


a)


02 04 06 08
q=OH/QO (b)


Figure 4: Effect of the separated flow configuration on the
frictional pressure gradient (a) and power (b) (Normalized
to the more viscous light-phase) in a horizontal pipe.

The dimensionless frictional pressure gradient, nf,
b) corresponding to Figure (3) is shown in Figure (4a). In
horizontal flows, the frictional pressure gradient is the total
pressure gradient, and it increases with increasing the
contact area (wetted perimeter) of the more viscous phase
(e.g., oil) with the wall (i.e., reducing ). The lowest
pressure gradient factors correspond to a configuration of
concentric CAF with the less viscous phase (e.g., water) in
the annulus. The highest pressure gradient factors
correspond to the opposite CAF configuration, namely,
concentric CAF with the oil in the annulus. Since the
pressure gradient is normalized to that of the single phase
oil flow, values of FIn < 1, indicate a lubrication effect ,
whereby the introduction of a less viscous phase (i.e.,
water) to the flow of the viscous oil results in a reduction of
the pressure gradient. The lubrication effect becomes more
significant as the less viscous phase contact with the wall is
increased (e.g., > 1.5f7) In horizontal flows, the nz
values for fully eccentric CAF and concentric CAF provide
bounds on the pressure drop reduction that can be achieved
by introducing a lubricating to the flow of viscous oil in
various eccentric CAF configurations.
Referring to Figure 4a given a flow rate of the light and
more viscous phase QL, the introduction of a less viscous
phase affects initially a decrease of the two phase pressure
gradient as long as the less viscous phase is in contact with
the pipe wall (5* > 0). However, eventually, increasing the
flow rate of the lubricating phase (i.e., increasing






Paper No


q=-' ')), results in an increase of the two-phase pressure
gradient, indicated by H. values exceeding one. As

shown in the figure, for each interface curvature or
eccentricity of the (viscous) core, a point of minimal
pressure gradient is obtained. Hence, this minimal point
indicates the maximal achievable reduction in the pressure
gradient by introducing a lubricating phase with the
specific viscosity ratio. In horizontal flows, the maximal
achievable pressure gradient reduction is of the order of the
viscosity ratio.
The results of the pressure gradient can be used to obtain
the pumping power requirement (normalized with respect
of that required in single phase of the more viscous phase,
Eq. 9.4). Power savings due to the less viscous phase
lubrication effect correspond to PoL <1. Here too, for
each interfacial curvature or core eccentricity, the point of
minimal power factor (i.e., maximal reduction in the
viscous phase pumping power requirements) can be
identified.


1 E+00
ll-
1 E-01




1 E-04

SE-06
1 E-05 7


1E
1 E+00
1 E-01
S1 E-02
-











S1E-03
1 E-04
EO-






1 E-06
1 E-07
-

' 1 E-06
-








1 E-07
-E0


*=2___x __E1 (a)


E= CAF AH

E=0


Y=0
Less-Viscous Heaw Phase in the Annulus


E=0


*=2 EI(b)

CAF -AH




Y=0
Less-Viscous Heaw Phase in the Annulus


1 E-07 1 E-06 1 E-05 1 E-04 1 E-03 1 E-02 1 E-01 1 E+00
gH/p4.
Figure 5: Maximal pressure gradient (a) and power
(b) reduction in horizontal separated flows.

Figures (5a,b) shows the effect of the viscosity ratio on the
maximal achievable pressure gradient and power
requirement reduction in horizontal flows. As expected,
when the viscous phase flows in a concentric core, the
lubrication effect increases with the viscosity ratio (i.e.,
with the core viscosity when using a particular
lubricant, pH ). In this flow geometry, the n and PoL are

proportional to / and of the same order of magnitude.
The required lubricant flow rate for achieving the maximal
pressure gradient reduction is about 50% of the viscous
phase, and it corresponding holdup is about 0.5. Maximal
power saving are achieved with a lubricant flow rate being
about 30% of the viscous phase, and its holdup is about
0.38 (see Figures 3&4).
When the viscous phase gets in touch with the wall at one


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

point (FE CAF, = 2ir, E =1)), the minimal pressure
and power reduction factors saturate at about 2.5%
(97.5% reduction) for high viscosity of the core phase.
The flow configuration in this case corresponds to a very
thin annulus (and very low flow rates) of the lubricating
phase, with q = q,/qo =q,/q -> 0, and a= EH -> 0.
However, from the practical point of view, a certain
amount of lubricating phase should always be added. For
2 -> 0 the optimum is rather flat, whereby close to
optimal operational conditions are as well reached with
q =0.05 0.1.
With stratification, the potential for lubrication by
introducing a less viscous phase is further reduced. With
a configuration of stratified flow with a plane interface, the
lubrication potential becomes already insignificant (see
Figure 5a). The minimal pressure gradient factor is 0.71
(pressure gradient reduction of 29%), and this value is
approached when the viscosity ratio (oil-to-lubricant) is
larger then 100. The corresponding lubricant (water)
holdup is about 0.1. However, the required lubricant flow
rate steeply increases with increasing the oil viscosity, and
consequently, power saving are no longer achievable with
stratified flow with a plane or convex (*

00
C


0 018
0016
0014
. 0 012
" 001
2 ooo
0008
0006
0004
0002


-0 001
790
0-o


0 0005
q=QH/QL


Figure 6: The effect of the separated flow configuration on
the variation of the holdup with q in upward inclined flow.

The effect of inclining the pipe upward is shown in Figure
(6). The holdup variation with the flow rate ratio is shown
for a (negative) inclination parameter (Y=-50) and the same
A/ = 0.1 used in Figure 3 for the case of horizontal flow.
The results can be conveniently interpreted by associating
the constant value of Y to a case of a constant pipe
inclination and a constant upward flow rate of the viscous,


-07 1 E-06 1 E-05 1 E-04 1 E-03 1 E-02 1 E-01 1 E






Paper No


light phase. Then, the variation of q represents changes in
the flow rate of the heavy (less viscous) phase. As shown in
Figure 6, for some region of holdups, only counter-current
flow is possible. Generally, in counter-current flow, q<0,
there are always two possible configurations of different
holdups for specified flow rate ratio (and interface
curvature), up to the flooding point beyond which no
solution is obtained. The concurrent region, q>0, correspond
to upward flow of the two phases. In this region, at least one
solution can be always obtained. However, in a certain
flow-rates ratio interval, three different holdup values
correspond to the same flow rates can be obtained (see
enlargement of the q= 0 region in Figure (6b) where the
additional two solutions of lower holdups are shown). The
existence of multiple solutions (M-S) regions for the case of
stratified flow with a flat interface in the counter-current and
co-current regions was discussed in Ullmann et al, (2003a,b),
where the multiplicity of holdups was also verified
experimentally. As can be seen in Figure (6b), M-S
characterizes also stratified flows with curved interfaces and
CAFs. The multiplicity of solutions and the non systematic
variation of the holdup with the assumed separated flow
configuration (i.e., 0 *,E) complicate the prediction of the
holdup in inclined flows.
The n ,in the upward inclined flow of the light phase
(Figure 7a, Y<0,) reaches much lower values than in
horizontal flow (Figure 4a). As shown in Figure 7a, the
Hn values may attain even zero and negative values.
However, most of the region associated with Hn <1
corresponds to counter-current flow. This figure also reveals
a very different variation of 1Hf with q compared to that
obtained in the case of horizontal flow (Figure 4a). In
particular the shape of the 1Hf curves is different in cases
where the lubricating phase is in contact with the wall
0* > 0, compared to those obtained when pipe surface is
entirely wetted by the more viscous phase.
As shown in Figure (7a), in the counter current region,
where two solutions are obtained for the holdup (for
specified flow rates ratio and curvature), the mode of the
lower frictional pressure gradient corresponds to the lower
holdup mode of the heavy and less viscous phase,
independently of the interfacial curvature. It is worth
noting that, in the counter current region, the frictional
pressure gradient may change its direction in one or both of
the solutions. Negative frictional pressure gradient indicates
a dominancy of the wall shear stress of the heavy phase
flowing downwards. This dominancy becomes more
pronounced as the heavy phase wetted perimeter increases
(q* -> 27T) and/or its velocity gradient at the wall increases.
The reason for the negative Hn values is quite obvious.
Figure (7b) demonstrates the velocity profile at the pipe
center line associated with a case of H, < 0 for concentric
CAF. A negative n is a result of back (downward) flow
of the heavy phase near the wall, in the opposite direction to
that of the light, viscous phase. Note that some frictional
pressure gradient reduction (ni, <1) can be observed also in
the (M-S) region of the concurrent up-flow, for flow


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

configurations with the less viscous phase flowing in the
annulus.
20
CAF-A=L
15 Y=-50


-5


5
10
1 CAFA H (a)
E-=0 7 E=E- 2E=-0 CAF-A=H
-100 -80 -60 -40 -20 0 20
q=QH/QL


-80 -60 -40 -20 0 20 40
u/Uis


-3 -2 -1 0 1 2 3 4
q=QH/QL

Figure: Frictional pressure gradient (normalized to the
more viscous light-phase) vs. q in inclined flow, a. /i = 0.1,
c. Fi = 10, b. velocity profile at minimum pressure gradient.

It is not surprising that the frictional pressure gradient can
be reduced by introducing a less viscous phase. Figures
(7c) shows the results obtained for the dimensionless
frictional pressure gradient for Ai = 10. This means that
the heavy phase introduced to the (constant) flow of the
light phase is 10 times more viscous. In cases of f > 1, the
contact of the viscous phase with the wall increases with 0*.
Therefore, one may expect that I, will increase as well

with 0* (opposite to the case discussed in Figure 7a). This
is indeed the trend observed at high flow rates of the heavy
phase in the concurrent region. However, in the gravity
dominated region, a reduction of the frictional pressure
gradient may result, which is even more significant than in
Figure (7a), where the heavy phase introduced is less
viscous; Namely, for a given flow rate of the light phase, the
frictional pressure gradient decreases by introducing an
additional flow rate of a more viscous and heavier phase.






Paper No


Surprisingly, the frictional pressure gradient is the lowest for
the configurations with the more viscous phase flowing in
the annulus and completely wets the pipe wall. Here too, the
negative Fn are obtained in the countercurrent region, as

a result of the downward (back flow) of the heavy phase.
Some reduction of Hi, to values lower than 1 can be
obtained also in the concurrent up-flow region (see Figure
7a). This as a result of the backflow of the heavy phase near
the wall in the upper and middle holdup solutions.
In inclined flows, not only the frictional pressure gradient
should be considered, but also the hydrostatic pressure
gradient. Examining the total pressure gradient factor
obtained for Y=-50 (with, <: 2) indicates that the minimal
value of the pressure gradient factor is 1 both in the case of
/ = 0.1 and fi = 10. This means that for these two upward
inclined systems, pressure gradient reduction for the light
(either more or less viscous) phase flow is impossible. The
reason is the increase of the hydrostatic pressure gradient
associated with the introduction of a second heavier phase to
the flow of the light phase.
0 9 RllH/ =10
08 Y/X (dpfl
07 -(a)
.06

05 33,5/2
= 04
03 o, E=
02E
0 1 E=_0_


20 40 60
Co-current Up 1/q=QL/QH


0 10 20 30 40 50
1/q=QL/Qn


Figure 8: Effect of the separated flow confi;
a. holdup and b. frictional pressure gradient
the more viscous heavy-phase) in an upward

Since the hydrostatic pressure gradient is
the loss of the possibility to reduce the press
introducing a second lubricating phase to t
viscous phase, Figures (8a,b) examine the
reducing the pressure gradient phase by int
viscous and lighter phase. In this case the ui
of the heavy (viscous) phase is
(corresponding to Y/X2 = -50 ). The
introduced is 10 times less viscous ph,
Figure (8a) shows the holdup variation wit
light phase flow rate. Obviously, with th
flowing upward, countercurrent flow is ii
irrespectively of the flow configuration


80 100 120


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

entire holdup curve corresponds to concurrent up-flow.
As shown, multiple solutions for the holdup are obtained
for CAF with the lubricant (the lighter phase) flowing in
the annulus.
The corresponding dimensionless frictional pressure
gradient FIH is shown in Figure (8b). Surprisingly, the

results show that when the lubricating phase wets the wall,
the frictional pressure gradient increases by an order of
magnitude and in fact attains the maximal values. The
minimal values of N, are obtained when the more

viscous phase flows in the annulus and completely wets the
wall. Here again the reason for the rather counter-intuitive
result is the gravity effect. This can be understood upon
examining the velocity profiles.

08 Light & Less Viscous (a)


02
o0 Heavy & Viscous Phase /5=10
-02 QL/QH=28
-04 3ack Flow

-08 Light & Less Viscous
-1 I.
-10 0 10 20 30 40 50 60
uUHS Up Flow
U/UHs


S Heavy & Viscous Phase (b)

04
0 2
0 Light & Less Viscous Phase
-02
-04
-0 6 Y/X 50
-08 Heavy & Viscous Phase HILL= 10
QL/Qn28
_1
-10 10 30 50 70 90 110 130 150 170 190 210 230
uHUHS Up-Flow

Figure 9: Velocity profile at the maximum (a) and
minimum (b) pressure gradients of Figure 8b.


CAF-A=H Figure (9a,b) show the velocity profiles at the pipe center
I line associated with the maximal and minimal FN values
60 70 80
in Figure (8b). When the less viscous light phase flows in
the annulus (Figure 9a), the heavy phase flows backward in
duration on the the core, and only near the core interface it is dragged
(normalized to upward by the light phase (resulting in a net upward flow).
inclined pipe. This results in increased velocity gradients in the annular
phase, and consequently, increased frictional pressure
responsible for gradient in the two- phase flow compared to single phase
ure gradient by flow of the viscous core phase. On the other hand, when the
he flow of the heavy and more viscous phase flows in the annulus (Figure
possibility of 9b), backflow of the heavy phase is obtained near the wall,
reducing a less resulting in a relaxation of the wall shear stresses. When
ward flow rate examining the total frictionall + hydrostatic) dimensionless
kept constant pressure gradient affected by introducing a less viscous and
lighter phase light phase in this inclined flow (Figure 10), a region of
ase ( = 10 ). lubricated flow, lf, <1 is obtained for all the separated
h changing the flow configurations. The preferred configuration in this
e heavy phase respect is still that of flow of the less viscous phase in a
impossible, and concentric annulus. However, the flow rate ratio associated
considered, the






Paper No


with minimal Hn indicate that relatively large flow rates of

the light phase should be added to the flow of the viscous
heavy phase. Consequently, when the dimensionlesss) power
requirements are considered, the minimal PoH value is 1,
hence no power saving are achievable in this case.


S09

S08
Sos

007

( 06


0 50 100 150 200
l/q=QL/QI
Figure 10: Effect of the separated flow configuration on the
total pressure gradient (Normalized to the more viscous
heavy-phase) in concurrent upflow.


07
S06

-04
C


*=0 9*=0.57 1 9*=l7 1 *=1.57 9*=2.7 E=0.7 E=0.5 E=0.2 E=0
Less-Viscous Heavy Phase
Configuration in the Annulus


1
09
08
S07

2
0 5
04
0 03
02
01


*=0 9*=0.57 9*=71 4*=1.57 9*=227 E=0.7 E=0.5 E=0.2 E=0
Less Viscous Heavy Phase
Configuration in the Annulus


Figure 11: Effect of inclination parameter on the minimal
dimensionless total pressure gradient (a) and power
requirement (b) (normalized to the light & viscous phase)
achievable when a less viscous (heavy) phase is added in
STF and CAF (the less viscous & heavy phase is in the
annulus).


In fact, the maximal achievable pressure gradient reduction
is very sensitive to the system inclination. Figure 11 shows
the effect of Y on the minimal dimensionless total pressure
gradient and power requirement achievable when a less
viscous and heavy phase is added to (a constant) flow of 100
times more viscous phase (t= 100). The phases' density
ratio considered is 2. With the stratified flow configuration,
starting form a fully eccentric core of the less viscous phase,
larger are associated with increased portion of the wall


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


wetted by the lubricating phase, up to *= 2c which
corresponds to fully eccentric core of the viscous phase. The
subsequent CAF configurations correspond to CAF with the
heavy lubricant flowing in the annulus, ranging form fully
eccentric to concentric viscous core. With Y=-0.1, the
minimal pressure gradient and power are by an order of
magnitude larger than in horizontal flow. With Y=-1, they
are increased by more than two orders of magnitude, and for
Y of the order of 10, the option of lubrication is practically
lost. Moreover, the minimal values shown for Y=-1,-5
correspond to the middle holdup solution, which may not
easily be established in practice.


1
09
08
07
S06
S05
S04
03
02
01
0

1
09
08
S07
0 06
S 05
04
O 03
02
01
0


4*=0 4*0.57 4*=1-l 4*=1.57 9 4*=27 E=0.7 E=0.5 E=0.2 E=0

STF Configuration CAF
Figure 12: Effect of the density ratio on the minimal
dimensionless total pressure gradient (a) and power
requirement (b) (normalized to the light & viscous phase)
achievable when a less viscous (heavy) phase is added in
STF and CAF (the less viscous & heavy phase is in the
annulus).

In Inclined flows, the values of the minimal achievable
dimensionless total pressure gradient and power requirement
are also sensitive to the density ratio. As shown in Figure
(12), when a constant Y is considered (e.g, Y=-1), the
achievable lubrication effect decreases significantly when
the density ratio decreases and approaches a value of 1.
Obviously, for maintaining a constant Y with reduced values
of p = H/PpL one has to compensate by increasing the pipe

inclination, and/or reducing the flow rate of the viscous light
phase, and/or increasing the pipe diameter. All these options
increase the role of gravity, hence the contribution of the
hydrostatic pressure gradient, whereby the potential for
pressure and power reduction by adding a less viscous,
heavy phase diminishes. Indeed, by adding a lubricating
phase which is lighter then the viscous phase (Figure 13), a
more pronounced total pressure reduction is achievable in
inclined flows, even with various stratified flow
configurations and high Y/X2, since in this case both the
frictional and hydrostatic pressure are reduced. However,
comparison of Figure 13b and lib shows that when the


4*=0 p*=0.57 9*=71 4*=1.57 9*=2.7 E=0.7 E=0.5 E=0.2 E=0

SR=1.05

PH/PL=R

R-2 E


R=5
Y=-1
AH/HL=0 01 (b)






Paper No


power requirement are considered, the potential of reducing
the pumping power in inclined flows by adding a lighter
lubricating phase is always lower than that achievable by
adding a heavier lubricating phase, due to the higher
required flow rates of the lubricating phase in the latter.


1
09
o8
07
S06
Sos
2 05-
S04
O
03
02
0 1
0



1
09
08
S07
0 06
- o
004
0 03
02
01


4*=27 4*=1.57 4.*=17 4*=0.57 4*=0 E=0.7 E=0.5 E=0.2 E=0
Less Viscous Light Phase
Configuration inthe Annulus

Y/X=-50

(b) vix2 5
Y/X o 1


PH/P==2
IIH/=l 00 X2=-0 1
Y y=0


0 -
4*=27 4 *1.57 4*=lx 4*=0.57 9 4 F0 E=0.7 E=0.5 E=0.2 E=0
Less-Viscous Light Phase
Configuration intheAnnulus

Figure 13: Effect of inclination parameter on the minimal
dimensionless a. total pressure gradient and b. power
requirement (normalized to the heavy & viscous phase)
achievable when a less viscous light phase is added in STF
and CAF (the less viscous & light phase is in the annulus).

Wall and interfacial shear stresses in STF

Local shear stresses- Using the solution for the
dimensionless velocity profiles UH,L obtained in Eqs. (3),
the dimensionless local wall shear stresses are given by:

- 7-H = V fi





Ls, 4H, -9 )04 (22)

1 cosh cos J ,,c osod )
L I
c 4 cos-t--- r0 o ,J co s wo d w
4 sin 0o L

where Ls (= 4pLUL, /R ) is the superficial wall shear
stress of the light phase, and H = sin o /(cosh cos 0) is
the Lame' coefficient. The interfacial shear stress (exerted on
the upper phase) reads:
f, ( 8H ), (23)
I Ls 4H )= 4H g


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

Scos(* -)coshco cos o,
T .= 4- --- ---- : --
1 cosh cos *
cosh cos* C @4 sin o --H osh[o, )oscod


Note that with the above normalization of the shear stresses,
a negative sign for the wall shear stresses indicates that the
wall shear is in the opposite direction of the light phase flow.
Average shear stresses- The dimensionless average wall
and interfacial shear stresses are obtained by integration of
the expressions obtained for the local shear stress over the
corresponding wetted perimeter ( -co < < oc and
specified q) and dividing by the wetted perimeter length:


THL= J ',L,,ds S L,, ; ds = RHd (24)

The following relation of the inverse cosine Fourier
Transform can be applied on Eqs (21-23) to obtain analytical
expressions for the average interfacial and wall shear
stresses:
2"
SfJJg()cos() )dwd = lim g(m) (25)
S0 0 oa0
Using Eq.(25), the following closed-from expressions were
obtained for the average interfacial shear, f in the case of


a curved or a plane interface, respectively:

(T) = S{ -sin. [( *- l)(-+ T(- )}

{simn (o- *)sinm* [-PHo +f(+PL ( -)]+

sin jio ( [- ) (- g* +"o+f) +L/ji o *)

FiP, (sin 4 + (r o) cos ) P (sin 0 cos )


(26.1)



2f(6 ?2


Equation (25) can be applied also to derive a closed form
solution for the average wall shear stresses, J .

However, given an explicit closed-form expression for T,
the integral momentum balances on the heavy and light
phases can be used to obtain the expressions for the average
wall shear stresses in the two phases, whereby:


= I[ += IH [7T(l -e)P ] (27)
SH I L
Sear stresses at the triple points An interesting aspect of
the wall and interfacial shear stresses is their limiting
behavior in the triple points (TPs) at +o0, where the
fluid-fluid interface meets the pipe wall. It is worth noting
that the computation of the local shear stresses requires
numerical integration of the Fourier integrals in the bipolar
coordinates (over 4), which can be carried out for some
large, however finite 4. Due to accuracy limitations of
numerical integration, the computation is carried out to
values of ma = {10}. The corresponding polar distance

r" T 0 from the TPs is = e The exact TP limits for
the wall and interfacial shear stresses can be determined
analytically by using residue calculus. The analysis
(Godstein, 2010) represents a generalization of that
presented in Moalem Maron et al., ( 1995) and Brauner et al.,
(1996a) for horizontal flows with curved (or plane)
interfaces, and that presented by Biberg and Halvorsen,


/





Paper No


(2000) for inclined flows with a plane interface. It indicates
that the characteristic behavior of the wall and shear stresses
when approaching the TPs is dependent on the values of a
and i Note that for specified 00 and

,a = -( .The results of the analysis are
summarized in Table 1. These predicted TP limits in the
different cases considered were validated by comparison
with the computed results using the expressions obtained for
the local shear stresses.

Table 1: Shear Stresses Limits in the Triple Points (TPs)

Viscosity Contact Wall Shear Wall Shear Interfacial
Ratio Angle Lower Phase Lower Phase Shear
F P=/pl a (~#- ) f (J- o) ? (J-o o)
l< < 0 a< r/2 0 0 0
0< <1 7i/2 a i1 a= r/2 Finite 0 Finite 0
F1 a= r/2 fH, L L fH 0
l< S< <1 0 FP-oo T/2 a pF 0 a< r/2 0 O 0
p- -> 0Oa< r/2 0 0 0
fP->0 42/2 a S= 1 0 f = f, f = f, cos a
1 0O! P1 a= =, = f ~ =,L
1 0o!_ a=o0 =r O = H f=- r

If the sharp contact angle is formed in the more viscous fluid,
the wall (and interfacial) shear stresses approach a zero value
in the triple points. This result remains valid also for/ -> oo.
Hence, in such cases the TPs represent stagnation points in
the flow. The approach to a zero value is however moderated
for smaller values of a and the zero values are realized for
larger 4.
The interfacial shear stress remains zero in the triple points
when the wetted angle is a = /-/2. The wall shear stresses,
however, attain the following finite (generally non zero)
limits when the interface is perpendicular to the wall.


a = i/2


0* 7;"


Hi- ) (+i)P H JL)+2Y cos0 I% sin2,*+-O* 1
()T +) sin3 2


(28.1)


+ P 2 COP P+2 -'sin2 *+-'
(i+ 1) ;' isin 0* 2


and for a = / 2, = T orY 0:

+ +1
HH .= ( 1); +)= ( ) (28.2)
Hence, in this case the wall shear stresses in the TP vary
discontinuously across the interface, and the jump in the wall
shear increases as the miss-match of the phase viscosities
becomes larger.
If the contact angle with the wall is blunt in the more viscous


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

phase, the wall and interfacial shear stresses tend to infinity
in the TPs, however they remain integratable.
While for a = / /2 Eqs. (28) remain valid also in the
limit of -> o the case of a > /2 and
Ji -> oo requires a separate treatment which yields:

TL,, = 0; THi = KHe(1 2) o (29)
Hence, the shear stress as seen by the less viscous fluid is
zero in the triple points in the high and low viscosity ratio
limits 2 -> 0 and /2 -> oo, irrespective of holdup and the
interface curvature. The corresponding wall shear stress
in the more viscous fluid, however, is generally zero in the
triple points, if the sharp contact angle is formed in the more
viscous fluid. The TPs wall shear stresses attain a finite,
generally non-zero limit if the interface is perpendicular to
the wall, and tend to infinity if the contact angle with the
wall is blunt in the more viscous phase.
In the special case of equal fluids' viscosity, = 1, the
two-phase effects arise only in inclined flows, due to density
differences between the two fluids, resulting in P, PL.
The analysis shows that in cases of i =1 the wall shear
stresses are continuous across the TPs, with:
,H L( ) =( PH + P(1- )) (30.1)

[(a- ir)sin2 0*-siin sin( sin(a+( ) -(r- 0 )sin2 a
S- sin2 *


(30.2)
(30.3)


t {E -> = fHL = COs a


The above results do not necessarily apply for the limiting
configurations of fully eccentric core of either the light, or
the heavy phases, where the two TPs merge into a single TP.
In the case the fully eccentric core is formed by the less
viscous phase (e.g., FEC of a light less viscous
phase, 1 < fi < oo, a = 0), the TP shear stress attains the
following finite limit:


(,o(1 -- (1 ) -4 (l 2
HL~a= / OD)=L c2


(31.1)


where (R0 = I-e.
In the case the fully eccentric core is the more viscous phase,
(e.g., FEC of a heavy viscous phase, 1 <5 < o, a = Z ) the
TP shear stress generally attains a finite limit (which -> o
for -> oo):


kH,L,a= (4 C)= -,a,= ( o =
J.P(IR/i^-^'+^-ii-^+i] =^


(31.2)


More details on the analysis of wall and interfacial shear
stresses profiles and their variation upon approaching their
TP limits were presented in Goldstein et al., (2009).

Closure relations for STF Two-Fluid models

In order to obtain closure relations for stratified flows with a
curved interface, the approach used in Ullmann et al., (2'11 4)
to derive closure relations for TF models is followed. The
exact analytical expressions obtained for the average wall
and interfacial shear stresses are used to obtain the values of
the correction factors, which are to be applied on
single-phase-based closure relations in order to reproduce
the corresponding exact values.






7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


FH H -
PHfH i IH -

FH 2,- F
PHfHIUH\(UL-U)' L


27L
?Lf UL\IUL


resulting in a reduction of the wall shear stress by a factor of
(32.1) =2 (i.e.,FH z0.5).


27
PLfL UL(U -UH) (32.2)


16 PR UJ D 44 -A
= Re' D + -D
Rej P (S +S, SJ +S,


(32.3)


where j=H, L. These F- terms evolve from the interaction
between the phases, which is not accounted for in the
commonly used single-phase-based closure relations. A
significant interaction is indicated by a large deviation of the
F- terms from a value of 1. Negative values of FH,L indicate
a major impact of the backflow, which results in a reversed
average wall shear and miss-prediction of the wall shear
direction by single-phase-based closures.
In gas-liquid flows, the common practice is to account for
the phases interaction in the closure used for the liquid wall
shear stress by modifying the definition of the liquid
hydraulic diameter (used for calculation of the liquid
Reynolds number). The gas phase is considered to be the
faster phase, which drags the liquid; hence, the liquid
hydraulic diameter is defined by considering the interface as
free (non-wetted). This assumption may not be generally
valid for gas-liquid flows; for liquid-liquid flows, it is not a
priori evident which of the phases is faster. Accordingly,
an adjustable definition of the hydraulic diameter was
suggested (Brauner & Moalem Maron (1989).
Figure (14b) shows the FH interaction factor corresponding
to the heavy phase wall shear in the case of upward
inclined flow of the light phase (Y=-5). The light phase is
10 times more viscous (/i=0.1). For convenience, the
solution obtained for the holdup vs. X2 (i.e., the flow rates
ratio) is shown (Figure 14a). In the countercurrent flow
region, the two values obtained for FH for a specified flow
rate ratio correspond to the two different solutions obtained
for the holdup. Both values are positive, indicating the wall
shear direction is, as anticipated, upward, opposite to the
downward flow of the heavy phase. For sufficiently high
downward flow rates of the heavy phase (X<0 ) both of the
FH values are of the order of 1. However, upon reducing the
QH, FH cowith -q=-QH/QL 0.
In the concurrent up-flow region (q>0), the FH values shown
in Figure (14b) are for the high holdup solution shown in
Figure (14a), which is the only solution obtained for high
values of q. For this solution branch, FH -> -o for

q -> 0. The FH factors corresponding to the two additional
(medium and low holdup) solutions in the 3-s region of the
concurrent up-flow (obtained for small positive q), are also
shown in Figure (14b) (enlargement of the q = 0 region,
between points 2 and 3 in figure 14a). As shown, the middle
holdup branch also corresponds to FH -co for q 0,
while the lowest holdup branch corresponds to FH =0.5.
Such values of FH can be obtained by modifying the
definition of the lower phase hydraulic parameter, by
considering the interface as non-wetted by the lower phase
flow. Indeed, for the low value of the heavy phase holdup
(along the lowest holdup solution branch) SH S, ; hence by
assuming S, =0 in Eq. (32.3.), the liquid hydraulic
diameter is increased by a factor of about of 2 ,


1
09
08
07
. 06
05
2 05-
04-
03
02
01


-04 -03 -02 -01 0 01 02 03 04 05
X2 IHQH/ LQL
1 2 3


2





0


-0


0

0

0



2

15

1

05

0

-05

-1


Y=-5


15
2 @ =3/2r7



15 3
0
15 (b)
-1 2 1 o a 1

-03 -02 -01 0 01 02 03 04 05
X 2'QH/LQL



1




6 o \ -0001 00004 00009 00014
Y=-5 x-QdrgLQL

0=3/2 (C)















(d)


-03 -02 -01 0
X2 =IQH/IILQL


igure 14: Upward inclined flow of the light phase (Y=-5): a.
holdup curve for STF (/ = 0.1, =1.5r ), b. the heavy
phase wall shear interaction factor, FH c. the light phase
wall shear interaction factor, FH, d. the interfacial shear
interaction factor, F,H and FH. Corresponding points in the
figures are marked by numbers.
While the value of FH, which corresponds to the lowest-
holdup solution can be modeled by changing the definition
of the hydraulic diameter of the heavy phase, this is not the
case for the middle and upper solution branches. The large


Paper No


01 02






Paper No


negative values of FH in the concurrent upflow region
indicate reversed wall shear in the heavy phase due to its
backflow (downward) near the wall. This is demonstrated
in Figure (15a), which shows the velocity profiles at the pipe
centerline corresponding to the (nonzero) middle and high
holdup solutions in the case of q = 0 (hence
QH > o and UH ->0 ). As shown, although the average
velocity is zero, the velocity gradient at the wall (hence the
wall shear stress) is not zero. Due to gravity, the heavy phase
flows downward near the wall, and is dragged upward by the
light phase near the interface, resulting in a circulating flow
of the heavy phase with a zero net flow rate. As shown in
(Figure 14b), the zero values of wall shear, hence FH =0, are
obtained at some positive (non-zero) q, which are not the
same for the high and middle holdup solutions. The large
deviations of the FH from the desired value of 1 indicate that
in the gravity dominated region, the single-phase flow based
closure for the wall shear stress of the heavy phase in
concurrent inclined flows may largely fail to represent the
wall shear in the two-phase flow system.
Figure (14c) shows the FL interaction factor for the same
two-phase system. While the wall shear stress of the light
phase changes more regularly, values of FL &1 are
obtained only along the lowest (heavy phase) holdup branch,
where the flow of light phase can be practically regarded as
single phase flow. For high flow rates of the heavy phase,
the FL approaches a value of 0.5, indicating that for low
holdups of the light phase, the interface should be better
considered as non-wetted for defining its hydraulic diameter.


1=-5 (b)
9 H/ I pow
8 QH/QL 0
7 r=3/2n
6
5 Light & Viscous Phase
4
3
2
SHeavy & Less Viscous Phase
-5 -3 1 1 3 5 9


Figure 15:
Figure 14a.


Velocity profiles: (a) point land (b) point 2 in


For the modeling of the interfacial shear, the common
practice in modeling gas-liquid systems is to evaluate the
interfacial friction factor based on the wall friction factor of
the gas (light) phase. However, in the general case, the


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

interfacial shear can be modeled based on the wall shear of
either of the phases, replacing the phase average velocity by
the phases' velocity difference (Eqs. 32.2). The larger is
the deviation of F, from 1, the more significant is the error
introduced by ignoring the interaction between the phases.
Therefore, the preferred model for 7 is the one
corresponding to F, closer to 1.
The values obtained for the interfacial shear interaction
factors are demonstrated in Figure (14d). The blue curve is
the interaction factor FH which result when the interfacial
shear is modeled based on the flow of the heavy and less
viscous phase. The red curve is the interaction factor FL
when the interfacial shear is modeled based on the flow of
the light viscous phase. Along the major part of the holdup
curve in the countercurrent and concurrent regions, the
FH values are closer to the value of 1, indicating that the
interfacial shear should be better modeled based on the flow
of the heavy and less viscous phase. The exception is the
lowest holdup branch in the concurrent up-flow region,
where FL z1 implying that the interfacial shear is
dominated by the flow in the light phase.

Conclusions

A complete set of analytical solutions for laminar separated
flow configurations in inclined systems has been obtained.
The solution can be used to investigate the effect and
significance of the specific flow configuration (e.g., interface
curvature, core eccentricity) and the operational parameters
(e.g., flow rates, properties, pipe diameter and inclination) on
the local and average flow characteristics (i.e., velocity
profiles and local and average wall and interfacial shear
stresses, holdup, frictional and total pressure gradient).
In particular, the possibility of reducing the pressure gradient
of a viscous fluid in inclined flows by adding a lubricating
phase is investigated. Obviously, in horizontal flows the
frictional pressure gradient of a viscous fluid can be reduced
by adding a lubricating less viscous phase, which partially or
entirely wets the pipe wall. It is shown that in inclined pipes,
the frictional pressure gradient can also be reduced when
adding a more viscous (and heavier) fluid in contact with the
pipe wall. However, in upward inclined flows, the effect of
hydrostatic pressure gradient adversely affects the possibility
to reduce the pumping power by introducing a second fluid.
In the gravity dominated flows, -Y, or -Y/X2 >0(10), the
power requirements for pumping a viscous liquid cannot be
significantly reduced by introducing to the flow a heavier or
lighter lubricating phase.
The exact solutions are used to investigate the local and
average shear stresses and their behavior when approaching
the contact Triple points (interface wall intersection). The
analysis of the TPs shear stresses indicates the major role of
the contact angle, a. The characteristics of the wall and
interfacial shear stresses profiles at the vicinity of the TPs
are determined by the fluids viscosity ratio and the contact
angle. In fact, the contact angle and the system EoD number
determine the interface curvature, which is relevant for a
particular two-phase system.
Closed-form analytical expressions for the average
interfacial and wall shear stresses were obtained for all STF
configurations. These can conveniently be used to test the






Paper No


validity of closure relations for Two-Fluid (TF) models. The
latter are widely used for engineering design of two-phase
pipelines. It is shown that commonly used closure relations
for the wall and interfacial shear stresses, which are based on
single-phase theory/correlations, are not representing
correctly the fine balance between the gravity body forces
and viscous shear in inclined flows. Consequently, the TF
models may yield poor predictions in inclined co-current and
counter-current flows. Following the route suggested in
Ullmann & Brauner (2006), the exact solutions are being used
to obtain the two-phase interaction factors, to be introduced
into the single-phase based closure relations for obtaining
more reliable closure relations for Two-Fluid models.

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7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

Goldstein, A. Analytical Solution of Two-Phase Laminar
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