7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Interface drag on plugs in downward sloping pipes
I.W.M. (Ivo) Pothof*, H. Kuipers:, G Reitsma: and F.H.L.R. (Francois) Clemenst
Deltares I Delft Hydraulics, department Industrial Hydrodynamics, Delft, Netherlands
Department of Applied Physics, University of Groningen, Netherlands
t Department of Water Management, Delft University of Technology, Delft, Netherlands
Ivo.Pothof@deltares.nl
Keywords: gas bubbles, gas transport, plug flow, bubbly flow, pipe slope
Abstract
Experimental data on the cocurrent twophase flow in downward sloping pipelines is scarce (Pothof 2008), despite the
numerous offdesign conditions in the water industry, process industry, hydropower and nuclear industry that are dominated by
cocurrent twophase flow. If the water discharge is sufficiently large, all gas bubbles and elongated pockets are transported in
downward direction along the pipe invert in a plug flow regime. The plug flow regime in downward sloping pipes is very
similar with that in horizontal pipes (Ruder and Hanratty 1990). A further increase of the water discharge marks a transition to
bubbly flow.
New measurements were performed with a highspeed camera on the twophase flow of water and air in the plug flow regime,
including the transition to bubbly flow. The experiments in the plug flow regime allowed for the measurement of interfacial
area, plug width and plug velocity, from which the plug drag coefficient could be determined.
This paper presents the acquired experimental data and derives an experimental model to predict relevant parameters such as
plug frequency, drag coefficient, plug drift velocity and plug velocity. It will be shown that both the gas void fraction and the
turbulence level determine the plug frequency and drag coefficient.
Introduction
In a pipeline for liquid transport with downward sloping
sections, air may have several negative consequences: 1) the
transport capacity of wastewater mains reduces significantly
when air accumulates in downward slopes (Escarameia
2006, Lubbers 2007a, Pothof 2008), 2) inadmissible
pressure shocks may occur due to the sudden release of air
from hydropower tunnels (Wickenhauser 2008) or water
pipelines (Malekpour and Karney 2008), 3) vibrations may
occur in pumps or turbines and 4) allowable pressures may
be exceeded during filling of storm water storage tunnels
(Vasconcelos and Wright 2009) or during pump start or
valve stroking operations (Falk et al. 2004).
A number of researchers has investigated the behaviour of
single elongated air pockets in downward sloping pipes in
order to determine the required liquid discharge to clear air
pockets from a pipeline (Escarameia 2006, Gandenberger
1957, Kent 1952). Despite the operational risk associated
with air in pipelines, there is a lack of knowledge on the
cocurrent airwater flow in downward sloping pipes; only a
few researchers have addressed this subject to some extent
(Bendiksen 1984, Kalinske 1943, Lubbers 2007a,
Wickenhauser and Kriewitz 2009). These authors have
shown that the motion of air pockets is governed by the pipe
Froude number or flow number F, defined as
F PP= s(1)
Pw Pg
where us, g, D, pw, pg are the superficial phase velocity
(m/s), gravitational acceleration constant (in ), internal
pipe diameter (m), liquid density and gas density. The
approximation in Eq. (1) holds for low pressure airwater
systems where pg << pw. The flow regimes of downward
cocurrent airwater flow relate to small superficial air
velocities (1 mm/s) and moderate superficial liquid
velocities (1 m/s). Even at airwater discharge ratios of
0.001 air accumulates in the downward slope, creating an
additional head loss due to the presence of air pockets. If
the water discharge is just large enough to prevent air
accumulation, the air moves along the pipe invert as a
regular series of plugs (Figure 3). The regular pattern of air
plugs was observed with a high speed camera.
The objectives of these measurements was to determine
important features of the plug flow regime, such as:
Bubble velocity
Air plug frequency
Typical bubble dimensions
Drag coefficient on the plugs
Transition to bubbly flow
Nomenclature
A Area (m2)
a Plug flow model parameter ()
b Plug flow model parameter ()
Co Drift flux coefficient ()
Cd Drag coefficient of a plug ()
D Internal pipe diameter (m)
d Bubble diameter (m)
Eo E6tvos number ()
f Force (N)
F Flow number, pipe Froude number ()
f Plug frequency (s')
g Constant of gravitational acceleration (ms 2)
dH/ds Hydraulic gradient ()
h Plug height (m)
p Presssure (Pa)
Q Discharge (m3 s')
r Plug radius
T Time after start of measurement (s)
u Velocity (m s')
w Maximum plug width (m)
X Plug flow variable for turbulence and void
fraction ()
Greek letters
a Void fraction ()
0 Pipe angle (rad)
p Density (kg m 3)
o Surface tension (N m 1)
r,, Wall shear stress (Pa)
Subscripts
b I
d I
g
i I
r I
s
t
w
bubble, plug
)rag, drift
3as, air
interface
Relative
Superficial, start
Tangential
Water
Theory
Assuming a constant bubble velocity, the drag force }d is in
equilibrium with a tangential component of the buoyancy
force due to Archimedes' principle ;t. Since the buoyancy
force is determined from the pressure distribution around
the air plug, the head gradient dH/ds due to the flowing
liquid in the pipe affects the tangential component of the
buoyancy force .t. The latter is obtained as follows:
= A,hp g
S=Ahpwg sin 0
dH)
ds
where A,, h, pw, g and 0 are the bubble interface area (m ),
bubble height (m), liquid density (kg/m3), constant of
gravitational acceleration (in i and downward pipe angle
(rad). Since the gas density is two orders of magnitude
smaller than the liquid density (p << p ) the gas
density has been neglected in equation (2). A formula for the
drag force is:
d = dAb w2
2
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
width (m), relative velocity (m/s), average liquid velocity in
the pipe cross section (m/s) and the bubble velocity (m/s).
From the fact that Fd and Fbt are equal to each other, the
following formula for the drag coefficient is obtained:
Cd = 2A 2 sin  d
(u, u b dsb
The above analysis shows that the following parameters
should be measured from the camera images in order to
establish the bubble drag coefficient Cd:
o Plug area A, of each individual bubble in the plug flow
regime
o Bubble velocity ub of each individual bubble.
o Maximum bubble width wb.
A number of other flow characteristics were determined
from the measurements. The plug frequency fb is measured
from the start frames of plug 40 and plug 1:
b = 39/(~ 40 1) (5)
The average bubble height h per flow combination is
calculated from the total air discharge, the plug frequency
fb and the average interface area A,:
h =  (6)
4Afb
where the bars denote the average value per flow
combination.
The plug height is influenced by both the pipe dimensions
and surface tension. Therefore, it is not clear whether the
plug height should be scaled to the pipe diameter or to a
typical bubble radius. The airwaterperspex (PMMA)
contact angle is approximately 90. The bubble nose is a
stagnation point, where the plug curvature is most
pronounced. In the centre of the plug, the interface is
practically flat, which implies that the local plug radius is
large and the interfacial pressure is negligible. Since the
pressure inside the air plug is constant, the plug curvature
at the nose should match the hydrostatic liquid pressure
around the plug nose. This analysis yields a minimum plug
height hm,, based on the surface tension CT and contact
angle. Assuming that the curvature of small plugs acts
mainly in two direction, the differential interface pressure
Ap, equals:
20 20
Ap, 2 (7)
rb hmin
where the last equality is due to the 900 contact angle. The
interfacial pressure balances the hydrostatic pressure
Ap = pghmn cos 0 such that the following expression
is obtained for the minimum bubble height:
Cdwbh w (U b
where Cd Ab Wb, Ur u, and ub are the bubble drag
coefficient, frontal bubble area (m2) maximum bubble
hmin
i pg cos
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Following equation (8), the minimum bubble height for a
waterair mixture in a transparent perspex pipe is 3.9 mm
at a 100 downward slope, slowly rising to 5.4 mm at a 600
downward slope.
Nicklin et al. (1962) have proposed the drift flux model for
the motion of elongated bubbles in vertical pipes.
Ub
Bendiksen (1984) found Co = 0.98 for negative pipe angles
up to 300. The drift flux model is assumed to be applicable
to plug flows as well, such that the drift flow number Fd
can be derived from the measurements.
The void fraction is computed as the ratio of the superficial
air velocity and plug velocity: a = Fg /F .
The transition from plug to bubbly flow has been
investigated by many researchers, including (Andreussi et
al. 1999, Hinze 1955). Andreussi et al. (1999) found a
Edtvos criterion (Eo = 5) based on a maximum bubble
diameter:
g p,o p )(1 a)d ax
Eo = =
g(p, Pg)(1a) 0.11(1+aab) 2 (10)
S(p/p 1/3 T I
This correlation expresses the influence of wall turbulence
(r,) and void fraction on the transition from plug flow to
dispersed bubble flow. For an airwater mixture at low void
fractions this transition is governed by the following
parameter group:
(l+aab) (1+aab)
Eo~ X
r. F2
The parameter group X will be found useful to determine
the plug drag coefficient and plug frequency.
Experiments
The experiments were conducted in the 0150 mm
transparent pipe loop for airwater flows in the hydraulics
laboratory of Deltares (formerly Delft Hydraulics) (Figure
1). The 4.5 m (30D) long downward sloping pipe section
was positioned at an inclination of 100. The camera was
positioned after 3 m (20D) in the downward sloping pipe
section and captured a pipe length of 0.38 m (2.5D), which
allowed a reliable assessment of the average bubble velocity
in the plug flow regime.
Figure 1: Overview of 0 150 mm airwater flow facility
Air was injected in the horizontal section. The air and water
flow rates were measured upstream of the injection point.
Further details of the facility are summarized in (Lubbers
2007b, Pothof and Clemens 2010).
A small air pocket was present at the upstream mitre bend. A
continuous array of air bubbles was ejected from the mitre
bend. These bubbles quickly evolved to a more or less
regular pattern of air plugs, that were observed by the
camera. The first author has captured images at 50 fps of 40
 100 bubbles per flow combination in the plug and bubbly
flow regimes; 40 bubbles were sufficient for averaging the
results. The flow regime transition from the blow back into
the plug flow regime was determined prior to these
experiments (Pothof and Clemens 2010). A set of 13
airwater flow combinations was observed (Figure 2).
The video images were analysed as follows:
o Each video was stored as a series of individual TIFF
images with a serial number and a reference grid
plotted digitally onto the images (50 fps)
o In our analysis, a common picture viewer (Explorer on
Windows, Preview on Mac OSX) was used to
determine the first and last shot of the trajectory of a
plug (between the third gridline from right and left) in
order to secure a common path length for all plugs.
6 
S PBlow back flow
o Plug flow
80
0 Bubbly flow 10
4
Blowback to plug
.9
E Plug to bubbly Plug flow
2
o /6 *8
< 1 3 4 +
0 Bubyflow
0.6 0.8 1 1.2
Water Flow number, Fw []
Figure 2: Airwater flow combinations and observed flow
regimes.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Combinations 2 and 7 contained plug and bubbly flow
behaviour such that only the bubble velocity could be
determined. Due to the mixed flow behaviour the plug
frequency could not be determined in combinations 2 and 7.
Therefore, the remaining 8 combinations were analysed in
more detail.
S0.06
S0.04
0.02
* Plug height
...... Minimum plug height/ D
SPlug width
.9*
43
..... ...
[]2
0.75
0.5
0.25 0
figure J: Example ot picture oetore ana after analysis with
GIMP. Also shown is the Histogram function which shows
the number of (black) pixels in the new layer, i.e. the size of
the superficial area; water and plugs flow from left to right.
o The GNU Image Manipulation Program (GIMP) was
used to determine the interfacial area of each plug.
Note the following:
o As a plug does not consist of one piece, but a
big plug and a number of smaller bubbles
attached to the plug, the interfacial area of the
total of the plug and the accompanying
bubbles was determined. During the passage
of the recorded section, the bubbles of a
single packet coalesce and split into smaller
bubbles, so it would make no sense just to
measure the area of the large plug. For the
maximum width and the velocity of the plug,
this remark holds as well.
o The interfacial area A, was determined by manually
drawing a selection contour around the plug. Next step
was to make a new layer in the image and coloring the
selected area black. The histogram gives then the
number of pixels within the new layer, i.e. the number
of pixels within the selection. See also Figure 3. The
maximum width wb was determined by measuring the
upper en lower limit of the plug in pixel coordinates
(cursor position is readable in the lowerleft corner of
the image window).
o The conversion to centimeters was determined from
the width of the pipe in pixels and the actual outside
diameter (16 cm).
o In order to calculate the bubble velocity, the length of
the measurement trajectory between the
start/endgridlines was measured in pixels.
All this info was stored in a spreadsheet for further analyses.
Results and Discussion
Figure 2 shows that 3 combinations predominantly revealed
a bubbly flow regime in which the plugs were broken into
many small bubbles. A continuous array of small bubbles
moved along the pipe invert and as suspended bubbles.
0 0.1 0.2 0.3 0.4
Interface area, A /A []
(b)
0.8
> 0.6
S 0.4
S0.2
0_
0.5 0.6
* Plug velocity
* Plug frequency
.7
* *.
,43
2
Qa/Qw*1000 []
Y 3
2
10 1
o 1
0
4
Figure 4: Overview of results. The color of the diamond
symbols show the combinations with the same water
discharge, the color code of the open square symbols show
the combinations with the same gas discharge. (a) Plug
height and width as a function of the interface area; (b) Plug
velocity and frequency plotted against the discharge ratio.
The plug height and interface area are linearly correlated
over the investigated range of air discharges (Figure 4a,
R2 = 0.98):
b= 0.074 A'+0.034
D A
The plug heights, derived from the measurements, are at
least 50% greater than the minimum plug height, based on
surface tension and hydrostatic pressure considerations (eq.
(8)). If the 900 contact angle between airwaterPMMA
would be neglected, then the minimum plug height would
be twice the bubble radius (hm,, 2rb) and the contact angle
would be 1800. In that case, the minimum plug height
would accurately match the smallest measured plug heights
in airwater combinations 1, 4 and 5. Apparently, the
minimum plug height should be based on a contact angle of
1800, as if a small water film is always present between the
pipe wall and the plug. The interface area correlates very
well with the square of the maximum plug width (R2
0.98):
1.42jj 2 0.0467
D
The shape of the individual plugs varies from triangular to
diamond shape, implying that the interface area is
proportional to the square of the maximum width, which is
confirmed by the experimental correlation (13). It is
surprising to note that the plug shape (A,, hb, Wb) can be
predicted from simple correlations on any of the main plug
dimensions.
The measured plug velocities vary from 35% to 65% of the
superficial water velocity (Figure 4b). The plug velocity
decreases in the airwater discharge ratio. The plug
velocity cannot be directly assessed from the airwater
ratio, because the plug velocity increases in the water
discharge at a fixed discharge ratio; see combinations 3, 4,
5 and 7 in Figure 4b. The experimental drift velocity Fd is
computed from equation (9). If Fd is plotted as the function
of the discharge ratio, the data points collapse to a single
curve (Figure 5, R2 = 0.98).
0256
F = 0.52 10
0.8
S0.6
o
E
0.4
0.2
10
5 4
3
Drift Flow number
Power (Drift Flow
0 1 2 3 4 5
Discharge ratio, Qg*1000/Qw []
Figure 5: Experimental correlation of drift flow number
The measured plug frequency varies from 1 to 1.7 s' The
plug frequency increases as the tendency to plug breakup
increases; combinations 5 and 8 with the largest plug
frequencies are the closest to the flow combinations with
temporary bubbly flow.
0.4
U
0.3
u
a0
M
a
_^O
*3
*6
*8
4
0 1 2 3
Qg/Qf*1000 []
4 5
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Figure 6: Experimental plug drag coefficients as a function
of the discharge ratio
Figure 6 shows the experimental plug drag coefficients,
that are found to vary within a rather small range from 0.28
to 0.37. Figure 4b and Figure 6 suggest that both the plug
frequency fb and drag coefficient Cd are influenced by
the liquid turbulence and void fraction. It is therefore
useful to investigate whether fb and Cd can be predicted
from the parameter group X in equation (11). A reasonable
fit for plug frequency and drag coefficient is found for a
single set of parameters a and b inX: a = 0.7, b = 0.5.
(1+22.1"05)
X =
F
The drag coefficient (R2= 0.92) and plug frequency (R2
= 0.92) are predicted from X by the following correlations
(Figure 7):
C = 0.095X + 0.145
S(16)
f, = 2.72 0.717X
'J
"i
0 0.2
C
o
0
CU
16
16
* Drag coefficient
* Plug frequency
2
U
C
cn
1
Parameter group, X []
Figure 7: Prediction of plug drag coefficient and frequency
from plug flow parameter X
Conclusions
New measurements with a highspeed camera have been
presented and analysed on the twophase flow of water and
air in the plug flow regime in a downward sloping pipe. The
investigated range of airwater discharge ratios was 0.410
to 410 The acquired data shows that the plug dimensions
(height, width, interface area) are strongly correlated to
eachother and can all be predicted from simple correlations
with the airwater discharge ratio. The dimensionless plug
drift flow number is also accurately predicted from the
airwater discharge ratio, such that the plug velocity and
void fraction are derived from the discharge ratio. The
prediction of the plug frequency and drag coefficient is
more complicated. Both variables are influence by
turbulence (modeled via the wall shear stress) and the void
fraction. Surprisingly enough, the plug frequency and drag
coefficient are predicted with reasonable accuracy from a
single plug flow parameter X (eq. (15)).
Acknowledgements
The authors wish to thank the CAPWAT participants, who
cofunded this research.
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