Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 9.3.4 - Accuracy of a monofiber optical probe
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00225
 Material Information
Title: 9.3.4 - Accuracy of a monofiber optical probe Experimental Methods for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Vejrazka, J.
Sechet, P.
Vecer, M.
Orvalho, S.
Ruzicka, M.
Cartellier, A.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: optical probe
bubble-fiber interaction
void fraction
measurement error
chord distribution
 Notes
Abstract: An optical probe is a sensor, which indicates whether liquid or gas phase is present at its tip. The most common use of the optical probe is the measurement of local void fraction in bubbly flows. In that case, the probe tip is placed in a point of interest and the local void fraction of the gas is determined as the total time spent in the gas divided by the total measurement time. In this work, the accuracy of the void fraction measurements by an optical probe is investigated experimentally. An interaction of a freely rising bubble with a tip of a mono-fiber optical probe is studied using a high-speed imaging. The dwell time (period of time, for which the probe tip stays within the bubble) is obtained from both the optical probe signal and visualization. The dwell time is then compared with an ideal value expected for a non-perturbed bubble. In this way, an error due to intrusive nature of the probe is evaluated. The dwell time error depends on the piercing location. If the probe tip pierces the bubble near its pole, the bubble decelerates. The indicated dwell time is then shorter than in a hypothetical case of an unperturbed bubble. Oppositely if the probe tip pierces the bubble near the equator, the bubble shrinks horizontally. Due to a corresponding change of its added mass, the bubble accelerates. The probe tip also deforms under the action of capillary forces. All these effects lead to a shortening of the dwell time when compared with an unperturbed bubble. In a typical bubbly flow, the piercing locations are random. Proper averaging of the dwell time error yields an estimation of the void fraction error. The void fraction is always underestimated, and the error is observed to depend mostly on a modified Weber number, M. This number expresses the ability of the inertia associated with the bubble motion to overcome the surface tension forces coming from the probe tip. The error on void fraction measurement is small if M is large enough. The value of the modified Weber number is hence also a criterion for the selection of an appropriate optical probe. A correction for the measurement error of the void fraction is proposed. This correction requires the knowledge of M. Two ways for estimating M a posteriori from the optical probe measurements are proposed. These estimations are applicable, however, only to monodispersed bubbly flows.
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
 Record Information
Bibliographic ID: UF00102023
Volume ID: VID00225
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 934-Vejrazka-ICMF2010.pdf

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



Accuracy of a monofiber optical probe


Jiri Vejrazka1, Philippe Sechet2, Marek Vecer3, Sandra Orvalho1, Marek Ruzicka1 and Alain
Cartellier2

1 Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic
Rozvojova 135, 165 02 Praha, Czech Republic
2 Laboratoire des Ecoulements Geophysiques et Industriels (UMR 5519),
Domaine Universitaire, BP 53, 38041 Grenoble Cedex 9, France
3 Faculty of Metallurgy and Materials Engineering, VSB-TU Ostrava
17. listopadu, 70833 Ostrava Poruba, Czech Republic
vejrazka@icpf.cas.cz, philippe.sechet@hmg.inpg.fr


Keywords: Optical probe, bubble-fiber interaction, void fraction, measurement error, chord distribution



Abstract

An optical probe is a sensor, which indicates whether liquid or gas phase is present at its tip. The most common use of the optical probe
is the measurement of local void fraction in bubbly flows. In that case, the probe tip is placed in a point of interest and the local void
fraction of the gas is determined as the total time spent in the gas divided by the total measurement time.
In this work, the accuracy of the void fraction measurements by an optical probe is investigated experimentally. An interaction of a
freely rising bubble with a tip of a mono-fiber optical probe is studied using a high-speed imaging. The dwell time (period of time, for
which the probe tip stays within the bubble) is obtained from both the optical probe signal and visualization. The dwell time is then
compared with an ideal value expected for a non-perturbed bubble. In this way, an error due to intrusive nature of the probe is evaluated.
The dwell time error depends on the piercing location. If the probe tip pierces the bubble near its pole, the bubble decelerates. The
indicated dwell time is then shorter than in a hypothetical case of an unperturbed bubble. Oppositely if the probe tip pierces the bubble near
the equator, the bubble shrinks horizontally. Due to a corresponding change of its added mass, the bubble accelerates. The probe tip also
deforms under the action of capillary forces. All these effects lead to a shortening of the dwell time when compared with an unperturbed
bubble.
In a typical bubbly flow, the piercing locations are random. Proper averaging of the dwell time error yields an estimation of the void
fraction error. The void fraction is always underestimated, and the error is observed to depend mostly on a modified Weber number, M. This
number expresses the ability of the inertia associated with the bubble motion to overcome the surface tension forces coming from the probe
tip. The error on void fraction measurement is small if Mis large enough. The value of the modified Weber number is hence also a criterion
for the selection of an appropriate optical probe.
A correction for the measurement error of the void fraction is proposed. This correction requires the knowledge of M. Two ways for
estimating M a posteriori from the optical probe measurements are proposed. These estimations are applicable, however, only to
monodispersed bubbly flows.


Introduction

In experimental studies of two-phase flows (limited to a
liquid containing gas bubbles in this study), the
experimenter often needs information about the phase
presence in a particular point of interest. For this purpose,
one defines a phase-indicating function K(t), which is (for
example) y(t) = 0 if the point of interest is occupied by the
liquid at time t and yu(t) = 1 if there is gas. The duration of
periods, during which the point of interest is within the gas
(referred thereinafter as gas dwell time r) can be extracted
from 4(t). A probability-density-function of gas dwell time
is one of the inputs of the statistical treatment, using which
the bubble size distribution or Sauter mean size can be
estimated (Revankar & Ishii 1992, 1993, Cartellier 1999).
The most basic parameter, extracted from the phase indicator
function, is the local void fraction


a =, yq(t)dt = (1)
S0 mnt
where r is the mean dwell time and t,, is the mean
interval between bubbles.
A very common sensor for the experimental
determination of the phase indicator function is the optical
probe (Miller & Mitchie 1970, Cartellier & Barrau 1998,
Saito & Mudde 2001). This probe consists of an optical
fiber with a sharp tip, which is placed in a point of interest.
The other end of the fiber is split in two branches: one
branch brings light from a source, and the other directs the
light reflected from the tip to a photodetector. As the
amount of reflected light depends on the refraction index of
the fluid surrounding the tip, the photodetector produces a
signal, which can be easily reduced into the phase-indicator
function.
As any experimental method, the measurements by
optical probe are a subject of accuracy issues. In the case of






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


Figure 1: Experimental setup.

optical probes, the major source of error is the intrusive
nature of the sensor. As the probe tip is introduced in the
point of interest, it modifies locally the flow and the bubble
behavior. The phase indicator function y0op(t), seen by the
probe, therefore differs from its ideal counterpart y((t),
which would be found by a hypothetical sensor free of
intrusive effects.
The experimental error of optical probes and sources of
errors has been studied several times (Barrau et al. 1999,
Kiambi et al. 2003, Julia et al. 2005, Chaumat et al. 2007,
Vejrazka et al. 2010). All these studies suggest that the local
void fraction, indicated by the optical probe, is
underestimated. Julia et al. 2005 in their investigation of the
optical probe error observed that the indicated dwell time is
longer (compared to the dwell time deduced from
high-speed imaging in the same flow without the probe), if
the bubble is pierced near its center. Oppositely, the
indicated dwell time was shorter for a piercing near the
perimeter. The bubble volume, determined from the optical
probe signal, was always smaller than its real value, and this
error was more important for smaller bubbles.
In this study, we examine the error of the optical probe
measurement in bubbly flows by a method similar to that
used by Julia et al. 2005, i.e. by comparing the probe signal
to high-speed visualization of the bubble-probe interaction.
The error of the dwell time is evaluated for various piercing
locations. By its averaging, the error of the local void
fraction is found. The measurement error is determined in
dependence on the bubble size and rising velocity, yielding
a dimensionless criterion for its estimation. The error
estimate allows correcting the void fraction measurements,
if a dimensionless number (containing the bubble size and
velocity) is known. Because this number is usually not
known a priori, two ways of for its estimations are
proposed.
More complete treatment of presented experiments can
be found in Vejrazka et al. 2010.


Experimental setup and procedures

The experimental setup used in this study is depicted in
Figure 1. The bubble-probe interaction was studied in a
rectangular glass vessel with a square bottom of inner size
llxll cm and height of 26 cm. A bubble generator was
placed in the vessel bottom. The bubble generator is a
device that produces bubbles of prescribed size by imposing
their detachment by a rapid motion of a capillary, through


Figure 2: Considered bubble geometry.

which gas is injected (Vejrazka et al. 2008). The optical
probe was inserted into the vessel from the top. It was the
"Type 1C Probe" delivered by A2 Photonic Sensors Ltd.,
Grenoble, France. The probe consists of a metallic body
(1.5 mm in diameter), from which the optical fiber
protrudes by 15 mm. The fiber diameter was Do = 0.13 mm
and its tip, produced by etching process, has an apex angle
of 30. The probe was fixed to a 2D traversing device, by
which its position was adjusted in the horizontal plane. The
probe tip was located typically 2 cm above the tip of bubble
generator's capillary.
The interaction of the optical probe tip with a bubble
rising in a stagnant liquid was recorded by a high-speed
camera (Redlake HS-4) with a frame rate of 10 000 fps and
with resolution of at least 170x512 pixels. The illumination
system consisted of a 500W halogen lamp, two diffusers
(opaque glass plates, not shown in Fig. 1) and an
improvised infrared filter (also not shown) made by another
glass vessel filled with water. A computer instrumented with
a multifunction data acquisition board recorded the output
signal from the optical probe and also the camera
synchronization signal. A care was taken about proper
synchronization of the probe signal recording and the
high-speed imaging.
The experiments were performed with air bubbles in
pure water and in two glycerin solutions. Experiments in
water were carried out at 30.10C. Corresponding density
was p = 995.6 kg/m3, viscosity p = 0.797 mPa.s and surface
tension a = 71.2 mN/m. The solution denoted "glycerin 8"
consisted of 60% (by weight) glycerol and water, and its
properties at 25.90C were p = 1152.0 kg/m3, p = 8.87 mPa.s
and = 65.3 mN/m. Finally, the solution denoted
"glycerin 20" consisted of 70% (by weight) glycerol and
water, and its properties at 27.10C were p = 1174.2 kg/m3,
/ = 13.6 mPa.s and r = 60.5 mN/m.
When experimenting, a bubble was produced by the
bubble generator, the optical probe signal was recorded and
the high-speed movie of the bubble-probe interaction was
acquired synchronously. The experiments were carried out
for several bubble sizes (ranging from 0.7 to 3 mm) in each
liquid. For a given size and liquid, the experiment was
repeated with different bubble-probe offsets (distance x
depicted in Fig. 2).
The high-speed movies of the bubble-probe interaction
were then treated using single-purpose software, which was
written in Matlab and which used its Image Processing
Toolbox. The treatment of each image frame consisted of
two major steps: A) Image processing, during which the
points laying at the bubble boundary were detected, and B)
boundary fitting, during which the detected boundary was
fitted with a smooth curve, and the bubble size and position
of its center-of-mass was found for this smoothed boundary.





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


c)


0


top top2 t


/2/U1
S/ul ref

to treft tpf2 t


Figure 3: Reconstruction of phase-indicator functions for a) optical probe; character of the signal S(t) is exaggerated, b) rigid
probe, c) reference probe.


The boundary found in step A) was used for constructing
the phase-indicator functions Vyg and which will be
described below. For the step B), a double-ellipse curve
(curve consisting of two halves of an ellipse, described by ax,
azl, az2 and 0, see Fig. 2) was found as the best fit of bubble
boundaries. This fitted boundary was then used for inferring
the sizes (ax, a,, and az2), the aspect ratio X= 2., .. + a.2),
inclination angle 0, the center-of-mass coordinates and the
bubble volume-equivalent diameter D. The bubble velocity
u was calculated by differentiating the center-of-mass
position. Finally, the bubble-probe offset x (Fig. 2) was
found as a difference between the horizontal probe position
and the horizontal position of bubble center-of-mass in the
last frame before contact.
On basis of the experimental data, three different
phase-indicating functions Vy(t) are constructed. One of them
(yop) is based on the optical probe signal, while the two
others (ylg and i are created using the information
provided by high-speed imaging. To construct any of these
phase-indicator functions, the time of bubble first contact
with the probe tip ti and the time of last contact t2, when
yl(t) should switch, have to be determined. The dwell time
of the measuring location within the bubble is found as a
difference of both, r = t t1.
The optical probe phase-indicating function yiop is
obtained from the photodetector voltage signal S(t) by
comparing it with a threshold value (Fig. 3a), which
was set slightly above the noise level of the "in liquid"
signal. Thereinafter, we use denomination optical probe
signal directly for y0op(t).
The phase-indicating function Vyg is constructed on
basis of the high-speed imaging in the same way as by Julia
et al. 2005. For each image, the distance d (Fig. 3b) between
the probe tip and the nearest point of bubble boundary is
measured. If the probe tip is inside the bubble, d is
considered negative. The first and the last contact times tgl
and tg2 are found by interpolating for zero d. The distance d
is evaluated toward the initial position of the probe tip
(which displaces during the interaction with the bubble) and
fyg is hence free of effects of probe deformation effects. The
function lVg could be produced by an imaginary probe,
which has the same intrusive character in terms of interface
deformation and bubble trajectory modification as a real
optical probe, but which does not deform. This imaginary


probe will be thereinafter referred to as a rigid probe. (We
expect that an optical probe with non-deformable tip would
produce the same signal).
The last phase-indicating function yVref is based only on
images obtained before the first bubble contact with the
probe. It is assumed that the approaching bubble maintains
its initial shape, velocity and direction. The shape is
determined from the last frame (taken at time to), in which
the probe tip is still observed outside the bubble. The
distances 11 and 12 (see Fig. 3c) are measured and the
corresponding instants of the first and the last contact are
calculated as tre = to + 111ul and tre = to + 12/Ul, respectively,
where ul is the bubble velocity at the first contact. As ref is
based only on images acquired before any contact between
the bubble and the probe, the bubble is considered as
unperturbed. The function yref is hence used as a reference
for judging on the probe intrusive nature, and it also
denoted as reference probe signal thereinafter.


Results and discussions
Visual observations
The bubble disturbance from the probe is documented
in Figure 4, where the initial and terminal bubble shapes are
compared. The background photos were taken at time t,2p,
when the bubble would have the last contact with the
reference probe (which is non-intrusive). The dotted
contours show the shape and position of the non-disturbed
bubble at this time, i.e. it is the bubble contour observed at
time to, but shifted by distance 12 (Fig. 3c) in direction of
bubble motion. This dotted contour would overlap with the
observed bubble shape if the bubble is not affected by the
probe. This is not the case, as bubble changes its velocity
and shape during its interaction with the probe.
In agreement with Julia et al. 2005, two different
bubble behaviors are observed in dependence of the
bubble-probe offset x. In the case of small x (first column in
Fig. 4), the bubble is decelerated during the interaction. This
is apparent from the bubble position, which is lower than
the expected position of non-disturbed bubble. As a result of
this deceleration, the probe tip is still located inside the
bubble at the ideal last-contact time tref. The deceleration is
hence delaying the last-contact time top2 of the optical probe,


b)


)


a)




4 )






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


Figure 4: Comparison of the initial and the terminal bubble shape. Snapshots are taken at time tre,. White dotted lines show
the bubble contour at time to, which was displaced by distance 12 in direction of velocity vector ul (see Fig. 5c). Arrows
indicate the position of the probe tip before the first contact with the bubble. Observations in water, bubble sizes are 1.12,
1.48, 1.78 and 2.23 mm (from the top).


compared to the reference probe, t,f2.
In the case of larger x (third and fourth columns in Fig.
4), the vertical bubble position corresponds roughly to an
expected position of an unperturbed bubble. A significant
bubble deformation is observed in the part of interface,
which is touching the probe. Due to this horizontal
shrinking, the probe tip is already located outside the bubble
at time tre2. This suggests that the bubble deformation
advances the last-contact time to,2, indicated by the probe,
when compared to the reference probe.
Finally, a deformation of the optical probe is observed
if the bubble is pierced near its equator. This is visible in the
last two columns in Fig. 4, where the initial probe position
is shown by an arrow.

Probe signals and indicated dwell time
The typical time series S(t) of optical probe output is
shown in Figure 5 for three bubble-probe offsets.


Differences in the duration of the "gas level" part of signal
are the only noticeable changes when the offset x is
modified (compare Figs. 8a and 8b).
The three phase-indicating functions deduced from
either the optical probe signal (yo,,) or high-speed imaging
(t/g, i are also compared in Fig 5. The first contact time
of the three phase-indicator functions is almost the same.
Oppositely, the three last contact times significantly differ.
Previously discussed effects of delaying to,2 relative to tre,
in the case of small x (Fig. 5a) and of its advancing for large
x (Figs. 5b and 5c) are reflected.
The comparison of signals of the optical probe (yo,,)
and rigid probe (yg) is quite revealing. For most bubble
probe offsets (Fig. 5a), they are very similar and their minor
difference can be attributed to the experimental uncertainty
(error on time information in Vyg is related to the time
interval between movie frames, which was about 0.1 ms).
The similar character of yo, and ,g is expected, as both


a) (case bl) b) (case b3) c) (case b4)







I I-------- ,-- I----- ----

0 1 2 3 t(ms) 0 1 2 3 t(ms) 0 1 2 3 t(ms)

Figure 5: Optical probe signal S(t), and phase-indicating functions uo,,(t), ylg(t) and t) for three cases shown in Fig. 5.
Arrows show the falling time of yu, (i.e. last contact t,2) evaluated with respect to the final probe position.






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


3.0
2.5
2.0
t 1.5
1.0
0.5
0.0



12

1.0

0.8
10.6
0.4

0.2
0.0


0 -0.5 0.0
X(-)


-1.0 -0.5 0.0
X(-)
1.2

1.0

0.8

0.6
0.4

0.2

0.0 .
-1.0 -0.5 0.0
X(-)
1.2
1.0

0.8

t0.6

0.4

0.2
0.0


0.5 1


1.2
1.0
0.8
0.6
0.4
0.2
0.0
.0 -1.0


-0.5 0.0
X(-)


2.0
c)
1.6

1.2

S0.8

0.4 -

S.... 0.0
0.5 1.0 -1.0 -0.5 0.0
X(-)


1.6
e) 1.4
1.2
1.0
S 0.8
0.6
0.4
0.2 a
0 10 0.0
0.5 1.0 -1.0


-0.5 0.0
X(-)


-1.0 -0.5 0.0
X(-)


0.5 1.0


Figure 6: Comparison of dimensionless dwell times Top, Tg, Tref and eq. (2). Experimental liquid and equivalent bubble
diameter D are (a) water, 1.12 mm; (b) water, 1.48 mm; (c) water, 2.23 mm; (d) glycerin 8, 1.45 mm; (e) glycerin 8, 3.01
mm; (f) glycerin 20, 1.84 mm; (g), glycerin 20, 2.80 mm. Figures (a), (b) and (c) correspond to the first, second and fourth
row in Fig 4, respectively.


indicate whether the probe tip is inside or outside the bubble.
Only when the probe touches the bubble near its equator
(Fig. 5b and 5c), an important difference between yuop and Vug
appears. Both functions give the same result on the first
contact time, but they noticeably differ in the last contact
time. Most of this difference can be attributed to the probe
deformation. The probe bends after the contact with the
bubble, and thus the probe sensitive tip is displaced
outwards the bubble (see also last two columns in Fig. 4).
Function yug was evaluated with respect to the probe initial
position. If it is reevaluated by taking in account the final
probe position instead, the last-contact time tg2 occurs


sooner (it is indicated by an arrow in Figs. 5b and 5c) and it
corresponds well to top2. From these observations, we can
draw two conclusions: (i) The rigidity of the optical probe
tip is of concern for proper measurements in bubbly flows
and it increases the dwell time error for piercing near the
equator. This is of a large importance, because piercing a
bubble near its equator is more probable than piercing it
near the pole. (ii) If the probe deformation is taken in
account, the phase-indicator function y1g corresponds well to
uop. This suggests that yuop indicates correctly which phase is
present at the probe tip.
It is obvious that the imperfections of the optical probe


TI. a)
-A- T a)
A Tg
of
- eq. (2) ,
--4

0^^^ ^

^y'-4


0.5 1.0


0.5 1.0


0.5 1.0






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


0.5 1 1.5 2
D (mm)


2.5 3


0



-0.2



"1-0.4



-0.6


10
M (-)


Figure 7: Error on voidage measurements, (a) as a function of bubble size, (b) as a function of modified Weber number


(intrusiveness and deformability) introduce an error on the
determination of the last-contact time t2 and thus also on
determination of the bubble dwell time r. In Figure 6, the
dwell times determined from the three phase-indicating
functions are compared for bubbles of several sizes and in
different liquids. The dwell time is plotted in a
dimensionless form, T= 7,,, + az2), against
dimensionless bubble-probe offset, X = xlax. The
dimensionless dwell time Trf, obtained from the reference
probe signal follows well the relationship

Tref = -X (2)
which is expected from geometrical considerations about
the observed ellipsoidal bubble shape (Fig. 2).
Figures 6a 6c show the profiles of dwell time across
the bubble for three different bubble sizes in water. Both Tr
and Tg substantially differ from the ideal dwell time profile
Tr.f for bubble sizes 1.12 mm (Fig. 6a) and smaller. An
important overestimation of dwell time near bubble pole is
evident. With increasing bubble size, To and Tg approach to
the reference dwell time Trf, but some deviations are always
observed (Figs. 6b and 6c). Coherently with the previous
observations, ro and Tg overestimates the dwell time when
piercing the bubble near its pole (X~ 0) and underestimates
it near the equator (X- -1 or +1).
Figures 6d 6g also show the dwell time profiles, but
obtained in glycerin solutions. Behavior is similar to that
identified for water, except that bubbles of bigger size are
now needed to get a rzo, which approaches to rzf. This
difference is related to the change in the bubble rise velocity.
The difference between Top and Tg is more pronounced in
glycerin solutions than in water. The error due to probe
deformation is hence more important in glycerin solutions.
It is perhaps a consequence of longer dwell time, yielding to
stronger impulse of forces bending the probe.

Mean dwell time and error of the void fraction
measurements
The mean dwell time can be evaluated by averaging of
the dwell time for various piercing locations. This can be
done by an integration of the dwell time profiles shown in
Fig. 6. It is obtained for the mean dwell time of the optical
probe signal (in a dimensionless form)

T = JXT (X)dX (3)
-1


and similarly for the rigid probe signal

Tg= X\T, (X)dX. (4)
-1
The factor X\ in the integrand appears because the
probability of piercing at offset X is increasing linearly with
[X], and hence appropriate weighting must be considered
when averaging. For the mean dwell time indicated by the
reference probe, equation (2) can be considered and the
integration analogous to (3) or (4) can be done directly,

Tej X X dX = 2. (5)

Dwell times Top and Tg differ from the reference dwell
time Crf, and a difference is expected also for their mean
values. As a consequence, also the void fraction indicated
by the probe differs from the reference value (cf. Equation 1
for the relation between mean dwell time and local void
fraction). The error of the local void fraction indicated by
either the optical or rigid probe can be obtained by
comparing the corresponding means,

op= -1= o- 1 -I XT (X)dX -1 (6)
aref ref 2
and

S 1 = -1 = IXT(X)d-, (7)
arf ref 2 1
where a,o, ag, aref are the local void fraction deduced from
,op, Vg and respectively.
The void fraction error, obtained for the optical and
rigid probes by integrating dwell time profiles (Fig. 6) using
last two equations, is shown in Figure 7a as a function of
the bubble size. Nevertheless, it is useful to express the
error as a function of a dimensionless quantity. A suitable
dimensionless number is the modified Weber number
(Vejrazka et al. 2010)

M = (8)

which characterizes the ability of the liquid momentum
associated with the bubble motion (~ pD3ul) to overcome
the impuls of surface tension forces from the probe tip
(~ aDopD/ul). Figure 7b shows the dependence of the void
fraction error on M. The data of both errors collapses more
or less to a single curve, supporting the suitability of
dimensionless description using M. In the case of error of


0



-0.2



-0.4



-0.6


a) U
A O A Ater
gA*


A 0
A o 0",P, water
A 0 o, glycerin 8
2 Ac, glycerin 20
water
5 Ec, glycerin 8
A -, glvcerin 20
. . * . . **.. .






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


2.00

1.75

3 1.50

1.25

1.00

0.75


10 100
Y'3CM (-)


1000


X2!3 CM (-)


100 1000


Figure 8: Change of (a) bubble velocity and (b) dwell time during the bubble-probe interaction at zero bubble-probe offset.


the rigid probe, this dependence is empirically fitted with
g = -0.5M -04. (9)
The error of the rigid probe, Eg, is caused by the intrusive
nature of the optical probe, which affects the bubble motion
and shape. The error of the optical probe, Sop, is further
enlarged by the probe deformation. We do not fit the cop data,
as a description of the probe deformability is not available.
The equation (9) can be used to correct the results of
the void fraction measurements by rigid probes, if the
modified Weber number M is known. The validity of Eq. 9
is limited to monodispersed bubbly flows.

A posteriori estimation of the modified Weber number
When an optical probe is used for measurements, the
value of modified Weber number M is usually not known a
priori, and Equation (9) cannot be therefore used for the
correction of void fraction error. There are two ways for
estimation of M from the measurements a posteriori.
The modified Weber number can be estimated by
taking in account the bubble deceleration in the case of
on-axis piercing (with offset x = 0), for which the longest
dwell time (rop.ma or rg,max) is detected (we assume a
monodispersed bubbly flow). After the first contact, the
bubble decelerates under the action of surface tension forces
(scaling as - zDop), which counteract against the liquid
inertia, i.e. the added-mass force (- zD3/6-C-duldt).
Considering these two forces, the bubble deceleration is
easily expressed (Vejrazka et al. 2010). The decrease of the
velocity between the first and last contact times (trf, and
tref,) in the case of zero offset is

u2 = 1 2/3 I C (10)

where C is the added-mass coefficient of the bubble. The
dwell time indicated by the optical probe is longer than the
reference time and it holds for on-axis piercing

Sgmax op max Zref mal X2/3M (11)

Last two equations are compared with experimental data in
Figure 8. The agreement is good for / Cn > 10.
For a rigid probe in a monodispersed bubbly flow, the
mean and maximum dwell time can be compared. From


equation (5), it follows rfrI /Tre = 3/2. Combination of
equations (7), (9) and (11) hence yields

= X22/3C 1 0.5M-04) (12)
3g, 1 grax
In a monodispersed bubbly flow, the ratio of mean and
maximum dwell time is hence a monotonically increasing
function of M. Because both mean and maximum gas dwell
time are known from the optical probe measurements, this
dependence can be used for estimation of M, if the aspect
ratio y is guessed. This procedure is applicable only to
monodispersed bubbly flows, in which all bubbles move
with a constant velocity.
There is another way for estimation of the modified
Weber number M a posteriori. The shape of the dwell time
profile of the rigid probe differs from the ideal profile (2)
and the difference is larger for smaller values of M. The
profile can be fit quite well with

T = k -X"),. (13)
The exponent would be n = 2 in the case of an ideal probe
(cf. Equation 2), which is not perturbing bubbles after the
contact. Experimentally observed values of n (obtained by
fitting Eq. 13 to the dwell time profiles shown in Fig. 6) are
lower. The exponent n depends on the modified Weber


1.6


1.4


1.2


1.0


0.8


M (-)
Figure 9: Dependence of the shape exponent n on the
modified Weber number.









2.5

2.0

S1.5

1.0

0.5

0.0
0.00 0.25 0.50 0.75 1.00
T(-)
Figure 10: Probability-density-function of the
dwell-time for dwell-time-profiles characterized by
equation (13) and k = 1.

number M and this dependence is shown in Figure 9.
In a monodispersed bubbly flow, the non-ideal shape of
the dwell-time profile implicates also a change in the
probability-density-function (p.d.f.) of the dwell time, p(r).
It is easily derived (Vejrazka et al. 2010) that for the dwell
time profile in form of equation (13), the p.d.f is
2
2 T [u K u )l n 1



U1
T k(ai +az2) k(ai +az2)
PrWz> = ,,zi +z2
for < k


0 otherwise;
(14)
The predicted p(r) is shown in Figure 10 for k = 1 and a few
values of n. The shape of p(r) is controlled by n, while the
other parameters (k, o,, i. + az2)) only stretches it.
The exponent n can be hence obtained by fitting the
dwell-time p.d.f. (obtained experimentally) by Equation (14),
and M can be then estimated using the plot shown in Figure
9. This procedure is again applicable only to monodispersed
bubbly flows, in which all bubbles move with a constant
velocity.
Finally, it is remarked that the knowledge of M is not
necessarily required for estimation of the error of void
fraction measurements. The dwell time p.d.f, which is
obtained experimentally, can be converted into the dwell
time profiles r(x) (see equation 21 in reference Vejrazka et
al. 2010). The profile r(x) is then known and equation (6)
can be used directly for the estimation of the void fraction
error.


Conclusions

The measurement error of an optical probe is caused
mostly by its intrusive nature. A buble, which comes in a
contact with the probe tip, gets perturbed and this
perturbation is modifying the dwell time of the bubble in the
measuring location. A supplementary measurement error is
caused by the probe deformation.
The dwell time error depends on the piercing location.


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

If the probe tip pierces the bubble near its pole, the bubble
decelerates. The indicated dwell time is then shorter than in
a hypothetical case of an unperturbed bubble. Oppositely if
the probe tip pierces the bubble near the equator, the bubble
shrinks horizontally. The probe tip also deforms under the
action of capillary forces. These effects lead to a shortening
of the dwell time when compared with an unperturbed
bubble.
The void fraction is always underestimated, and the
error observed to depend mostly on a modified Weber
number, M. A correction for the measurement error of the
void fraction is proposed. This correction requires the
knowledge of M. Two ways for estimating M a posteriori
from the optical probe measurements are proposed. These
estimations are applicable to monodispersed bubbly flows.


Nomenclature

ax horizontal bubble semi-axis, Fig. 2 (m)
azi upper bubble semi-axis, Fig. 2 (m)
a.2 lower bubble semi-axis, Fig. 2 (m)
C added-mass coefficient of the bubble (-)
d distance between the probe tip and projection of
bubble boundary, Fig. 3 (m)
D bubble diameter (m)
Dop diameter of the probe tip (m)
k coefficient determining shape of dwell-time
profile, see eq. (13) (-)
1 distance between the probe tip and projection of
bubble boundary, Fig. 3 (m)
M modified Weber number, see Eq. (8), (-)
n exponent determining shape of dwell-time
profile, see eq. (13) (-)
p probability-density function (1/s)
S output of the optical probe (V)
t time (s)
T dimensionless dwell time (-)
x bubble-probe offset, Fig. 2 (m)
X dimensionless bubble-probe offset (-)

Greek letters
a void fraction (-)
E void fraction error (-)
X bubble aspect ration (-)
V phase-indicator function (-)
P viscosity (Pa. s)
Ur surface tension (N/m)
0 bubble inclination angle (-)
T dwell time (s)

Subsripts
g relevant to the rigid probe or its signal
max maximal value, or for on-axis piercing
op relevant to the optical probe or its signal
ref relevant to the reference probe or its signal
0 relevant to conditions before contact of the
bubble with the probe
1 relevant to the contact of the bubble with the
probe (except ai and az2)
2 relevant to the last contact of the bubble with the
probe (except a,1 and az2)






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


Acknowledgements

This research was supported by the Grant agency of ASCR
(project no. IAA200720801). Indispensable equipment (the
optical probe, high-speed camera, optical equipment and
data acquisition system) was acquired in the past for
projects supported by Grant Agency of the Czech Republic
(projects no. 104/07/1110 and 104/06/P287).


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