Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 8.3.3 - Bayesian Design of Magnetic Resonance Studies of Multiphase Flows
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 Material Information
Title: 8.3.3 - Bayesian Design of Magnetic Resonance Studies of Multiphase Flows Experimental Methods for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Holland, D.J.
Tayler, A.B.
Sederman, A.J.
Blake, A.
Gladden, L.F.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: multiphase flow
buble size measurement
velocity measurement
 Notes
Abstract: This paper presents two new magnetic resonance (MR) techniques that enable measurements of systems that evolve too quickly or in which the signal-to-noise ratio is too low to be imaged using conventional techniques. The first approach that will be described is known as compressed sensing and exploits the sparsity of the image being acquired to reduce acquisition times. This technique will be demonstrated using velocity mapping of single phase flow of both liquid and gas through packed beds. The technique can be used to reduce image acquisition times by a factor of 4 over conventional imaging techniques, without sacrificing the quality of the image and has been used to image gas at a spatial resolution an order of magnitude greater than has previously been achieved. The second technique utilises a Bayesian framework to directly access parameters of the system under study. In the example demonstrated here, measurements of a bubble size distribution are presented using 2 orders of magnitude less data than would be required using a conventional imaging approach. This new methodology enables measurements of systems that would otherwise be impossible to measure using MR.
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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Volume ID: VID00203
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 833-Holland-ICMF2010.pdf

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


Bayesian Design of Magnetic Resonance Studies of Multiphase Flows


D.J. Holland*, A.B. Tayler*, A.J. Sederman*, A.Blake! L.F. Gladden*

Department of Chemical Engineering and Biotechnology, University of Cambridge, Cambridge, CB2 3RA, UK
t Microsoft Research Cambridge, 7 J. J. Thomson Avenue, Cambridge CB3 OFB, UK.
djh79@cam.ac.uk
Keywords: multiphase flow, bubble size measurement, velocity measurement




Abstract

This paper presents two new magnetic resonance (MR) techniques that enable measurements of systems that evolve
too quickly or in which the signal-to-noise ratio is too low to be imaged using conventional techniques. The first
approach that will be described is known as compressed sensing and exploits the sparsity of the image being acquired
to reduce acquisition times. This technique will be demonstrated using velocity mapping of single phase flow of
both liquid and gas through packed beds. The technique can be used to reduce image acquisition times by a factor of
~4 over conventional imaging techniques, without sacrificing the quality of the image and has been used to image
gas at a spatial resolution an order of magnitude greater than has previously been achieved. The second technique
utilises a Bayesian framework to directly access parameters of the system under study. In the example demonstrated
here, measurements of a bubble size distribution are presented using 2 orders of magnitude less data than would be
required using a conventional imaging approach. This new methodology enables measurements of systems that would
otherwise be impossible to measure using MR.


Introduction

Magnetic Resonance (MR) imaging is increasingly be-
ing used to study multiphase flows as it has the ability
to non-invasively study density, concentration and ve-
locity (Fukushima (1999)). However, most MR studies
of multiphase systems have been restricted to "magnetic
resonance friendly" systems; i.e. those which evolve on
a sufficiently slow timescale and for which the system
under study has sufficiently high signal-to-noise, usu-
ally associated with relatively long nuclear spin relax-
ation times. In this paper, two techniques are described
to reduce the acquisition times of MR measurements.
The reduced acquisition times of these techniques en-
able measurements of systems that evolve too quickly,
or in which the signal-to-noise ratio is too low, to be im-
aged using conventional techniques.
One of the advantages of MR is its ability to study
both the distribution of a fluid and the local velocity
with which it is moving without the use of a tracer.
This ability has led to an increase in the use of MR to
study multiphase flows, including, for example, the rhe-
ology of complex fluids (Callaghan (1999)), the flow of
liquids through packed beds (Sederman et al. (1998)),
and granular flows (Fukushima (2006); Holland et al.


(2008)). However, the time taken to acquire a veloc-
ity image can be prohibitively long, particularly when
the velocity distribution varies with time or the signal-
to-noise ratio (SNR) is low. To overcome these limita-
tions many studies have explored methods to increase
the temporal resolution of velocity-encoded imaging
(e.g. Kose (1991a,b); Sederman et al. (2004); Galvosas
& Callaghan (2006); Jung et al. '2' I,,. These "ultra-
fast" (< 1 s) techniques have all been demonstrated to
provide a substantial improvement in the temporal reso-
lution. However, despite these advances, each has lim-
itations in terms of the systems that can be studied and
the trade-off between spatial and temporal resolution.
The first technique described in this paper, com-
pressed sensing (CS), provides a method of reducing
data acquisition times by utilising ideas that arose from
image compression (Candes et al. (2006); Donoho
(2006)). CS utilizes the fact that images are highly
compressible, and certain linear transforms render them
sparse, that is, they can be accurately represented by
only a few non-zero elements. Sparsity in a trans-
form domain allows good reconstruction from an under-
sampled set of measurements in k-space. In this paper
we demonstrate that CS can be used in combination with
MR to produce maps of the velocity distribution of a liq-











uid or gas. For gas phase imaging, we use CS to obtain
images at a higher spatial resolution than has previously
been practicable. A particular advantage of CS is that
it is applicable to a wide range of systems and can be
applied in conjunction with both conventional and ultra-
fast velocity imaging sequences, thus in future it will
enable velocity measurements of systems evolving over
very short time scales.
The second technique described in this paper uses
much more detailed knowledge of the system studied
to vastly simplify and accelerate the data acquisition.
In conventional imaging, a complete image is acquired
and following this the required information is extracted
using an image processing algorithm. For example, in
bubble sizing of a gas-liquid bubbly flow a series of im-
ages would be segmented to identify the bubbles in each
image and a size distribution reconstructed from these.
However, the end result of this process is the size distri-
bution the images themselves are of little or no inter-
est. This means that most of the information acquired is
redundant. The second technique described in this pa-
per poses the bubble sizing question in a Bayesian sense
to estimate the size distribution directly and thus signif-
icantly reduce the number of data points required and
eliminate the need for complex, ultra-fast data acquisi-
tion techniques.
This paper briefly describes the theory for both CS
and the Bayesian approach used. Results are then pre-
sented using each technique in a specific application. In
the case of CS, results are shown for single phase flow
of liquid and gas through a packed bed. These results
demonstrate that the imaging time can be reduced by
a factor of almost 4 with negligible effect on the im-
age quality and that this enables the measurement of ve-
locity of a gas at a resolution that is an order of mag-
nitude greater than had previously been achieved. The
Bayesian data analysis approach is demonstrated for the
measurement of bubble size distributions in gas-liquid
bubbly flows from only 128 data points, nearly two or-
ders of magnitude less than that required using a con-
ventional approach.

Theory

Compressed Sensing CS allows accurate reconstruc-
tions of 'incoherently' under-sampled k-space data sets
by utilising the principles behind image compression
(Candes et al. (2006); Donoho (2006)). In this con-
text, 'incoherent' means that the under-sampling causes
incoherent artefacts, or more formally that the sampling
operator must not be easily (sparsely) represented in
some chosen transform domain of the image (Candes et
al. (2006)). These 'incoherent samples' are fused with
the prior knowledge that the image is sparse in the cho-


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


sen transform domain, and the image is then recovered
using a non-linear reconstruction method (e.g. Donoho
(2006)). A variety of transforms exist in which im-
ages can be sparsely represented and in which the
under-sampling leads to incoherent artefacts. Exam-
ples of these transforms include spatial finite-differences
(i.e. computing differences of neighboring pixels) and
wavelet transforms. Each of these transforms will be ap-
propriate for different types of images, though the spa-
tial finite differences has been found to be most useful
for the velocity images studied in this work.
The image reconstruction algorithm that we have used
is a variant of basis pursuit (Lustig et al. (2007)), which
uses the 1l-norm as a surrogate for sparsity. The 1 -norm
formulation is attractive because it leads to a convex op-
timization problem which can be solved by efficient al-
gorithms and because it has allowed strong theoretical
analysis of the quality of reconstruction (Candes et al.
(2006)). A summary of the CS reconstruction we have
used is presented in the following paragraphs, for a more
detailed description see Holland et al. (2010).
Consider the case that the image to be reconstructed is
stacked as a vector x, T is the operator that compresses
the image from pixel representation to a sparse repre-
sentation (e.g. the wavelet transform), F is the under-
sampled Fourier transform mapping the image domain
to k-space and y is a vector containing all the k-space
measurements. The reconstruction is then obtained by
solving the following constrained optimisation problem:


min II|x||I
s.t. I||Fx y||ll2 < ,
||(1 M). *x||12 <


where ei and 62 are thresholds that can be set to the
expected noise level. The 1l-norm, IIxrl1 i I x acts
as a proxy for sparsity i.e. minimising the above objec-
tive produces an image which has the sparsest represen-
tation in the transform domain while remaining consis-
tent with the acquired measurements. The second con-
traint in this optimisation utilises additional prior knowl-
edge of the distribution of the signal which is often avail-
able when imaging the velocity. In this case, the vec-
tor M takes the value 1 where signal is expected and 0
where only noise is expected. In practice we solve Eq. 1
in its unconstrained Lagrangian form:

argmin I|Fx yl2+A II Tx 1+A2 1|(1 M) x2 .
(2)
We follow the approach of Lustig et al. (2007) which
uses projected conjugate gradients to solve Eq. 2. Since
the objective is convex, the algorithm is guaranteed to
find the global minimum of the function. The relative
weight of each term in the reconstruction is controlled by











adjusting the parameters A, and A2; the values of these
parameters being determined by the characteristics and
SNR of the data.
The final point to note is that in MR imaging, veloc-
ity measurements are typically achieved by encoding the
velocity of the fluid, V, over an observation time, A,
using a pair of magnetic field gradients applied with a
strength g and for a duration 6, such that the phase of the
signal is proportional to the velocity. Thus the signal, S,
in a pixel m, n is given by:


Sm,n = p(m, n) exp (-i-g6AV),


where 7 is the gyromagnetic ratio of the observed nu-
clei. Thus, the velocity is calculated from the phase of
the complex signal in each pixel of the image. The re-
construction described by Eq. 2 is equally applicable to
both real and complex images. However, it is difficult
to apply CS to the phase data of a complex image di-
rectly. Instead the real and imaginary intensity images
are reconstructed using CS and the phase of the signal is
calculated after reconstruction.
Bayesian Bubble Sizing The measurement of bubble
size is based on the characteristics of the Fourier trans-
form of an image. As is well established, in the Fourier
domain of an image, the size and shape of an object are
described by the magnitude of the signal, while its posi-
tion is given by the phase of the signal (Nixon & Aguado
(2002)). This information can be used to formulate a
Bayesian method for rapidly and robustly estimating the
size of an object, such as a bubble, using MR.
In Bayesian analysis the state of a system 0 is inferred
from a set of observations y from the posterior probabil-
ity density function p(-' .i):


where p( .,, i' is the likelihood function and p(O) is the
prior knowledge. In this work we are attempting to de-
termine the size distribution of bubbles, which corre-
sponds to 0, given a set of measurements, y, of the signal
intensity in k-space. Therefore, if the likelihood func-
tion and prior distribution are known then the posterior
probability distribution can be obtained. The posterior
distribution can be used to infer the most probable state
of the system, in this case, the most probable bubble size
distribution.
The likelihood model for the signal intensity in k-
space is obtained by considering the distribution of the
signal as a function of the bubble size. Let the projection
of the bubble onto the x-axis be defined by a function
h (r, x), where r is the characteristic size of the shape
and x is the position. If we have a system where there
are N of these objects, each located at different posi-
tions, the overall signal will be given by:


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


N
f (x) h (rj,x x), (5)
j31
where rj is the size of the jth shape, centred on posi-
tion xj. Defining the Fourier transform of the function
h (r, x) as H (r, k), then the Fourier transform of Eq. 5
will be given by:

N
F (k) H (rj,k) exp (-i27kx). (6)
j-1
For a particular value of k, assuming {xj } is indepen-
dent and identically distributed, then it can be shown that
the signal magnitude is given by a Rayleigh distribution:


pF(F(k)l)
p (IF (k) I) exp
a (k) \


F (k) 2 '
2u (k))


where a2 0.5NE (|H(r, k)|2). Equation 7 defines
the likelihood function for the Bayesian bubble sizing
algorithm. For the ideal case of spherical bubbles of a
uniform size, a2 is given by:


2 N sin (27kr) 27krcos (27kr) (
2 2k3 (8)

If a distribution of bubble sizes is present then
E (IH(r, k) ) can be obtained by Monte-Carlo simula-
tion. This model of the likelihood distribution, Eq. 7,
can be used in Eq. 4 to determine the posterior distri-
bution p(-l .i) and hence the most probable bubble size
distribution given the acquired data, y.

Experimental

Compressed Sensing. CS experiments were per-
formed on a Bruker DMX 200 spectrometer with a ver-
tical 4.7 T superconducting magnet. The experiments
were performed using a 64 mm diameter radiofrequency
(r.f.) coil operating at 199.7 MHz for proton (1H) and
188.3 MHz for fluorine (19F). The magnet was equipped
with a three-axis shielded gradient set producing a max-
imum gradient strength of 0.136 T m 1 in the x, y, and
z directions.
Velocity images of single-phase liquid and gas flow
were acquired using the same pulse sequence, and us-
ing the same packed bed. The bed used was a cylindri-
cal column of inner diameter 27 mm, randomly packed
with 5 mm diameter glass spheres. The column was of
length 1 m. Velocity images were obtained using a spin-
echo sequence that was designed to minimise the total
echo time (Sankey et al. (2009); Holland et al. (2008)).


p('11'i) cc p( ,I' ip(),











The velocity was encoded using half a period of a sine-
shaped gradient waveform. Two images were acquired,
each with flow encoding gradient strengths of the same
magnitude but opposite direction.
The experimental set-up for acquisition of the liquid-
phase velocity images was as follows. Deionised water
was pumped in a closed circuit by a Verder VG330-10
gear pump and controlled by a PC-operated Bronkhorst
Cori-flow (model M55C4-AAD-11-K-C) mass flow
controller; the flow rate of water was varied from 0 60
kg h 1. The T1 relaxation time constant of the water was
reduced to 50 ms by adding gadolinium chloride to the
water at a concentration of 1.3 mM. The flow-encoding
gradients were applied for 6 = 1.19 ms and separated by
an observation time (A) of 3.3 ms. The gradient strength
used was varied for different water flow rates to opti-
mise the dynamic range of the velocity measurement.
First, a full k-space acquisition was performed. Veloc-
ity images were obtained with a field-of-view of 30 mm
x 30 mm at an in-plane resolution of 178 pm x 178
pm with a 1.5 mm slice thickness. The repetition time
of the experiment was 300 ms and a 4 step phase-cycle
was used, giving a total acquisition time of about 7 min.
CS measurements were then performed using the same
pulse sequence, with the phase-encoding gradient set to
only acquire the desired lines of k-space; in these acqui-
sitions 28 % of the k-space raster was incoherently sam-
pled (for full details of the sampling scheme see Holland
et al. (2010)).
Gas-phase velocity images were acquired on the same
packed bed as used for the liquid-phase studies. Sulphur
hexafluoride (SF6) gas was supplied in a closed circuit
using a DILO Piccolo compressor (model B022R01).
The gas from the compressor was stored in a pressure
vessel of volume 6 L. Gas was drawn from the pres-
sure vessel; the flow rate was measured using a rotame-
ter (Brooks Sho-rate 1357/D2B5D1B00000) to be (15
1) x 10 6 m3 s 1 A bypass line ensured excess gas
was returned to the feed of the compressor. The pres-
sure in the column was regulated using a back pressure
regulator set to 4.9 bar absolute. As for the liquid veloc-
ity measurements, in implementing the pulse sequence
the flow-encoding gradient strength was adjusted to op-
timise the dynamic range of the measurement. A flow-
encoding gradient duration and observation time of 0.5
ms and 1.9 ms, respectively, were used. Gas velocity im-
ages were acquired with a field-of-view of 29.4 mm x
29.4 mm with a spatial resolution of 230 pm x 230 pm
and a slice thickness of 1.5 mm. The repetition time was
34 ms and 608 scans were acquired, giving a total imag-
ing time of 90 min for a full k-space raster. This was
chosen as the practical upper limit for the duration of a
gas-phase velocity imaging pulse sequence. CS velocity
mapping was performed at the same resolution of 230


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


pm x 230 pm but only sampling 30 % of the data points
in the phase-encoding direction. Thus, 2048 scans were
acquired in the same total imaging time of 90 min.
Bayesian Bubble Sizing. Bubble size distribution
measurements were performed on a Bruker AV400 spec-
trometer with a vertical 9.4 T superconducting magnet.
The experiments were performed using a 38 mm diame-
ter r.f. coil operating at 400.13 MHz for 1H. The magnet
was equipped with a three-axis shielded gradient set pro-
ducing a maximum gradient strength of 0.28 T m 1 in
the x, y, and z directions.
The bubble size distribution was studied using both
an optical technique and the MR technique proposed in
this work on a system where the gas hold up (voidage)
was small. The apparatus used consisted of a 31 mm
ID column filled with deionised water containing 16.8
mM dysprosium chloride. Gas bubbles were introduced
using either a porous stone (sparger 1) or porous rubber
(sparger 2) gas-sparger. Gas was supplied to the system
using an Omega FMA3206ST gas thermal mass flow
meter. To compare the MR measurement of the bub-
ble size distribution with optical measurements, a flow
rate of 100 ml min 1 was used. At this gas flow rate
the voidage in the system was ~ .' with both spargers.
For the optical comparison experiments, 0.1 mg ml 1
sodium dodecyl sulphate was added to the liquid to sta-
bilise the bubble size distribution. Photographs of the
system were obtained using a Canon Powershot A630
digital camera. To examine the evolution in bubble size
with height up a column a gas flow rate of 400 ml min 1
was used with no surfactant to encourage bubble coales-
cence.
The Bayesian approach to bubble size measurement
eliminates the conventional requirement that every point
in k-space is sampled with the same bubbles located at
the same positions, only that all samples are obtained
for the same bubble size distribution. As a result, data
were acquired using a single point imaging (SPI) pro-
tocol. An SPI protocol was used due to its simplicity
and to minimise the effects of velocity attenuation on the
signal. The same imaging protocol was used for all ex-
periments, as the bubble sizes in each case were similar.
The experiment was performed in two steps: (1) a single
point was acquired to determine the voidage at the given
gas flow rate, then (2) 128 data points were acquired for
k-space points ranging from -716 m 1 to 716 m 1, but
excluding points for I|k < 40 m-1. This approach in-
creases the signal-to-noise ratio of IF(k)I for large k,
which is critical to identifying the correct bubble size.

Results

Compressed Sensing. Figure 1 shows a compari-
son of (a) conventional and (b) compressed sensing MR







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


velocity
b) (mm s-1)
135

l 90

S45

go0

27 mm 27 mm -45


Figure 1: Measurements of the velocity of water flowing through a packed bed of 5 mm diameter spheres acquired
using (a) full k-space sampling and (b) CS with only 28 % sampling. The velocity was measured in the
axial direction (i.e. coming out of the plane) and is given by the colour bar. The total imaging time was (a)
430 s for the fully-sampled image and (b) 120 s for the under-sampled image. (c) Comparison of the total
flow measurement using full k-space sampling and CS reconstruction with the flow rate of water measured
by the mass flow controller.


measurements of the flow of water through a packed
bed. The velocity distributions measured using the two
techniques are indistinguishable, despite the compressed
sensing image having been obtained from only 28 % of
the full k-space data set. The difference between the two
images can be quantified using the relative norm error
defined by:



f2-error (9)


where Vi and VCS are the velocities in the ith pixel ob-
tained from the conventional and CS reconstructions of
the velocity respectively. The mask image, Mi, excludes
any errors in the measured velocity where there is no
signal. The 2 -error measured for the image shown in
Fig. l(b) was 11 %.
To further confirm that the velocity distribution mea-
sured with CS was quantitative, the total flow through
the packed bed was calculated and compared to the mea-
sured input flow rate. These total flow measurements are
shown in Fig. l(c) for slices located at 3 different axial
positions for images obtained using full k-space sam-
pling and the CS reconstruction from only 28 % of the
full k-space data set. The results show excellent agree-
ment, both between the CS and conventional MR data
sets and with the known input flow rate measured using
the mass flow controller. The root mean squared error
for these measurements was 3 %.
The results shown in Fig. 1 demonstrate that CS can
be successfully used to acquire velocity images from
only 28 % of the full k-space data set. This is equivalent
to reducing the acquisition time of the velocity image


27 mm


27 mm


velocity
(mm s"1)
263

175

88


-88


Figure 2: Images of the velocity distribution for SF6
flow through a packed bed of 5 mm diameter
spheres using (a) full k-space sampling and
(b) compressed sensing at a resolution of 230
pm x 230 /m.


by a factor of ~4. At present CS has only been demon-
strated for conventional imaging of the velocity and thus
the reduction in imaging time was from ~7 min to ~2
min. In the future, this technique will be used in combi-
nation with ultra-fast imaging techniques to monitor the
evolution of rapidly evolving systems.
CS was also applied to velocity measurements in
the gas phase. In this case the goal of the experi-
ment was to improve the spatial resolution that could
be achieved. Figure 2 shows gas-phase velocity mea-
surements for SF6 flow through a packed bed of 5 mm
diameter spheres using a conventional full k-space ac-
quisition and a CS reconstruction where only 30 % of
the full k-space data set was acquired. The reduction
in the number of k-space points sampled allows greater
signal averaging and thus improved signal-to-noise ra-
tio, when compared with the full k-space sampling ap-
proach. The image reconstructed using CS, Fig. 2(b),







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


34 mm 34 mm
(a) (b)


0 1 2 3 4 5 6
Diameter (mm)


Figure 3: Photographs of a swarm of gas bubbles obtained using (a) sparger 1 and (b) sparger 2. The gas flow rate
was 100 ml min 1, giving a total voidage of ~ 2' in each case. Note the larger bubble size obtained from
sparger 2. (c) Comparison of the size distribution obtained in these systems using both the optical (solid
lines) and MR (dashed lines) techniques. The size distribution obtained from the optical and MR techniques
are in excellent agreement.


shows significantly less noise than the fully-sampled im-
age (Fig. 2(a)) without any ,igiilik.il reduction in the
resolution of the image. The signal-to-noise ratio in the
image reconstructed using CS was 21, compared with
5.2 in the fully-sampled image. The 2 -error of the ve-
locity images, given the sampling fraction and signal-to-
noise ratio, was estimated to be 15 % and 10 % for the
fully-sampled and CS image, respectively (Holland et al.
(2010)). Furthermore, the error in the total flow mea-
surement decreased from 8 % to 3 %, which is within
the experimental uncertainty of the rotameter measure-
ment. Thus, CS has enabled measurements at a resolu-
tion that corresponds to an 11-fold decrease in the voxel
size of the image compared with the highest resolution
gas-phase velocity images we have reported previously
(Sankey et al. 121' I".,

Bayesian Bubble Sizing. Figure 3 shows two pho-
tographs of gas bubbles in a 31 mm ID column using
spargers 1, Fig. 3(a), and 2, Fig.3(b). As seen in Fig. 3,
sparger 2 produces larger bubbles than sparger 1. As
the gas hold up in these systems was very low (~ 2'. ,
the bubble size distribution could be measured directly
from the photographs. These are shown by the step-wise
function in Fig. 3(c) (solid lines). The dashed lines in
Fig. 3(c) show the bubble size distribution estimated us-
ing the k-space approach outlined in this letter. In this
calculation, it was assumed that the bubble size distri-
bution was given by a log-normal distribution, as the
log-normal distribution is commonly found in practice
(Saxena et al. (1988); Junker (2006)). The function r2
was therefore obtained from a Monte-Carlo simulation,
where E (H(r, k)l) was calculated from 1000 images
of the projection of 30 bubbles with sizes given by a log-
normal distribution. The bubble size distributions mea-


sured for each system using the optical and MR tech-
niques were in excellent agreement, confirming the ac-
curacy of the proposed bubble sizing approach.


The bubble sizing approach was also used to track the
evolution of bubble size with height up a column. The
mean bubble size estimated using the Bayesian bubble
sizing approach is shown in Fig. 4 at a gas flow rate of
400 ml min which corresponds to a voidage of ~2.5
%. In these measurements the mean bubble size was
calculated as a function of height above the distributor.
As can be seen in Fig. 4 the bubble size was found to
increase with increasing height, presumably due to coa-
lescence between nearby bubbles. In future these mea-
surements will be used to validate numerical models of
bubble coalescence.


The Bayesian bubble sizing approach described here
has enabled accurate characterisation of bubble size dis-
tributions from only 128 data points. By comparison for
a conventional imaging approach to at a similar reso-
lution would require 128 x 128, or 16384 data points.
This is 2 orders of magnitude more points than were
required with the Bayesian approach described in this
paper. The reduction in the number of data points re-
quired to measure the bubble size distribution eliminates
the need for complex ultra-fast imaging techniques. Fur-
thermore, the approach is equally applicable to both high
and low voidage systems and to the measurement of any
approximately spherical droplets or bubbles, as might be
found in emulsions or sprays. Thus, this Bayesian bub-
ble sizing approach opens exciting new possibilities for
research.






















0 20 40 60 80 100 120
Height (cm)

Figure 4: Measurements of the evolution of the bubble
size with height up the column at a gas flow
rate of 400 ml min1. Measurements were
made using (o) 1 scan and (A) 32 scans.


Conclusions

In this paper two recent developments in MR measure-
ments of multi-phase flows have been demonstrated.
The first technique, compressed sensing, is a generic
technique that is applicable to almost any MR imaging
approach. It was demonstrated for velocity mapping of
single phase flow of liquid and gas through a packed bed
and was shown to allow a decrease in imaging time by
almost a factor of 4 without ,igiiliik.i loss of image
quality. The second technique presented utilised more
specific knowledge of the system studied to devise a
bubble sizing approach using Bayesian analysis. This
approach was demonstrated to provide an accurate es-
timate of the bubble size distribution in a system with
sufficiently low voidage (~ 2 %) that it could also be
measured optically. The Bayesian bubble size measure-
ment technique was then used to track the evolution in
bubble size distribution with height up a column.
The new imaging techniques presented in this paper
are applicable to a wide range of systems. In the case
where little is known about the system under study, the
compressed sensing approach is likely to reduce acquisi-
tion times by approximately a factor of 4. In cases where
the system can be well characterized and specific infor-
mation is sought, the Bayesian approach can decrease
required data rates by several orders of magnitude and
can enable measurements of systems that would other-
wise be impossible to study.


Acknowledgements

The authors would like to acknowledge the contribution
of Dmitry Malioutov to the compressed sensing work
presented in this paper.


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


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