Analyzing
clustering in bubbly flow using ultra-fast X-ray tomography

Y.M. Lau, K.
Müller & M. Schubert

Helmholtz-Zentrum
Dresden-Rossendorf e.V., Institut für Fluiddynamik

Bautzner
Landstraße 400 | 01328 Dresden

Tel: +49 (0) 351
260 3766, Email: y.m.lau@hzdr.de

Abstract

Bubbly
flows can be found in numerous fields in engineering, e.g. bio-oxidation
processes for waste-water treatment, production of pharmaceuticals, penicillin,
citric acid, etc. An important feature occurring in bubbly flow is the
appearance of clusters of bubbles (Figueroa-Espinoza
& Zenit (2007), Tagawa et al. (2012)).
These clusters have a very different dynamical behavior than clouds of
uniformly distributed bubbles. Clustering could influence the overall flow
structure, introducing enhanced velocity fluctuations and hydrodynamic
interactions. Nevertheless, the complexity of clustering in bubbly flow makes
fundamental research still necessary. There are
several mathematical tools available for quantifying clustering, ranging from
pair correlation, counting box methods, Lyapunov exponents, Voronoï diagrams
and Minkowski functionals.

In
this work, we evaluate several of the mathematical techniques and one method we
adopted is Voronoï diagrams. We apply Voronoï analysis using ultra-fast X-ray
tomography to identify and measure the extent of clustering in bubbly flows in
aerated columns. Ultra-fast X-ray tomography is based on the scanning electron
beam principle (Fischer & Hampel (2010)) and dual slice images are reconstructed
from radiographic projections. Measurements are carried out with a frequency of
1000 Hz. The segmented images are stacked in the time domain and thereby
three-dimensional bubble objects are constructed with two spatial dimensions
and one temporal dimension. A representation of the extracted data per scanning
plane is given in Figure 1. Based on these types of data, a Voronoï analysis is
applied to quantify the bubbles clustering.

Figure
1: Extracted bubble objects from a single plane measurement of ultra-fast X-ray
tomography. The bubbles? interface is constructed from segmented images.

Given
a set of bubbles, the corresponding Voronoï diagram (Monchaux et al. (2010)) is
the unique decomposition of the 2D/3D space into independent cells associated
with each bubble. One Voronoï cell is defined as the ensemble of points that
are closer to one bubble than to others. From the definition, it appears that
the area/volume of a Voronoï cell is the inverse of the local concentration of
bubbles. Therefore the investigation of the Voronoï area/volume field is
strictly equivalent to that of the local concentration field. As the mean value
of the Voronoï areas/volumes is nothing but the average concentration, a
normalized Voronoï areas/volume is defined by normalizing the area/volumes by
their mean value. Figure 2a shows an example of the constructed 2D Voronoï
cells, on which the probability density functions (PDF) of the normalized Voronoï
areas are computed. Clusters can be differentiated by comparing these to PDF of
randomly distributed bubbles.  

Figure
2: Analyzing clusters in X-ray tomographic images of bubbly flow using Voronoï
diagrams. (a) Two-dimensional Voronoï
plot of a tomographic image of a bubble column with a diameter of 100 mm and a
superficial gas velocity of 2 cm/s. The blue dots indicate the centroids of the
bubbles within the Voronoï
cells. (b)  PDF of two-dimensional Voronoï analysis: (black) measurement data,
(blue) fit of the data and (red) random Poisson distribution.

Experiments
are performed with different liquid properties (de-ionized water and polymer
solutions) and inlet conditions (range of superficial velocities). Figure 2b
gives an example of a PDF comparison between experimental (data) and random
(rand) bubble distribution as functions of the normalized Voronoï area. The
experimental data (black line) is fitted by a form of Poisson/Gaussian function
(blue line), while a random Poisson PDF is shown by red line. The
fitted-experimental PDF and random PDF intersects twice, leading to much larger
PDF values of experiments in a low value area zone and slightly larger ones in
a high value area zone. These low and high value area zones are corresponding
to large and small local bubble concentration, respectively. The much larger
experimental PDF values at high concentration and slightly lower ones at low concentration
indicate clustering of bubbles and voids, respectively.

In
the presentation, a comprehensive analysis of the clustering in bubble columns
depending on operating conditions will be shown by applying different
mathematical approaches to the tomographic images. Furthermore, the
experimental technique proposed in this work allows us to determine other
important parameters of the clusters such as bubble sizes and velocities.

References

Figueroa-Espinoza,
B and Zenit R. Clustering in high Re monodispersed bubbly flows. Physics
of fluids 17, 091701, 2005.

Fischer,
F. and Hampel, U. Ultra-fast electron beam X-ray computed tomography for
two-phase flow measurement. Nuclear Engineering and Design 240 (9),
2254?2259, 2010.

Monchaux,
R., Bourgoin, M. and Cartellier, A. Preferential concentration of heavy
particles: A Voronoï analysis,
Phys. Fluids 22, 103304, 2010.

Tagawa,
Y., Martínez Mercado, J., Prakash, V.N., Calzavarini, E., Sun, C. and Lohse, D.
Three-dimensional Lagrangian Voronoï analysis for clustering of particles and
bubbles in turbulence. J. Fluid Mech., vol.
693, pp. 201?215, 2012.