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The 16th Conference of ILASS-Asia December 18-19, 2013, Nagasaki, Japan
-87-
RECONSTRUCTION OF LOCAL VOLUME-WEIGHTED
DROP-SIZE DISTRIBUTIONS OF A SOLID CONE SPRAY
USING ADAPTIVE TOMOGRAPHIC TECHNIQUE
Songrit TANCHATCHAWAN1,2*, Pumyos VALLIKUL2, Pisit YONGYINGSAKTHAVORN2, Christophe DUMOUCHEL3
1*Engineering Department, Thailand Institute of Scientific and Technological Research, songrit@tistr.or.th
2Department of Mechanical and Aerospace Engineering,
King Mongkut’s University of Technology North Bangkok, Thailand
3UMR6614-CORIA-Universitet INSA de Rouen, France
This paper reports the development of an adaptive tomographic reconstruction algorithm to measure local properties
of a solid-cone spray generated from a pressure swirl atomizer. The technique has been applied to measure local
drop-size distributions and local extinction coefficients of the spray simultaneously and at multi-resolutions. Input
data to the algorithm are the measurement data of the extinction and of the forward scattered light from laser beams
travelling through the spray. The laser beam scans across the spray’s cross-section; the distance of each scan is less
than or equal to the laser beam-width. Since the reconstructed local extinction coefficient relates directly to the liquid
volume concentration and that the area of the local region is a priori known, the amount of the liquid contained within
this region can be determined. In this paper, the local drop-size distribution and the amount of liquid volume within a
local region are combined and presented in the form of liquid-weighted drop-size distribution. It has been shown
from the reconstruction results that the centre core region of the spray contains small drops and the drops are larger
when moving toward the edge of the spray. The amounts of the liquid volume, on the other hand, are less both at the
centre core and outer edge regions than that at the intermediate regions. The shape of the volume-weighted drop-size
distribution changes significantly when moving from the inner core to the intermediate regions of the spray. As
shown in the adaptive reconstruction results, the changes of the profiles of the distributions can be well resolved.
Keywords: Adaptive Tomographic Technique, Extinction Coefficient, Volume-Weighted Drop-Size Distribution,
Multi-Resolution, Solid-Cone Spray
1. INTRODUCTION
When a cross-sectional plane of a liquid spray is considered,
there are two pertinent quantities that characterize the spray: the
volume based drop-size distribution and the liquid volume
concentration. The volume based drop-size distribution is a
volume-fraction density function in the diameter space. The liquid
volume concentration is the ratio of liquid volume of all drops to the
volume that contains the drops.
For the non-homogenous spray, the liquid drop-size
distribution has been defined within a local region on the
cross-sectional plane which is called the volume-weighted drop-size
distribution [1,2]. It is equal to the product of the local
volume-fraction density function with the spatial liquid
volume-fraction. The distributions are reconstructed from the
tomographic data.
The shape of the volume-weighted drop-size distribution is
similar to that of the volume based drop-size distribution but the
area under which is smaller and being equal to spatial liquid
volume-fraction. The size of the local region has to be small such
that the intensive properties within it are uniform.
Reconstruction technique to refine the size of the local region
when the width beam used being large had been introduced by [3].
The technique resolved the drop-size distribution into the local
regions of smaller size than that of the width of the beam. Another
technique introduced by [4] resolved the local regions into the
smaller local regions moreover the technique allowed the size of the
local regions to be adaptable. Both techniques, however, applied to
the volume based drop-size distribution from the measured
diffraction data. They have not been applied to reconstruct the local
extinction coefficient or local liquid volume concentration within
the spray such that the local volume-weighted drop-size distribution
could be determined at multiple resolutions.
In this paper we will focus on the theoretical work and the
experimental setup to measure the local extinction coefficients in a
water spray from their line-of-sight transmission data. The
reconstructed results together with the previous measurement
drop-size-distribution data [4] will be used to determine local
volume-weighted drop-size distribution. Verification of the
reconstruction process and results will be demonstrated in details.
W
L
I
L
I
K
)(yK
W1W2W3W4
LdxK
LeII 0
0
LdxK
LeII 0
0
0
I
0
I
x
y
L
Fig. 1 Modeling of line-of-sight intensity,
L
I
, narrow (left)
and large (right) beam width
The 16th Conference of ILASS-Asia December 18-19, 2013, Nagasaki, Japan
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2. STRIP INTEGRATION OF
LOCAL EXTINCTION COEFFICIENT
A monochromatic light beam of incoming intensity, ,
travelling through an absorbing medium of path length , leaves
the medium with the intensity, , following the Beer-Lambert law,
(1)
where is the extinction coefficient. The dimension of the
extinction coefficient is (per path length) and its value depends
on temperature and concentration of the medium along the
travelling path [5]. For the beam of very small cross-sectional area,
the coefficient can be assumed constant across the width of the
beam. The transmitted intensity, , of a narrow beam is shown
schematically in Fig. 1 (left).
Since the width of the beam used in this work is finite then
the extinction coefficient varies across the width of the beam. To
take into account the effect of the variation of the coefficient across
the width, it is proposed in this work to use average extinction
coefficient to calculate the out coming intensity, . The spatial
average of the extinction coefficient over the width of the beam is
calculated from
(2)
Substitute Eq. (2) into Eq. (1) and rearrange the equation in term of
the measurable and unknown quantities result in:
(3)
Equation (3) constitutes a tomographic problem. The extinction
coefficient function on the left hand side is unknown to the problem
while the variable and the parameter on the right hand side, which are
the transmission,
, and the width of the beam, , are known
from the measurements.
The integrand in the left hand size of Eq. (3) is a spatially
average quantity which contains information of the local extinction
coefficients distributed over the line of sight whose width is equal to
that of the beam. To pan out the unknowns of the above problem,
the line-of-sight integration of the average extinction coefficients
can be written in the form of strip integration of the local extinction
coefficients as following
(4)
The tomographic problem (3) in continuous form is then
(5)
where is the local extinction coefficient and is the measured
transmission over the line-of-sight measurement.
3. MEASUREMENT OF LINE-OF-SIGHT
TRANSMISSIONS
The reconstruction algorithm has been applied to measure local
extinction coefficient within a solid cone spray of water generated
by a pressure-swirl atomizer. The measurement was conducted on a
cross-sectional plane of the spray below the nozzle exit. Two
simultaneous measurements: diffraction and absorption
measurement were performed under this setup.
The diffraction measurement data was analyzed and the
drop-size distributions within the spray at multiple resolutions were
determined [4]. The absorption measurement data, in the present
work, has been used to reconstruct extinction coefficient.
Figure 2 (left) shows schematically the measurement setup in
[4]: two scanned modes had been performed, between 0 and 30 mm,
each measured line of sight is 2.5 mm apart and beyond that each
measurement was 10 mm apart. Figure 2 (right) shows the
line-of-sight measurement data of transmission. The transmission is
lowest for the line-of-sight passing the centre of the spray, having
the value 60% and being highest for the line-of-sight at the outer
edge of the spray. These data together with the width of the laser
beam are used as the input data for the reconstruction algorithm to
determine the local extinction coefficient, see Eq. (5).
4. DISCRETIZATION OF
THE TOMOGRAPHIC PROBLEM
This sub-section shows the discrete form of Eq. (5). As
mention earlier, the spray used in this experiment is an
axi-symmetrical spray and contains N-1 concentric rings and a
centre core. Within each region, the spray’s local extinction
coefficient is assumed constant and having the value equal to
as shown in Fig. 3
z = 70 mm
y = 05 15 25 35 45 55 mm
Laser beam
Swirl Atomizer y
z
Scan trajectory
Fig. 2 Measurement setup (left) and measured line-of-sight transmission (right)
0
10
20
30
40
50
60
70
80
90
100
010 20 30 40 50 60
T(y)
y (mm)
measurement
The 16th Conference of ILASS-Asia December 18-19, 2013, Nagasaki, Japan
-89-
AM,1/2
AM,j+1/2
AM,j+2/2
AM,N-1/2
AM,N/2
AM,j/2
yM
rN
rN-1
rj+2
rj+1
r1
rj
KM,1
KM,j
KM,j+1
KM,j+2
KM,N-1
KM,N
yiy1
... ...
W
Fig. 3 Local extinction coefficients and local areas
of the strip.
There are three values of total local regions used in this work.
For the coarsest grid reconstruction, the number of the local regions
is 6. Within a certain regions where the rate of spatial change
of the extinct coefficient is high, the local reconstruction grids are
adapted to be finer grids, from 6 to 7 and to 8 local regions
respectively.
Figure 3 also shows as an example of a line-of-sight past
and the width of the beam is . The intensity of the beam is
absorbed along its travelling line-of-sight; the local extinction
coefficients and the intersection areas between the beam and the
local region it pasts are and respectively.
Note that the denominator 2 appears from the fact that overall
line-of-sights being twice of that shown in the Fig. 3. Using the
definition, the mathematical model of the line-of-sight
transmissions, based on Eq. (5) can be written into the
discrete form as
(6)
The parameters on the left hand side of Eq. (6) are and ,
which are the characteristic matrix of the problem representing the
local intersection areas and the unknown local extinction coefficient
The parameters on the right hand side of Eq. (6) are the measured
line-of-sight transmission and the width of the laser beam,
being 10 mm.
5. RESULTS AND DISCUSSIONS
The reconstructed local extinction coefficients have been
reported in the form of local liquid concentrations which is the ratio
of the liquid volume to the measurement probe volume. The relation
between the extinct coefficient and the liquid volume concentration
appears elsewhere [1] as
(7)
The Sauter mean diameter, , and the scattering cross-section,
, of each local region is known a priory from previous works
[4,6].
Representing the local extinction coefficient in the form of
liquid volume concentration has advantages in that the amount of
liquid contained within some interested local regions can be
elaborated e.g. the liquid volume in different local regions can be
compared or overall liquid volumes from different local regions can
be determined.
Fig. 4 Reconstruction results of local liquid concentrations at
different resolutions: six (top), seven (middle) and eight
(bottom) local reconstruction regions.
0
10
20
30
40
50
60
70
80
020 40 60
Cv,j (ppm)
r (mm)
r = 00-10 mm r = 10-20 mm r = 20-30 mm
r = 30-40 mm r = 40-50 mm r = 50-60 mm
0
10
20
30
40
50
60
70
80
020 40 60
Cv,j (ppm)
r (mm)
r = 00-08 mm r = 08-16 mm r = 16-24 mm
r = 24-32 mm r = 32-40 mm r = 40-50 mm
r = 50-60 mm
0
10
20
30
40
50
60
70
80
020 40 60
Cv,j (ppm)
r (mm)
r = 00-08 mm r = 08-14 mm r = 14-20 mm
r = 20-26 mm r = 26-32 mm r = 32-40 mm
r = 40-50 mm r = 50-60 mm
The 16th Conference of ILASS-Asia December 18-19, 2013, Nagasaki, Japan
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Figure 4 shows reconstruction results of local liquid volume
concentrations at different resolutions. Local liquid volume
concentration results show on the top, middle and bottom of the
figure are the reconstructed at six, seven and eight local regions
over the spray cross-section respectively. The refinement has,
however, been advanced only on the inner regions; the two outer
most regions remains at their initial resolutions
Since the liquid volumes have been distributed to a more
numerous local regions then most concentrations in the finer grid
reconstructions are lower than those of the coarser reconstruction,
except at the local regions where the liquid volume concentration
being maximum. The maximum liquid volume concentrations for
the coarse, the first and the second refine reconstructed grids are
located at the local regions between 30-40 mm, 32-40 mm and
26-32 mm respectively. Thus, a finer reconstruction results are
obtained. The liquid concentrations of the two outer most rings do
not show any significant changes.
Figure 5 shows the accumulation volume of liquid along the
radius of the reconstructed plane. The volume is calculated by
integration liquid volume within the infinitesimal volume of the
local regions, or
(8)
Figure 6 shows the volume-weighted drop-size distributions
for all local regions when being reconstruct at different resolutions.
As can be seen from Fig. 6 (top), the changes in sizes of the first
four inner regions distributions are noticeable and becoming the
target of adaptations of the reconstruction resolutions.
Increasing the number local regions to 7 and 8 regions, result
in the new distributions shown in Fig. 6 (middle) and (bottom)
respectively. It can be realized from the multi resolution
reconstructions that the new distributions show gradual changes
among the adjacent distributions. This means that some amount of
liquid volumes in the adapted zone are reallocated and smeared over
to the additional newly generated local regions. For the two outer
most regions, neither the size of the regions nor the values of the
weight coefficients are affected by the reconstruction adaptation.
Fig. 5 Liquid volume accumulation from 0 to 60 mm.
Fig. 6 Local volume-weighted drop-size distribution at different
resolutions: six (top), seven(middle) and eight (bottom) local
reconstruction regions.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
020 40 60
Q (r) (mm3)
r (mm)
six local regions
seven local regions
eight local regions
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
525 45 65 85 105 125
wj fv (mm-1)
D (mm)
r= 00-10 mm
r= 10-20 mm
r= 20-30 mm
r= 30-40 mm
r= 40-50 mm
r= 50-60 mm
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
525 45 65 85 105 125
wj fv,j (mm-1)
D (mm)
r = 00-08 mm
r = 08-16 mm
r = 16-24 mm
r = 24-32 mm
r = 32-40 mm
r = 40-50 mm
r = 50-60 mm
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
525 45 65 85 105 125
wj fv,j (mm-1)
D (mm)
r = 00-08 mm
r = 08-14 mm
r = 14-20 mm
r = 20-26 mm
r = 26-32 mm
r = 32-40 mm
r = 40-50 mm
r = 50-60 mm
The 16th Conference of ILASS-Asia December 18-19, 2013, Nagasaki, Japan
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The volume-weighted drop-size-distributions from all regions
can simply be combined to give volume based
drop-size-distributions of the whole reconstructed plane; this is
shown in Fig. 7. It can be seen from the figure that the overall
volume based drop-size-distributions obtained from the
reconstruction at low resolution appears oscillatory particularly at
the region of small drops. These oscillations reduce as the
reconstruction resolution increases. The large drop distribution
remains the same and independent from the reconstruction
resolutions.
Fig. 7 Drop-size distribution over the whole plane.
6. CONCLUSIONS
Local extinction coefficients and liquid volume concentrations
on the cross-sectional plane of the water spray have been
reconstructed by using the tomographic technique.
Reconstruction at multiple resolutions allows large amount of
liquid volume within the lower resolution regions to be distributed
on those higher resolution region. The gradual changes of the
concentration between two adjacent regions have been realized.
The volume-weighted drop-size distributions represented the
two important characteristics of the spray: the drop-size-distribution
and the liquid volume concentration very well.
The reconstructed local volume-weighted drop-size
distributions can be simply used to synthesis the drop-size
distribution over the whole plane of the spray; the more local
regions are used, the more accurate of the total drop-size
distribution is obtained.
ACKNOWLEDGEMENTS
The first author is being supported under the PhD scholarship
program by the Ministry of Sciences and Technology (MOST),
Thailand and Thailand Institute of Scientific and Technological
Research (TISTR). Research facilities are partially provided by
the Joint Graduate School of Energy and Environment, King
Mongkut’s University of Technology Thonburi and the Graduate
School, King Mongkut’s University of Technology North Bangkok.
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Fungtammasan, B., Deconvolution with Maximum Entropy
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187-198, 2009.
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Evaporation Time and Rate of Water Sprays from Their Local
Drop-Diameter Distributions and Liquid Volume Concentration,
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Dumouchel, C., An Adaptive Tomographic Technique to
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0
0.005
0.01
0.015
0.02
0.025
0.03
525 45 65 85 105 125
fv (mm-1)
D (mm)
six regions
seven regions
eight regions
