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The 16th Conference of ILASS-Asia December 18-19, 2013, Nagasaki, Japan

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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

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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.

REFERENCES

1. Yongyingsakthavorn, P., Dumouchel, C., Vallikul, P. and

Fungtammasan, B., Deconvolution with Maximum Entropy

Solution to Determine Local Extinction Coefficient and Local

Volume Concentration Values from Laser Diffraction Data,

Particle and Particle Systems Charaterization, Vol.26, pp.

187-198, 2009.

2. Yongyingsakthavorn, P., Vallikul, P., Dumouchel, C.,

Fungtammasan, B. and Tuntivoranukul, K., Prediction of

Evaporation Time and Rate of Water Sprays from Their Local

Drop-Diameter Distributions and Liquid Volume Concentration,

Atomization and Sprays, Vol.21, pp. 159-165, 2011.

3. Boyaval, S., and Dumouchel, C., Deconvolution Technique to

Determine Local Spray Drop Size Distributions – Application to

High-Pressure Swirl Atomizers, ILASS-Europe, Zurich, 2001.

4. Tanchatchawan, S., Vallikul, V., Yongyingsakthavorn, P. and

Dumouchel, C., An Adaptive Tomographic Technique to

Reconstruct Local Drop Size Distribution of Liquid Spray at

Multi-Resolution, Proc.25th Annual Conference on Liquid

Atomization and Spray Systems-Americas (ILASS-Americas),

2013.

5. Ingle, J. D. J. and Crouch, S. R., Spectrochemical Analysis,

Prentice-Hall, 1988.

6. Dobbins, R. A. and Jizmagian, G.S., Optical Scattering Cross

Sections for Polydispersions of Dielectric Spheres, The Optical

Society of America, Vol. 56, pp. 1345-1350, 1966.

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