Fitting of Different
Drop Size Distribution Functions
to Spray Data
from a Y-jet Type Airblast Atomizer
Tuomas Paloposki (Aalto University)
Takao Inamura (Hirosaki University)
Contents
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•
•
•
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Motivation
Experimental data of Inamura and Nagai
Scope of current work
Results
Conclusions
Why drop size distribution functions?
• A good way to describe a large amount of information
(can be applied both to experimental results and
to data related to numerical calculations).
• Might improve our theoretical understanding of sprays.
Left:
Right:
small-scale model experiment of black liquor spraying
(A. Kankkunen / Aalto University)
experimental and numerically simulated diesel sprays
(O. Kaario et al. / Aalto University)
Why statistical methods?
Probability density [1/mm]
• Is the fit good enough?
• Considering that 4583 drops were measured?
• Considering that 2 adjustable parameters are deployed?
0.04
Inamura & Nagai (1995)
Run #01
0.03
Log-normal distribution
0.02
Experimental data
0.01
0.00
0
40
80
Drop size [mm]
120
The work of Inamura and Nagai
• A new type of Y-jet airblast atomizer was developed
during the early 1990s.
• The objective was to produce drop size distribution
which would be independent of liquid flow rate.
• The way to achieve this was to use a fluid amplifier.
Drop size distribution data
• Experimental data for several different atomizer
configurations were collected by Inamura and Nagai.
• The measurement method was the
classical ”capture drops on a slide
and measure using a microscope”.
– This measurement method has errors,
but they are probably different from
the errors of modern optical techniques.
• The samples were reasonably large
(approximately 3000 – 8000 drops).
=> Ideal data for statistical analyses!
Scope – experimental conditions
• Three different atomizer configurations.
• Seven different liquid flow rates for each atomizer.
=> 3 x 7 = 21 data sets
The best configuration
Scope – Distribution functions
• Four different drop size distributions were fitted
to each data set.
Log-normal
Nukiyama-Tanasawa
Log-hyperbolic
A sum of 2 log-normal distributions
code
LN
NT
LH
LN2
=> 4 x 21 = 84 cases to be studied.
number of
adjustable
parameters
2
3
4
5
Fitting technique:
find parameter values
which minimize this number
Probability density [1/mm]
Summary for Run 01
0.04
Inamura & Nagai (1995)
Run #01
Log-normal distribution
Nukiyama-Tanasawa distribution
Log-hyperbolic distribution
Sum of two log-normal distributions
Experimental data
0.03
0.02
0.01
0.00
0
40
80
Drop size [mm]
120
Summary for all runs (LN, NT, LH & LN2)
+ Run 01 highlighted
100
LN
Test statistic c02 [ ]
01-07 11-17 21-27
Run 01
80
60
LN2
LH
NT
01-07 11-17 21-27 01-07 11-17 21-27
01-07 11-17 21-27
40
Theoretical
values for c2
P = 0.01
20
0
P = 0.05
Log-normal distribution
and behavior of parameters
25.0
3.0
Runs 11-17
Best-fit values
Runs 11-17
Parameter s [ ]
Logarithmic mean size [µm]
• The log-normal distribution function:
20.0
15.0
10.0
5.0
0.0
Best-fit values
2.0
1.0
0.0
0.0
4.0
8.0
Liquid flow rate [g/s]
12.0
0.0
4.0
8.0
Liquid flow rate [g/s]
12.0
Regression lines for configuration (a)
We can compute linear regression between
the parameters of the log-normal distribution
function ( and ) and the liquid flow rate.
A value of 0.08 is obtained for b1 and the 95 % confidence interval
extends from –0.09 to +0.25. Thus, the value of b1 appears not to differ
from zero in a statistically significant manner. However, a fundamental
assumption (normality of arrays) is not satisfied.
3.0
20.0
Parameter s [ ]
Drop mean size [µm]
25.0
15.0
10.0
Experimental data
Linear regression
5.0
2.0
1.0
Experimental data
Linear regression
Runs 11-17
Runs 11-17
0.0
0.0
0.0
4.0
8.0
Liquid flow rate [g/s]
12.0
0.0
4.0
8.0
Liquid flow rate [g/s]
12.0
Nukiyama-Tanasawa distribution and
behavior of parameters p and q
• The Nukiyama-Tanasawa distribution function:
0.3
Runs 11-17
Best-fit values
parameter q [ ]
parameter p [ ]
50.0
25.0
0.0
-25.0
Runs 11-17
Best-fit values
0.2
0.1
0.0
-0.1
-0.2
-50.0
-0.3
0.0
4.0
8.0
Liquid flow rate [g/s]
12.0
0.0
4.0
8.0
Liquid flow rate [g/s]
12.0
Nukiyama-Tanasawa distribution
function and correlation between p and q
• Model for correlation between parameters p and q:
p = C∙qk for q > 0;
–p = C∙(–q)k for q < 0
For all data
analyzed
in this study,
C ≈ 1 and k ≈ –1.
Parameter p [ ]
50
25
Runs
Runs
Runs
Runs
Runs
Runs
1 - 7 observed
1 - 7 regression
11 - 17 observed
11-12, 15-17 regression
21 - 27 observed
21 - 27 regression
0
-25
-50
-0.4
-0.2
0.0
Parameter q [ ]
0.2
0.4
The log-hyperbolic distribution function
and the sum of two log-normals
• Better fit than with the log-normal and
Nukiyama-Tanasawa distribution functions.
• Behavior of the parameters quite unstable.
• No trends could be detected.
• Mathematically complex functions
=> considerable effort had to be invested in
carrying out the calculations.
Conclusions
• Best fit to experimental data was obtained with
a sum of two log-normal distributions.
• The next best was the log-hyperbolic distribution;
then the Nukiyama-Tanasawa; and finally log-normal.
• Even the sum of two log-normal distributions failed
at a 5 % significance level in 7 cases out of a total of 21.
• In general, the parameters of the distribution functions
behaved strangely. The parameters p and q of the
Nukiyama-Tanasawa distribution function appear to be
strongly correlated.
Thank you for your attention!