7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Drop size spectra in sprays from pressureswirl atomizers
Gunter Brenn and Andreas Tratnig
Graz University of Technology, Institute of Fluid Mechanics and Heat Transfer
Inffeldgasse 25/F, 8010 Graz, Austria
brenn @fluidmech.tugraz.ac.at
Keywords: Conical sheet atomization, Sauter mean drop size, drop size spectrum, gamma distribution
Abstract
A study on the characterization of sprays from Newtonian liquids produced by pressureswirl atomizers is presented. The
global drop size spectra of the sprays are measured with phaseDoppler anemometry, and global mean drop sizes are derived as
moments of the spectra for varying atomizer geometry, liquid flow rate, and physical properties of the liquids. Dimensional
analysis provides a correlation for the nondimensional global Sauter mean diameter. A relationship between the global Sauter
mean drop size and the global drop size RMS is established. A method is developed for predicting the global drop size spectra
in the sprays, using easily accessible experimental input parameters. The basis for the function defining the spectrum is a
gamma distribution, which is known from the literature as physically relevant for ligamentmediated sprays.
1. Introduction
Liquid sprays produced by pressure atomizers are important
for many technical processes, such as energy conversion,
coating, and cooling, to name but a few. A group of widely
used pressure atomizers are the prefilming atomizers,
which turn the emerging liquid into sheets propagating from
the atomizer orifices. The flat fanshaped or conical sheets
break up into droplets due to the KelvinHelmholtz
instability, which leads to ligament formation, and due to the
capillary instability of the ligaments, which forms the final
droplets. Due to the wide use of this type of atomizers,
researchers have been investigating since decades the spray
drop ensembles resulting from the sheet breakup as
functions of the influencing parameters. The highest detail
of information on a spray drop ensemble is represented by
the spectra of the drop size and velocity. It has been an aim
of spray research since a long time to find a reliable way of
modeling and/or predicting at least the spectrum of the drop
size produced by sheet breakup, since this spectrum
influences most transport properties of the spray relevant to
the technical process. It is the aim of the present work to
provide a reliable prediction of the spray drop size spectrum
as produced by the breakup of conical sheets from
pressureswirl atomizers.
Starting from the pioneer work by Squire (1953), many
researchers analyzed the stability behavior of parallelsided
or attenuating liquid sheets in their gaseous environments
(e.g., Dombrowski & Johns 1963, Clark & Dombrowski
1972, Li & Tankin 1991, Senecal et al. 1999). These works
concentrate on determining the wavelength and growth rate
of the most unstable ("optimum") disturbance. The optimum
wavelength, together with the sheet thickness at breakup,
determines the resultant diameter of ligaments detached
from the sheets. The second step is to deduce a
representative drop size, assuming breakup of the ligaments
by the mechanism analyzed theoretically by Weber (1931).
The so obtained drop size is interpreted as a Sauter mean
drop size in the spray. The drop sizes from the model
calculation and from measurements, however, often do not
agree very well (Dombrowski & Johns 1963).
It has been the subject of research since a long time to find
an appropriate mathematical function that characterizes the
drop size spectra of sprays. There exist two different
approaches: the first one uses mathematical functions
assumed suitable for representing the shapes of the spectra
(Mugele & Evans 1951, Xu, Durst, & Tropea 1993, Bhatia
& Durst 1989, Paloposki 1994). An alternative approach
uses physically based functions for representing the drop
size spectra. One model assumption is that drops in
ligamentmediated sprays are produced by coalescence of
liquid blobs which make up the ligaments (Villermaux et al.
2004). This model leads to a function for the drop size
spectrum which involves the gamma function and is
therefore called a gamma distribution (Marmottant &
Villermaux 2004, Bremond et al. 2007, Villermaux 2007).
Basically the same kind of function is obtained by
Dumouchel (2006) using a new formulation of the
maximum entropy formalism.
The approach with the gamma distribution is favored in the
present context and used for predicting the drop size spectra
in hollowcone sprays on the basis of experimental data.
The paper is organized as follows: the next section presents
experiments characterizing the drop sizes in the sprays.
Section 3 presents the experimental nondimensional results.
In section 4 we show that the approach to represent the
measured mean spray drop sizes with the Dombrowski and
Johns model leads to unsatisfactory results. As an
alternative, Section 5 develops a modified gamma
distribution as a function to predict the drop size spectra
accurately enough to represent their moments well. In
section 6 the conclusions from the work are drawn.
2. Experiments on sprays from pressureswirl
atomizers
2.1 The test rig
The experiments were carried out using the test rig sketched
in Figure 1. This rig consists of a spray box, a lowpressure
and a highpressure pump, a spray nozzle, an exhaust
ventilation, and devices for process monitoring as well as
drop size and velocity measurements. The liquid is pumped
with the help of a lowpressure pump to the Coriolis mass
flow meter Foxboro CFS 10, where, further to the mass flow
rate, the liquid density and temperature are measured. The
highpressure pump feeds the atomizer. The liquid is
sprayed into the spray box, where it is collected. The
exhaust ventilation suppresses spray drop recirculation,
which would disturb the measurements of the spray
properties. The position of the atomizer may be changed by
means of the twoaxes traverse.
Traverse
Figure 1: Sketch of the spray test rig.
The atomizers used in the experiments were Delavan SDX
pressureswirl atomizers. They consist of a swirl chamber,
an end plate, and an orifice disk, as sketched in Figure 2
(sketch by Nestl6 Product Technology Centre Konolfingen,
Switzerland). The swirl chambers used in the study were
types SB, SC, SD, and SF, each with a single inlet slit. The
values of the swirl chamber diameter dsc and height hsc, and
the inlet slit width bCh, are summarized in Table 1. Orifice
plates with different exit hole diameters, termed types 30, 40,
50, 60, and 70, were used.
The type numbers denote the orifice hole diameter dor in
thousandths of an inch. The hole diameters in mm are listed
in Table 2. The uniform orifice plate thickness is 3.16 mm,
and the inlet curvature radius is 1.85 mm throughout. Thus,
the cylindrical section of the orifice has a length of 1.31 mm
S End plate
Swirl chamber
Orifice disc
Figure 2: Sketch of the Delavan SDX atomizer. The liquid
enters the swirl chamber through the single inlet slit (sketch
by Nestl6 PTC Konolfingen).
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Table 1: Dimensions of the Delavan SDX swirl chambers.
Swirl Swirl chamber Swirl Swirl chamber
chamber height hsc chamber inlet width
type [mm] diameter bch [mm]
dsc [mm]
SB 1.23 11.85 1.61
SC 1.36 10.71 2.45
SD 1.88 10.71 2.45
SF 3.77 10.71 2.45
Table 2: Exit hole diameters in the Delavan SDX orifice
plates.
Orifice type Orifice diameter dor [mm]
30 0.762
40 1.016
50 1.270
60 1.524
70 1.778
in all orifice plates. Since the swirl chambers may be
combined flexibly with the orifice plates, there is a wide
geometric variability of the atomizers.
2.2 Techniques for characterizing the spray flows
The present series of experiments on the formation of sprays
from pressureswirl atomizers aimed at a detailed
characterization of the sprays. Two important global
geometrical parameters, the sheet opening angle and the
sheet breakup length, were determined from photographs of
the sprays by simple image processing (see Figure 3). The
photographs were taken with a digital camera, using
flashlight illumination from the front in order to have
reliable images for determining the sheet breakup length.
Furthermore, the mean drop sizes in the sprays, especially
the global number mean and Sauter mean drop sizes,
D10,global and D32,global, were determined as functions of the
various influencing quantities. The global Sauter mean drop
size represents a drop with the same volume to surface area
ratio as the entire spray. It is thus an important quantity for
characterizing the atomization process for applications with
transfer processes across the drop surface. The measurement
of properties of the spray drops requires a measuring
technique such as the light diffractionbased Malvern
technique or phaseDoppler anemometry (PDA). The
advantage of the latter is that it provides local information
about the spray drops with high resolution, so that PDA was
chosen for the present study. Global spray properties are
derived from the local PDA data by a postprocessing
routine, which will be discussed below.
A phaseDoppler anemometer measures size and velocity of
drops passing an optically defined probe volume. The laser
light source of the present DANTEC PDA system is a
continuouswave ArgonIon laser Coherent Innova 90C3.
The probe volume of the present standard PDA system is
formed by the intersection of four laser beams, depicted as
Figure 3: Photograph of a water spray with the opening
angle a, produced by the spray nozzle type SC30 at the
mass flow rate of 20.8 kg/h (corresponding driving pressure
difference: 7 bar).
Figure 4: Sketch of a spray cross section with the points of
measurement and the transmitting and receiving optics units
of the PDA system. The points of measurement are
equidistant.
grey lines in Figure 1. The diameter and velocity
measurement bases on the analysis of laser light scattered
from the probe volume by the spray drops. The refracted
and reflected components of the scattered light may be used
for PDA. The dominance of one of these two scattering
modes, which is a prerequisite for correct PDA
measurements, is determined by the optical configuration of
the PDA system and the refractive index of the drop liquid.
An important parameter of the optical configuration of the
PDA system is the offaxis angle (scattering angle) (p where
the receiving optics unit is placed (see Figure 4). The liquids
used in the present investigations are optically transparent
watersucroseethanol solutions. As a consequence of this,
the dominant scattering mode chosen was the refractive
mode, which yields high scattered light intensities in
forward directions. The appropriate scattering angle for this
mode was 500 in all measurements. The phase factor
relating the drop size to the Doppler signal phase shift was
0.575 /gm throughout. The maximum measurable drop
size was 450.9 gm. The optical signals are converted into
electrical signals by the photodetectors. For the signal and
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
data processing, the Dantec PDA processor 58N50 and the
Dantec BSA Flow Software in the version 2.12 were used.
For obtaining global spray properties, local drop ensembles
were measured with the PDA at 31 points in each spray
cross section. The distance of the plane of measurement
from the atomizer orifice was 80 mm in all measurements.
This distance was found by visualization of the drops as the
best compromise for the measurements of all experiments,
closest possible to the sheet breakup zone to ensure that the
measurements capture the drop formation process only,
without a notable influence from evaporation and
coalescence, and still ensuring spherical drops for the PDA
measurements. The points of measurement were arranged
equidistant along one radius of the cross section, as sketched
in Figure 4. Axial symmetry of the sprays was verified in
preliminary tests. The outermost point was determined by
the condition that the frequency of drop arrival at the probe
volume (the data rate) there should be 30 Hz. The distance
between the points of measurement varied between the
different sprays. Sufficient statistical reliability of the PDA
measurement results was ensured by acquiring 20,000 drops
in each local measurement.
For every local measurement, the drop sizes were Saffman
corrected (Saffman 1987), and the drops were grouped in
100 size classes by a homedeveloped Matlab procedure.
For deducing global spray information, the local data were
weighted with the local drop number flux and with the
annular part of the spray cross section for which the data are
representative (see the illustration in Figure 4). The most
important global spray property, the global Sauter mean
drop size D32,global, is given by the equation
J I J I
D3(r)i(r,D1)27r Ar Z D3(r))n(r,D1)r,
D32,global J I J I
D2)nf(r) ,D1)27r Ar D(r(rD,)r
J11 1J1 1 1 1=1
(1)
In this equation, D, (r) is the mean drop size of size class i at
measuring point j, and i (rj,D1) denotes the number flux of
the drops with sizes in size class D, at the measurement
position rj, which was obtained by dividing the drop rate by
the validation rate achieved in the PDA measurements.
Validation rates of the order of 60% were reached, which is
an indication for good measurements, given the partly high
drop concentrations in the spray zones where the
measurements were carried out. The term 27rjAr, which is
the product of the mean circumference of the annular part j
of the spray cross section and the width of the annulus,
accounts for the positiondependent area of the annulus for
which the local measurements are representative. The factor
Ar was cancelled from the equation due to the equidistant
arrangement of the measuring positions.
The above formulation of D32,global implies the replacement
of integrals over the drop size range and the spray cross
section by sums of discrete rectangle areas in the sense of
a numerical integration by a quadrature formula. The error
in this approximation depends on the radius of the cross
section, the radial step width Ar, and the maximum variation
of the functions 27xrf(r,d)d3 and 27rfi(r,d)d2 with the
radial coordinate. For experiment #23, the resulting maxi
003
0025
002
E
0015
LJ.
S001
0005
0
*PDF
 D1D0,global
  D32,global
0 50 100 150
d [pm]
Figure 5: Global Probability Density Function of the drop
size and global mean drop sizes from spray experiment # 1
listed in Table 3.
mum error is estimated in the order of 8% of D32,global. In
experiments with smaller spray angle, the accuracy is even
better.
The drop size distribution of the entire spray is given by the
global Probability Density Function (PDF). This function is
given as
J
Sl(r ,D,) r,
1 = (2)
PDF(D1) (2)
ZZ i(rj,D,) rJ
=1 1=1
where AD is the width of the 50 size classes used for
computing the global size PDFs. The global drop size PDF
of the spray from experiment #1, specified in Table 3, is
depicted in Figure 5 as an example. Next to the PDF, the
global number mean and Sauter mean drop sizes, D10,global
and D32,global, are depicted in Figure 5 by solid and
dashdotted lines, respectively.
2.3 The test liquids
The test liquids used were watersucroseethanol solutions
with varying composition. By changing the contents of the
three mixture components, the liquid viscosity [ and the
surface tension c may be adjusted independently (Dorfner et
al. 1995). The liquid density, however, cannot be controlled.
The liquid properties were varied in the ranges 0.00869 Pas
< K < 0.1714 Pas, 0.0465 N/m cT < 0.072 N/m, and 1201
kg/mi3 p 1315 kg/m3.
The test liquids were characterized as follows: the dynamic
viscosity was measured with capillary viscometers from
Schott, types 501 20 and 501 23, since the liquids were all
Newtonian in flow behaviour. The surface tension against
air was measured with a ring tensiometer Krtiss K 8. Since
the test liquids under investigation do not contain
macromolecular, surfaceactive components, the equilibrium
values of the surface tension obtained by these
measurements occur in the atomization process also and are,
therefore, relevant for the characterization. When measuring
mixtures with an ethanol content, the measurements were
carried out quickly to keep evaporation losses of the ethanol
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Table 3: Data defining the 30 spray experiments liquid
properties, flow rate, pressure drop, and measured D32,global
and sheet opening angle a.
Experi Atomizer Dynamic Liquid Surface
ment# configura viscosity density tension
tion 9 p a [mN/m]
[mPas] [kg/m3]
1 SB 30 16.26 1240 72
2 SB 30 18.67 1247 72
3 SB 70 12.13 1220 72
4 SD70 10.60 1213 72
5 SF 30 14.24 1232 72
6 SF 50 13.24 1226 72
7 SF 70 8.69 1201 72
8 SB 70 40.25 1276 72
9 SC 70 48.66 1282 72
10 SD 50 41.56 1277 72
11 SD 70 52.85 1285 72
12 SF 30 54.00 1288 72
13 SF 50 32.50 1267 72
14 SF 70 46.79 1281 72
15 SB 70 127.83 1315 72
16 SC70 171.41 1310 72
17 SD 50 140.00 1314 72
18 SD 70 110.00 1306 72
19 SF 50 116.16 1307 72
20 SF 70 101.80 1304 72
21 SB 30 63.45 1277 51.4
22 SB 70 56.00 1269 48.5
23 SC 70 57.84 1275 57.9
24 SD 50 43.50 1257 52.0
25 SF 50 65.57 1280 49.0
26 SF 70 52.10 1250 52.1
27 SB 70 166.90 1300 53.1
28 SC 70 146.00 1290 49.7
29 SF 70 157.60 1290 46.5
30 SC40 152.10 1297 51.5
low. The density p of the liquids was measured with the
flow meter sketched in Figure 1, which was also used for
measuring the liquid mass flow rate through the atomizer.
The various physical properties of the test liquids, together
with the physical conditions of the experiments and the
atomizers used, are put together in Table 3. The global
Sauter mean drop sizes D32,global and the cone angles a of the
liquid sheets measured in the experiments are also listed in
Table 3. The ambient medium in the spray experiments was
always the air of the laboratory at a pressure of 1.013 bar
and a temperature of (20 + 1)C, resulting in an ambient air
density of 1.204 kg/m3. This quantity, which has an
influence on the spray formation due to the
KelvinHelmholtz instability of the conical sheets, was not
varied in the experiments and therefore does not appear as
an influencing quantity in our study.
2.4 Measurement program
The experiments were planned using the method of the
factorial experimental design. Basically, this design defines
combinations of extreme values of the parameters to be
chosen. In addition, intermediate states between the extreme
Experi Mass flow Driving Global Sheet angle
ment # rate pressure Sauter ca
m [kg/h] difference mean []
Ap diameter
[bar] D32,global
______ ___[pm] ____
1 151 123 58.86 55
2 173 152 52.92 50.5
3 141 19 130.63 80
4 187 22 113.06 72
5 200 84 58.17 60
6 234 38 92.62 55
7 153 7.5 171.49 63
8 152 22 112.61 61
9 279 46 90.74 59
10 312 71 72.74 50
11 303 30 110.97 52
12 200 80 66.46 15
13 243 35 77.54 49
14 351 29 100.02 47
15 444 87 89.09 43
16 423 89 73.60 42
17 494 139 61.74 33
18 388 46 91.62 45
19 410 80 80.05 25
20 538 58 80.91 42
21 200 105 73.60 45
22 211 30 103.58 54
23 279 40 89.31 63
24 312 73 63.88 64
25 380 72 71.00 38
26 391 39 79.01 56
27 400 69 90.05 48
28 400 54 91.43 47
29 480 40 88.76 42
30 250 65 88.04 23
ones may be included to enhance the information about
influences from the various parameters on the process
results. The parameters varied in the factorial design are the
swirl chamber height, the orifice diameter, as well as the
dynamic viscosity and the surface tension of the liquid.
Mass flow rates were varied between 141 and 538 kg/h.
Depending on the geometry of the atomizers and the liquid
properties, such as the dynamic viscosity, this led to
pressure drops across the atomizers between 7.5 and 152 bar.
These spray conditions are relevant for the industrial
production scale. Overall, 30 different spray experiments
were carried out.
3. Characterization of the sprays resulting from
sheet breakup
The results of the experiments essentially consist in the
global Probability Density Function (PDF) of the spray drop
size, and in moments or a ratio of moments of this PDF,
such as the global number mean and Sauter mean drop sizes,
D10,global and D32,global. The PDA measurements, of course,
provide much more detail, such as two velocity components
of the drops and the times of arrival at and transit through
the probe volume. The latter information, however, was not
analyzed here, since the interest of the present investigation
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
is focused on the spray drop size spectrum. Geometrical
spray properties determined by image processing in this
study were the sheet breakup length L and the opening
angle a.
A dimensional analysis of the formation of the global Sauter
mean drop size in the sprays was carried out. The list of
parameters relevant for the drop formation includes the
global Sauter mean drop size D32,global itself, the liquid
density p, dynamic viscosity [, and surface tension CT
against air, the diameter dsc, height hsc, and inlet slit width
bch of the swirl chamber, the orifice diameter dor, and the
driving pressure difference Ap.
The purpose of this analysis is to develop a universal
representation of the global mean spray drop size D32,global as
a function of the relevant influencing parameters. With the
three basic dimensions m, s, and kg involved, the above set
of nine parameters results in six independent
nondimensional groups characterizing the atomization
result. The groups found by the analysis are the
nondimensional D32,global, three length scale ratios
representing the atomizer geometry, and the pressurebased
Reynolds number as well as the Ohnesorge number of the
process, which read
D32,global dor
dsc dsc'
App p dsc
Rep ,
1^
hsc bch
dsc dsc
Oh= .do
Vu dor,
Seeking a correlation for the nondimensional D32,global in the
form of a product of powers of the nondimensional groups,
times a constant, the exponents and the constant were
determined by nonlinear regression to the experimental
data based on the leastsquares method. The correlation
found reads
D32,global
dsc c
32,global
/ h~00574
= 3.074 Re 8505 Oh0 7538 hs
dsc
/ 03496 00426
.(dor, bch
Ydc) l )
0 02
measurement
trend line
0015
R = 09566
001
0 005
0 0005 001 0015 002
go32,global /
dsc c
Figure 6: Nondimensional representation of the measured
global Sauter mean drop size of the watersucroseethanol
solution sprays listed in Table 3 by correlation (3).
In Figure 6, the values (D32,global/dsc)c from correlation (3) are
depicted together with the measured values from the 30
experiments given in Table 3. The correlation represents the
experimental data with a coefficient of determination R2=
0.9566, which is an excellent result.
In the following, the sensitivity of the global Sauter mean
drop size given by equation (3) against uncertainties in the
measured quantities entering the righthand side was
analyzed. The geometric parameters of the nozzle may be
measured with high accuracy and therefore bring in very
low systematic errors. In contrast to this, the driving
pressure difference and the liquid density and surface
tension against air may be subject to stochastic errors due to
the measuring techniques applied. Process parameters may
furthermore be subject to fluctuations, which cannot be
avoided. The total uncertainty of the global Sauter mean
drop size as described by equation (3) can be written as
AD32,global
D32,global
0.4253 A(A2 + 0.3769 2
Ap I o a
+ 0.0484 + 0.09674M
The relative uncertainties of the measured driving pressure
difference A(Ap)/ Ap and the surface tension are of the order
of 1%. The relative uncertainty of the measured liquid
density Ap/p is of the order of 0.25%, as given by the
producer of the Foxboro flow meter. The dynamic viscosity
was measured with an uncertainty of 1%. The total
uncertainty of the measured global Sauter mean drop size
AD32,global/ D32,global then results from equation (4) as
0.58%, which indicates a very high reproducibility of this
quantity. The maximum deviation of an individual data
point from correlation (3) is 11.6% of the measured value.
Characterized by the coefficient of determination with the
value of 0.9566, the correlation represents the experimental
data very well. An additional parameter for characterizing
the goodness of the fit is the coefficient of variation Cv,
which is defined as
s(x)
Cv (5)
Xmean
Here, s (x) denotes the sample standard deviation, and xmea
the sample mean value of a quantity x. In the present
experiments, the coefficient of variation of the global Sauter
mean drop size is calculated with the differences between
measured and modeled values as the variable x. In view of
the significance of the quantity c, for the present study, the
mean measured D32,global is taken as the quantity xmean. The
coefficient Cv according to this definition for the data set in
Figure 6 has the value of 2.86 %, which is an excellent
result.
Another spray property of interest is a measure of the width
of the global drop size PDF, i.e. the degree of variation of
the drop size in the spray. For quantifying this, the
rootmean square (standard deviation) of the drop size in the
global spectra, RMSglobal (D), was determined. For
plausibility reasons and from experimental evidence we can
expect a relation between a global mean drop size and the
RMSglobal to exist, which may represent an increase of the
RMSglobal with the mean drop size. The local standard
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
deviation of the drop size RMS (D) is defined as
RMS(D)= 1fnl(Dl D0)2
N I
where N is the total number of drops in the ensemble, n, the
number of drops in size class i, and I the total number of
size classes. The global standard deviation of the drop size
RMSglobal(D) is calculated from the local PDA data as per
RMSgloba (D)
Z n(Di,rj).2 jAr (D D10,global)
j=1 i=1
I n(Di,rj).2 rjAr
j=l i=l
D20,global D1h,global
(7)
(Sowa 1992). The symbols Dlo,global and D2o,global denote the
global number mean and areamean drop sizes, respectively.
The size class summation extends over all global size
classes i; the summation over all annular parts j of the spray
cross section covers the whole cross section. By linear
regression of the experimental data, the correlation
RMSglobal (D) = 7.3504 + 0.1268 D32,global
(8)
+0.0008 D32,global
was found, where the values of D32,global must be entered in
microns to yield the RMSglobal in microns. In Figure 7, the
correlation is depicted as a trend line to the experimental
data in a diagram of the global standard deviation of the
drop size as a function of the global Sauter mean drop size.
The coefficient of determination has a value of 0.9757; the
coefficient of variation is 3.28 %, where the arithmetic mean
of the measured values of the drop size RMS is the
reference quantity xmean of equation (5). It is seen that the
global standard deviation of the drop size increases with the
global Sauter mean drop size, which confirms the
expectation. It is interesting to note that, according to
relation (8), there exists a least scatter in the drop size in the
70
measurement
trend line
R2 = 0 9757
10
0 50 100 150 200
D32,globa l [pm]
Figure 7: Global standard deviation of the drop size as a
function of the global Sauter mean drop size for the
watersucroseethanol solution sprays listed in Table 3.
sprays, even when the Sauter mean diameter becomes very
small. This relation seems to exist between the dimensional
quantities; it is unnecessary to nondimensionalize the data,
despite the strong variations of liquid properties, atomizer
geometry, and operation conditions. A further investigation
of this finding is beyond the scope of the present study.
Having carried out this experimental survey of spray
formation with pressureswirl atomizers, in the following
sections a theoretical basis is presented for establishing a
relation of the relevant parameters of the spraying process
not only with moments of the drop size spectra, but also
with the shape of the spectra.
4. Prediction of the atomization result by the
Dombrowski and Johns model
The basis of the prediction of the atomization result by the
approach of Dombrowski & Johns (1963), which is widely
accepted in the literature, is a stability analysis of the liquid
sheet and of the ligaments formed by the sheet breakup. For
this analysis it is important to know the sheet velocity and
thickness in the breakup zone, and the physical properties
of the liquid and the ambient gaseous medium. For
determining the sheet velocity and thickness, various
methods are established in the literature (e.g., Lefebvre 1989,
Dahl & Muschelknautz 1992, Walzel 1998, Senecal et al.
1999, Schmidt et al. 1999). The results presented here are
based on Walzel (1998) and Schmidt et al. (1999), who
define a velocity number (p as the ratio of the liquid sheet
velocity U to the potential velocity:
U
T = U (9)
V2 Ap/p (9)
For given mass flow rate m driving pressure difference Ap,
sheet opening angle a, and geometry of the atomizer, the
smallest possible velocity number (pmi occurs when the
air core in the pressureswirl atomizer disappears, so that the
liquid fills the whole orifice cross section. This value of the
velocity number is given as
4m/n pd2r cos (a/2)1
(Pmin = (10)
(Schmidt et al. 1999). The largest possible velocity number
pmax arises when the pressure energy is converted into
kinetic energy without losses, i.e., when the liquid sheet
velocity is the potential velocity. Then (p exhibits the
value pmax = 1. As an alternative for the velocity number
p Lefebvre (1989) specifies a correlation for the discharge
coefficient cD, which turns out problematic for the present
work, since in some cases it predicts values of CD greater
than 1, which is unphysical. Schmidt et al. (1999) use an
expression which turns out inappropriate for the present
work, since it predicts no air core in the atomizer in some
experiments of this study, where this was not the case.
As an alternative approach to get a good estimate of the
velocity number p, we take the arithmetic mean of the two
extremes of p, i.e., we set
(P= ((max +Pmin)= i (1+(min) (11)
2 2
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
This relation represents the present experiments best, which
was found from comparisons of the sheet velocity following
from equations (9) (11) above with the velocities of very
large drops in the sprays measured with the PDA close to
the sheet breakup zone.
From the velocity number p, then, the sheet velocity U is
calculated. Next, for calculating the sheet thickness, we
assume a conical sheet with constant cross section normal to
the direction of the sheet velocity. This configuration is
sketched in Figure 8. The sheet thickness then comes out
proportional to the inverse of the distance from the pole of
the flow. Denoting the radial extension of the sheet (normal
to the symmetry axis of the atomizer) r, and the state at the
orifice with subscript Or, the thickness of the annular film
tor there is obtained from a formulation of the constant
liquid volume flow rate as
t dor 1
tor = 2
2
1 41i / p
2 d7 r Ucos(a / 2)
Figure 8: Sketch of the liquid flow through the atomizer
orifice and the conical sheet with its cone angle a and
thickness t(x).
The constant sheet cross section and velocity lead to the
formulation of the sheet thickness at a sufficient distance
from the orifice
t(x) p (13)
n U[dor tor +2x sin(a /2)
In equation (13), the x coordinate is oriented in the direction
of the sheet motion, as sketched in Figure 8. The origin x =
0 is located at the orifice. The sheet thickness t is measured
normal to the direction of the sheet velocity U. Its value in
the breakup zone is important for determining the ligament
diameter, and, consequently, the drop size.
In a next step, the mechanism of drop formation in sprays is
discussed. Stability analyses of liquid sheets formed in
atomization processes with prefilming atomizers were
carried out by a number of researchers, e.g., Squire (1953),
Dombrowski & Johns (1963), Li & Tankin (1991), and
Senecal et al. (1999). Dombrowski & Johns considered the
KelvinHelmholtz instability of liquid sheets under the
influence of surface tension and liquid viscosity. They
derived a dispersion relation for the growth rate of long
waves, using a onedimensional model of the liquid sheet.
However, neglecting variations of the shear stress across the
thickness of the liquid sheet, the determined dispersion
relation turned out incorrect for viscous liquid sheets, since
it cannot represent the continuity of the shear stress across
the interface (Senecal et al., 1999). Li & Tankin (1991)
considered these variations, but assumed dominant growth
of longwave disturbances, which leads to inaccuracies in
the prediction of sheet instability for conditions found with
many prefilming atomizers. Senecal et al. (1999) pointed
out that the liquid sheet disintegration, characterized by
values of the gas Weber number
Pg U2tor
Weg 2= (14)
S20
above the threshold of 1.69, is dominated by shortwave
disturbances. In the gas Weber number, pg denotes the gas
density and C the surface tension of the liquid against the
ambient gas.
In the present study we investigate twodimensional viscous
incompressible liquid sheets of thickness t = 2h, as
described by Li & Tankin (1991), as well as by Senecal et al.
(1999). The sheet moves in xdirection through a gaseous
medium which is assumed to be quiescent, inviscid, and
incompressible. The orientations of the coordinate axes are
depicted in Figure 9. Deformations T of the liquid sheet are
described in the exponential form
r = 91T[o exp(ikx + ot)] (15)
Here, ro is the initial wave amplitude, k = 2 7/0 is the (real)
wave number, and o is the complex frequency. Therefore,
the stability analysis carried out is temporal. The
disturbance responsible for the sheet breakup is the one
with a wave number Ks leading to the largest growth rate
s The wavelength of this disturbance, together with the
sheet thickness at breakup, determines the resulting
ligament diameter. This most unstable disturbance is found
h x
h (
(a)
Figure 9: Disturbance waves on plane liquid sheets: (a)
Antisymmetric or sinuous waves; (b) symmetric or varicose
waves (adapted from Senecal et al. 1999).
from the dispersion relation o = f (k) of the moving sheet.
This dispersion relation represents the jump condition of the
normal stress across the deformed liquidgas interface as a
result of capillary pressure in the sheet due to the
deformation.
The liquid normal stress at the interface follows from the
solution of the equations of continuity and momentum of the
liquid sheet flow, formulated in the disturbance quantities.
Decomposing the velocities and pressure into an irrotational
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
and a rotational part, the linearized equations of continuity
and momentum can be expressed in terms of a velocity
potential 4 and a stream function V Hence, the normal
stress in the liquid sheet can be formulated in terms of the
velocity potential based momentum equation (Li & Tankin
1991). The gas normal stress at the interface is determined
by the continuity and momentum equations for the gas,
which is assumed to be inviscid and at rest in its undisturbed
state. The pressure induced by surface tension on the
deformed interfaces is related to the curvature of the
interface.
The normal stresses in the liquid and the gaseous phases,
and the pressure induced by surface tension, are mutually
related by the liquid sheet dynamic boundary condition for
the normal stress at the interface, which yields the following
dispersion relation between the complex frequency co and
the disturbance wave number k for antisymmetric
disturbances:
p(o +ikU) + 2 k2 (k2 + s2) tanh(kh)
2 (16)
2
4 k3s tanh(sh) + pgC2 + k3 = 0
(Li & Tankin 1991). The corresponding relation for
symmetric disturbances is very similar to (16), with
tanh(kh) and tanh(sh) replaced by coth(kh) and coth(sh),
respectively. Liquid sheets with antisymmetric and
symmetric deformations are depicted in Figure 9. In both
cases, s represents the parameter
s= k2 +p(ma +ikU)/g (17)
An order of magnitude analysis shows that the terms of
second order in viscosity in (16) can be neglected against
the other terms (Senecal et al. 1999). With this
simplification, and by rearrangement, the growth rate for the
antisymmetric disturbances is given by
2 2tk2 tanh(kh) 1 /(4 2k4 tanh2 (kh) 2
o, =+
p [tanh(kh) + Q] tanh(kh) + Q
Q2U2k2 [tanh(kh)+Q](QU2k2 +k3 /))12
(18)
(Senecal et al. 1999). Here, or denotes the real part of the
complex frequency, which is the rate of growth of the
disturbances, and Q is the ratio of the gas to liquid densities,
p / p, which is O (103).
Senecal et al. (1999) point out that sheets under the
influence of KelvinHelmholtz instability in a gaseous
environment are subject to short wavedominated
atomization, if the gas Weber number exceeds the value of
1.69. Consequently, the dominant wave numbers are large,
and the values of the functions tanh(kh) and coth(kh)
converge to unity. If additionally the smallness of the
density ratio Q<<1 is accounted for, equation (18) reduces
to
2 k2 4 [42k4 k3
r + 2 +QU2k2 k (19)
p p P
which is also obtained for the symmetric mode. By
requiring the derivative of or with respect to k to vanish
at the maximum of the function, the wave number Ks (and
the related wavelength As) corresponding to the maximum
growth rate Is0 is obtained. The equation determining Ks
is of third order and must be solved numerically.
Based on these disturbance parameters, the sheet
disintegration process can be quantified following the
mechanism proposed by Dombrowski & Johns (1963).
Since the growth rate of short waves does not depend on the
sheet thickness, as seen in equation (19), the sheet breakup
length L is calculated as follows:
L = UT = ln (20)
Is To
Here, U denotes the sheet velocity, which is at the same time
the relative velocity between the liquid and the quiescent
gas. t denotes the breakup time, and rb is the value of the
sheet deformation at breakup. The quantity ln( rIb / r ) has
the value of 12 (Senecal et al. 1999). The breakup length L
plays the role of the xcoordinate in equation (13) and
determines the sheet thickness tb = t(L) at breakup, which,
together with the wavelength of the dominant disturbance,
defines the crosssectional area of the ligaments detached
from the sheet. The diameter of the ligaments dL separated
from the liquid sheet in the breakup zone is found from a
volume balance as
dL 8tb (21)
SKs
In this balance it is assumed that ligaments are formed from
tears in the sheet once per wavelength, which is typical for
short wave atomization (Senecal et al. 1999).
The last step of the spray formation process is the breakup
of the ligaments into drops. The diameter of the drops dD
formed by this process may be calculated using Weber's
result for the wavelength of the fastest growing disturbance
on a cylindrical viscous liquid jet, which leads to the
equation
dD
for the drop size
Ohnesorge number
1.88 dL (1+3 Oh)1/6 (22)
(Weber 1931). Here, Oh denotes the
(23)
Oh [
VpdL
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The calculation of the drop size based on these assumptions
leads to an unsatisfactory result, which is shown by the
comparison with the measured global Sauter mean drop
sizes D32,global (Dombrowski & Johns 1963) depicted in
Figure 10. The inaccuracy is quantified by the coefficient of
variation Cv with the arithmetic mean value of the
measured global Sauter mean drop sizes as the reference
quantity xmean of 27.2 %, which represents large deviations
of the model data from the measured values. This finding is
in agreement with the data in Dombrowski & Johns (1963),
which also show unsatisfactory agreement between the
measured and the calculated drop sizes, although with less
scatter than presently seen. Problematic is also that the
model predicts very large sheet breakup lengths in a
number of cases, which is in conflict with the breakup
lengths measured by photographic visualization in our study,
as presented in Table 4. These discrepancies clearly indicate
the need for an alternative approach.
5. Prediction of drop size spectra in the sprays
An alternative to the above described method of predicting
mean diameters of the droplets in the sprays is to predict the
global drop size spectrum, from which the mean diameters
can then be deduced. This approach avoids the calculation
of the drop size invoking the Dombrowski and Johns model
and the hypothesis that the ligaments formed in the sheet
disintegration break up into droplets through the capillary
mechanism described by Weber (1931).
The drop size spectrum in a spray is given by a probability
density function. This function is obtained experimentally
by measurements with phaseDoppler anemometry (PDA).
The function is represented in a discrete form by
measurement data grouped in size classes. The aim of the
prediction of this function is to find a model function
suitable for representing the spectrum most accurately with
a minimum number of free parameters. The suitability of the
model function is judged by comparison of the function
with the measured data, and also of moments of the function
with moments measured, such as the number mean and
Sauter mean drop sizes. The model function searched for
should be derived from physical principles. One approach is
to apply the maximum entropy formalism. The requirement
that the Shannon entropy of the spray should be a maximum
leads to the threeparameter generalized gamma distribution
of the spray drop size spectrum (Dumouchel 2006):
Hfd
*
50 100
dD [pm]
150 200
Figure 10: Comparison of the measured global Sauter mean
drop size D32,global and the drop size dD calculated by the
Dombrowski and Johns model. The underlying experiments
are listed in Table 3.
In this function, which is a probability density function, d is
the drop size, V and q denote nondimensional parameters,
and dqo is a constraint drop size.
An alternative to obtain a suitable mathematical
representation of the drop size PDF is to consider the
formation of spray drops by the breakup of ligaments, as in
the present case. The form of the drop size distribution can
then be determined by the model suggested by Villermaux et
al. (I2'" 4). In their model, the spray drops are formed by an
aggregation process of subdrops that make up the ligaments.
According to this model, the subdrops are arranged in
several sublayers of the ligaments, as sketched in Figure 11.
VV d qv1
PDF (d)= q exp
T(v) dqOs
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
In this process, the spray drop size is larger than
thickness of the ligaments, since the drops result
subdrop coalescence. This model feature correspond
experimental evidence. The timedependent distribution
the subdrop sizes is directed by the coalescence of
interacting subdrops in the adjacent sublayers. Unde
assumption of uncorrelated sizes of the intera
subdrops, the development of the subdrop size distribi
is directed by a convolution of the independent sub
size distributions. Since the sublayers are assumed t
mutually independent, the distribution of the final s
drop size is also directed by a convolution. The soluti
the corresponding evolution equation is the numberb
drop size probability PB, which is similar to (24) ai
given in nondimensional form as (Villermaux et al. 200
PB (x)= xv1 exp(v x)
F (v)
Here, x denotes the nondimensional spray drop size
with the dimensional spray drop size d and the liga
diameter dL, and the gamma distribution parameter v.
function satisfies the normalization condition
SPB dx = 1
0
Table 4: Properties of the 30 spray experiments releva
the liquid sheet instability. The gas
calculated with p, = 1.204kg/m3.
Weber numb
Experi Liquid Wavelength Measured Predicted
ment # sheet of the breakup breakup W
veloci dominant length length nu
ty disturbance Lme, [mm] Lpred [mm] W(
U [m/s] As [gm]
1 112.2 84.36 10 12.30 1
2 124.8 75.10 10 11.52 24
3 36.3 539.52 11 63.85 2
4 40.8 426.98 9 50.31 3
5 115.5 76.74 10 10.94 3'
6 62.9 208.05 13 26.55 1(
7 26.0 951.09 17 106.24 2
8 37.1 713.87 22 101.49 2
9 56.3 392.15 18 62.91 6
10 82.3 202.96 15 34.14 1
11 48.8 508.39 24 80.59 6
12 103.4 162.30 18 30.82 24
13 60.3 299.35 21 44.62 9
14 50.4 460.70 31 71.57 7
15 77.8 386.91 33 86.24 1:
16 77.6 458.05 33 110.61 1:
17 115.7 230.64 24 59.00 3
18 60.0 524.05 30 104.38 9
19 90.6 295.18 23 66.67 2
20 71.9 385.12 30 78.78 1
21 115.8 128.21 11 28.54 4(
22 44.8 480.03 21 82.16 5
23 54.0 402.50 17 69.68 8
24 86.0 164.32 12.5 30.34 2
25 87.5 191.83 23 40.47 3
26 59.3 315.17 23 55.39 1i
27 70.4 456.95 37 115.87 1i
28 64.7 464.93 30 112.53 1
29 61.7 505.23 30 125.69 2(
30 83.8 334.05 16 87.44 2
the
from
ds to
n of
f the
r the
acting
ution
drop
o be
Figure 11: Illustration of a ligament consisting of subdrops
in three sublayers (adapted from Villermaux 2007).
;pray The coalescence process leads to a drop size distribution
on of characterized by an exponential decrease in the range of
)ased large drops (Villermaux 2007). Consequently, also the drop
nd is size distribution of the resulting spray exhibits this
14) characteristic. This feature is represented correctly by the
gamma distribution (25).
(25) A disadvantage of the formulation of the gamma distribution
PB in equation (25) can be seen in the role of the ligament
d/dL, diameter dL. The formation of ligaments is believed to be
ment due to the detachment of liquid portions from the end of the
The liquid sheets deformed by surface waves, as discussed in
section 4. Both the sheet breakup length and the ligament
diameter deduced from this analysis are subject to
(26) uncertainties, as pointed out in section 4. It is therefore
desirable to avoid the ligament diameter as a reference
length scale. As an alternative, an empirical gamma
distribution PB,E is specified as dependent on a modified
nt for
r nondimensional spray drop size x, containing the
er is
wavelength of the dominant disturbance A, and a
parameter K. The wavelength As, as calculated in section
ias 4, is presented in Table 4 for the 30 experiments of this
eber study and can be regarded as a reliable sheet property. The
mber parameter K is introduced as a scaling parameter
g [] (Villermaux et al. 2004). Thus, the gamma distribution PB,E
is written by modification of equation (25) as
9.87 v" d 1 d
1.74 PB,E = ) exp v (27)
.63 v(v) A As
.82 The dimensional form of the gamma distribution PB,E in
7.98 equation (27) is obtained by dividing the gamma
0.49 distribution by the wavelength As and the parameter K.
.02
.42 The so obtained numberbased drop size PDF reads
.85 1d d
6.87 PDF(d)= exp v (28)
.53 ASK F(v) AK) AK
1.58
.97
.92
3.88
3.07
5.05
.82
3.03
6.95
).15
.97
.50
6.98
2.15
5.32
5.89
6.02
).74
3.25
This gamma distribution also satisfies a normalization
condition. A typical gammadistributionbased drop size
spectrum is depicted in Figure 12 together with the
experimental data from spray experiment #23 defined in
Table 3. The values of the parameters of the gamma
distribution were found by fitting the function to the
experimental data using the LeastSquares Method. The
value of As is determined by the conditions of experiment
#23. The coefficient of determination R2 has the value of
0.9918, which indicates an excellent representation of the
measured data.
At first appearance in Figure 12a, the representation of the
whole measured drop size spectrum by the modeled PDF is
excellent, although, for small drop sizes up to 20 gmn, the
model seems to slightly overpredict the measured data, and
for drop sizes above 50 gm the measured data are slightly
underpredicted. For a model function with only two
independent parameters (v and K), however, the achieved
agreement is excellent. Nonetheless, the representation of
this set of data with logarithmic scaling of the ordinate axis
in Figure 12b shows deviations in the range of the large
drop sizes. Although the values of the PDF in this range are
small, these deviations lead to considerable differences
between measured and modeled higherorder moments of
the spectra as, e.g., in the global Sauter mean drop size
D32,global. This is due to the fact that this mean diameter
depends on the cubed and squared drop size. Since large
drops contribute strongly to the overall liquid volume and
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
This finding indicates that, for a more accurate prediction of
higherorder moments of the drop size spectrum, the
probability density of large drop sizes must be represented
better than by the function (28). This improvement can be
achieved by an appropriate modification of the gamma
distribution. This was done empirically by adding an
exponential term to the function, which raises the large drop
size
* measurement
model
R2=0 9918
0 10 20 30 40 50
Dlo,gobal,calc [Pm]
0 50 100 150 200
D32,gobal,.clc [Pm]
(a) (b)
Figure 13: Comparison of (a) the global number mean drop
size D10,global and (b) the global Sauter mean drop size
0 100 200 300 400 500 D32,global from the measurements and from the gamma
d [pm] distributionbased Probability Density Function (28) for the
(a) experiments in Tables 3 and 5. The individual values of the
parameters v and K of the gamma distribution were
Smeasurement obtained by fits to the corresponding experimental data.
model
R=09918
1 OOE04 **,
1 OOE05 \
1 00E06 \ *,
1 OOE07 
0 100 200 300 400 500
d [pm]
(b)
Figure 12: (a) Measured and gamma distributionmodeled
global Probability Density Functions of the drop size from
spray experiment #23. (b) The same data in a
semilogarithmic representation.
surface area of the spray, even small numbers of large drops
influence the Sauter mean diameter strongly. This is
confirmed by the comparison of the global number mean
and Sauter mean drop sizes, D1o,global and D32,global, as derived
from the measured PDA data and as modeled with the
gammadistributionbased Probability Density Function (28),
shown in Figure 13. While the comparison of the global
number mean drop size Dio,global in Figure 13a shows
reasonable agreement between measurement and prediction
by the model, there is a substantial underprediction of the
global Sauter mean drop size D32,global, shown in Figure 13b.
This is quantified by the coefficients of variation of the data
set, which have the values of 6.5 % for D10,global, and 12.6 %
for D32,global. The reference quantities are the respective
mean values of the measured global mean drop sizes from
all the 30 experiments. Consequently, the present gamma
distribution model as given by equation (28) turns out
unsuitable for predicting higherorder moments of the drop
size spectra.
probability density, but changes the probabilities of the
small sizes only slightly. The resulting modified Probability
Density Function PDFMod reads
PDFMod (d) =
(1+CC2) AK F(v) AK]
(29)
Sd d d
exp v d +C dexp d 
IA,K A2 A,C2
In this equation, the additional exponential term contains the
drop size d, the wavelength of the dominant disturbance
As and two additional nondimensional parameters C1 and
C2. The factor in front of the square brackets follows from
the normalization condition for the modified Probability
Density Function. As a consequence of this modification,
the function has the four adjustable parameters v, K, C1, and
C2. It is well known that an increase of the number of
parameters improves the goodness of fits to experimental
data. However, the mathematical form of the function is also
important for getting a physically plausible representation of
the experiment. It is believed that this requirement is met by
the present modified function better than with many
functions studied in earlier publications (e.g., Bhatia &
Durst 1989, Xu et al. 1993, Paloposki 1994).
The drop size spectrum defined by equation (29) was
computed to represent the experimental data of spray
experiment #23 specified in Table 3. The data of this
experiment were presented in Figure 12 above already in
comparison with the model function given by equation (28).
For computing the modified function, the gamma
distribution parameter v and the parameter K were taken
from the function (28); only the parameters C1 and C2 were
0035
003
0025
002
S0015
0.
001
S005
0
1 OOE01
1 OOE02
1 OOE03
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
newly determined using the leastsquares method for best fit
to the measured global drop size PDF (an attempt to
redetermine the whole set of parameters revealed less
agreement). The results are shown in Figure 14, where the
solid lines indicate the Probability Density Function (29),
and the dashed lines characterize the contribution of the
exponential term added to the original gamma distribution
from equation (28). These data show far better agreement
between the modeled and measured spectra, both in the
linear and in the semilogarithmic representations, than seen
in Figure 12. This is confirmed by the slight increase of the
coefficient of determination R2 to a value of 0.9943. As an
effect of the modification, further to the fairly good match
of the spectra in the range of large drop sizes, the
overrepresentation of small drops seen in Figure 12a is also
reduced. To support the present approach, Figures 15 and 16
show the data from experiments #1 and #2.
Similar to Figure 13, a comparison of the global number
mean and Sauter mean drop sizes, D10,global and D32,global, as
obtained from measurements and modeled with the
modified gammadistribution (29), is shown in Figure 17.
Here, especially the significant improvement of the overall
agreement between the measured and the calculated global
Sauter mean drop sizes over the data in Figures 10 and 13b
is noteworthy. The calculated global number mean drop
sizes are also in better agreement with the measurements.
The coefficient of variation of the data set of the global
number mean drop size D10,global is 3.6 %, and for the global
Sauter mean drop size D32,global it is 4.8 %. Thus, an accurate
prediction even of higherorder moments of the drop size
spectra is achieved with the modified gammadistribution
based Probability Density Function PDFmod found in this
study. For the comparisons in Figure 17, the values of the
four parameters of the function (29) were computed
individually for each of the 30 experiments. These values,
however, cannot be determined a priori for a given
experiment. It is therefore of big interest to relate the values
of the parameters to characteristic data of each experiment,
which are easily accessible.
Therefore, in a further step, the four parameters of function
(29) were modeled in a universal manner. The gamma
distribution parameter v was modeled as a function of the
ratio of the orifice diameter of the atomizer to the liquid
sheet thickness at the orifice, dor/tor, and the gas Weber
number Weg. It was found that v can be expressed as a
product of powers of these two parameters, where the
constant and the exponents were determined by nonlinear
regression analysis of the experimental data. The correlation
found for the gamma distribution parameter v reads
0.4040
v = 0.4867We0.2068K dor (30)
,tor )t
The parameters K, C1, and C2 were modeled as functions
of the nondimensional wavelength As /d,, of the dominant
disturbance. Once again, the constants and the exponents
were obtained from a nonlinear regression analysis of the
30 experiments with the test liquid sprays. The three
correlations read
K= 0.005491 A
Sdsc
0.7211
(31)
1 00E01
1 00E02
1 00E03
E
1 OOE04
1 00E05
1 00E06
1 00E07
measurement
extended model
 model extension
R2=09943
0 100 200 300 400 500
measurement
extended model
 model extension
R2=0 9943
0 100 200 300 400 500
d [pm]
(b)
Figure 14: (a) Measured and extended gamma
distributionmodeled global Probability Density Functions
of the drop size from spray experiment #23. (b) The same
data in a semilogarithmic representation.
measurement
extended model
 model extension
R2= 09799
0 50 100 150
d [pm]
1 OOE01
1 00E02
100E03
1 OOE04 0
a 1 OOE05
100E06 
1 OOE06
1 00E07
0 50 100 150
d [pm]
200 250 300
measurement
extended model
 model extension
R2=0 9799
200 250 300
(b)
Figure 15: (a) Measured and extended gamma
distributionmodeled global Probability Density Functions
of the drop size from spray experiment #1. (b) The same
data in a semilogarithmic representation.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
0 50 100 150
d [pm]
1 OOE01
1 OOE02
100 E03
E
1 OOE04
S1 OOE05
1 OOE06
1 OOE07
0 50 100 150
d [pm]
(b)
measurement distribution parameter v is represented by this model with
extended model limited accuracy, as indicated by the coefficient of
 model extension determination with a value of only 0.6464, the
R2=09935 representation of the other three parameters by the
correlations is supported by the high values of the
coefficient of determination R2 of 0.9216 for K, 0.8177 for
C1, and 0.9128 for C2.
In Table 5, the values of the four parameters are given
together with the global number mean and Sauter mean drop
200 20 30 sizes, Dio,global and D32,global, measured in the experiments.
The values of the four parameters were found by individual
fits to the experimental data. data in a semilogarithmic
representation.
measurement As a concluding investigation, the global number mean and
extended model Sauter mean drop sizes of the sprays, D1, global and D32,global,
 model extension
were determined by the modified gammadistribution based
R= 09935 Probability Density Function PDFmod, applying the
distribution parameters as calculated from the respective
correlations (30) (33). Comparisons of the global number
mean drop size D10,global and the global Sauter mean drop
size D32,global, as derived from the PDA data and modeled
with the extended gammadistribution based Probability
200 250 300 Density Function, are depicted in Figures 19a and 19b,
respectively. As can be seen in these figures, there is good
overall agreement between the measured and the modeled
Figure 16: (a) Measured and extended gamma
distributionmodeled global Probability Density Functions
of the drop size from spray experiment #2. (b) The same
data in a semilogarithmic representation.
0 10 20 30 40 50 0 50 100 150 200
DlO,gobal.,.al [pm] D32,go al,.al P [pm]
(a) (b)
Figure 17: Comparison of (a) the global number mean drop
size D10,global and (b) the global Sauter mean drop size
D32,global from the measurements and from the modified
gamma distributionbased Probability Density Function (29)
for the experiments in Tables 3 and 5. The individual values
of the gamma distribution parameter v and the parameters
K, C1, and C2 of the modified gamma distribution were
obtained by fits to the corresponding experimental data.
S 0.7274
C, =79.299 s
0.5362
C2= 0.0160 ds
dsc )
The individual data of these four distribution parameters,
together with their representation by the correlations, can be
seen in Figure 18 for each experiment. While the gamma
Table 5: Measured global mean drop sizes and calculated
drop size distribution parameters of the 30 experiments.
Experi Dio,global D32,global V K C1 C2
ment # [[tm] [ptm] [] [] [] []
1 21.73 58.86 1.5517 0.2328 2 0.22
2 18.13 52.92 1.552 0.2196 2 0.22
3 34.00 130.63 1.301 0.0531 6 0.1
4 29.27 113.06 1.2334 0.06 6.19 0.11
5 18.11 58.17 1.2718 0.2185 3 0.23
6 20.57 92.62 1.3262 0.088 4 0.16
7 45.44 171.49 1.0267 0.0393 11.75 0.067
8 35.43 112.61 1.487 0.0473 9 0.07
9 26.42 90.74 1.3904 0.062 6 0.095
10 20.18 72.74 1.7783 0.0901 4 0.14
11 36.11 110.97 1.5014 0.07 8 0.09
12 20.42 66.46 1.3956 0.113 3 0.16
13 25.39 77.54 1.381 0.0807 6 0.11
14 26.37 100.02 1.284 0.0531 7 0.09
15 26.24 89.09 1.4716 0.0645 5 0.1
16 22.89 73.60 1.777 0.0391 10.86 0.077
17 20.35 61.74 1.861 0.0782 7 0.119
18 25.80 91.62 1.55 0.0439 11 0.07
19 18.66 80.05 1.6874 0.0616 7 0.1
20 23.79 80.91 1.6597 0.0557 10 0.08
21 18.90 73.60 1.5182 0.1437 2 0.21
22 30.51 103.58 1.7125 0.0593 7 0.1
23 27.02 89.31 1.6805 0.0641 7 0.1
24 18.43 63.88 1.8488 0.1029 5 0.14
25 21.77 71.00 1.74 0.1055 3 0.15
26 20.42 79.01 1.4857 0.0617 7 0.1
27 26.58 90.05 1.782 0.0532 9 0.08
28 27.26 91.43 1.8199 0.0537 9 0.08
29 25.27 88.76 1.3628 0.0458 11 0.07
30 22.53 88.04 1.3575 0.0546 7 0.1
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
mean drop sizes, which is, however, less than
individually determined parameters as presented
17. This is indicated by the increase of the coe
variation from the value of 3.6 % in Figure 17a t
Figure 19a for the global number mean drop size,
the value of 4.8 % in Figure 17b to 9.8 % in Figu
the global Sauter mean drop size. An exception is
of experiment #7, which exhibits the biggest
D1O,global and D32,global, representing at the same
largest deviations from the model, which are 25.8
global number mean drop size D10,global, and 22.5
global Sauter mean drop size D32,global, each valu
respective measured value as the reference qua
reason for this deviation can be found in
conditions of this experiment, which are character
gas Weber number of 2.02, as given in Table 4. S
wavedominated atomization, which is presum
present experiments, presupposes gas Weber nun
above 1.69, the closeness of Weg = 2.02 to this th
experiment # 7 indicates the importance of consid
long wave disturbances in this particular experim
the determination of spray properties for this e
based on shortwave disturbances may be the reas
inaccurate representation of this result by the mode
In general, the good agreement between means
calculated mean drop sizes supports the high va
developed model function for representing drop si
in hollowcone sprays from pressureswirl atomic
present type.
* measurement 0 25
 trend line
02
\* 015
0 6464
7. R2
Sme
tre
0 0 02 0 04 00
A. / d.. [
15 R= 0 8177
10
5
.. measurement
trend line
0 0 02 0 04 0 06 0 08 01
As / ds []
03
* mea
 tren
with the
in Figure
efficient of
o 10 % in
and from
re 19b for
the result
measured
time the
% for the
% for the
e with the
ntity. The
the spray
rized by a
ince short
ed in all
ibers well
reshold in
ering also
ent. Thus,
experiment
on for the
0 10 20 30 40 50
Dioglobal calc [pm]
0 50 100 150
D32,gobal,calc [pm]
200
Figure 19: Comparison of (a) the global number mean drop
size D1o,global and (b) the global Sauter mean drop size
D32,global from the measurements and from the modified
gamma distributionbased Probability Density Function (29)
for the experiments in Tables 3 and 5. The four distribution
parameters were calculated by the correlations (30) (33).
6. Conclusions
l. Sprays from Newtonian liquids produced by pressureswirl
sured and atomizers of the Delavan type SDX were characterized
lue of the experimentally for their drop size spectra and global mean
ze spectra drop sizes. For measuring drop sizes, phaseDoppler
zers of the anemometry was used. The measured global Sauter mean
drop sizes were represented by a universal correlation
obtained by dimensional analysis. The correlation is
reproducible with an uncertainty of 0.58 %; the
nd li.n.et experimental data are represented by the correlation with R2
= 0.96. The global RMS of the drop size in the sprays was
R = 0 9216 found to increase with the global Sauter mean drop size, i.e.,
the larger the biggest drops, the wider the drop size spectra.
The drop size spectra of the sprays investigated were
S modeled using a gamma distribution. For representing even
higherorder moments of the experimental spectra well, the
6 008 01 original version of the model distribution found in the
literature had to be modified empirically so as to represent
the large drops in the sprays better. Correlations of the four
parameters of the modified function with easily accessible
properties of the spray experiments enable the drop size
surement spectra of the sprays and their moments to be predicted with
d Ine high accuracy. Input data needed are the liquid mass flow
rate, the driving pressure difference, the sheet opening angle,
R= o 9128 density, dynamic viscosity, and surface tension of the liquid,
the ambient gas density, and the swirl chamber and orifice
diameters of the atomizer. The findings of this work allow
for the prediction of drop size spectra of hollowcone sprays,
which is of importance for many applications.
0 002 004 006 008 01
A. / d.. []
Acknowledgements
Figure 18: Correlations for the four parameters of the
modified gamma distribution. (a) gamma distribution
parameter V, (b) parameter K of the gamma distribution,
and parameters (c) C1 and (d) C2. The quantity vc represents
the righthand side of equation (30).
Financial support of the present work by the Nestl6 Product
Technology Center Konolfingen (Switzerland) in the frame
of an R&D cooperation is gratefully acknowledged.
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