7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Analysis of the Liquid Atomization out of a Full Cone Pressure Swirl Nozzle
Nicolas Rimbert and Guillaume Castanet
SNancy University, LEMTA,
2 av. de la Foret de Haye, F54504 VandoeuvrelsNancy cedex
nicolas.rimbert@esstin.uhpnancy.fr and guillaume.castanet @ensem.inplnancy.fr
Keywords: Liquid atomization, Spinning, jets, Instability theory, Turbulent atomization, L6vy stable statistics
Abstract
A thorough numerical, theoretical and experimental investigation of the liquid atomization in a full cone pressure swirl nozzle is
presented. The first part is devoted to the study of the inner flow. CAD and CFD software are used in order to determine the most important
parameters of the flow at the exit of nozzle: axial velocity U, angular velocity c, kinetic turbulent energy k, turbulence dissipation s and the
related length scales. An important conclusion is the existence of two flow regions: one in relatively slow motion (the boundary layer) and a
second nearly in solid rotation at a very high angular rate (about 100 000 rad/s) with a thickness of about 4/5th of the nozzle section. Then, a
theoretical and experimental analysis of the flow outside the nozzle is carried out. In the theoretical section, the size of the biggest drops is
successfully compared to results stemming from linear instability theory (Kubitschek and Weidmann, 2007). However, it is also shown that
this theory cannot explain the appearance of the numerous small drops observed in the stability domain whose size are close to the
Kolmogorov and Taylor turbulent length scale. The experimental setup involves a highspeed (HS) camera and a Phase Doppler Particle
Analyser (PDPA) used to characterize the droplet size and velocity distribution. Images from the HS camera reveal that the full cone is
made of a fully atomized core surrounded by a slow outer liquid sheet. This configuration directly results from the inner flow since the
atomization is much more efficient for the inner region of the flow in solid rotation due to a very high centrifugal force. Length and width
of the outer liquid sheet are determined by processing the images of the HS camera. They are shown to be in good agreement with
WeberDombrovski empirical law. The spatial distribution of the average liquid axial velocity and mean diameter is determined from the
PDPA measurements. Full size and velocity PDF can be obtained at various positions and in this study the emphasis is put on the near
nozzle area. The evolution of the marginal size PDF on the vertical axis is then more qualitatively investigated. Due to centrifugal force, the
smaller droplets tend to prevail on the spray axis and a peak close to the Taylor length scale appears progressively in the PDF when
increasing the distance from the nozzle. It is then assumed that these droplets are the results of a turbulent cascading atomization process
and the near nozzle PDF is therefore fitted using a logstable law (Rimbert and SeroGuillaume, 2004). The value of the stability index is
found to be 1.35 very close to a previous experimental result of 1.39 (Rimbert and Delconte, 2007) but far from a known theoretical value
of 1.70 (Rimbert, 2010). This let think about a slightly different underlying process due to the helical nature of the turbulence.
Introduction
This work is an extension of preceding studies
by Rimbert and Delconte [1]. It aims at predicting the
spraying behaviour of a pressure swirl atomization from
first principles.
4 bar 5 bar i bar 7 bar
0 20 40 600 0 2 40 60 0 20 40 0 20 40 so
Figure 1: Spraying behaviour of a Danfoss OD 1.87Kg/h
EN 601 500 0.50 USgal/h 45S Nozzle under different
applied gauge pressure. Images are obtained from
highspeed imaging using a FASTCAMultima APX RS
Photron camera at 75000 frames per second (scale is given
in pixels 1 pixel 30 unm)
Instability theory is used to describe the large
scale behaviour of the spray. While our first attempts made
use of classical RayleighTaylor theory, Kubitschek and
Weidmann [2,3] made important contribution to this field
by making the full linear stability analysis of a helical
column of fluid in zero gravity and by successfully
comparing their prediction to experimental data collected
through highspeed imaging. However how successful their
work may be, there is still a long way to practical injection
system. For instance, the pressure swirl nozzle studied in
the present works leads to angular velocities three orders of
magnitude higher (i.e. one thousand times). Therefore
instability thresholds are way behind and the spraying
behaviour is mostly governed by turbulence. While modern
reviews on atomization can mostly be divided into two
categories: turbulent atomization based on Kolmogorov
work [ 4 ] vs. linear instability/ligament mediated
atomization [5], the present work is a part of a set of
experiments [6] aiming at proving that large droplets are
mainly governed by classical linear instability theory while
small droplets are mainly produced by the turbulence
developing in the mixing zone. Experimental data also seem
to indicate that a turbulent ligament mediated mechanism
may be at work in the present case. While akin to [7,8], the
resulting scenario make use of classical turbulent quantities
(turbulent kinetic energy density k and dissipation e) rather
than to new empirical parameters. Moreover the shape of
the droplet size distribution is shown to be a logarithmically
stable law (cf. [9]) which can be related to turbulent
intermittency modelling (cf. [10,11,12,13,14]).
This work starts with a description of the
experimental facility as well as the geometry of the simplex
pressure swirl nozzle under investigation. It then shown that
a good order of magnitude of the angular velocity of the
spray can be obtained from first principle and that better
quality results can be obtained from Computational Fluid
Dynamics. These results can then be used as starting point
for Kubitschek and Weimann linear stability analysis and it
will be shown that size of the largest droplets can be related
to this mechanism. Unfortunately, droplets smaller than 80
num cannot be produced by this mechanism and there are
experimental evidences that they are numerous. Therefore
their size distribution is studied in the last section of this
paper in relation to turbulent intermittency modelling.
Nomenclature
Latin letters
g Gravitational acceleration (m.s 2)
A Aggregation frequency (s')
a Centrifugal acceleration (m. s2)
z Vertical axis coordinate (m)
P Pressure (N.m 2)
d Droplet diameter (m)
V Air velocity (m.s ')
v Droplet velocity (m. s')
k Turbulent kinetic energy density (m2.s 2)
k Wavenumber (m1)
d1o Mean diameter (m)
d32 Sauter mean diameter (m)
e Eccentric length (m)
r Radius (m)
n Number density (part/m3)
m Moments of the number density (variable)
CD Drag Coefficient
In Neperian logarithm
log Decimal logarithm
Re Reynolds number
We Weber number
Oh Ohnesorge number
L Hocking number
Greek letters
a Stability index of the L6vy law
p Skewness parameter of the L6vy law
o Scale parameter of the L6vy law
6 Shift parameter of the L6vy law
gt Dynamic Viscosity (Pa.s)
v Kinematic Viscosity (m2.s')
p Density (kg.m 3)
y Surface tension (kg.s 2)
e Turbulent energy dissipation density (m.s3)
T Characteristic time (s)
co Angular velocity (rad/s)
X Wavelength (m)
x Taylor microscale (m)
T Kolmogorov scale (m)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Subscripts
max Maximum
turb Turbulent
drop Droplet
rot Rotation
int Integral
marg Marginally stable
i Inflow
o Outflow
z Vertical axis component
G Gas property
L Liquid property
0 Initial state
1 Final state
Experimental Facility
Experimental setup is depicted in figure 2. It
consists in a Danfoss OD 1.87Kg/h EN 601 500 0.50
USgal/h 45S nozzle fed by a pressured water tank. Data
were collected using a Dantec Dynamics PDPA (Phase
Doppler Particle Analyser) and a green argon continuous
laser (wavelength: 514.5 nm). The PDPA equipped with a
classic receiver, has been used in refraction mode with a
diffusion angle of 500. According to the manufacturer
droplets size can be measured with confidence in a dynamic
interval ranging from lx to 40x (or 1.6 decades). This
means that bigger droplets have a tendency to saturate the
photomultipliers while smaller ones may not trigger it.
However data were collected over two decades. This is not
however in the present case, a major drawback.
Nozzle
N70
 Z=5
y x
z
jAA AAAAi
Figure 2: sketch of the experimental facility. A PDPA
gives both size and velocity distribution of the droplets. Red
triangles represent the different measurement points.
Figure 3 and figure 4 present the inner geometry
of the nozzle: a funnel is fed by three square channels
giving to the flow an angular momentum respective to the z
axis. The 10:1 contraction ratio of the funnel thereafter
enhances the rotation speed. This is a full cone nozzle since
the outflow radius is small enough not to allow some
outside air to penetrate through the z axis inside the nozzle
(as in hollow cone pressure swirl nozzle)
/  "
AI
Figure 3: sketch of a simplex pressure swirl nozzle. The
funnel is fuelled by three channels forming a 1200 angle.
These channels are off axis conferring a zangular
momentum to the liquid. This rotation is amplified by the
funnel. Arrows shows the path of the liquid.
Figure 4: CAD representation of the four parts of Danfoss
OD 1.87Kg/h EN 601 500 0.50 USgal/h 45S Nozzle.
CFD analysis and Angular Momentum
Conservation
Figure 5: sketch of the computational domain used in
CFD analysis. Distance between vertical axis and
midline issued from the channel is the eccentric length e.
By defining the axial angular momentum around
vertical z axis (cf. figure 5) as
J = J.e, = (OMA pV).ezdv
V
and by balancing inflow and outflow vertical angi
momentum, one gets:
d 8(OM ApV).ez
dt 8t c
+ (OM A pV).ez (V.n)ds(M)
Jz,,
(OM A pV).e (V.n) ds (M)
Jz
So that for steady flows:
*
J,=J,"
Table 1: geometric parameters of the nozzle
Parameters Value
Outlet radius 106 /m
Inlet channel side length 108 /m
Inlet velocity 18 m/s
Eccentric e 260 nm
Since
J,, = 3epA,V2
the
the
the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
J,o = PJ'ro4 zV, (5)
with the parameters given in table 1, this leads to the value
3eA V
co = ' 200,000 rad.s1 (6)
2;Tr
The flow is therefore rotating at a tremendous speed. At
such speed boundary layers in the nozzle cannot be
neglected any more and a full computation resorts to
Computational Fluid Dynamic. This has been done using
Fluent as described and more thoroughly detailed in [1].
Qualitative results of these computations can be found in
figure 6 where some sample pathlines are depicted. More
quantitative results are summarized in table 2.
Figure 6: Sample pathlines obtained from the numerical
simulation of the nozzle inflow (colour is a tag of the
different particles).
It can be seen in table 2 that the mean angular
velocity obtained in numerical simulation is half the
predicted velocity. It both means that inner frictions are
important inside the nozzle (this leads to a couple on the
ular nozzle which may act on the screw) and that the angular
velocity is still very high resulting in an radial acceleration a
of the order of 1,000,000 m/s or 100,000g at the exit. This
important radial acceleration is the main atomizing
mechanism in this kind of nozzle. The appearance of a large
(2) boundary layer is an inhibiting factor for atomization since
it slows down nearly 25% of the liquid. This boundary layer
forms a rotating water sheet at the nozzle exit as can be seen
in figure (1). The cone is not hollow as it contains a mist of
droplets resulting from the atomization of the mainvortex
core.
Table 2: outlet parameters of the Danfoss swirl nozzle,
inlet pressure is 4 bars
Parameters Value
Axial velocity 18 m/s
Angular velocity 100,000 rad.s1
Turbulent kinetic energy 14 m2/s2
Turbulent dissipation 106 m2/s3
However, at such high speed of rotation, turbulence
cannot be ruled out as a secondary mechanism and using
values of table 2, it is possible [1,15] to compute an order
of magnitude of important turbulent length scales:
Lit C = 168pm (magnitude 2.23) (7)
(4) 8
for the integral scale
I]
l_ k
A, 15v 14pm (magnitude 1.15) (8)
for the Taylor scale (average size of the dissipating eddies)
/ 3 4
@l51 lp\m (magnitude 0.00) (9)
for the Kolmogorov length scale. The observed boundary
layer thickness 0 being close to 25 gtm one gets the
following relation
doe 0+L + nt +0, (10)
which can be related to the fact that the boundary layer is
surrounding the main vortex core. Note that these lengths
are also given in decimal magnitude (i.e on a decimal
logarithmic scale, 0 standing for 1 pm). These orders of
magnitude will be used in the forthcoming analysis.
Instability Theory
The non dimensional parameters governing the
atomization mechanism are the Weber number and the
Ohnesorge number defined by:
pU2d
We = 1PG
7
Oh = L 8.103
This means that droplets are aerodynamically stable (i.e.
that breakup mechanism cannot be the air friction) and that
viscosity is not an impediment.
In [1], it was supposed that a very slight
modification of RayleighTaylor analysis can be used to
determine the maximum amplified wavelength. The
centrifugal acceleration a supersedes in this analysis the
terrestrial gravitational acceleration g classically used. By
doing so, one gets:
S 3 1/2
\PzaL)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
of that reference).
Using these results, the maximum amplified wavenumber is
found to be:
k,,m= 5.5 (14)
which leads to maximum amplified wavelength of
2Hr
; ax = 114pm (magnitude 2.06) (15)
kmax
Interestingly enough, this is still of the same order of
magnitude as the Rayleigh Taylor crude estimate. But more
interestingly, the marginally stable wavenumber can be
estimated, and one gets:
kma = 10 (16)
so that the marginally stable wavelength is
marg 63um (magnitude 1.80) (17)
marg
meaning that shorter wavelength cannot be amplified.
Therefore the appearance of droplets of size close to the
micrometer cannot be explained by this mechanism (Note
that neither droplets of the scale of the Kolmogorov length
nor of the Taylor microscale can be predicted by the linear
instability theory). Also note that these wavelength are not
equivalent to the resulting droplet diameter and one easily
gets the following relation
d = (6r22)/3 (18)
which introduce a shift in the magnitude i.e. one gets 196
upm (magnitude 2.29) for the diameter of fastest growing
droplet and 160 Ajn (magnitude 2.20) for the marginally
stable one (here r =105 pm, assuming r = L,n/2 leads to the
respective value of 169 and 139 /jn or magnitude 2.20
and 2.14).
PDPA results
93pm (magnitude 1.97)
While this seems to give a good order of magnitude of the
bigger droplets, this analysis is based on a plane interface
assumption whereas the radius of curvature of the columnar
interface is actually close to 100 pm. Since the curvature of
the interface is of the same order of magnitude as the
wavelength, it cannot actually be neglected. Moreover
acceleration is kept constant in this crude description while
it is actually increasing as the square of the distance to the
rotation axe in the rotating reference frame. Therefore it
should be interesting to use a sound description of this
instability mechanism. As previously stated this has been
done by Kubitschek and Weimann [2]. In their analysis they
use two non dimensional numbers: the rotational Reynolds
number
r2
Reo = (12)
and the socalled Hocking parameter [16]
L = (13)
pr3 c
which is actually an inverse Bond number. In our case Rerot
S1000 while L z 7.103. These values are very close to some
results given in [2] for Reo = 1000 and L 102 (cf. Fig. 2
5 0 5
Figure 7: Spatial extension of the spray. Centrifugal forces
tend to select the biggest droplets on the border of the spray.
Length are given in mm and mean droplet diameter dio
range according to the colorbar from 30 to slightly over 60
pm; Maximum mean velocity is found to be 17 m/s.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
High speed imaging (cf. figure 1) enables the
measurement of the outer cone breakup length. It leads to
the value 1.8 mm. This is close to with Dombrowski and
Hooper empirical law [17] which gives
U 17
L =12 12 ; 2mm. (19)
CD 100,000
However, this value may be underestimated because the
outer sheet is rotating slower than the inner core, thanks to
inner friction inside the nozzle. Moreover, highspeed
imaging has shown the sheet breakup length to be a very
fluctuating quantity. Therefore to avoid spurious reflection
of the laser on the outer sheet, measurements have been
made starting 6 mm downward. The PDPA moved on a grid
as depicted in figure 2 and results are given in figure 7. Near
the exit, the average droplet velocity can be estimated to be
17 m/s close to the CFD estimate of 18 m/s. There is a clear
segregation of large droplets on the border of the cone as
they are more influenced by the centrifugal acceleration than
by the air friction
PDF jointe talle vitesse nombre de gouttes en echelle logarthmlqu
25
I25
2
20 40 60 80 100 120 140 160 180
Diametre (microns
Figure 8: Joint sizevelocity PDF of the spray 30 mm
downward. Injection gauge pressure equals 6 bar. Colour is
related to the observed number of droplets (in decimal
logarithmic scale)
This segregation effect, which is related to the fact
that smaller droplets are more affected by their fiction with
ambient air, can be seen more precisely on the isocontour of
the joint sizevelocity PDF depicted in figure 8. Note that
the colour scale is logarithmic and that the measurement
point is located 30 mm under the nozzle exit. Small droplets
on the ten micrometers scale have an average velocity close
to 10 m/s whereas larger droplets are flowing with the inlet
speed i.e. 17 m/s.
Another illustration of this effect can be seen on
figure 9. The size marginal PDF stemming from the data of
figure 8 is shown on a semi logarithmic scale. Reference
values of the different characteristic scales of turbulence and
instability theory are recalled for comparison. It can be seen
that most droplets are actually located between the value of
the Kolmogorov scale (1 Gun, magnitude 0) and the nozzle
diameter (210 Lnm, magnitude 2.3). Two peaks are present,
the first one being located close to the estimated value of the
Taylor microscale and the second one close to the value
estimated out of instability theory.
F p t d Ma"' .. .,..
Figure 9: Size PDF 7 mm (top) and 30 mm bottom )
downward. The bigger droplets are of the magnitude of the
linear instabilities while the most common one (mode) are
close to the Taylor microscale (purple). This emphasizes the
effect of differential segregation due to different particle
relaxation times. Data are collected on 50,000 droplets.
Note the difference between the top chart (data
collected 7 mm downward) and the bottom chart (data
collected 30 mm downward). There seems to be an increase
in small droplets but this is a relative increase as the PDF
are normalised. This is another illustration of the
aforementioned segregation effect, larger droplets being
ejected outward.
Droplets slowdown in the surrounding air
To make a proper assessment of the droplet PDF
stemming from the primary atomization of the spray, it is
necessary to minimize the segregation effect. This can be
obtained by collecting data as close to the nozzle exit as
possible. Figure 10 illustrates the segregation effect
obtained 6mm downward (which is the closest measurement
point to the exit). To obtain this representation the size
velocity PDF has been sampled with 80 different size bin.
For each size bin, a Gaussian PDF has been fitted to the
conditional velocity PDF. The resulting average velocity is
depicted by black triangle while the standard deviation is
represented by red x. To obtain smoother PDF, two billions
droplets are statistically taken into account. Comparing
figure 8 and 10, it can be seen that the segregation effect is
lessened but not absent.
Mean and standard deviation of the velocity of droplets as a function of their size
1 ~ 1 I 1     
: 
12
1o
61
4  .
2
20 40 60 80 100 120 140 160 180
Diameter (micron)
Figure 10: Estimate of the crest location of the joil
sizevelocity PDF, black A are experimental mean data, re
x stands for the standard deviation around this value and th
dashed line is the theoretical equation (). Data are collect
6mm downward from 2,000,000 droplets. Injection gauge
pressure equals 6 bar.
To obtain the full extend of this segregation effec
let us resort to a simple analytical model based on th
dynamical equation of the droplet trajectory:
dv PG 1
dz C, (Re)v2
dt p, + p, d
dz
= v (2(
dt
vz (0,0)= v 17m.s1
Integrating this system yields:
d1 1 1
v,1 1 (tl3to) (21
Co (Re)v2dz = +1l/2)d (
Which can be easily integrated using Stokes' drag law CD
24/Re. This leads to:
where
V,1 = V 0 exp 
drop
d2 1
r 18v pG L +1/2
At time instant t1, droplets of different sizes are not located
at the same place. To recover the shape of the crest of the
surface depicted in figure 7, let us consider the droplets
that are located in the PDPA measurement volume at
abscissa zi. Setting both to = 0 and zo = 0 and integrating
(22) leads to
Z1 = vTdrop 1exp (24)
So that the v(d) dependency can be written:
Vz,1 0 1 1 z1 (25)
(p, /,G +1/2) d2 18v
Setting zi = 6mm, writing d in micrometer and considering
that v,o = 17 m/s, leads to
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
288
v1z, 17 (26)
So that droplets of diameter close to 4 gtm are fully stopped
thank to air friction. Curve (26) is depicted in figure 7 as
the dashed line and partly explains the comma shape of the
experimental curve. A better agreement could be obtained
by considering more complex drag law (for instance
Schiller and Naumann [18]) but analytical computation
will be trickier. Of particular interest is the halfvelocity
cutoff which is located at:
Vz,0 288
= vz, d0 (27)
2 cutof
So that doto = 6 pm (magnitude 0.76). As a crude
assumption, in next section, droplets larger than this value
will be considered unaffected by the surrounding air while
droplets smaller will be considered to have been slowed
down enough not to be taken into account in the statistical
description without introducing a bias.
Droplets' log stable size distribution
nt
:d
le
:d
'e
In this section, thanks to preceding remarks, the size
:t; PDF is analysed in the magnitude range [0.8, 1.9]. i.e. away
le from both the segregation effect and the instability
wavelength range. Using 2,000,000 droplets ensures that the
resulting PDF is very smooth as can be seen on figure 11.
The black line is the result of the fitting of the PDF with a
logstable law. While these kind of law as been shown both
to be possible asymptotic solution to a self similar
fragmentation equation [9] and useful in modelling
turbulent intermittencies, no clear relationship between
atomization and turbulence have been drawn yet. Let us try
to do so. In [14], logstable intermittency theory is put on a
firmer ground thanks to a selfavoiding random vortex
stretching mechanism. On the one hand, among the
consequence of this study (see also [13]), comes the fact
1) that the scale parameter of the logstable turbulence PDF is
given by
lnIn a= 1.70 (28)
2) 27J
which in the present case leads to the value
on, = 1.77 (29)
Fitting of the size distribution with a Levy PDF of stability index 1 3548
08 1 12 1.4 16 1 8 2
Size Magnitude (0 stands for 1 micrometer)
Figure 11: Marginal size PDF of the spray 6mm downward.
Fit of a log stable law. Resulting parameters are a = 1.35, /f
= 1, ologd= 0.38 (ohd 0.88), 6logd = 0.93. Data are collected
on 2,000,000 droplets.
On the other hand, the fitting of the PDF leads to
the following value of the droplet size scale (for the four
parameters of the L6vy stable law see caption of figure 11):
U= 0.88 (30)
So that:
n =1i
oind 2 0ns
Intriguingly enough this relation has also been obtained in
another atomization mechanism (without rotation) [6]. This
can be easily explained if the following relation is supposed
between the droplet diameter and the local turbulent energy
dissipation, seen as random variables:
d oc c (32)
This is somewhat different from Hinze modelling which
equates turbulent dynamic pressure to surface tension [19]
and state that the maximal droplet size for an isolated
droplet in a isotropic turbulent flow filed is given by
3/5
dm = 0.725 ( 2/5 (33)
\PG )
In our case this leads to dm.a equals to 500 /jn. In this
modelling size and dissipation seems to be inversely
correlated (hence the minus sign in 2/5). Therefore, due to
this minus sign, no refinement of this modelling can explain
the observed dependency (32).
However a possible scenario can be devised by
considering that resulting droplets are issued from the
reorganisation of filament. While this is known for a long
time [20], this has been emphasized recently in [5,7,8]. In
these works the agitation of the filaments, whose width
distribution is assumed to be exponential (thanks to
maximum entropy formalism [8]), introduces a reordering
of the matter in a selfconvolution process: i.e. the resulting
droplets diameter is the random sum of the diameters of a
given set of elementary blobs. The number of elementary
blobs being related to the roughness of the filament. Since
convolution of several exponentials leads to gamma
distributions, the author adequately fit their data with
gamma PDF. Note however that the speed involved in these
experiments may not be high enough for turbulence to be a
driving mechanism and that their experimental size PDF are
quite narrow. The last point may also be related to their
sizing technique involving highspeed camera imaging
(which has a poor size range). Anyway this modelling
cannot be directly applied to the present case, firstly because
gamma PDF are not adequate to fit our data; secondly
because there is, to our knowledge, no way to estimate the
unknown parameters of the gamma PDF in our case.
However, keeping these physical insights in mind, we will
try to build an agglomeration mechanism leading to (32)
Let us suppose that coherent structures, or
filaments, developing inside the water are able to resist to
the mixing with the surrounding air but ultimately recess
thanks to an agglomeration process. A possible way to
obtain the size distribution n(d,t) of droplets can be given
by Smoluchowski's equation [21]:
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
an(d,t) 1d
n (d, ) (, ) n (do, t) n
(34)
n(d,t) 4A(d,4)n(f ,t)d4
where A is called the aggregation kernel. IfA is supposed
constant this equation has an analytical solution [22] and
the moments of the PDF are given by:
n(t)=mo(t)= 2 (35)
2+A,, t
m, (t)= do (no mo (t)) (36)
So that the average length reads:
d (t) dAnt (37)
m, (t) 2
do is the size of the elementary blobs, no is number of
elementary blobs (hence their product can be considered as
the length of the filament). The overall aggregation time t
can be assumed to be the lifetime of large turbulent eddies
i.e. their eddy turnover time or integral time scale
L
rt = 42ps (38)
k2
whereas, the aggregation frequency A can be thought to be
inversely proportional to either the Taylor time scale
rT = 15 15ps (39)
or to the Kolmogorov time scale
K V = I 4,s (40)
Actually both are proportionals. Using, either (39) or (40),
and (38) in (37), leads to the desired dependency (32). The
main hypothesis made in the present model is that
aggregation is a quick enough process so that it happens at a
very high speed 1/rK during an overall time Tnt During this
aggregation steps the size of the resulting droplet increases
as value of turbulence dissipation increases. Other turbulent
structures, possessing different values of the turbulence
dissipation will lead to different size of droplets. The
resulting droplet size will be proportional to the inverse of
Kolmogorov time scale i.e. to the square root of the
turbulence dissipation. In a way this can be considered as a
turbulent micro mixing mechanism [23].
Conclusions
In this work an analysis of the atomization
mechanism of a turbulent simplex pressure swirl full cone
atomizer has been developed. The eccentric length of the
feeding channel with respect to the vertical axis creates
some angular momentum which is thereafter amplified by
the contraction ratio of the funnel. Momentum conservation
predicts a very high angular momentum which is confirmed
by CFD computation. The latter bring evidences that a large
boundary layer forms inside the nozzle. This slow boundary
layer forms a conical sheet outside the nozzle. The
behaviour of the sheet is closer to a pressure swirl hollow
cone atomizer and the breakup length of the sheet is
adequately predicted by Dombrowski and Hooper empirical
law. The inner vertical core atomize due to centrifugal force.
It destabilizes according to linear instability theory.
However comparison with prediction of Kubitschek and
Weidmann is not conclusive. This mechanism cannot predict
the appearance of droplets smaller than one hundred
micrometers. Since they are quite numerous another
mechanism seems to be at work. By taking care to set aside
small particles which have been strongly slowed down in
the near nozzle area, the statistics of these small droplets
have been shown to follow a logstable PDF. The scale
parameter of the Levy stable law for the droplet magnitudes
has been shown to be equal to half the value of the scale
parameter of turbulence intermittency. Since this seems not
to be unique, a mechanism based on the turbulent
agglomeration and reorganization of elementary filaments
has been devised in a tentative to explain this relationship.
However what is still unexplained is the value of the
stability index of the Levy stable law. It is equal to 1.35
which is very different from the value 1.70 established for
non helical homogeneous turbulence. Since the flow is
strongly helical, it may suggest that another mechanism
(different from the selfavoiding random vortex stretching
mechanism) may be at work. Actually to cite [24]:
"All in all, we would expect turbulence in a rapidly
rotating system to be very different from conventional
turbulence"
Acknowledgements
The authors would like to thank Steen Gaardsted
and Norskov Preben from Danfoss Burner Components,
Nordborg, Denmark, for their help.
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