7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The Instability of Particle Laden Spiraling Liquid Jets
S. Gramlich* and M. Piesche*
University of Stuttgart, Institute of Mechanical Process Engineering (IMVT),
BOblinger Str. 72, 70199 Stuttgart, Germany
gramlich@imvt.unistuttgart.de
Keywords: Atomization, Jet, Instability, Suspension, EulerEuler, Prilling, Rotating Disc Atomizer
Abstract
The breakup of a particle laden dilated viscous liquid jet, emitted from a cylindrical nozzle in a rotating cup is studied
by means of linear stability analysis and experiments. The motion of the liquid jet is divided into a time steady and
a time dependent part. The time steady motion implying the contour as well as the trajectory of the undisturbed jet
is solved by balancing forces and mass. A solution method to solve the resulting ordinary differential equation of
second order with only one boundary condition is shown. The time dependent motion of the disturbances is studied
by stability analysis of a particle laden liquid jet. It is based on a balance of mass and impulse in the Eulerian
formulation, hence each phase is treated as a continuum. The surrounding gas motion is treated as potential flow. Jet
stability is also investigated experimentally. The curvature, breakup length and the resulting drops are analysed using
shadow imaging. The experimental findings are compared to the numerical model.
Introduction
In many industrial applications like prilling, spray dry
ing or spray coating rotating disc atomizers are used
to disintegrate droplets of a liquid that can be particle
laden. Operating the atomizer in the regime of Rayleigh
type drop formation, the resulting drop sizes are nar
rowly distributed.
The analysis of the Rayleigh type disintegration of
straight and cylindrical jets of pure liquids have been
addressed by many authors. The resulting drop size
was determined by linear stability analysis for exam
ple in Lord Rayleigh (1879), Weber (1931), Chan
drasekhar (1981) and Keller et al. (1973) and non linear
stability analysis in Pimbley & Lee (1977) and Bogy,
(1979). The instability of viscous jets dilated by grav
itational force has been investigated by Nonnenmacher
et al. (2004). Piesche et al. (2005) derived a model to
analyse cylindrical particle laden liquid jets.
The instability of spiralling liquid jets with Newto
nian and nonNewtonian viscosity have been addressed
amongst others by Kitamura et al. (1977), Wallwork et
al. (2002), PArAu et al. (2007) and Decent et al. (2009).
For the determination of the resulting drop size, the exact
time steady solution of the radius at the point of breakup
is necessary, since the drop has the same volume as the
cylindrical piece of jet with the wavelength A and the
radius rs. The equation of motion describing the time
steady motion of the jet is underdetermined. Sauter &
Buggisch (2005) found a method to solve this equation
for the contour of vertical jets exposed to gravity.
For the breakup and drop formation of particle laden spi
ralling liquid jets no sufficient experimental or numeri
cal fundamentals can be found. The knowledge about
the influence of a disperse and high concentrated phase
on the breakup behaviour rely on product specific expe
rienced data and are gained by empirical studies. Physi
cal and mathematical models as well as numerical meth
ods to predict the drop diameter for Rayleigh type disin
tegration of particle laden spiralling liquid jets are miss
ing.
The objective of this work is to derive a model to pre
dict the drop size from Rayleigh type disintegration as a
function of particle size and concentration. The analysis
include temporal as well as spatial stability behaviour.
Besides the theoretical modelling, experimental investi
gation is presented, to prove the accuracy of the physical
model.
Nomenclature
Roman symbols
d Diameter (m)
f Drag ratio
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
F
k
K
n
p
r
t
u
v
w
z
CDI
CDs
We
Re
Ro
D
H
S
T
Greek
6
a
(T
p
t
A
wc
(2
Force (N)
Curvature radius (m)
Spatial wave function
Position vector (m)
Pressure (Pa)
Radial coordinate (m)
Time (s)
Radial velocity (m s 1)
Velocity vector (m s1)
Axial velocity (m s 1)
Axial coordinate (m)
Drag coefficient of jet
Drag coefficient of particles
Weber number
Reynolds number
Rotation number
Diameter ratio
Viscosity behaviour
Dispersion parameter
Time
symbols
Inclination angle
Radial Displacement (m)
Angular coordinate
Angle of incident of gas flow
Density (kg mn3)
Volume concentration
Constant
Surface tension (N m 1)
Viscosity (Pa s)
Wavelength (m)
Angular velocity (rad s 1)
Temporal wave function
Subscripts
O Condition at orifice
1 Substitution
cor Coriolis
crit Critical
max Maximum
min Minimum
ref Reference
single Single particle
c Centrifugal
D Drop
D Drap
g Gas
i Imaginary
1 Liquid
p Particle
r Real
s Solid
T Derivative with respect to T
Z Derivative with respect to Z
Figure 1: Physical model.
Physical Model
The physical model is based on the drawing in figure 1.
It shows the contour of an oscillating phase interface of
a liquid jet which is spiralling around a cylinder with ra
dius no rotating with the angular velocity cw. The jet is
assumed to be rotationally symmetric. The liquid con
tains particles with diameter dp of volume concentration
c, and emerges from a cylindrical nozzle with radius ro
with the velocity wo. Entering the spiralling trajectory
around the rotating cylinder, the axial velocity wl in
creases. The solid particles are moving with the same
velocity as the liquid w, = wl w. The surrounding
air is treated as potential flow.
The liquid has the viscosity p/ and the density pi, the
solid has the density ps and the gas has the viscosity p,
and the density p, respectively. On the interphase be
tween liquid and gaseous phase acts the surface tension
(.
Because of small disturbances on the jets surface, the jet
radius rs changes locally and temporally. Thus, the mo
tion of the liquid, solid and gaseous phase can be split
up into a time independent basic motion describing the
contour and trajectory of the liquid jet and a time depen
dent oscillation induced by capillary forces which grow
along the jet and lead to breakup. Therefore, the jet ra
dius rs can be decomposed as rs(z,t) rs(z) + 6(z, t)
with a time steady jet radius rs and a time dependent os
cillation of the jets' surface 6(z, t).
In the following, the time steady motion of a spiralling
liquid jet is presented and then the stability analysis
is shown, giving the critical wave number of the jet
breakup.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Time steady motion of a spiralling liquid jet
The time steady motion of a spiralling liquid jet can be
accessed by balancing forces on an infinitesimal part of
the jet as shown in figure 2. The equations of motion
of the jet are derived in a cylindrical coordinate system
r, z moving along the jets' axis. The global coordinate
system n, p is also cylindrical but stationary giving the
actual position of the moving jet coordinate system. The
curvature radius of the jet is k. The acting forces are
inertial force, viscous force, pressure force, surface ten
sion force, centrifugal forces, coriolis force and drag
force. This results in the equations of motion in axial
direction:
dW 1 1 dR, 3H 1 d ( dW
W + __ (R2
dZ WeR dZ Re R dZ \ dZ
+ RoNsin a Vg2cos (1)
Pl 7R,
and normal to the jets axis:
W2 RoNcosa+2W (Ro\ P c Vg, sin V
K VD pi Rs
(2)
with the dimensionless quantities:
W =
W
Vg
Rs =r
my
Z =
,0
ro
K=
CD,1
a
time steady axial velocity,
the gas velocity vector,
the dimensionless jet radius,
the dimensionless axial coordinate,
the position vector,
the radius of the curvature,
the drag coefficient,
the inclination angle,
( the angle of incidence of gas flow
and the characteristic numbers:
We = p7ro the Weber number,
Re = P oo the Reynolds numb
Ro = roo the rotation number
D = the diameter ratio a
2 the density ratio.
er,
r,
nd
The shear thinning behaviour of the viscosity of a parti
cle laden liquid expressed by the CarreauYasuda model
can be included in the function H. For Newtonian be
haviour the viscosity function reduces to unity H 1.
Equation (1) is an ordinary differential equation, where
Figure 2: Force balance.
only one starting condition is known, namely the axial
velocity of the jet at the orifice Wz=o 1. The gra
dient in axial direction serving as the second boundary
condition  Kz=o is not accessible. The problem can
be solved applying the method of Sauter & Buggisch
(2005) to the presence of a centrifugal field, which is
presented in the following:
Applying the time steady equation of continuity WR =
1 to equation (1) it can be written as
P cDlwi Vg cosC (3)
P1 7 dZ (3)
d dWa
Treating the gradient as constant K = , and solv
dW dZ
ing equation (3) for the axial velocity gradient d and
taking the derivative with respect to W an approximate
solution can be obtained for K. In the direction field of
equation (3) (figure 3) all vector lines mark possible tan
gents to the solutions trajectory. In positive W direction
the vector field is diverging. The dashed line shows the
approximate solution which is a good approximation for
high velocities. However, for low velocities the approxi
mation diverges from the exact solution (solid line). One
can obtain the actual gradient Tz taking the approx
imate solution at a position of higher velocity W and
integrate backwards to the actual velocity, since the di
rection field is converging in negative W direction.
With the obtained solution for the gradient dz, the
integration in Zdirection can be carried out giving the
new axial velocity W z+dz.
The calculation of the axial velocity described above is
06 5. ,
05
1 2 3 4 5 6
Axial Velocity V
Figure 3: Direction field with approximate and exact
solution: We = 4, Re=0.2; Ro=2, H=l, Z=0.
coupled with equation (2) from which the radius of the
curvature K can be solved and a set of equations for the
position N, the angular coordinate p and the inclination
angle a:
dN sin a
dZ D
dcp cos a
dZ ND
da cos a 1
dZ ND K
This set of equations (2), (3) and (46) determines the
time steady motion of a spiralling liquid jet. The influ
ence of the surrounding gas is included by the drag force.
The influence of the particulate matter is captured by a
viscosity function H derived from the CarreauYasuda
model.
Stability Analysis
The oscillations occurring on the liquid/gas interphase
are investigated by stability analysis. In order to access
the problem mathematically and reduce the complexity
of the problem, several assumptions are made:
The material properties are constant.
The motion is rotationally symmetric.
The solid particles are spherical.
The temporal and local change of the solid concen
tration is neglected.
The motion of the liquid is laminar.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The equations of motion for particle laden liquid flow
are given in Eulerian formulation. Hence, each phase is
treated as a continuum. The equations of motion result
from balances of impulse and mass. The coupling of the
liquid and solid phase is given by the interaction terms
in the equations for impulse. A drag coefficient for the
particles is defined postulating creeping flow around the
spherical particles.
The theory implies the assumption that the single phases
are influenced by random or forced disturbances which
are either growing or damped depending on the liquid,
solid and gaseous properties and the geometric and
operating conditions. The investigation of this physical
process starts with the NavierStokes equations in cylin
drical coordinates for the liquid and solid phase. The
flow is divided into a time steady flow and an oscillating
time dependent flow with the relations for the axial and
radial velocity Wk Wk + w%, Uk uk + .' the
pressure p p + p' and the jet radius r, = r + 6
resulting in the linear perturbed differential equations in
dimensionless notation.
Liquid phase:
(4) OU U,[ aW1
OR R aZ
(5)
(6)
+W
OT 8Z
OP' 1 ( 2U11 1 OUl'
18 6es
f (UR U,)
Re DR
R2 + 9Z2
awl wl
+W
OT OZ
aP' 1 2Wi' 1 WV/' O a2 W
OZ Re 9R2 R OR 9Z2
18 6 f(W,'
f (l W')
Re D2
Solid phase:
au + u aw 0
OR R aZ
au^+ au
OT OZ
pi OP' 18 pi 1 t
^+^^
ar az Rp
S+ W a
P1 OP' 18 p 1 e
+ fZ(W
p, 8Z Re p, D2
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
with P = the dimensionless pressure and f =
cD,s the ratio of the drag coefficient for a parti
CD,s, single
cle collective and the drag coefficient of a single sphere.
T wo/rot is the dimensionless time.
The solution of the set of partial differential equations
(712) requires the specification of boundary conditions.
At the liquid/gas interphase the pressure drop is given by
P' = P, + R2 __1
Pi Re )OR
1 A (2A 13)
We + (13)
with the dimensionless radial displacement A
The radial velocity of the disturbance at the jet r
Ul R= R equals the motion of the radial displacem<
U OA WOA
u'1 = + W
iOT iOZ
The shear stress at the interphase is zero:
awl, + 1 0.
dR O Z
For solving a set of partial differential equations des
ing an oscillation a suitable separation approach is
U'(R, Z, T) Uk(R) exp [i (K(Z) Q2(T))]
W'(R, Z, T) = Wk (R) exp [i (K(Z) Q2(T))]
P'(R, Z, T) = P(R) exp [i (K(Z) Q(T))]
A(R, Z, T) = A(R) exp [i (K(Z) Q(T))]
with the complex functions describing a spatial wav
K(Z) = K + iK,
and a temporal wave
Q(T) =QR + i i.
The real part of the wave functions describe the os(
tion whereas the imaginary parts describe the temi
or spatial growth or decay of a disturbance.
amplitude of a disturbance is a function of the radius
Treating equations (712) with the previous appr(
equations for the amplitude functions can be obtain
Liquid phase:
i U, + iKzW 0
dR R
i (WKz QT) UL =
i (WKz QT) W1 =
1 1 d2WI 1 ,ii /
KzP + dR2 R dR
Re D(i )
K Zi V)
Solid phase:
, + iKzRi,
dR R
(14) i (WKz QT) U,
(14)
P1 dP' 18 pi 1 e
Ps dRt Re p D2
i (WKz QT) WS
SKzPi p 18 pi 1 s
iKzP + R D 4f
Ps Re ps D2 f v
6u)
w)
(16) where the indices Z and T denote the spatial and tem
poral derivative.
(17) For simplicity the following variables are introduced:
(18) 18 ei Ps
a i= (WKz 2T) f (26)
(19) Re D pi
ve 18 ,
b f (27)
Re D
C= i (WKz QT) (28)
18 6s P (29)p
d f (29)
Re D2 ps
cilla
poral The following substitutions are necessary in order to re
The duce equations (2025)
is R.
)ach, u1 + + d (30)
ed: c(a + b) dR
a+d
WVn =1 W+ a d iKzP (31)
c(a + b)
(20) to a set of three Bessel's differential equations
d2U11 1 i 11
dR2 R dR
(2 + S2) J1
0 (32)
dPi 1 [d 2iU1
dR Re dR2
18, f
ReD2
1 i, U1
R dR R2
d )11 1 ii 11
dR2 R dR
d2P 1 dP
dR2 R dR
_ 2W11i 0
Znf, 
Z
K 11
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
with S2 =K + Re(a + b).
From potential theory the pressure oscillation at the
gas/liquid interface can be determined as:
SKTK + Us + WKz) A. (35)
with K1 Ki(RsK) the modified Bessel function of
the second kind of order one and its derivative with re
spect to R: K = Ko + K Ki.
The result is the dispersion relation for a particle laden
dilated viscous liquid jet taking into account the sur
rounding gas.
a+b 2 1 S2+K~Io(KzRs)
a+d Kz S2 K I(Kz Rs)
pl KzKK R,UT "I"K)
1 1
S2 K (K Io(KzR,) 1
S2 K K Ii(Kz R,) R)] (36)
Distinction is made between temporal and spatial devel
opment of the disturbances. In temporal stability anal
ysis, a spatial constant wave with wave number Kz,r
grows or decays in time. The imaginary part of the
spatial wave function is zero. The temporal growth
rate QT,i can be determined from the dispersion relation
(36). Since the growth rate for a specific wave number
changes along the jet because of the elongation of the
stationary contour, the growth must be integrated over
the residence time in temporal analysis. If the temporal
growth is positive
rT
Jo
QT,i dT > 0
the wave is temporally unstable. The most unstable
wave shows the largest integral growth Qi,max and has
the critical wave number Kz,r,crit.
In spatial stability analysis a temporal constant wave
with angular frequency QT,r grows or decays in space
with the growth rate Kz,i. A wave is unstable if
Ki .
Jo
Kz,i dZ < 0
and the most unstable wave is found at Ki,min and has
the wave number Kz,r,crit.
The resulting drop size can be determined evaluating the
(2We)0 5+3We/Re
Figure 4: Modified Breakup length for various pure
and particle laden liquids.
wave length A = /ro 27/Kz,r and calculate the
volume equivalent drop diameter from the time steady
solution:
dD 1 127 (37)
2 2 R (37)
2ro 2 V Kz, ,. it
for spatial as well as for temporal instability.
Experiments
The experimental setup consists of a rotating cup with
two possible radii of no 60mm or no 120mm.
A single cylindrical orifice of also two possible radii
ro 1= mm or ro 0.5mm is mounted in the cylin
drical side wall of the cup. The angular velocity of the
cup is varied from w 30 90rad/s. The axial veloc
ity is varied between wo 0.2m/s and wo 1.2m/s.
A centrally mounted liquid dosage supplies the cup with
the fluid. Shadow imaging technique is used to anal
yse pictures of the jet breakup recorded by a high speed
camera (Redlake HG100K) operating at 1000 frames
per second. The backlight was provided by a LED
panel with diffuser plate. The images were analysed
using a self customized image analysis tool based on
MATLABTM. The fluids chosen for the experiments
Table 1: Liquid Properties (T = 293K)
Specification Pt (Pas) pi (kgmn ) a (Nm 1)
Water. 0.001 998 0.072
60% aq. glyc.. 0.010 1156 0.067
90% aq. glyc.. 0.193 1235 0.066
were aqueous solutions of glycerol. The chemical prop
'... .. .
water
c 10
n
.N
,)
a.
0
0
a
1
10

S Experiments We 10
S Experiments We 400
3 2 1
10 10 1
Rotation Number Ro
Figure 5: Experimentally and numerically determined
drop sizes versus Rotation number Ro.
erties of the liquids are given in table 1. As particu
late matter, two different collectives of glass spheres are
used with volumetric mean sizes of dpi,5o,3 = 6.55pm
and dp2,50,3 = 22.5pm with a volume concentration of
c = 0.04 and a density of pp = 2G'i;l,'l/rnm3.
Results
The experimental findings regarding the modified
breakup length Zod = 2 (1 + 2(2Ro)025) of
pure and particle laden liquids are plotted versus the
breakup parameter (2We)05 +3We/Re in figure 4. The
modification with the rotation number for elongated jets
was introduced by Nonnenmacher et al. (2004). It can
be seen that the particles have a negligible effect on the
breakup length.
In figure 5 the calculated and experimentally deter
mined drop sizes of pure liquid jets are plotted against
the Rotation number Ro with the Weber number We as
parameter. The jet length, serving as boundary condition
is calculated based on the experimental findings. The ra
tio of VWe/Re is kept constant. With increasing rota
tion number, the jet becomes more accelerated and thus,
the jet radius at the point of breakup decreases. This re
sults in smaller drop sizes. Further, with increasing We
ber number, the drop size decreases as well. An increase
of the Weber number means also an increase of inertial
force, and therefore, a jet becomes longer. Thus, the jet
is exposed to the centrifugal force for a longer time and
its radius at the breakup point becomes smaller.
In the following, experimental findings for particle
laden liquids are compared to numerical results. The
breakup length serving as boundary condition was di
rectly taken from the experiment. The constant experi
mental conditions are given in table 2. Figure 6 shows
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
o 0
q
00
S10
D) 0 0
N 0 0Experiment
Temporal Analysis
o Spatial Analysis
Q.
0
_0
0 20 40 60 80 100
Angular Velocity (. [s ]
Figure 6: Experimentally and numerically determined
drop sizes versus angular velocity c.
Table 2: Experimental Conditions.
Variable Value Units
Pi 1155 kg/m3
/i 10 mPas
ro 1 mm
no 60 mm
dp,1 6.55 pmt
cd,2 22.5 pmt
the drop sizes scaled with the diameter of the orifice
versus the angular velocity c) at a constant exit velocity
wo 0.5m/s for a liquid with particles of size dpi. The
result of the spatial as well as the temporal analysis is
plotted with the experimental finding. The drop size de
crease with angular velocity for the same reason as for
pure liquids. The drop sizes determined by the spatial
and the temporal analysis are in satisfactory accordance
with the experiment. The drop size of the spatial analy
sis is always higher than the one determined by temporal
analysis. However, Keller et al. (1973) found the spatial
solution to be more realistic than the temporal one.
Finally, the experimentally and numerically determined
drop sizes are plotted against exit velocity wo for two
different particle sizes dpi and dp, at an angular fre
quency of c 30s '. The influence of the particle
size seems to be negligible, for the examined param
eters. With increasing exit velocity, the drop size in
creases slightly.
Conclusions
In this work, we have shown a physicalmathematical
model to determine the drop size obtained by rotational
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
1
10
0
2
04 06 08 1
Exit Velocity w0 [m/s]
Figure 7: Experimentally and numericall
drop sizes for two different particle sizes v
locity.
atomization in the Rayleigh regime of jet
for particle laden liquid jets. The time stead
a spiralling jet was determined by a force
a solution method was derived by study
tion field. An approximate solution could
serving as a starting condition for the ex;
integration. The determination of the drop
on linear stability analysis of the equations
the Eulerian formulation accounting for ti
solid phase. The influence of the surroundi
included in the model. The comparison bet
mental findings and numerical solutions sh
els' capability of calculating the drop siz
laden liquids. The influence of particle si
to be negligible in the examined range.
Since a volume concentration of cs I .
cient, further investigations on the concent
dency of the breakup length and the result
is mandatory.
 . 
  n pa
a Expe
Tem
....... Spat
Expe
Tem
Spat
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