DISPERSIVE AND DISTRIBUTIVE MIXING
CHARACTERIZATION IN EXTRUSION EQUIPMENT
Winston Wang and Ica Manas-Zloczower
Department of Macromolecular Science
Case Western Reserve University
Abstract
Mixing is a key step in almost every polymer
processing operation. The traditional methodology for
improving machinery performance has relied more on
users' experience and trial and error experiments.
Recently, the fast development of advanced computing
resources has enabled the use of numerical modeling as an
alternative and more efficient approach in studying the
influence of design and processing conditions on
equipment mixing performance. In this work using
numerical simulations we record the flow history of a
number of tracers in the equipment and use them in
conjunction with dispersion kinetics models to evaluate
minor component size and concentration distributions.
Background
One of the principal roles played by polymer
processing equipment such as extruders is that of mixing.
In order to measure the effectiveness of a given extruder at
mixing, a benchmark criterion is necessary. There are two
types of mixing that need to be evaluated: distributive
mixing and dispersive mixing.
In dispersive mixing, solid agglomerates or liquid
droplets held together by interfacial tension must be
subjected to mechanical stress in order to reduce their
length scale. Therefore, the most important flow
characteristics determining dispersive mixing efficiency
are the magnitude of shear stresses generated and the
quality/strength of the flow field (elongational flow
components).
In distributive mixing, repeated rearrangement of the
minor component enhances system homogeneity. In
continuous mixing processes, composition uniformity at
the emerging stream is directly related to the material
residence time distribution.
Numerical simulations of polymer processing
equipment provide an effective means of analyzing the
capability of a given machine.
Description of Method
Flow Field Simulations: In our group we carried out
three dimensional, isothermal flow simulations for various
batch and continuous mixing equipment (1-14).
In this project, we first looked at the flow patterns in a
twin-flight, single screw extruder. A fluid dynamics
analysis package-FIDAP, using the finite element method
was employed to solve the 3D, isothermal flow of a
Newtonian fluid. No slip boundary conditions on the screw
surfaces and barrel walls were used. The operating
conditions were selected such that 1 clockwise revolution
of the screw was made per unit time, and a pressure
difference applied across the inlet and outlet surfaces
satisfied the condition of Qp/Qd = -1/2.
The equations of continuity and motion for the steady
state, isothermal flow of an incompressible Newtonian
fluid were solved:
∇⋅V = 0
(1)
[ ]
∇ ⋅ ρ VV = ∇V − ∇ ⋅ τ
(2)
The Cray T-90 at the Ohio Supercomputer Center
running FISOLV was used to solve the field equations.
Flow Field Characterization: To evaluate the flow
field quality we looked at the shear stresses generated and
elongational flow components. One simple way to quantify
the elongational flow components is to compare the
relative magnitudes of the rate of deformation, D , and the
vorticity, ω , tensors. The parameter λold, defined as:
λ old = D D + ω
(3)
can be used as a basic measure of mixing efficiency for the
machine design and/or operating conditions. λold is equal to
one for pure elongation, 0.5 for simple shear and zero for
pure rotation.
A more rigorous method to quantify the elongational
flow components is by employing a flow strength
parameter, Sf, that is frame invariant:
(
S f = 2 tr D 2
)2
o2
tr D
(4)
o 2
where D is the Jaumann time derivative of D i.e. the
time derivative of D with respect to a frame of reference
rotating with the same angular velocity as the fluid
element. The parameter Sf ranges from zero for pure
rotational flow to infinite for pure elongational flow. A
simple shear flow corresponds to a Sf value of one. For
consistency we normalize the flow strength parameter and
call it λnew:
λ new = S f 1 + S f
(5)
Like λold, λnew is equal to one for pure elongation, 0.5 for
simple shear and zero for pure rotation.
Particle Tracking: In work previously done by our
group(3, 15), an algorithm was developed for tracking
massless points that affect neither the flow field nor other
particles. Since the flow field is completely deterministic,
the location of the particles can be found by integrating the
velocity vectors:
We use a coordinate system that rotates with the same
angular velocity as the screw, so that one finite element
model could be used for all the calculations. We also use a
periodic boundary condition to simulate an extruder longer
than the finite element model. Due to the no-slip boundary
conditions, a particle which runs into, or overshoots a wall,
is considered stuck there and no longer moves.
Parent Agglomerate Size Distribution: Recording
particle flow histories allows one to calculate the minor
component (agglomerate or droplet) size distribution at the
exit from the extruder.
From work previously done in our group, erosion
kinetics for relatively sparse agglomerates in simple shear
flows can be described by (16):
(7)
where R is the radius of the agglomerate, t is the time, β is
a proportionality factor that reflects the fraction of the
overall hydrodynamic force bearing on the fragment, K is a
proportionality factor related to the structure of the
agglomerate, Fh is the hydrodynamic force inducing
erosion and
Fc is the cohesive force resisting
fragmentation. The hydrodynamic and cohesive forces can
be written as:
Fc = κ Rm
2
]
(10)
where Ro is the initial agglomerate size and the parameter
Θ is given by:
Θ 2 = 5πµγÝ
β R02 2 Fc .
(11)
For denser agglomerates (m = 1), the integration of the
rate of erosion equation gives:
[ (
(
))]
(12)
where,
where X(t1) is the location of a particle at any time t1, X(t0)
is the location of the same particle at initial time t0, and
V(t) is the corresponding velocity vector of the particle.
The location of each particle is calculated every time step.
Fh = 52 µγÝ
πR
[
(6)
t0
− dR dt = k (βFh − Fc ) ,
)
1 − R R0 = 1 − be bt b + aR0 e bt − 1
t1
X(t1 ) = X(t0 ) + ∫ V(t)dt
)(
(
Θ 2 − 1 1− e −2ΘKFc t / R 0
R
1−
=
R0 Θ (1+ Θ) + (1 − Θ )e −2ΘKFc t / R 0
(8)
(9)
with µ the fluid viscosity and γ&the shear rate. Parameter
κ is a scaling factor that gauges the strength of the
individual bonds between the interacting particles and m is
a measure of the agglomerate packing structure. Parameter
m varies from 0 for sparse agglomerates to 2 for dense
agglomerates (16).
Integration of the rate of erosion equation for sparse
agglomerates (m=0) gives:
a = 5K βµγÝ
π 2
(13)
b = Kk
(14)
Since the kinetic models in Equations (10) and (12)
were developed by considering erosion in simple shear
flow, we need to modify them to account for the different
flow field strengths that a particular agglomerate may
experience. Based on work by Bentley and Leal (17) and
Kharkhar and Ottino (18), showing that the critical
capillary number for droplet breakup in different types of
flow ranging from simple shear to pure elongation,
depends on the shear rate as well as on a flow strength
parameter, we modify the expression for the hydrodynamic
force on a spherical agglomerate in the principal strain
direction to read:
Fh = 5λµγÝ
πR 2
(15)
where γÝ is the shear rate and λ is the flow strength
parameter as defined in Equation (3). Using Equation (15)
for the hydrodynamic force will modify the parameters in
the erosion kinetic models (Equations (10) and (12))
according to:
Θ 2 = 5λπµγÝ
βR02 Fc
(16)
and
a = 5λk βµγÝ
π.
(17)
Particle flow histories were used in conjunction with
the erosion kinetic models to calculate the parent
agglomerate size distributions at various axial cross
sections of the extruder. To calculate the dynamics of
particle size distribution, we account for the reduction in
size of the parent agglomerate during each time step ∆t, by
taking into consideration the actual shear stress and flow
strength experienced by the particle during that time step.
Also, in the kinetic models for erosion, the initial
agglomerate size Ro is adjusted at each time step.
Distributive Mixing Characterization: Distributive
mixing involves movement of fluid elements, blobs of
fluid or clumps of solid from one spatial location in a
system to another. An exact description of distributive
mixing involves specification of the location of the
interfaces as a function of space and time. Owing to the
enormous number of minor component particles involved
in a mixing process, only a statistical definition of the state
of distributive mixing can be derived. As in work
previously performed in our group (6, 7, 15),
characterization of distributive mixing of the minor
component was done by examining the spatial distribution
of a number of massless tracers in the system.
Pairwise correlation functions: To quantify the
distributive mixing of a minor component, we use pairwise
correlation functions. The basis of the pairwise correlation
is quantifying the distance between pairs of particles.
Since our particles are represented as points, we use the
discrete version of the pairwise correlation function:
f (r) =
(
)()
2
∑ δ ri′ + r δ ri′
N(N − 1) i
(18)
where f(r) is the coefficient of the correlation function for
a separating distance between particles ranging from r-∆r/2
to r+∆r/2; the summation is preformed over all possible but
non-repeating pairs of particles, i.e. the number of pairs for
a system containing N particles is N(N-1)/2, which is
exactly the inverse of the normalizing factor in f(r) and
δ(r) is defined as 1 if a particle is present and 0 if a particle
is absent. The vector r originates from the vector r’. The
coefficient f(r) can therefore be interpreted as the
probability of finding a neighbor particle at a displacement
ranging from r-∆r/2 to r+∆r/2. For small enough ∆r, f(r)
can be represented as:
f (r) = C(r) *2π r∆r
(19)
where C(r) is the coefficient of the probability density
function of the correlation function f(r). In addition the
following identity is obtained:
r max
∫
C(r )* 2πrdr = 1
(21)
r= 0
where rmax is the largest dimension in the system such that
C(r) is always zero for r>rmax; i.e., there is no correlation
between points further apart than rmax.
In this work, the pairwise correlation function was
calculated using the cylindrical coordinates of the particle
positions. This allows for evaluation in a domain that is
unhindered by the existence of the screw.
The effectiveness of particle spatial distribution at
different axial cross sections was measured by a parameter
ε, which defines the difference between the actual
distribution C(r) and the corresponding ideal distribution
C(r)ideal as follows:
ε=
r =r max
∫
r=0
C(r ) − C(r)ideal rπ dr
(22)
Parameter ε ranges between 0 and 1.
Results and Discussion
Numerical Model: The finite element mesh shown in
Figure 1 contains 16200 8-node brick fluid elements, and
8340 4-node quadrilateral boundary elements, for a total of
20535 nodes. FIMESH, a mesh generator part of FIDAP,
was used to create the computer model.
The physical parameters of the extruder are given in
consistent dimensionless length units. The radius of the
barrel was 4.009623, and the radius of the screw was
3.534884, the flight clearance was 0.0096, and the flights
were 0.44 thick. The extruder is squared pitched, so one
pitch corresponds to approximately eight length units.
The flow field calculations required 395 user CPU
seconds over a total elapsed time of 860 seconds and 11.8
MWord of memory.
Particle Tracking: In this project, 655 tracers were
grouped into 5 clusters of 131 particles and were initially
placed in the center of the numerical model. Figure 2
shows the initial position of these tracers, which were
followed for different time units with a time step of
0.0001. The tracers are given an initial size Ro = 1.55 mm
and their dispersion is recorded along the extruder. We
assume a constant and continuous feed of particles at their
initial positions, such that our results pertain to steady state
conditions.
Dispersive Mixing: Using data from previous
experiments in our group studying dispersion of silica
agglomerates having a density of 0.14 g/cm3 (sparse
agglomerates) in poly(dimethyl siloxane) of viscosity 60
Pa.s in a cone-and-plate device (simple shear flow) yielded
the following parameters for the kinetic model described
by Equation (10) : Fc/β = 0.00631 N; KFc = 1.102x10-5 m/s
(19). Results from similar studies with carbon black
agglomerates of density 0.4 g/cm3 (medium dense
agglomerates) (20) were fitted to the following parameters
of the kinetic model described by Equation (12): a =
64.157 m-1s-1; and b = .055456 s-1.
Figure 3 shows agglomerate size distributions for the
case of sparse agglomerates (m=0) and medium dense
agglomerates (m=1) at axial positions corresponding to the
exit planes of extruders 2 and 8 pitches long. The sparse
agglomerates eroded quickly and 22% reached a radius of
less than 0.3 mm within 2 pitches. After 8 pitches close to
half of the agglomerates were that size. In this model, the
minimum agglomerate size is largely dependent on the
shear rate/stress experienced. The agglomerates were
unable to erode smaller than 0.294 mm because they did
not travel through regions of high enough shear stress. As
we would expect, due to the larger cohesive force the
denser agglomerates experienced slower erosion and after
8 pitches they were still fairly large.
Figure 4 shows the evolution of the mean agglomerate
size along the extruder length, as well as the evolution of
the standard deviation. The medium dense agglomerates
have initially a higher standard deviation than the sparse
ones, indicating that they are more sensitive to variations
in particle flow history. However, as they become smaller
and progressively harder to erode, the standard deviation
decreases.
Distributive Mixing: Figure 5 shows that the position
of the particles at the exit of an 8 pitch extruder. Visual
inspection of these distributions can reveal some
characteristics of the mixing process. Particles injected
near the flights appear poorly distributed and tend to stay
close together while cycling around the channel and
exhibit very poor distributive mixing. Particles injected in
the center of the channel appear to be better distributed in
the cross section.
For a meaningful and unbiased comparison among
different operating conditions and/or different designs, a
reference is needed to base the comparison. For this, as in
our previous work (6, 7, 15) we define an ideal distribution
to be one that consists of the same number of particles
(655) randomly distributed over the extruder cross section.
An example of an ideal distribution is shown Figure 6.
Figure 7 illustrates the evolution of the parameter ε
along the axial length of the extruder. As expected,
distributive mixing is improving along the extruder length,
however no significant changes can be observed beyond an
axial length of 6 pitches.
Conclusions
Towards the goal of developing new mixing criteria
for process control and equipment scale up, we have
presented the use of particle tracking as a method of
capturing the dynamics of the mixing process. By tracking
the motion of particles in the flow field, we have obtained
the flow history for each particle (flow strength and shear
stress experienced.) When used along with a dispersion
kinetic model, this allows for the calculation of
agglomerate size distribution and average agglomerate
size. This illustrates one possible avenue of developing a
new dispersive mixing criterion.
The spatial distribution of a minor component at
various axial cross sections of the extruder, reported in
terms of the probability density function of a pairwise
correlation function, can form the basis for evaluating
distributive mixing. Via the ε parameter, we compare the
evolution of the pairwise correlation function along the
extruder with an ideal case where the minor component
particles are randomly distributed throughout the cross
section. The deviation of the actual minor component
distributions from the ideal case illustrates one possible
way of quantifying distributive mixing.
Acknowledgements
The authors would like to express their gratitude to the
National Science Foundation for supporting this research
project under the grant DMI-9812969.
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Figure 1: Finite element model for a twin-flight single
screw extruder
Figure 2: Initial positions of the 5 clusters of 131 tracers
each.
Figure 5: Evolution of particle spatial distribution
calculated for 655 particles after 8 pitches.
Figure 3: Parent agglomerate size distributions for 655
agglomerates after 2 and 8 Ls (pitches) of the extruder
Figure 6: Ideal particle spatial distribution for 655
particles.
Figure 4: Evolution of mean parent agglomerate size and
the standard deviation calculated for 655 agglomerates
along the extruder length.
Figure 7: Evolution of parameter ε along the axial length
of the extruder.
Key words:
Dispersive Mixing characterization, distributive Mixing
Characterization, Agglomerate size distribution, single
screw xtruder.