Estimating aggregation efficiency from observation of flow behaviour in a high
shear mixer
Amol M. Nilpawar1, Gavin K. Reynolds2, Agba D. Salman1 and Michael J. Hounslow1
1
Department of Chemical and Process Engineering, University of Sheffield, Mappin Street,
Sheffield, S1 3JD, UK
2
Pharmaceutical and Analytical Research and Development, AstraZeneca, Macclesfield, SK10
2NA UK
Abstract
High speed images of the bulk motion in a high shear granulator have been analysed using particle
image velocimetry (PIV) to obtain surface velocity fields. The surface velocity field is found to
contain a systematic component and an apparent random component induced by particle-particle
collisions. This random component is found to exhibit a Gaussian distribution, allowing the kinetic
theory of granular flow to be used to extract the granular temperature from the velocity field. The
granular temperature is then used to estimate the collision frequency of granules within the bulk
flow. During a model granulation experiment, temporal granule size distributions were measured
offline. A binary aggregation population balance equation was fitted to the temporal granule size
distributions using a discretised population balance model. For corresponding time points high
speed images were recorded of the granular bed and collision frequencies estimated. Using the
collision frequencies and the fitted aggregation rates, aggregation efficiency was estimated for the
duration of the granulation experiment. Aggregation efficiency was found to decay with
granulation time and to be of the order of 210-8.
Keywords: High shear mixer, Granulation, Granular flow, Population balance modeling,
Aggregation
1. Introduction
High shear granulation is used in a range of industries such as pharmaceuticals and home and
personal care products. Typically a powder bed is agitated by an impeller and with the addition of a
liquid binder, granules are formed. Despite the widespread use of high shear granulation, it still
remains extremely difficult to predict the granule size distribution. The principal rate processes
influencing the granule size distribution during granulation can be summarised as wetting and
nucleation, consolidation and growth, and attrition and breakage [1]. Here, we prefer to describe
growth as aggregation, namely where two granules collide, becoming one entity. Knowledge of the
rates of these processes can allow prediction of the granule size distribution. Population balance
models have been used to fit aggregation rate constants to experimental data [2-4]. However,
ideally it should be desired that the rates of these processes can be calculated rather than fitted. Tan
et al. [5] showed, using the kinetic theory of granular flow (KTGF), that in a fluidised bed
granulator the aggregation rate constant can be related to the granule collision frequency using an
efficiency factor. Nilpawar et al. [6] measured the surface velocity field in a high shear granulator.
They found that the velocity fluctuations exhibit a Gaussian distribution, consistent with KTGF.
Using this observation, they estimated the granule collision frequency.
Here, a granulation experiment is performed, whereby the granule size distribution is sampled at
several different times. At corresponding times the bed surface velocity is obtained by PIV analysis
of high speed images of the bulk flow. A binary aggregation population balance equation is fitted
to the sampled granule size distributions in order to estimate the aggregation rate. Additionally the
granule collision frequency is estimated from the velocity field. Comparison of these two rates
allows a discussion to be made on the observed aggregation efficiency.
2. Experimental
The granulation experiment was carried out using a 10 litre high shear mixer (Roto Junior,
Zanchetta). This is a vertical shaft mixer with a base-mounted three-bladed impeller. The mixer
also has a lid mounted chopper. A heated jacket is used to control the granulator temperature. A
Photron DVR high speed camera was used to record high speed optical images of the moving
powder bed.
Materials used for granulation were comminuted calcium carbonate (Durcal 40, Omya, France) and
polyethylene glycol (PEG) with an average molecular weight of 4000 Daltons. The grade of PEG
had a melting range of 50-55°C. The Durcal 40 was preheated in the mixer to a temperature of
60°C. The binder was heated separately to 70°C, and poured onto the powder bed over a period of
2.5 min. The granulation time used for the sampling was started at the end of the binder addition.
The temperature of the mixture during granulation was approximately 64°C. The mixer was
operated with an impeller speed of 200rpm and a chopper speed of 1400rpm using a 2.8kg load of
material. The binder was added to make an overall liquid to solid ratio of 0.13. Nilpawar et al. [7]
have showed that during granulation the powder accumulates near the periphery of the mixer due to
the centrifugal force exerted by the impeller. The high speed camera was set-up as previously
described [7] in order to capture optical images of the powder bed surface. The mixer lid was
opened during image capture at times of 1, 4, 7, 10, 15 and 20 min after binder addition. Images
were capture for a period of approximately 3 seconds, which is equivalent to approximately 10
revolutions of the impeller. At the corresponding times, the mixer was briefly stopped and a small
sample of material was retrieved and cooled for offline size distribution measurement using a
Camsizer (Retsch). A schematic diagram of the arrangement is shown in Figure 1.
Figure 1 Schematic diagram of high speed camera configuration for capturing optical images of the powder bed
surface.
3. Data Analysis
The one-dimensional binary aggregation population balance equation is:
v

n( , t ) 1
Eq (1)
   (u, v  u, t )n(u, t )n(v  u, t )du  n(v, t )   (u, v, t )n(u, t )du
t
20
0
where n is the number probability density function of granules of size ν at time t and β is the
aggregation rate kernel. Following from the work of Nilpawar et al. [6], a suitable aggregation
kernel would be the equipartition of kinetic energy kernel which assumes particles collide as a
consequence of their random component of velocity [3]:
1 1
Eq (2)
 li , l j   li  l j 2  3  3
li l j
The population balance equation was solved using a discretised population balance model described
by Hounslow et al. [8], where the rate of change in the number of particles in the size interval i is
given by:
imax
i 2
i 1
dN i
1
j i 1
2
j i
Eq (3)
 N i 1  2
 i 1, j N j   i 1,i 1 N i 1  N i  2  i , j N j  N i   i , j N j
dt
2
j 1
j 1
j 1
In order to fit this model to the experimental granules size distributions, the aggregation kernel was
decomposed into time and size dependent parts [9]:
 li , l j , t    0 (t )   * (li , l j )
Eq (4)
For this analysis, Eq. 4 is used for β*. The aggregation rate constant, β0, was found by minimising
the overall sum of square errors between the experimentally measured granules size distribution and
the discretised population balance model solution. The fitted time dependent aggregation rate
constant was found to be (in kg m-0.5 s-1):
 0  9.4  10 10  7.0  10 13 t
Eq (5)




Figure 2 Mass weighted granule size distributions for the experimentally measured samples () and the solution of the
binary aggregation population balance equation (–) using the time dependent aggregation rate constant given in
Equation (5).
The powder bed surface velocity field was obtained from the high speed images using particle
image velocimetry. Specifically a MPIV [10], a PIV toolbox for MATLAB was used with an
interrogation area of 88 pixels. This corresponds to an area of the powder bed surface of
approximately 33 mm. A typical image from the high speed camera is shown in Figure 3. A
central spatial region was chosen to sample the temporal velocity field. A typical temporal velocity
sample for velocities in the direction of the impeller motion, here termed the X component, is
shown in Figure 4. The systematic component of the velocity field was obtained using a SavitzkyGolay low-pass filter [11]. The fluctuating velocities can then be calculated by subtracting the
temporal velocities from the filtered velocity field. The frequency distributions of the temporal
velocity fluctuations for both orthogonal components of the powder bed surface were found to fit a
Gaussian distribution. Figure 5 shows a typical frequency distribution of temporal velocity
fluctuations.
Figure 3 Typical image from high speed camera showing the powder bed surface, some mixer features and the spatial
location of the sampled velocity data.
Figure 4 Temporal velocity distribution for a fixed spatial location 1 minute after binder addition. The line shows a
Savitsky-Golay filter of the raw data.
(i)
(ii)
Figure 5 Frequency distribution of (i) X and (ii) Y component temporal velocity fluctuations for a fixed spatial
location 1 minute after binder addition. The raw data is shown as discrete points and the continuous line is a Gaussian
distribution fit to the data.
The mean and fluctuating motion of particles within a continuous granular medium can be
described in KTGF [12]. This theory is a continuum model, an extension of the classical kinetic
theory of dense gases, which takes non-ideal particle-particle collisions and gas-particle drag into
account. KTGF assumes ‘molecular’ chaos, as in the classical kinetic theory. This implies that all
particles are homogeneously distributed within an ensemble, the particle velocity distribution for all
particles is isotropic and the velocities of two particles involved in a collision are independent of
each other.
In KTGF the actual particle velocity c (m/s) is decomposed into a local mean velocity v (m/s) and a
random fluctuating velocity C (m/s) [13]:
Eq (6)
c  vC
Associated with the random motion of the particles, a pseudo temperature θ (termed Granular
Temperature), analogous to the temperature of gases in the classical kinetic theory of dense gases, is
defined as one-third of the mean square of the fluctuating component:
1
  C C
Eq (7)
3
In the ideal case for KTGF, the distribution of speeds should be Maxwellian and the velocity
fluctuations in each direction should be Gaussian (Eq. 7). In this case, the granular temperature is
the variance of the Gaussian distribution.
 C X,2 Y,Z 
1
Eq (8)
f (C X, Y,Z ) 
exp 

2X, Y,Z
 2 X, Y,Z 
Goldschmidt [12] derived an expression for the collision rate between two particles from KTGF.
From this, Tan et al. [5] derived an aggregation kernel:
3 S
li  l j 2 13  13
 i , j  g i , j
Eq (9)

li l j
Here,  is the aggregation efficiency (i.e. the fraction of collision events that result in aggregation),
gi,j is the radial distribution function, and S is termed the mixture granular temperature. This is
synonymous with the description of temperature in the kinetic theory of gases:
1
 S  mn C  C  mn
Eq (10)
3
where mn is the average mass of a particle.
Comparison of Eq. 8 with Eq. 2 and Eq. 4 shows that the size dependent part of this expression is
clearly the EKE, with the time dependent term given by:
3 S
 0  g i , j
Eq (11)

By calculating the collision frequency as:
3 S
rcoll  g i. j
Eq (12)

the aggregation efficiency can be estimated from:
 
0
Eq (13)
rcoll
Following Nilpawar et al. [6] the rate of collision was calculated from the granular temperature
obtained by fitting a Gaussian distribution to the velocity fluctuations from the temporal velocity
distributions at each granulation sample time. Both orthogonal directions of the velocity field on
the bed surface were used in calculating granular temperature. The average mass of a particle
require in Eq. 9 was based on a density of 2700 kg/m3 (consistent with the materials used), and the
4-3 mass median granule size calculated from the granule size distributions.
4. Results and Discussion
Aggregation efficiency was calculated for the six sample times during the granulation experiment
using Eq. 12. The collision frequency was converted into consistent units using a bulk density of
1000 kg/m3. Figure 6 shows how the aggregation efficiency changed with granulation time. This
shows relatively high aggregation efficiency a short time after binder addition (1 minute).
However, the efficiency quickly approximately halves from about 5.510-8 to about 2.510-8 by 4
minutes after binder addition. The aggregation efficiency then continues to steadily reduce during
the 20 minute wet massing period.
Figure 6 Aggregation efficiency as a function of granulation time
The aggregation kernel presented by Tan et al. [5] (Equation 8) is an excellent platform for
developing a predictive high shear granulation model. Observations of the powder surface velocity
give every indication that the particle velocities can be described using KTGF, and hence that the
EKE kernel is a reliable physically based size dependent aggregation kernel. Combining this with
granular temperature measurements (or indeed estimates from continuum or discrete numerical
models) allows collision frequencies to be calculated. This only leaves a suitable model for the
aggregation efficiency to be found in order to build a predictive high shear granulation model. It
should be expected that aggregation efficiency will depend on the granule properties (e.g. the
amount of binder available) and perhaps the impact velocity (again, impact velocity can be
estimated from granular temperature [14]).
The change of the aggregation efficiency during granulation can be conceptually described as
follows from the perspective of granule composition and impact velocities. Soon after addition of
the binder, the aggregation efficiency is relatively high. There is plenty of surface liquid available
for granules to be ‘sticky’ enough to aggregate. Additionally the average granule size is relatively
low, suggesting that impact momentum will be relatively low and hence impacting particles will be
more likely to be captured than rebound. As granulation progresses, the granule surfaces become
less wet due to collecting drier smaller primary particles or granules. As a consequence of repeated
impacts granules become more dense and internal liquid tends to be squeezed back to the granule
surface. This mechanism is often described as consolidation [1]. The liquid on the surface allows
the granules to aggregate, but at an overall reduced rate compared to the initial binder distribution
period, due to the internal binder transport within the granules. Additionally with increasing
granulation time the granules become more dense and larger, suggesting that impact momentum
will increase and the chance of granules rebounding after impact will increase, leading to a net
reduction in the observed aggregation efficiency.
Figure 7 Number weighted granule size distributions for the experimentally measured samples () and the solution of
the binary aggregation population balance equation (–) using the time dependent aggregation rate constant given in
Equation (5).
Some discussion should also be made on some of the limitations of this analysis. This analysis is
based on the velocity field at the powder surface. Although two orthogonal components of velocity
have been taken into account, the third dimension is not considered. Additionally the flow
behaviour within the bulk flow could be different to that exhibited on the surface, but it seems that
the velocity fluctuations observed at the surface should be consistent with those beneath the surface.
Additionally this analysis does not take into account breakage rate processes. Figure 7 shows the
number weighted granule size distributions. Despite a general increase in granule size, a noticeable
small shoulder of material at about 50m appears after about 10 minutes. This is clearly evidence
of an attrition/breakage process during granulation. As this breakage is not taken into account in
the population balance model used in this paper, it can be expected that the fitted aggregation rate
constant is underestimating the true underlying aggregation rate. This in turn would suggest that the
aggregation efficiency is possibly underestimated to a certain extent. Despite this, it appears that
the aggregation only model is a reasonable description of the overall granulation experiment, and
certainly this small extent of attrition is not observable on the mass weighted granule size
distribution (Figure 2).
5. Conclusions
Aggregation rates were found by fitting a binary aggregation population balance model to granule
size distributions sampled at a number of different times during a granulation experiment. At the
corresponding sampling times, high speed optical images of the powder bed were recorded. Particle
surface velocities determined from the high speed images showed that the particle velocities had a
systematic and a random component, with the random component exhibiting a Gaussian frequency
distribution. It was found that the kinetic theory of granular flow allowed estimation of the
collision frequency of the granules from the powder velocities. By comparing the fitted aggregation
rate constant with the collision frequencies, an estimate of aggregation efficiency was made.
Aggregation efficiency was found to be initially high after binder addition, but decreased during
granulation time. This appears consistent with a conceptual description of the binder nucleation and
wetting and consolidation processes of granulation. This analysis provides a platform from which a
predictive high shear granulation model can be built by separating the contributions of granule
motion and composition to the overall change in the granule size distribution.
6. Acknowledgements
Loan of the Photron DVR high speed camera from the EPSRC Engineering Instrument Pool is
gratefully acknowledged.
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