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Towards a complete population balance model
for fluidized bed spray granulation:
Simultaneous drying and particle formation
M. Peglow, S. Heinrich, E. Tsotsas
Faculty of Process and Systems Engineering,
Otto-von-Guericke-University Magdeburg
Abstract: This paper concerns the simultaneous
processes of agglomeration and drying. In order to predict
temperatures and moisture content in gas and particle phase,
heat and mass transfer mechanism and particle size enlargement
has been considered simultaneously. The presented model takes
heat and mass transfer phenomena between particle phase,
suspension gas and bypass gas into account. The disperse
phase is modelled by a three-dimensional population balance
(PBE), which can be reduced to a set of three one-dimensional
PBE. The latter are coupled with heat and mass transfer balances
of the gas phase. Furthermore some simulation and experimental
results are presented.
INTRODUCTION
The process of particle size enlargement can change properties
of granular materials significantly. By agglomeration fine
sized primary particles are transformed into free-flowing
and dustless consumer products. The fluidized bed technology
offers a possibility to combine the process of agglomeration
and drying in a single apparatus. The product to be dried
is fluidized by passing hot air through it. A rapid drying
rate is known to be the advantage of fluidized beds, since
the mixing of solids and gas is very efficient. In the literature,
many attempts have been made to describe the process of particle
formation in a fluidized bed in terms of population balances.
The population balance approach describes the temporal change
of particle property distributions (PPD). As a result, one
obtains the temporal change of the particle number distribution
with respect to selected particle properties. The latter
are named as the internal coordinates. Frequently, the particle
size or volume have been considered as the only significant
internal coordinate of the disperse phase. Thus, a one-dimensional
population balance equation (PBE) for growth, agglomeration
and breakage of particles has been applied to numerous processes
in chemical engineering such as crystallization, granulation
and agglomeration. see (Bramley et al., 1996; Ding et al.,
2006). The concept of one-dimensional PBE faces several problems
and limitations, see (Iveson, 2002). Therefore, the number
of significant internal coordinates has to be increased.
As soon as the number of internal coordinates is extended,
the mathematical and numerical effort increases rapidly.
In the framework of Adetayo et al. (1995), and Hounslow et
al. (2001), a high shear granulation in rotating drums in
terms of population balances is investigated. Hounslow et
al. (2001) extended the internal coordinates by the particle
tracer mass. The resulting two-dimensional PBE was reduced
to a set of two one-dimensional PBE assuming that particles
of the same size contain the same amount of tracer mass.
Tan et al. (2004) applied this model to fluidized bed melt
agglomeration.
For the more general case of liquid spray granulation, one-dimensional
population balance models can be found, see e.g. Saleh et
al. (2003), but studies on the simultaneous description of
agglomeration and drying in a population balance model are
missing. The impact of operating conditions on the PPD has
been investigated by various authors, see (Adetayo et al.,
1995; Watano et al., 1996; Schaafsma, 2000). Watano et al.
(1996) observed that the moisture content in solids is one
of the most important particle properties to control the
agglomeration process. This leads to the conclusion that
properties such as particle size and moisture content have
to be considered simultaneously in a population balance model.
Our study presents a novel model that is capable to eliminate
the missing link between the processes of agglomeration and
drying.
THEORY AND NUMERICAL METHODS
A two-dimensional particle property distribution is defined
as f(t,v,c), where v and c are two distinct properties, usually
the granule volume (size) and any extensive property of the
particles such as moisture of particles or enthalpy. In a
first step, we want to follow the idea of Hounslow et al.
(2001). Beside the granule volume v they considered the tracer
mass c as the second internal coordinate, see (Hounslow et
al., 2001). The total number of particles N in a domain D
is given by
(1)
It should be noted that the granule volume v contains volume
of tracer mass and volume of particles, that is c ≤ v.
The two-dimensional population balance equation can be obtained
by extending the classical one-dimensional PBE to two-dimensional
space as in (Hounslow et al., 2001),
(2)
Hounslow et al. (2001) reduced the two-dimensional population
balance model to a set of one-dimensional population balances
using the marginal distribution approach. The reduced population
balance for the particle number distribution n(t,v) can be
deduced as
(3)
The temporal change of tracer distribution m(t,v) is given
by
(4)
Thus, the above continuous system of equations may be applied
for the computation of temporal change of particle number
and tracer mass distribution. It is difficult to solve this
system analytically, so numerical methods have to be applied
for the solution. One way to solve such systems numerically
is to discretize the size domain into small sections and
to reduce the system into a system of ordinary differential
equations.
Hounslow et al. (1988) developed a discretized method for
population balance equation as given in equation (3). A discretization
scheme of reduced PBE (4) is suggested by Hounslow et al.
(2001). In both approaches the particle volume domain is
divided into small sections and the density is assumed to
be constant within each section. The final set of equations
preserves particle numbers and mass. However, for certain
application the discretization given in Hounslow et al. (2001)
is inconsistent with intensive properties of solid phase.
Therefore, Peglow et al. (2006) suggested a modified discretization
of Hounslow et al. (2001). The modified method which preserves
the mass and the number of the system as before and predicts
the intensive properties such as temperature of solids or
concentrations within a granule as well. The final set of
discretized equations for a grid of type
(5)
is given by
(6)
for the evolution of number distribution and
(7)
for evolution of tracer distribution. The correction factors
are given by
(8)
An extended formulation of equation (7) for an adjustable
geometric grid of type
(9)
is presented in Peglow et al. (2006). Now in the next section
we want to derive a population balance model for simultaneous
drying and aggregation. For this purpose we are going to
apply the reduced set of population balances as given in
equations (3) and (4). Furthermore a selected simulation
result will be presented.
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