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Population Balance Modeling of Particle Size and Porosity in
Fluidized Bed Spray Agglomeration
Published as part of Industrial & Engineering Chemistry Research special issue “Marco Mazzotti Festschrift”.
Eric Otto,* Aisel Ajalova, Andreas Bück, Evangelos Tsotsas, and Achim Kienle
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Cite This: Ind. Eng. Chem. Res. 2024, 63, 17545−17556
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ABSTRACT: Fluidized bed spray agglomeration is a unit operation applied for the size
enlargement of solid granules by aggregation. End-use agglomerate properties crucially depend
on agglomerate size and porosity, which, in turn, depend on the operating conditions of the
fluidized bed process. In the context of plant automation in an inherently uncertain process
environment, the application of process control algorithms is crucial. To facilitate the
application of advanced control algorithms relying on accurate yet computationally efficient
models, we present a novel population balance model for the evolution of the agglomerate size
and porosity. In contrast to other models presented in the literature, the porosity is
incorporated by means of a power law relationship with the agglomerate size, which is parametrized by the agglomerate fractal
dimension. In order to test the new model, the fractal dimension of agglomerates, produced in a series of batch and continuous
experiments, is investigated and correlated to the fluidized bed process conditions. Additionally, based on the experiments, an
empirical model for the aggregation kinetics is proposed and kinetic parameters are estimated and also correlated to the process
conditions. The proposed model is validated by comparing measured particle size distributions from experiments with model
predictions. The results show good agreement; i.e., the relative error of the particle size distributions is below 7% for all validation
experiments.
■
INTRODUCTION
Agglomeration (also termed aggregation or granulation)
processes with a binding agent are widely used in different
industries for the formulation of solid particles, e.g., for the
production of food powders, pharmaceuticals, detergents,
fertilizers, and more.1 During the process, the liquid binder
in the form of a solution or suspension is sprayed on the
surface of a particle population. When particles collide at wet
surface spots, agglomerates may be formed, initially bound by
liquid bridges, which subsequently solidify into solid bridges
upon drying (see Figure 1a). When carried out in a fluidized
bed, the particles are fluidized by an upward-facing stream of
air, resulting in well-mixed behavior with high energy- and
mass-transfer rates. The process is realized in the fluidization
chamber, either in batch mode or continuously with particle
feed and withdrawal as presented schematically in Figure 1b.
Although the process is primarily used for size enlargement,
various other agglomerate properties are of interest. Besides
the particle size distribution, the agglomerate morphology, i.e.,
the internal structure, is relevant due to its influence on
properties such as dispersibility, solubility, bulk density, and
more.2−5 In general, the economic value of the product
agglomerates is determined by these end-use properties. To
produce agglomerates with predefined properties under
inherently uncertain process conditions, process control is a
crucial tool. Furthermore, from an industrial point of view, it is
© 2024 The Authors. Published by
American Chemical Society
mandatory to optimize the production process with respect to
product quality, throughput, and energy considerations such as
heat consumption for the heating of the fluidization medium.
Both process control6−11 and optimization,12,13 can be
enhanced by accurate yet computationally efficient dynamic
models, capturing the evolution of the desired properties
depending on the operating conditions.
In this contribution, we present a population balance model
for the fluidized bed spray agglomeration process that
considers particle size represented by volume or diameter,
respectively, and morphology represented by agglomerate
porosity. The literature provides some population balance
models for a fluidized bed and more general agglomeration
processes that include both properties. Iveson14 presents a
comprehensive, four-dimensional population balance equation
(PBE) for wet granulation processes with granules consisting
of two different solid fractions and a binder. The granule solid
phase mass, binder to solid ratio, granule porosity, and solid
Received: May 10, 2024
Revised: August 15, 2024
Accepted: September 18, 2024
Published: October 3, 2024
17545
https://doi.org/10.1021/acs.iecr.4c01660
Ind. Eng. Chem. Res. 2024, 63, 17545−17556
Industrial & Engineering Chemistry Research
pubs.acs.org/IECR
Article
Figure 1. (a) Stages of agglomeration: (1) primary particles, (2) wetted primary particles, (3) wet agglomerate after primary particle collision, and
(4) dried agglomerate. (b) Fluidized bed process including spraying, particle feed, and withdrawal.
phase mass fraction of the first component are the internal
coordinates. Verkoeijen et al.,15 Poon et al.,16 and Chaudhury
et al.,17 referring specifically to fluidized bed granulation,
restrict themselves to agglomerates with only one solid
component and use a PBE with the solid volume (vs), liquid
volume (vl), and air volume (va) as independent variables
describing an agglomerate. The porosity is then given by the
ratio of air plus liquid to total volume. Lastly, Barrera Jiménez
et al.18 formulate a PBE with agglomerate volume and porosity
(i.e., the ratio between pore and total volume) for a twin-screw
wet granulation process.
In the models of Verkoeijen et al.,15 Poon et al.,16 and
Chaudhury et al.,17 it is assumed that the respective volumes of
solid, liquid, and air components are conserved during
aggregation. With this assumption, the porosity of newly
formed agglomerates is uniquely determined by the
compositions of the parent agglomerates. Although using a
different formulation of the PBE, the same assumption is made
by Iveson.14 Barrera Jiménez et al.18 neglect any liquid
components and assume that the porosity of a newly formed
agglomerate is the weighted average of the parent agglomerates’ porosities. Speaking in terms of solid and air volumes, this
is equivalent to the volume-conserving approaches described
above. All of these contributions have in common that, besides
agglomeration, a so-called consolidation term is considered in
the population balance equation, which accounts for changes
in porosity independent of any aggregation (and breakage)
events.
Independently, from population balance modeling, the
porosity of aggregates in a population is often described by
means of a power law relationship, e.g., in Rosner and
Tandon,19 Logan and Kilps,20 Sorensen,21 and Singh and
Tsotsas,22 which is parametrized by the agglomerate fractal
dimension and in some cases a so-called prefactor. From this
relationship, as will be shown in the upcoming section, it
follows that, at least in batch and continuous steady-state
processes, the air (or pore) volume increases during a binary
agglomeration event; i.e., the aggregates become more porous
the larger their solid volume becomes. This nonconservation of
pore volume, which is supported by the experimentally
investigated agglomerates in this contribution and in Singh
and Tsotsas,22 contradicts the assumptions made in the models
presented above. In order to incorporate this observed
behavior into a model that can be used for process control,
we present a novel extension of the PBE, with the goal of more
accurately representing the morphogenisis of agglomerates in
fluidized beds. In contrast to the previously reported models,
we choose a one-dimensional particle size distribution (PSD)
with solid volume as the internal coordinate, which is possible
since the porosity and solid volume are not independent
particle properties due to their power law relationship. Since
the solid volume is conserved during aggregation, the
traditional birth and death terms are used in the PBE,23,24
where the agglomeration kinetics are described by the so-called
agglomeration kernel. The functional relationship between
solid volume and porosity, obtained from the above-mentioned
power law, is used to transform the PSD with solid volume as
an internal coordinate from the population balance equation to
a PSD with total volume as an internal coordinate, which is the
particle size distribution that can be measured online.
For the model application in a process control context, the
influence of the process conditions has to be modeled. In the
PBE model presented here, the agglomeration kinetics on the
one hand and the agglomerate fractal dimension on the other
hand depend on the gas inlet temperature and the binder
concentration. For both dependencies, empirical correlations
are proposed. To validate these correlations and the overall
PBE model, three batch and five continuous fluidized bed
spray agglomeration (FBSA) experiments have been conducted, and the model predictions of the particle size
distributions are compared to measurements.
Finally, a modification of the model, accounting for timevarying process conditions, is proposed. Initially, the dependence between porosity and solid volume, parametrized by the
fractal dimension, is given for batch and continuous processes
at steady state and therefore only for constant process
conditions. Thus, for the description of the transient phase
in continuous processes as well as generally continuous
processes with changing process conditions, we propose a
17546
https://doi.org/10.1021/acs.iecr.4c01660
Ind. Eng. Chem. Res. 2024, 63, 17545−17556