PBMs in granulation can track multiple granule properties. In the case where only a single property is evaluated, PBMs are called one-dimensional (1-D). The earliest works focused on 1-D models that tracked particle volume, since this property has the advantage of being additive; when two granules coalesce, the volume of the resulting granule is the sum of the volumes of the original granules.
One of the earliest population balance methods successfully applied to particle growth was proposed by Hounslow et al. [22]. They proposed a solution method based on discretisation of the population balance equations and applied the model to a crystallisation process. The equations included terms for nucleation, agglomeration and growth. Since then, PBMs have
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been extensively used in the field of granulation to predict granule properties and increase our understanding of the granulation process [65,67].
Although 1-D PBMs can be solved faster than multidimensional models, tracking a single property has its limitations [65]. Generally, properties such as porosity and liquid content are not homogeneously distributed over particle size classes. Depending on the application of the granules, it is imperative to be able to predict these properties. In such cases, tracking granule volumes alone is insufficient.
In a review in 2002, Iveson [24] confirmed such limitations and proposed a four- dimensional approach to population balance modelling. The author suggested balances for porosity and binder content in addition to granule volume, as well as a balance tracking composition in the cases of granulating with powder mixtures.
Verkoeijen et al. [68] expanded upon this concept and introduced a three-dimensional PBM. Instead of volume, liquid content and porosity, the model tracks volumes of solid, liquid and air. Although this method does not directly yield the properties Iveson [24] suggested, these properties can all be calculated when the three volumes are known. Similar to total volume in 1-D population balances, volumes in 3-D PBMs have the additional advantage of being additive, whereas changes in liquid content or porosity are much more complex to compute. As an example, a granule coalescing with another granule with a potentially different liquid content and porosity may yield a particle of equal, greater or lower liquid content and porosity, depending on the size difference of the original granules. Volumes of solid, liquid and air, however, are simply added, greatly simplifying calculations.
Darelius et al. [69] expanded upon the concept of the 3-D PBM proposed by Verkoeijen et al. and proposed a more mechanistic kernel. The authors assumed granules without any air, and added an additional balance for the liquid volume present inside the granules. The purpose of this change was to enable the calculation of the pore saturation, upon which the rate of compaction and the probability of coalescence were based. Their data was also compared to experimental data using a high-shear mixer and oversaturated granules. The model agreed reasonably well with the experiments for volume, pore saturation and porosity, demonstrating the usefulness of 3-D PBMs.
A major disadvantage of 3-D PBMs is their reduced computational efficiency. In the literature, several methods to reduce the complexity of multidimensional PBMs can be found. Both Hounslow et al. [70] and Biggs et al. [71] used a linked system of multiple 1-D PBMs. By assuming that all granules in a single size class have the same properties, such as liquid content, the model is reduced to two linked population balance equations. The model also assumes that time and size are the only parameters that influence the rates. This method will be referred to as the lumped parameter approach [72]. Both works show promising results, although Hounslow et al. point out several limitations of the assumptions. In the case where rates are interdependent, such as binder liquid content for aggregation kernels, the assumptions of lumped parameters are inappropriate.
Barrasso and Ramachandran [72] expand on the idea of a reduced order method by evaluating the validity of the assumptions of this lumped parameter approach. The authors numerically solve a four-dimensional PBM for a binary powder mixture with volume of solid, liquid, air and the additional solid component. Next, the lumped parameter approach is applied in several different ways, setting several parameters as bin properties. For some models, only a single parameter was considered lumped, whereas the most reduced model had three lumped parameters, with only the solid volume being a distributed parameter. The
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accuracy and computational time were compared for the models evaluated. Solution times varied from almost 34 hours for the 4-D model to merely 3 seconds for the 1-D model. The authors conclude that the liquid content should be kept as a distributed parameter due to its importance in the agglomeration of granules. Depending on the required accuracy, the lumped parameter approach is a useful technique for the order reduction of multidimensional PBMs.
An alternative order reduction technique was proposed by Chaudhury et al. [73], using a combination of tensor decomposition, separation of variables and single variable decomposition. This method is based on splitting up the functions depending on multiple granule properties into equations that are more readily solved. The authors show that this method hardly reduces accuracy while greatly reducing computational time.
In summary of this subsection, the number of dimensions evaluated in a population balance model for granulation is dependent on the desired accuracy and the available computational time. If accuracy is the focus of a study, multiple dimensions should be used. The most important properties that should be included are solid or total volume, and liquid volume or liquid content, as these are the key factors in coalescence. However, if computational efficiency is the focus, such as in the case of control, a lumped parameter approach should be used to save time.