The performance of population balance models greatly depends on the rate processes simulated. In the literature, a shift to more mechanistic models [25,26] and combinations with DEM simulations can be found [27-29]. In this section, the principles behind different representations of mechanisms and kernels are discussed.
Nucleation is a mechanism of ‘birth’ in population balance modelling. Hounslow et al. [65] describe birth according to the following equation (Equation 2.17):
𝑑𝑛
𝑑𝑡 = 𝐵0𝑓𝑁 (2.17)
where n is the number density function, t is time, B0 is the nucleation rate per volume per
second and fN is the nuclei size distribution. A common assumption is setting the distribution
to monodisperse, in which case the number density function is a Dirac delta function. Assuming nuclei are the smallest class of granules formed in the system, Equation 2.17 can be discretised according to Equation 2.18:
𝑑𝑁1
𝑑𝑡 = 𝐵0 (2.18)
where N1 is the number of particles per volume in the lowest bin. Even with this assumption,
determining the actual nucleation rate is a challenge.
Poon et al. [91] developed a nucleation and wetting kernel based on the drop-controlled regime. The model accounted for slow wetting and penetration times, but assumed no overlap of droplets and a preference for binder distribution over fines instead of existing granules. The kernel was successfully applied in a three-dimensional model.
In the literature, coalescence kernels are the most commonly studied type of kernel [107]. An example of a discretised agglomeration kernel [22] is shown in Equation 2.19.
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Table 2.1: Overview of solution methods discussed in Section 2.4.2.
Solution method Study
Linear discretisation Hounslow et al. (1988) [22]
Fixed pivot discretisation Kumar and Ramkrishna (1996) [75] Chakraborty and Kumar (2007) [79] Kumar and Ramkrishna [78]
Lee et al. [108]
Moving pivot discretisation Kumar and Ramkrishna (1996) [76] Cell averaged discretisation Kumar et al. (2006) [77]
Kumar et al. (2008) [81] Kumar et al. (2008) [82] Chaudhury et al. (2013) [83] Bertin et al. (2016) [84]
Finite volumes Filbet and Laurençot (2004) [85]
Qamar and Warnecke (2007) [86] Qamar and Warnecke (2007) [87] Two-tier hierarchical method Immanuel and Doyle (2003) [88]
Immanuel and Doyle (2005) [89] Pinto et al. (2007) [90]
Poon et al. (2008) [91] Poon et al. (2009) [92] Event-driven Monte Carlo Gantt and Gatzke (2006) [93]
Braumann et al. (2007) [94] Yu et al. (2015) [99]
Braumann et al. (2010) [95] Stochastic weighted approach Patterson et al. (2011) [38]
Lee et al. (2015) [97] Lee et al. (2015) [98] Quadrature method of moments Alopaeus et al. (2006) [101]
Rajniak et al. (2009) [102] Mortier et al. (2011)
Lattice Boltzmann Majunder et al. [103]
Majunder at al. [104]
Daubechies orthonormal wavelets Liu and Cameron (2001) [105]
𝑑𝑁𝑖 𝑑𝑡 = 𝑁𝑖−1∑ 2 𝑗−𝑖+1𝛽 𝑖−1,𝑗𝑁𝑗+ 𝑖−1 𝑗=1 1 2𝛽𝑖−1,𝑖−1𝑁𝑖−1 2− 𝑁 𝑖∑ 2𝑗−1𝛽𝑖,𝑗𝑁𝑗− 𝑖−1 𝑗=1 𝑁𝑖∑ 𝛽𝑖,𝑗𝑁𝑗 ∞ 𝑗=𝑖 (2.19) (1) (2) (3) (4)
where N is the number of particles per volume in a specific bin, i is the bin being currently investigated, j is the bin compared to i, βi,j is the coalescence kernel at bins i and j in volume
per unit of time and t is the time. The terms, numbered 1-4, can be explained as follows: Terms 1 and 2 show the increase in the number of particles in bin i due to the coalescence of a particle in the previous bin with a particle from any bin equal to or smaller than the previous bin size. Terms 3 and 4 represent the reduction of the number of particles due to agglomeration with a particle in a previous bin or a particle in the current bin or greater, respectively. It is important to note that the coalescence kernel, βi,j, is of great importance for
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the actual kinetics of the agglomeration process. Many studies focus on finding an expression for this kernel.
For example, Adetayo et al. [109] incorporated a mechanistic element into their empirical kernel by adopting a two-stage coalescence approach. The two stages were based on the non- inertial and inertial growth regimes, and were dependent on the viscous Stokes number. In later studies [110,111], a cut-off kernel based on granule size was introduced. Collisions with granules below the cut-off size W* were considered successful. With this model, seemingly contradictory experimental data could be explained, although the authors emphasised that W* was only understood qualitatively.
Liu and Litster [112] developed a more physically based kernel incorporating type I and II coalescence. Only collisions satisfying either type of coalescence were considered successful. It was found that type I coalescence favoured small granules, whereas type II coalescence showed a preference to larger granules. This result indicates that the effect of granule size on coalescence is also dependent on granule and binder properties, as the Stokes deformation number is partially responsible for coalescence behaviour. The model was not fitted to any experimental data. In spite of this, it agreed better with experimental data in the early stages of granulation than the experimentally fitted work by Adetayo et al. [109]. In addition, it showed good agreement with various experimental data sets in general.
A comparison of four different collision frequency kernels was made by Darelius et al. [113]. The kernels were combined with a collision efficiency kernel in order to develop a mechanistic population balance model. Collision frequency kernels compared were a size independent kernel [70], Smoluchowski’s shear kernel [114], a kernel based on equipartition of kinetic energy (EKE kernel) [65] and a kernel based on equipartition of fluctuating translational momentum (ETM kernel) [65]. The first kernel considers all collisions equally likely; the second bases collisions on powder flow in suspension; the third assumes that the average velocity does not depend on size, but a random component does; the fourth assumes full randomness of velocity, but a constant translational momentum for all granules.
The ETM kernel turned out to describe results for high impeller speeds best, whereas the EKE kernel fit the data for low impeller speeds better. Since the EKE kernel emphasises granule size more than the ETM kernel, this result implies that granule size is more important for growth at lower impeller speeds. It was also shown that a general trend could be observed for the collision efficiency by including the liquid saturation in the efficiency expression. Since liquid saturation plays an important role in growth behaviour, as demonstrated by regime maps [14,49], this finding appears to be consistent with theory.
In a semi-mechanistic model, Chaudhury et al. [115] decoupled coalescence and consolidation. The model was capable of capturing both steady growth and induction growth behaviour. Although the kernel is practical for modelling, it needs as few empirical parameters as input. Another semi-mechanistic induction growth kernel was proposed by Pohlman and Litster [116]. The model in this work is based on the types of coalescence and rebound discussed in Section 2.3.2, whereas the model proposed by Chaudhury et al. focuses on droplet spreading.
As with nucleation kernels, there are few breakage kernels to be found in the literature [107]. Hounslow et al. [70] used tracer data to propose a kernel based on bimodal breakage. The kernel used is shown in Equation 2.20:
37 𝑑𝑁𝑖 𝑑𝑡 = ∑𝑏𝑖,𝑗 𝑛 𝑗=𝑖 𝑆𝑗𝑁𝑗− 𝑆𝑖𝑁𝑖 (2.20)
where, Ni is the number of particles per volume in a specific bin, i is the bin being currently
investigated, j is the bin compared to i, Si is the breakage selection rate constant per unit of
time, bj,i is the number of fragments from bin j assigned to bin i, and t is the time. The
challenge here is finding definitions for both S and b. Particularly b, which determines the size of the newly formed granules, is particularly difficult to determine reliably.
The model assumed that the only variables involved were time and granule size. Using these assumptions, the model was capable of describing both granule size distributions and tracer-mass distribution accurately. The authors pointed out that it was unusual that some rate constants depended on time and implied that some other variable must change with time.
Vogel and Peukert [117] attempted to introduce a mechanistic breakage kernel by using a probability of breakage. The breakage selection rate constant S was set to be dependent on the kinetic energies of the particles, and the breakage function was determined by the sizes of the particles involved in a collision. Experimental data was used to determine particle properties.
A mechanistic breakage model was developed, incorporated in population balances and experimentally validated by Ramachandran et al. [107]. The method involved the three- dimensional approach proposed by Verkoeijen et al. [68], using solid, liquid and air volumes. The breakage kernel itself was based on a method similar to Stokes deformation criteria, using external stress applied and the intrinsic granule strength. For the external stress, particle-particle, particle-wall and particle-impeller collisions were taken into account. Capillary forces, viscous forces and frictional forces determined the intrinsic strength of the granules. The kernel performed well compared to empirical and semi-empirical kernels and accurately described breakage kinetics.
A commonly applied equation to determine densification in 3-D PBMs was introduced by Verkoeijen et al. [68]. This method, based on Equation 2.6 by Litster et al. [18] is shown in Equation 2.21: 𝑑𝑎 𝑑𝑡 =−𝑘𝑐( (𝑙 + 𝑎)(𝑠 + 𝑙 + 𝑎) 𝑠 − 𝜀𝑚̃𝑖𝑛(𝑠 + 𝑙 + 𝑎)2 𝑠 ) (2.21)
where s, l and a are the volume of solid, liquid and air, respectively, t is the time, kc is a
compaction rate constant and εmin is the minimum attainable porosity. The constant kc has to
be determined experimentally.
An example of an equation for the growth rate by Hounslow et al. [22] is shown by Equation 2.22: 𝑑𝑁𝑖 𝑑𝑡 = 2𝐺 (1 + 𝑟)𝐿𝑖 ( 𝑟 𝑟2− 1𝑁𝑖−1+ 𝑁𝑖− 𝑟 𝑟2− 1𝑁𝑖+1) (2.22)
where G is a growth rate in units of length over time, r is the ratio between the lengths of the current bin and the next bin, Ni is the number of particles per unit of length in bin i, and t is
the time. Again, G must be determined experimentally.
In general, describing a mechanism as a single set of equations does not represent the actual phenomena in a granulator. Breakage, for example, is much more likely to occur near
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walls or the impeller, and nucleation will take place in the spray zone. The rates mentioned above attempt to describe overall growth, birth and death rates, representing the granulator as a single black box, or a grey box in the case of semi-mechanistic models. Another option to this problem is compartmentalisation of the equations, which is the representation of different locations in the granulator by different sets of equations. Compartmentalisation is especially important when moving to more mechanistically accurate models.
Chaudhury et al. [118] investigated a compartmentalised model using a spray zone, a bulk high velocity zone, a bulk low velocity zone, and a shear zone. It was found that the PSDs varied between zones initially, but the granulator contents became more homogenous as time passed. This finding implies that a single-compartment model is suitable for longer granulation times, but does not accurately capture initial granulation behaviour.
Yu et al. [119] focused specifically on the spray zone of a high-shear mixer for the incorporation into compartmentalised population balance modelling. By comparison with experimental data, it was found that the spray zone is a well-mixed, thin surface layer of the powder bed.
Since twin screw granulation (Section 2.2.3) is mostly a linear process, it is especially suitable for compartmentalised models with well-defined boundaries based on the screw elements. Barrasso and Ramachandran [120] developed and evaluated a twin screw granulator PBM, combined with DEM to obtain residence times per compartment. Results agreed qualitatively with experimental data. The possibility of evaluating the effect of different screw configurations on granule properties, in particular, opens up quality-by-design approaches for twin screw granulation.
In summary, the equations that represent a granulation process in a PBM may vary greatly in complexity depending on the requirements of the model. If the model has to be accurate and mechanistically relevant, experimentally obtained fitted parameters are not sufficient. Although mechanistic representations of coalescence and to some extent breakage exist, further development of mechanistic nucleation, consolidation, layered growth and breakage equations are necessary. Furthermore, considering the granulator to be a single well-mixed compartment does not represent the actual situation, and may lead to inaccuracies when modelling early-stage granulation. On the other hand, if time is the most important factor, for example for on-line control, simple, fast models might be needed.