Model implementation

In order to evaluate the performance of the developed 1-D model, it was implemented using Mathematica software for the three different discretisation methods presented in Equation 7.22, Equation 7.23 and Equation 7.30, as shown by Figure 7.2. To supply data for values of initial liquid volume vL, critical-packing liquid volume fraction ϕcp, porosity ε, and

time needed to achieve critical packing tmax, experimental data was used. All available

granulation data from the experiments performed using the consolidation-only granulator (COG) with the lactose-100 cSt silicone oil system as described in Chapter 5 was combined. The combined data points were used to calculate the fitted line and the values for ϕcp and tmax,

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values used do not represent the actual physical values; after all, it was not possible to obtain the real values of ϕcp and tmax in the work performed using the COG. However, the behaviour

observed does not change, and the experimental data does conform to Hounslow et al.’s model for surface tension-driven growth [30], which serves the purpose of evaluating the population balance model. The results are shown in Table 7.1.

Table 7.2 shows the other parameters used to perform the simulation. The lowest bin size was set to the diameter of the initial liquid droplet, and the number of particles simulated was set to 1000. The latter has no real effect on the actual simulation, but serves as a useful tool to ensure the zeroth moment is conserved. For the size ratio between neighbouring bins r, two different values were used; a ratio of 21/3, and a ratio of 21/9. The former value was also applied by Hounslow et al. [22] in their discretisation of the growth term. The latter value was selected to increase the accuracy of the discretisation, at the cost of increasing the number of bins evaluated, as shown in Table 7.2. The actual number of bins was selected based on the bins size ratio r; the number of bins should be sufficient to account for all possible granule sizes. To ensure the final granule size was well within the limits of the simulated bin sizes, an additional number of bins was added to the minimum number of bins required.

The parameters shown in Table 7.1 and Table 7.2 were used for standard discretisation [22], Marchal et al’s discretisation [23] and Bertin et al.’s discretisation [84], as presented in Section 7.4.2.2, for a total of six simulations. All simulations were solved using Mathematica software’s NDSolve function for a total simulation time of 250 minutes to capture all experimental data, as well as to ensure the fits had reached the no-growth regime. The obtained time-dependent particle size distributions (PSDs) were then converted to time- dependent number-based average particle sizes in order to compare them with experimental data. The results of the first set of simulations, with r = 21/3, are shown in Figure 7.3.

Table 7.1: Parameters from experimental data used for the model in Equation 7.20.

Parameter (units) Value

vL (μL) 6.785

ε (-) 0.43

ϕcp (-) 0.13

tmax (min) 156

Table 7.2: Simulation parameters used for the model.

Simulation type Number of granules L1 (mm) r (-) Number of bins (-)

Simulation 1

1000 2.349 2

1/3

10

Simulation 2 21/9 20

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Figure 7.3: Comparison of particle size as a function of time for both the experimental and simulated data for all three discretisation methods with r = 21/3.

All three models shown in Figure 7.3 demonstrate an increase in particle size over time, but Marchal et al.’s discretisation method [23] shows the most unstable behaviour. After a rapid increase in particle size, the size fluctuates. This fluctuation is related to the number of granules present in each bin; inspection of the individual bins regularly shows a negative number of granules. Bertin et al.’s discretisation method [84] yields stable growth, but the growth rate is generally underestimated compared to the fit to the experimental data. The standard discretisation method [22] appears to yield the smallest deviation from the fitted curve, slightly overestimating both the growth rate and the final granule size. The theoretical maximum granule size as fitted to the experimental data is 5.49 mm, whereas the predicted maximum according to the standard discretisation method is 5.92 mm, a deviation of 7.8 %. This means the volume of the granule is overestimated by 25 %.

The overestimation of the final granule size is not surprising, as the final size is dependent on the last bin in which growth occurs. Creating narrower bins by changing r to 21/9 should improve the accuracy of the solution; this is demonstrated in Figure 7.4.

Compared to Figure 7.3, the modelled curves in Figure 7.4 more closely match the fit to the experimental data. Marchal et al.’s method [23] displays small deviations and appears to predict a final granule size similar to that of the standard discretisation method [22]. However, the method still displays negative granule numbers for bins, which is undesirable. Bertin et al.’s method [84], which greatly underpredicted the final granule size as well as the growth rate, appears to more closely predict the final granule size, with a deviation of 0.18 % in diameter, or 0.55 % in volume. The standard discretisation method appears to capture the kinetics of growth extremely well. However, the method still overestimates the final granule size by 7.8 %.

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Figure 7.4: Comparison of particle size as a function of time for both the experimental and simulated data for all three discretisation methods with r = 21/9.

This difference between the standard discretisation method and Bertin et al.’s method [84] is caused by their different approaches. The latter method is an upwind scheme, which takes into account the granule size of the next bin. This method inherently underpredicts granule growth, as growth decreases with increasing granule size. Therefore, the solution is likely to predict the maximum granule size as the bin size before the actual maximum granule size. The standard discretisation method, on the other hand, uses the previous bin to calculate growth. Therefore, it uses smaller granule sizes compared to Bertin et al.’s method, and consequently overpredicts growth. As a result, the standard discretisation method is likely to designate the bin size after the actual maximum granule size as the maximum.

Because of the inherent flaws of discretisation methods, no single method can accurately predict the final granule size. However, the kinetics of the growth process can be captured. Figure 7.5 shows how the solutions converge for the extreme case where the bin size ratio r is equal to 21/81. In this scenario, both standard discretisation and Bertin et al.’s method [84] show agreement with the analytical solution. However, Marchal et al.’s method [23] deviates from the solution even at this low value for r. Furthermore, the solution keeps fluctuating beyond the final time of 250 minutes, never fully converging to the analytical solution. Therefore, it is not recommended to use this method for the simulation of growth if the growth parameter G is not a constant.

From the simulations performed, it appears as though standard discretisation is the most reliable method for the prediction of the kinetics, especially considering the fact that real simulations are expected to run for much shorter granulation times. Although Bertin et al.’s method [84] shows good convergence to the analytical solution, it underestimates the actual granule size too much for shorter granulation times, particularly at large values of r.

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Figure 7.5: Comparison of particle size as a function of time for both the experimental and simulated data for all three discretisation methods for the extreme case r = 21/81.

In order to improve the accuracy of the prediction of the final granule size, the bins surrounding the final granule size should be narrow, i.e. the value for r should be reasonably small. However, reducing r increases the number of bins and, consequently, the computational load. Moreover, simulating other granulation mechanisms, such as agglomeration, in conjunction with growth would further increase the possible granule sizes, the number of bins, and the computation time. Therefore, it is not recommended to decrease r if simulation speed is important.

Since layered growth most likely never reaches the no-growth regime in simulations of industrial processes, the predicted growth rate is much more significant than the predicted final granule size. As such, upwind methods such as Bertin et al.’s discretisation method [84] are not recommended. Standard discretisation appears to better predict the kinetics, although it deviates significantly for large bins sizes. It should be possible to develop a discretisation method that better captures the kinetics by using a more representative granule size for each bin. Marchal et al.’s attempt [23] to average the flows in and out of bins resulted in instabilities and negative numbers of granules in some bins, which is undesirable. Different averaging methods might be more suitable.

It should also be noted that the case studied here was the most straightforward simulation of growth possible. No liquid addition was considered, and it was assumed that there was no consolidation. These two processes will greatly influence the complexity of the equations. Furthermore, nucleation, agglomeration, breakage and attrition were not considered. There was no mass balance for powder available for layering; it was assumed that, as with the COG experiments, powder was not the limiting factor. However, in practical granulation, this is not the case. It is likely that the layering rate decreases with a decreasing amount of fines. As such, the model evaluated here is just the first step in the mechanistic simulation of layered growth. The further development of models capable of simulating this type of growth could

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provide industry with the tools to better understand the granulation process, as well as facilitate easier granulation process design.

7.6 Conclusions

In the study presented here, two novel population balance models (PBMs) were developed for layered granule growth by finding a mechanistic expression for the growth rate in order to address the final objective of this thesis; to develop and evaluate a mechanistic population balance model. The first model was a 3-D PBM based on Verkoeijen et al.’s work [68], which tracks volumes of solid, liquid and air per size bin. The second model, a 1-D PBM tracking only particle number for each size bin, was discretised using three different discretisation methods and evaluated using Mathematica software.

Results showed that standard discretisation predicted the growth rate best for narrow bins, but that it could not accurately predict the final granule size due to the limits of discretisation. Other methods deviated more from the curve fitted to experimental data, especially for wider bins, which are more likely to be used for simulations when simulation times should be short. It is therefore recommended to either use standard discretisation, or develop a discretisation scheme that better represents the expected growth rate.

Overall, the results of the simulation showed that the developed model is capable of predicting granule growth behaviour. Future development of the population balance equations should focus on the inclusion of other processes such as consolidation, liquid addition, nucleation, agglomeration, breakage and attrition.

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