(208f) Applying Reconstruction Methods on the Solution of a Moment-Based Pharmaceutical Drying Process Population Balance Model

Moment-based solution methods are commonly used for Population Balance Models (PBMs) to reduce computational load. However, after the calculation of the moments it is difficult to interpret them as such with respect to actual changes in the distribution. Therefore, a reconstruction of these moments to recover the underlying distribution is appealing and in some cases needed in order to draw conclusions with respect to system behaviour. Several reconstruction techniques are available, using for instance linear or nonlinear inversion approaches. However, in most cases a large number of moments are required. Additionally, the reconstruction techniques often suffer from solution multiplicity and ill-conditioned problems. Only a limited number of reconstruction methods able to reconstruct the distribution from a finite number of moments have been described. Here, the Quadrature Method of Moments (QMOM) has been applied to describe the evolution of the moisture content for a population of wet pharmaceutical granules.

The objective of this contribution is twofold, (1) comparing different reconstruction methods and (2) comparing the distribution obtained after reconstruction of the moments with the result obtained by directly solving the PBM using a solution method that yields the entire number distribution. For the comparison of different reconstruction methods, comparison is made between parameter fitting methods and the method of splines.

The finetuning of the parameters for the method of splines was very important for the final result as well as for the computational time. An additional parameter, i.e. a different value for the first and the last interval for tol_red, was introduced to improve the result and speed up the calculation. Using the method of splines allowsto reconstruct the bimodal distribution occurring at the transition from the first to the second drying phase. Moreover, the method of splines requires no a priori knowledge about the underlying distribution. The number of iterations to obtain the final distribution is quite high, resulting in a high computational effort. As a consequence the method is not very useful when the distribution is required at different time steps.

In contrast, none of the parameter fitting methods was able to correctly predict several peaks in the final distribution, e.g. a bimodal distribution. The methods based on parameter fitting are also only useful when the underlying distribution is known, e.g. an exponential function.

However, none of the methods was able to reconstruct the distribution perfectly. As such, attention should be paid when the distribution is used to monitor the process, as the possibility exists that a deviation of the process from normal behaviour is detected. When the PBM-model is used for process follow-up, it is recommended to use a non-moment based solution method for the PBM. Only in the case that one is just interested in the final distribution, the combination of a moment-based solution method and the method of splines is recommended.