Model Reduction of Population Dynamics of Particles in Bi-Component Granulation Processes

Wet granulation in pharmaceutical plants is a broadly used process, in which binder droplets facilitate binding of the small drug particles together and larger granules formation. The governing equation of these dynamic systems is derived by a population balance study which turns in a complex mathematical model. However, in order to control these processes in real time, development of reduced order models for these processes is necessary. This work proposes a new model reduction method for two-component coagulation systems by exploiting the method of moments and orthogonal projection approach.

The two-component agglomeration process considered in this paper has no chemical reactions. The population balance analysis for this granulation system is in form of an Integro-Differential Equation (IDE). Also, the rate of coagulation namely the coagulation kernel is dependent on size and composition. This kernel considers two steps of agglomeration; collision and binding. Either of these two steps happens with a probability, called collision probability and collision success factor, respectively. However, there exists no explicit analytical solution of the IDE for this type of coagulation kernels.

In order to simplify the results of the method of moments, some papers such as [1], exploit Taylor expansion and derive a closed finite-dimensional ordinary differential equation set. However, this approach cannot be used for composition-dependent models such as the one used in [2]. To address this issue, this work proposes a new model reduction approach using the method of moments in conjunction with Laguerre polynomials. In this way, we expand the distribution function over the set of orthogonal Laguerre polynomials which are function of moments. Also, we evaluate our new reduced model with the results obtained from a Monte Carlo simulation as a bench mark. This model will be the foundation for efficient observer and controller design for such bi-component agglomeration processes.

References:

[1] N. Hashemian and A. Armaou, "Simulation, model-reduction and state estimation of a two-component coagulation process," AIChE J., 2016, 62, pp 1557-1567, DOI:10.1002/aic.15146.

[2] N. Hashemian, M. Ghanaatpishe and A. Armaou, “Bi-Component Granulation Processes via Laguerre Polynomials,” Proceedings of the American Control Conference, accepted, MA, USA, 2016.