(355a) On the Challenge of Solving 3-D Population Balance Equations

The interest in solving population balance equations (PBE) has been a long-standing one. However the recent activities in applying Process Systems Engineering concepts and tools on the design and operation of pharmaceutical processes has heighten the interest in understanding crystallization processes, with the aim of controlling both the size and shape of the crystal population. This necessitates the solution of a 3-D population balance which follows the evolution of three characteristic properties of the crystal, such as height, width, and depth. Methods based in discretizing each of the three size dimensions, through a finite difference or a finite volume concept, yield a very large number of ordinary differential equations (ODEs). A recent publication reported the need to follow in time more than 1.5 million ODEs to simulate the evaluation of the 3-D PBE. Even though computational facilities are more widely available and less expensive than before, such calculations are not appropriate for online applications aiming to control the crystal population at the end of the batch using a model based control approach such as model predictive control. This presentation will examine an alternative method to solve the PBE based in a variant of the Galerkin Method. The primary part of the solution is written in terms of one or two distribution functions, like the Weibull one, depending on whether the population is expected to have a unimodal on bimodal character. Such distributions have either two or three parameters whose variation in time allows a good approximation of the solution of the PBE. The solution can be further refined by an additive corrective term consisting of a linear combination of the initial members of an orthogonal set of basis function, such as Laguerre polynomials. It is demonstrated that the proposed method needs one or two orders of magnitude fewer ODEs to provide the same accuracy with prior methods.