Identification of the Breakage Rate and Distribution Parameters in a Nonlinear Population Balance Model for Batch Milling

Comminution is an important unit operation in a variety of industries including ceramics, minerals, and pharmaceuticals. For industries, such as the pharmaceutical, which must meet stringent product specifications, design and control over size reduction processes is essential.  Over the past half century, there has been extensive work to describe size reduction using mathematical models.  Population balance models (PBMs) which utilize the concepts of specific particle breakage rate and breakage distribution were applied to size reduction processes [1].   While the PBM was used with varying success, early investigators noticed non-random deviation between experimental data and predictions in the model.  This unexplained deviation became large at relatively long milling times and when accumulation of fine particles became significant. 

Recognizing this deviation, Austin and Bagga [2] attributed such non-linear particle breakage behavior to multi-particle interactions.  The traditional population balance model, however, is linear due to the fact that the specific breakage rate of a particle of size x is only dependent on size x and thus incapable of taking into account multi-particle interactions.  To explain the non-linear phenomenon, Austin and Bagga [2] and Austin et al. [3] introduced a time dependence on the specific breakage rate parameter.  While this new model was able to reconcile some differences between the experimental data and model predictions, it could not explain all types of deviation.  Bilgili and Scarlett [4] further explained that while some breakage phenomenon such as particle fatigue is truly time dependent, multi-particle interactions can only be explained through a non-linear population balance model and not a linear time-variant one.

Bilgili and Scarlett [4] recently introduced a population balance model to mathematically include non-linear effects arising from multi-particle interactions for rate processes.  In their model, the specific breakage rate is decomposed into an apparent breakage rate and a population dependant functional where the functional describes different types of non-linear kinetics.  Further novelty in the model extends from the fact that the non-linear model reduces to the linear model in the absence of multi-particle influence.  Numerical simulations of the non-linear model conducted by Bilgili et al. [4,5] were able to predict many types of deviation found in literature.  Due to the ability of the non-linear population balance model to encompass many different types of milling behavior, it may lead to better design, control, and optimization of milling processes. 

Because no analytical solution exists for the non-linear model, calculation of the breakage rate parameters, breakage distribution parameters, and the non-linear functional parameters remains a formidable challenge.  In this study, we present a full numerical method to solve the inverse problem for non-linear particle breakage.  Our method, developed in Matlab, uses a non-linear optimizer to minimize the sum of the squared relative residuals between the experimental mass fraction distribution and the modeled distribution.  The model distribution is solved using a numerical ordinary differential equation solver.  This method was first applied to generated data with random noise to test the efficacy of our method.  It was then applied to particle size distributions exhibiting non-linear behavior found in milling literature.  Fitting the non-linear PBM to experimental data will elucidate breakage mechanisms as well as reveal the effect of multi-particle interaction.  This will aid in the development and optimization of size reduction processes used in the pharmaceutical industry and will lead to better process control and higher quality products.      

 

REFERENCES:   

1.     Austin, L.  A review: introduction to the mathematical description of grinding as a rate process, Powder Technology. 1971, 5: 1-17.

2.     Austin, L., Bagga, P.  An analysis of fine grinding in ball mills, Powder Technology.  1981, 28: 83-90.

3.     Austin, L., Shah, J., Wang. J., Gallagher, E., Luckie, P.  An analysis of ball-and-race milling: Part1.  The Hardgrove mill, Powder Technology.  1981, 29: 263-275.

4.     Bilgili, E., Scarlett, B. Population balance modeling of non-linear effects in milling processes. Powder Technol. 2005: 153, 59–71.

5.     Bilgili, E. Yepes, J., Scarlett, B. Formulation of a non-linear framework for population balance modeling of batch grinding: beyond first-order kinetics. Chem. Eng. Sci. 2006, 61: 33–44.