(668d) 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.