(597c) Using Semidefinite Programming to Calculate Bounds on Particle Size Distributions
Many chemical engineering processes involve a population
of particles with a distribution of sizes that changes over time. For example,
crystallization, colloidal suspension formation, catalyst attrition,
polymerization, and aerosol formation all fit this general framework . In
each of these processes, the particle size distribution (PSD) can have a large
effect on macroscopic properties of engineering interest. For example, for
pharmaceutical crystals, the PSD affects the ease with which the crystals can
be filtered and compacted into tablets, thereby affecting the cost and
processing time of the pharmaceutical product [2, 3]. Moreover, once the drug
has been introduced to a patients system, the PSD affects its
bioavailability. Thus, the PSD is tied to both the pharmaceuticals efficacy
and safety [4,5].
Because of the importance of the PSD in these diverse
chemical engineering applications, many methods have been developed to model
how a PSD changes over time. Usually, this model is a PDE known as a
population balance model . In some cases, this PDE can be solved
analytically. However, it often must be solved numerically. Solving the PDE
has the advantage that the result is a full description of the final PSD in
terms of a number density function; the disadvantage is that obtaining this
solution numerically can be computationally expensive. For this reason, it is
very common to instead model only finitely many moments of the PSD,
which amounts to solving a system of ODEs [6,7]. Modeling only the moments
certainly reduces the computational burden, but this comes at a cost: moments are
only a summary description of the PSD, i.e, they do not contain enough
information to reconstruct all of its details. This is because there are, in
general, many PSDs corresponding to a given finite set of moments . Thus,
given only finitely many moments of an unknown distribution, there is no clear
answer to industrially relevant questions such as:
How many particles have size in the range a to b?
What is the D10 of the distribution?
What is the qualitative shape of the distribution?
Faced with these questions, one might be tempted to
apply one of the various methods available for constructing a number density
function with a specified finite set of moments [9,10,11,12]. With the
resulting number density function, answering the above questions would be
trivial. However, the problem with this strategy should be clear from the
foregoing discussion: the calculated number density function describes just one
of the many PSDs with the specified moments. Accordingly, it would provide
just one of the many possible (valid) answers to each of the above questions,
giving us a false sense of certainty in our knowledge of the distribution.
We take a more rigorous approach. Acknowledging that
reconstructing a PSD from finitely many moments is an ill-posed inverse
problem, we make no attempt to answer the above questions exactly. Instead, we
calculate provable bounds on the answers. These bounds require no a
priori knowledge of the shape of the distribution, no experimental data, and no
regularity assumptions on the number density function describing the PSD.
The bounding algorithms we will present are a natural
application of results from the mathematical literature regarding moments of
positive finite Borel measures (i.e. generalized distributions) . In
particular, we will calculate the proposed bounds using Semidefinite Programs
(SDPs) . While SDPs have been applied in chemical engineering in the
context of optimal control [15,16], to the best of the authors knowledge,
their natural application to particle size distributions has, until now, gone
Example of bounds calculated on the PSD cumulative distribution function (CDF)
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