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 [1].  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 patient’s system, the PSD affects it’s bioavailability.  Thus, the PSD is tied to both the pharmaceutical’s 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 [1].  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 [8].  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) [13].  In particular, we will calculate the proposed bounds using Semidefinite Programs (SDPs) [14].  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 unnoticed.

Figure 1: Example of bounds calculated on the PSD cumulative distribution function (CDF)

 

References

 

[1] Ramkrishna D. Population Balances. San Diego: Academic Press. 2000.

[2] Myerson A. Handbook of Industrial Crystallization. Woburn: Butterworth-Heinemann. 2002.

[3] Wibowo C, Chang W, Ng K. Design of integrated crystallization systems. AIChE Journal. 2001;47:2474-2492.

[4] Braatz RD. Advanced Control of Crystallization Processes. Annual Reviews in Control. 2002; 26(1):87-99.

[5] Nowee S, Abbas A, Romagnoli J. Model-Based Optimal Strategies for Controlling Particle Size in Antisolvent Crystallization Operations. Crystal Growth Des. 2007;8:2698-2706.

[6] Hulburt H, Katz S. Some problems in particle technology: A statistical mechanical formulation. Chemical Engineering Science. 1963;19:555-574.

[7] Randolph AD, A LM. Theory of Particulate Processes. New York: Academic Press Inc. 1988.

[8] McGraw R, Nemesure S, Schwartz SE. Properties and evolution of aerosols with size distributions having identical moments. Journal of Aerosol Science. 1998;29(7):761-772.

[9] de Souza LGM, Janiga G, John V, Thevenin D. Reconstruction of a distribution from a finite number of moments with an adaptive spline-based algorithm. Chemical Engineering Science. 2010;65(9):2741-2750.

[10] Hutton K, Mitchell N, Frawley PJ. Particle size distribution reconstruction: The moment surface method. Powder Technology. 2012;222:8-14.

[11] Cogoni G, Frawley PJ. Particle Size Distribution Reconstruction Using a Finite Number of Its Moments through Artificial Neural Networks: A Practical Application. Crystal Growth & Design. 2015; Published online.

 [12] Woodbury AD. A FORTRAN program to produce minimum relative entropy distributions. Computers and Geosciences. 2004;30(1):131-138.

[13] Lasserre JB. Moments, positive polynomials and their applications, vol. 1. World Scientific. 2009.

[14] Vandenberghe L, Boyd S. Semidefinite programming. SIAM Review. 1996;38(1):49-95.

[15] VanAntwerp JG, Braatz RD, Sahinidis NV. Globally optimal robust process control. Journal of Process Control. 1999;9(5):375-383.

[16] VanAntwerp JG, Braatz RD, Sahinidis NV. Globally optimal robust control of large scale sheet and film processes. In: American Control Conference, 1997. Proceedings of the 1997, vol. 3. IEEE. 1997; pp. 1473-1477.