(199a) Efficient Simulation of Population Balance Models By An Improved Method of Characteristics Approach

Particulate processes are commonly encountered in various systems (e.g., crystallization [1], aerosols [2], cell dynamics [3], and polymerization [4, 5]), where the particle size distribution (PSD) plays a key role on the product quality[1]. Recent advancements in particle technology have allowed on-line acquisition of PSD, and have inspired increasing efforts in modeling and computation of particulate systems in order to utilize such information for improved process and quality control [6].

The application of model predictive control (MPC) to particulate systems has been limited by the computational cost of numerically solving the population balance equations (PBEs) that describe the dynamics of these systems. Discretization techniques including the finite volume/difference methods can be applied, but are typically computationally expensive and exhibit problems with numerical diffusion or dispersion [7]. Although more advanced numerical schemes, such as the parallel high-resolution finite-volume method, provide much better tradeoffs between numerical accuracy and computational cost [8-10], the cost is still prohibitive for real-time MPC applications. As an alternative, some papers [11-13] have proposed approaches that combine standard method of moments or quadrature method of moments with the method of characteristics (MOCH). Such approaches require solving the moments equations as the first step to obtain the temporal information on supersaturation and nucleation rate during crystallization for subsequent computation of PSD using MOCH.

This work proposed an approach that utilizes the method of characteristics alone to solve population balance models together with physical constraints (for example, mass conservation), to accurately simulate the PSD by transforming the PBEs into a system of ordinary differential-algebraic equations (DAE). The resulted DAE is conveniently handled by available commercial solvers such as ode15s in Matlab. Four example systems are simulated and analyzed for demonstration purposes, including (i) growth-dominated crystallization with saturation control; (ii) growth and nucleation during cooling crystallization (both size-independent and size-dependent growth); (iii) a growth and agglomeration system; and (iv) evaluation of temperature-dependent growth in crystallization.

Compared with existing PSD simulation methods, the proposed approach is more advantageous for its straightforward procedures of model formulation, efficient computation, accurate results (no numerical diffusion or dispersion, with accuracy comparable to analytical solutions for those problems that have analytical solutions), and broad applicability. The proposed approach is sufficiently computationally efficient to enable the implementation of real-time MPC to particulate systems with aggregation and multiple internal and external dimensions, as well as facilitating the use of PBEs in robust parameter estimation, design, and control algorithms, which typically involve large numbers of PBE simulations.

Acknowledgements:This research was supported by Novartis pharmaceuticals.

References:

[1]        A. D. Randolph and M. A. Larson, Theory of Particulate Processes: Analysis and Techniques of Continuous Crystallization. New York: Academic Press, 1971.

[2]        T. E. Ramabhadran, T. W. Peterson, and J. H. Seinfeld, "Dynamics of aerosol coagulation and condensation," AIChE Journal, vol. 22, p. 840-851, 1976.

[3]        N. V. Mantzaris, J.-J. Liou, P. Daoutidis, and F. Srienc, "Numerical solution of a mass structured cell population balance model in an environment of changing substrate concentration," Journal of Biotechnology, vol. 71, p. 157-174, 1999.

[4]        T. J. Crowley, E. S. Meadows, E. Kostoulas, and F. J. Doyle III, "Control of particle size distribution described by a population balance model of semibatch emulsion polymerization," Journal of Process Control, vol. 10, p. 419-432, 2000.

[5]        X. X. Zhu, B. G. Li, L. B. Wu, Y. G. Zheng, S. P. Zhu, K. D. Hungenberg, S. Mussig, and B. Reinhard, "Kinetics and modeling of vinyl acetate graft polymerization from poly(ethylene glycol)," Macromolecular Reaction Engineering, vol. 2, p. 321-333, 2008.

[6]        Z. K. Nagy and R. D. Braatz, "Advances and new directions in crystallization control," Annual Review of Chemical and Biomolecular Engineering, vol. 3, p. 55-75, 2012.

[7]        R. Gunawan, I. Fusman, and R. D. Braatz, "Parallel high-resolution finite volume simulation of particulate processes," AIChE Journal, vol. 54, p. 1449-1458, 2008.

[8]        R. Gunawan, I. Fusman, and R. D. Braatz, "High resolution algorithms for multidimensional population balance equations," AIChE Journal, vol. 50, p. 2738-2749, 2004.

[9]        D. L. Ma, D. K. Tafti, and R. D. Braatz, "High-resolution simulation of multidimensional crystal growth," Industrial & Engineering Chemistry Research, vol. 41, p. 6217-6223, 2002.

[10]      S. Qamar, M. P. Elsner, I. A. Angelov, G. Warnecke, and A. Seidel-Morgenstern, "A comparative study of high resolution schemes for solving population balances in crystallization," Computers & Chemical Engineering, vol. 30, p. 1119-1131, 2006.

[11]      E. Aamir, Z. K. Nagy, C. D. Rielly, T. Kleinert, and B. Judat, "Combined quadrature method of moments and method of characteristics approach for efficient solution of population balance models for dynamic modeling and crystal size distribution control of crystallization processes," Industrial & Engineering Chemistry Research, vol. 48, p. 8575-8584, 2009.

[12]      S. Qamar, S. Mukhtar, A. Seidel-Morgenstern, and M. P. Elsner, "An efficient numerical technique for solving one-dimensional batch crystallization models with size-dependent growth rates," Chemical Engineering Science, vol. 64, p. 3659-3667, 2009.

[13]      M. J. Hounslow and G. K. Reynolds, "Product engineering for crystal size distribution," AIChE Journal, vol. 52, p. 2507-2517, 2006.