(167d) Efficient Solution of Multi-Dimensional Population Balance Equations Using Lattice Boltzmann Method

Population balances equations (PBEs) have become an indispensable tool for scientists and engineers in a wide range of disciplines [1]. PBEs are hyperbolic partial differential equations used to model processes, such as crystallization, where the entities of interest are distributed along property coordinates. Many crystallization processes require more than one property to characterize the crystals. The numerical solution of the resulting multi-dimensional PBEs can be computationally challenging. Recently, we presented lattice Boltzmann method (LBM) for solving 1D PBEs in crystallization, which was found to be very promising in terms of accuracy and efficiency [2].

In this paper, we first present LBM to solve multi-dimensional PBEs with growth and nucleation. The proposed technique handles processes with size independent growth efficiently. However, smaller time steps are required to ensure stability leading to larger computation time for process with size dependent growth rate. For such processes, a coordinate transformation scheme is proposed such that in the transformed coordinate, the problem can be treated as size independent one and thus computation time improves significantly without sacrificing accuracy [3]. LBM is further extended to processes with aggregation and breakage phenomena. Aggregation and breakage, which act as source terms in the governing PBE, are included as forcing terms in the LB formulation [4]. Several benchmark problems taken from literature are used to show that LBM provides significantly lower computation time, while maintaining the same level of accuracy, as the well-established high resolution (HR) method [5].

References

[1] D. Ramkrishna. Population balances: Theory and applications to particulate systems in engineering. Academic Press, San Diego, CA, 2000.

[2] A. Majumder, V. Kariwala, S. Ansumali, and A. Rajendran. Entropic Lattice Boltzmann method for Crystallization Processes. Chem. Eng. Sci., In Press.

[3] A. Majumder, V. Kariwala, S. Ansumali, and A. Rajendran. Fast high-resolution method for solving multidimensional population balances in crystallization. Ind. Eng. Chem. Res., 49(8):3862-3872, 2010.

[4] S. Succi. Lattice Boltzmann equation for fluid dynamics and beyond. Oxford University Press, New York, 2001.

[5] R. Gunawan, I. Fusman, and R. D. Braatz. High resolution algorithms for multidimensional population balance equations. AIChE J., 50(11):2738-2749, 2004.