(339l) Lattice Boltzmann Method: An Efficient Technique for Solving Population Balance Equations
Crystallization is widely used in chemical, pharmaceutical, food and semiconductor industries as a method of separation and purification. In industrial crystallization processes, the control of crystal shape and size is of key importance as these properties not only affect the product qualities, e.g. bioavailability, but also the efficacy of the downstream processes. Recent experimental results have demonstrated that model-based control offers significantly improved performance in comparison to model-free approaches . These model-based online control schemes require fast and accurate numerical techniques to solve the model equations.
Population balance equations (PBEs) are widely used to model the dynamics of crystallization and other particulate systems . In a batch crystallization process, these equations have to be coupled with mass balance equations to take into account the mass transfer from solution to solid phase. PBEs are hyperbolic partial differential equations which do not have an analytical solution except for few limited cases and hence need to be solved numerically. The accurate numerical simulation of the crystallization process can be challenging as the crystal size distribution (CSD) can be very sharp and various mechanisms may be present which affect CSD simultaneously. Over the years, researchers have developed various numerical schemes. The available methods show performance limitations as the dimensions of the PBE increases. This motivates the need to develop reliable and computationally efficient numerical methods to solve PBEs for application in online model based control.
In this work, we introduce lattice Boltzmann method (LBM) as an alternate tool for solving PBEs. In recent years, LBM has attracted significant attention from the fluid dynamics community. In this method, fictitious particles resembling groups of molecules are considered in a lattice with finite set of velocities [3, 4]. These particles collide at the lattice nodes and propagate in such a way that the macroscopic behavior of the system is recovered in the long-time limit. In principle, with an appropriate choice of mesoscale equilibrium distribution, any macroscale dynamics other than hydrodynamics can be simulated by lattice Boltzmann type of method. In spite of the ability of LBM to provide fast, accurate and easily implementable numerical scheme, to the best of our knowledge, it has not been used to solve PBEs.
In the current work, we consider crystallization with nucleation and growth. The governing PBE for a crystallization process with size independent growth rate is analogous to the advection equation, which can be handled efficiently by LBM [4, 5]. We further apply LBM to solve PBEs representing a crystallization process with size dependent growth rate. This is done by taking advantage of the functional form used to describe the size dependence of growth rate and proposing a coordinate transformation technique which simplifies the problem to a size independent one. Nucleation as a boundary condition in crystallization is handled in the same manner as open flow boundary condition in a fluid flow problem. Few benchmark problems drawn from literature are used to show the reliability of the solution by analyzing convergence and accuracy. It is found that for 1D problems, solution provided by LBM has the same level of accuracy and computation time as the well established HR method . For multidimensional problems, computation time required by LBM to maintain the same level of accuracy is much lower than that of HR method.
 Z. K. Nagy, M. Fujiwara, and R. D. Braatz. Modelling and control of combined cooling and antisolvent crystallization processes. J. Process Control, 18(9):856?864, 2008.
 D. Ramkrishna. Population balances: Theory and applications to particulate systems in engineering. Acadamic Press, San Diego, CA, 2000.
 S. Succi. Lattice Boltzmann equation for fluid dynamics and beyond. Oxford University Press, New York., 2001.
 I. V. Karlin, S. Ansumali, C. E. Frouzakis, and S. S. Chikatamarla. Elements of the lattice Boltzmann method. I: Linear advection equation. Commun. Comput. Phys., 1(4):616?655, 2006.
 A. Majumder, W. P. Yudistiawan, V. Kariwala, S. Ansumali, and A. Rajendran. Lattice Boltzmann method for solving population balance equations. In proceedings of International Conference on Chemical Engineering 2008, Dhaka, Bangladesh, 2008.
 R. Gunawan, I. Fusman, and R. D. Braatz. High resolution algorithms for multidimensional population balance equations. AIChE J., 50(11):2738?2749, 2004.