Numerical solution of population balance equations for nucleation, growth and aggregation processes

This article focuses on the derivation of numerical schemes for solving population balance models (PBMs) with simultaneous nucleation, growth and aggregation processes. Two numerical methods are proposed for this purpose. The first method combines a method of characteristics (MOC) for growth process with a finite volume scheme (FVS) for aggregation process. For handling nucleation terms, a cell of nuclei size is added at a given time level. The second method purely uses a semi-discrete finite volume scheme for nucleation, growth and aggregation of particles. Note that both schemes use the same finite volume scheme for aggregation process. On one hand, the method of characteristics offers a technique which is in general a powerful tool for solving linear growth processes, has the capability to overcome numerical diffusion and dispersion, is computationally efficient, as well as give highly resolved solutions. On the other hand, the finite volume schemes which were derived for a general system in divergence form, are applicable to any grid to control resolution, and are also computationally not expensive. In the first method a combination of finite volume scheme and the method of characteristics gives a highly accurate and efficient scheme for simultaneous nucleation, growth and aggregation processes. The second method demonstrates the applicability, generality, robustness and efficiency of high-resolution schemes. The proposed techniques are tested for pure growth, simultaneous growth and aggregation, nucleation and growth, as well as simultaneous nucleation, growth and aggregation processes. The numerical results of both schemes are compared with each other and are also validated against available analytical solutions. The numerical results of the schemes are in good agreement with the analytical solutions.

[1]  D. L. Ma,et al.  High-Resolution Simulation of Multidimensional Crystal Growth , 2002 .

[2]  John H. Seinfeld,et al.  Dynamics of aerosol coagulation and condensation , 1976 .

[3]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[4]  Panagiotis D. Christofides,et al.  Predictive control of particle size distribution in particulate processes , 2006 .

[5]  D. Ramkrishna.  The Status of Population Balances , 1985 .

[6]  Shamsul Qamar,et al.  Adaptive high-resolution schemes for multidimensional population balances in crystallization processes , 2007, Comput. Chem. Eng..

[7]  D. L. Ma,et al.  Optimal control and simulation of multidimensional crystallization processes , 2002 .

[8]  D. Ramkrishna,et al.  On the solution of population balance equations by discretization - III. Nucleation, growth and aggregation of particles , 1997 .

[9]  R. Braatz,et al.  High resolution algorithms for multidimensional population balance equations , 2004 .

[10]  M. Hounslow,et al.  A discretized population balance for nucleation, growth, and aggregation , 1988 .

[11]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[12]  S. Motz,et al.  Comparison of numerical methods for the simulation of dispersed phase systems , 2002 .

[13]  G. Pinder,et al.  Numerical solution of partial differential equations in science and engineering , 1982 .

[14]  Prashant Mhaskar,et al.  Predictive control of crystal size distribution in protein crystallization , 2005, Nanotechnology.

[15]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[16]  Gerald Warnecke,et al.  Solving population balance equations for two-component aggregation by a finite volume scheme , 2007 .

[17]  René David,et al.  Crystallization and precipitation engineering—IV. Kinetic model of adipic acid crystallization , 1991 .

[18]  H. Muhr,et al.  Crystallization and precipitation engineering-VI. Solving population balance in the case of the precipitation of silver bromide crystals with high primary nucleation rates by using the first order upwind differentiation , 1996 .

[19]  M. Hounslow,et al.  Adjustable discretized population balance for growth and aggregation , 1995 .

[20]  Shamsul Qamar,et al.  A comparative study of high resolution schemes for solving population balances in crystallization , 2006, Comput. Chem. Eng..

[21]  Bram van Leer,et al.  Upwind-difference methods for aerodynamic problems governed by the Euler equations , 1985 .

[22]  M. Hounslow.  A discretized population balance for continuous systems at steady state , 1990 .

[23]  R. Noyé,et al.  Numerical Solutions of Partial Differential Equations , 1983 .

[24]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[25]  En Sup Yoon,et al.  On the solution of population balance equations (PBE) with accurate front tracking methods in practical crystallization processes , 2002 .

[26]  Francis Filbet,et al.  Numerical Simulation of the Smoluchowski Coagulation Equation , 2004, SIAM J. Sci. Comput..