(155c) Numerical Solution of a Two-Dimensional Population Balance Equation for Aggregation

There are several particle properties which influence the particle density distribution in many aggregation processes. Therefore, a one-dimensional population balance equation (PBE) where the most appropriate particle property namely particle size is assumed to be the only particle property is not adequate to simulate such processes. The two dimensional PBE which is an extension of the one dimensional PBE is given as [1]

,

(1)

where  and  are the two extensive properties of the particle. The first term is corresponding to the birth of particles  due to aggregation of smaller particles. The last term describes the death of particles  due to collision and adhesion to particles.   

Numerical solution of the above PBE is difficult due to the double integral and non-linear behavior of the equation. Several numerical techniques can be found in the literature ([2] [3] [4], [5]) but all of them either have problems regarding preservation of properties of the distribution or they are computationally very expensive. In this work, we present a new technique which is based on taking averages of all newborn particles in a cell. The entire property domain is first divided into small sections. The division could be inhomogeneous, for example geometric or any other appropriate choice depending upon the problem can be taken.  Each cell will have its representative where all the particles of this cell are assumed to be concentrated. The center of each cell can be a simplest choice for representatives. The solution strategy follows two steps: one to calculate averages of the properties of the newborn particles in a cell and the other to assign them to the neighboring 4 nodes such that some properties of the distribution are exactly preserved.

For the validation of the scheme, numerical results are shown in Figures 1 and 2. Two different types of mono-disperse particles have been considered as an initial condition. The aggregation kernel is taken to be constant. The computation has been made at a high degree of aggregation 0.98. The number of grid points in each direction has been taken to be 20. Figure 1 shows the complete size distribution. The corresponding moments along with the analytical results have been plotted in Figure 2. The numerical results are in excellent agreement with analytical results.

Figure 1: Evolution of Particle Size Distribution	Figure 2:  Change of  Zeroth and First Moments

LITERATURE

[1]   A. A. Lushnikov, Evolution of coagulating systems. III. Coagulating mixtures, J. Colloid Interface Sci. 54, 94-101 (1976).

[2] C. D. Immanuel and F. J. Doyle, Efficient solution technique for a Multi-dimensional population balance model describing granulation process, Powder Technol. 156, 213-225 (2005).      

[3]   Y. Xiong and S. E. Pratsinis, Formation of agglomerate particles by coagulation and sintering ? Part I. A two-dimensional solution of the population balance equation, J. Aerosol Sci. 24, 282-300, (1993)   

[4]   T. Trautmann and C. Wanner, A fast and efficient modified sectional method for simulating multicomponent collisional kinetics, Atmos. Environ. 33, 1631-1640 (1999).

[5]   I. J. Laurenzi, J. D. Bartels and S. L. Diamond, A general algorithm for exact simulation of multicomponent aggregation processes, j. Comput. Phy. 177, 418-449 (2002).