The drop number density is assumed to be small enough for the coalescence between droplets to be negligible. Break-up is assumed to result from turbulent forces so that the breakage frequency is assumed to be a function of the local energy dissipation. The energy dissipation field arises from the k - &epsilon model for the flow. The population balance is however coupled to the fluid momentum equation through the appearance in the former of the energy dissipation and the position and size dependent turbulent diffusion coefficient for the particles. In addition the effect of fluid turbulence on the drop velocity is modeled through the steady state force balance. Breakage frequencies are allowed to vary with position, although the size distribution of broken fragments is assumed to satisfy a form of similarity assumed in the work of Narsimhan et al. (1984) that rids it of explicit spatial dependence. The breakage kernels used in this work are the result of a fresh fit to the data of Sathyagal et al. (1996) to yield the degenerate breakage frequency. We have used the method of Kumar and Ramkrishna (1996) to discretize the population balance equation. The proposed solution method takes the advantage of the structure of the breakage matrix in the discretized population balance equation. In light of Sylvester's theorem, the discretized version leads to an elegantly simple operator equation with a straightforward solution in terms of the spectral expansion of the operator. The spectral expansion of the modified diffusion operator is obtained using Arnoldi algorithm (Saad, 2003) implemented in MatlabŪ. Foregoing predictions will be compared to the detailed computations using the control volume approach.
References (1) Narsimhan, G.; Nejfelt, G., Ramkrishna, D. Breakage Functions of Droplets in Agitated Liquid-Liquid Dispersions. AIChEJ., 1984, 30, 457. (2) Sathyagal, A. N.; Ramkrishna, D., Narsimhan, G. Droplet Breakage in Stirred Dispersions. Breakage Functions from Experimental Drop-size Distributions. Chem. Eng. Sci. , 1996, 51, 1377. (3) Kumar, S., D. Ramkrishna. On the Solution of Population Balance Equations by Discretization-I. A Fixed Pivot Technique. Chem. Eng. Sci. , 1996, 51(8), 1311. (4) Saad, Y. Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia, 2003. (5) MATLAB; release 13; The Mathworks: Natick, MA, copyright 1984-2002.