(186k) A Simple Numerical Approach for Solving Population Balance Equations

Population balance equations (PBEs) are one of the fundamental tools in science and engineering for studying, understanding and modeling population dynamics.¹ In a few special cases PBEs can be solved analytically, but, more often, analytical solutions are not available and one has to revert to some sort of numerical technique. To this end, moments based methods,^2-3 the method of weighted residuals,⁴ the method of discretization proposed by Kumar and Ramkrishna,⁵ stochastic Monte Carlo methods⁶ or high resolution algorithms⁷ are some of the possible options.

Each method has peculiar advantages and disadvantages, but all require a mathematical reformulation of the original PBEs, which may turn out to be a non-trivial task to accomplish when complex phenomena need to be taken into account. Here, a new method is proposed which, compared to the existing methods, has the main advantage of being extremely simple to implement without losing generality and accuracy.

The original PBEs are solved for a limited number of internal coordinate (grid points) using a logarithmic shape preserving interpolation to estimate the value of the population distribution between the grid points, and evaluating the summations/integrals (for discrete and continuous distributions, respectively) appearing in the PBEs using a suitable numerical quadrature. The method is validated by comparing the time evolution of the population distribution for a number of different cases for which analytical solutions exist, including non-gelling and gelling aggregating systems, with and without breakage. As few as 50-150 grid points are sufficient to describe accurately the dynamics of populations with internal coordinates covering up to six orders of magnitude.

References

[1] Ramkrishna D. and Singh M.R. Annu. Rev. Chem. Biomol. Eng. 2014, 5, 123-146.

[2] Marchisio D.L. et al. AIChE J. 2003, 49, 1266-1276.

[3] Costa L.I. and Trommsdorff U. Chem. Eng. Technol. 2016, 39, 2117-2125.

[4] Canu P. and Ray W.H. Comput. Chem. Eng. 1991, 15, 549-564.

[5] Kumar S. and Ramkrishna D. Chem. Eng. Sci. 1996, 51, 1311-1332.

[6] Meimaroglou D. and Kiparissides C. Ind. Eng. Chem. Res. 2014, 53, 8963-8979.

[7] Gunawan R. et al. AIChE J. 2004, 50, 2738-2749.