Efficient solution of population balance equations with discontinuities by finite elements

Two refinements of Galerkin's method on finite elements were evaluated for the solution of population balance equations for precipitation systems. The traditional drawbacks of this approach have been the time required for computation of the two-dimensional integrals arising from the aggregation integrals and the difficulty in handling discontinuities that often arise in simulations of seeded reactors. The careful arrangement of invariant integrals for separable aggregation models allows for a thousandfold reduction in the computational costs. Discontinuities that may be present due to the hyperbolic nature of the system may be specifically tracked by the method of characteristics. These discontinuities will arise only from the initial distribution or nucleation and are readily identified. A combination of these techniques can be used that is intermediate in computational cost while still allowing discontinuous number densities. In a case study of calcium carbonate precipitation, it is found that the accuracy improvement gained by tracking the slope discontinuity may not be significant and that the computation speed may be sufficient for dynamic online optimization.

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