FCMOM (Finite size domain Complete set of trial functions Method Of Moments) is a variation of the method of moments to solve PBE (population balance equations). FCMOM was presented recently [1] and afterwards validated for mono-variate PBE. In FCMOM, the solution of the PBE is sought, instead of the [0,∞] range, in the finite interval between the minimum and maximum particle size; the evolution of the minimum and maximum particle size is tracked imposing moving boundaries conditions. The main advantage of FCMOM is that it provides the solution of the PBE both in terms of the moments and of the reconstructed particle size distribution. In all the applications developed for mono-variate PBE (constant, linear, diffusion-controlled particle growth; particle dissolution; particle aggregation with constant, sum and product kernels), the solution obtained with FCMOM is in excellent agreement with the analytical solution, while the computational effort is low (comparable with QMOM).

In this work, FCMOM is applied to bi-variate PBE and is validated by comparison with analytical solutions of particle growth (constant, linear, diffusion-controlled), particle dissolution and particle aggregation (constant kernel).

In the bi-variate case, FCMOM performs as well as in the mono-variate case. In fact, the algorithm is still simple and efficient, and the computational time is low (typically, twice the computational time required for the corresponding problem in the mono-variate case). Moreover, the particle bi-variate (volume-surface area) distributions obtained by FCMOM are again in very good agreement with the analytical solutions.

Finally, applications to bi-variate particle aggregation with kernels derived by the kinetic theory of granular flows and to simultaneous particle aggregation and coalescence are discussed.

[1] Strumendo M., Arastoopour H., A new approach in solving PBE, Proceedings of the Fifth World Congress on Particle Technology, Orlando, Florida (USA), April 23-27, 2006.