Comparison between the CQMOM and Monte Carlo

However, this set of equations does not result in the correct moment evolution, since only mo and m will be correctly predicted, whereas all the other moments (i.e. mt with k 2) will inevitably be wrong. 

For this specific example, consider the case of constant aggregation and breakage kernels, namelyPa,y,s, r = j o, cia,y = o, and a beta daughter distribution function 

Let us now investigate the evolution of the initial distribution undergoing aggregation and breakage with the CQMOM and with the MC method. The MC simulation was carried out using the constant-number MC method reported in Section 7.5 with 10000 notional 

A separate issue is that of the realizability of the quadrature approximation. For example, since for the case under investigation both and 2 (representing the size and the chemical composition of the particles) are positive, it must be ensured that a realizable quadrature is employed to sample the phase space. A realizable quadrature in this case has positive nodes that result in sensible values. For this specific problem, it turns out that the portion of phase space explored is fully realizable only if some mixed moments, such as m2,i, are included in the tracked moment set. This means that the option N = 2 and AI2 = 1 that does not include m2,i can potentially lead to problems (for example when mass transfer and chemical reactions are involved), whereas the option N = 3 and AI2 = 1 is a better choice. 

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