A new framework for numerical modeling of population balance equations: Solving for the inverse cumulative distribution function

The generalized quadrature method of moments

“…For Case 1, we can see in fig. 12 that using QMOM, the results converge when the number of moments increases to the value given in Peterson et al (2022). With gamma-GQMOM, only one additional quadrature point is used compared to QMOM.…”

“…14, we compared the mean value and variance obtained with QMOM, gamma-GQMOM and lognormal-GQMOM, using a large enough number of moments (16) so that the solution is converged. The reference solution is taken from Peterson et al (2022). Gamma-GQMOM and QMOM converge to the same solution, whereas lognormal-GQMOM converges to another one, closer to, but still different, from the reference solution.…”

“…We consider next the apparently very simple test case of symmetric binary breakup. Depending on the initial conditions, it has been shown that this test case can lead to some difficulties for QMOM, such as convergence to an incorrect solution (Peterson et al, 2022). Thus, let us consider the following PBE for the NDF f (t, v):…”

“…As in Peterson et al (2022), we look at the mean M 2 /M 1 and variance M 3 M 1 /M 2 2 − 1 of the volume distribution. For Case 1, we can see in fig.…”

“…Case 2 was shown in Peterson et al (2022) to be difficult for QMOM, i.e., the method converges to the wrong solution. In fig.…”