New finite volume approach for multidimensional Smoluchowski equation on nonuniform grids

Abstract

This present work is based on developing a deterministic discrete formulation for the approximation of a multidimensional Smoluchowski (Coalescence) equation on a nonuniform grid. The mathematical formulation of the proposed method is simpler, easy to implement, and focuses on conserving the first‐order moment. The new scheme resolved the issue of mass conservation along individual components in contrast to the existing scheme which focuses only on conservation of the total mass of all components. The validation of the new scheme is conducted against the existing scheme by considering some classical tests. The comparison reveals that the new scheme has the ability to compute the higher‐order moments with higher accuracy than the existing scheme on a coarse grid without taking any specific measures. For the higher‐dimensional population balance equations, the mixing of components quantified using χ2 parameter is also computed accurately and efficiently using a very coarse nonuniform grid.

Investigating Chemical Heterogeneity in Inclusion Populations: A Multivariate Population Balance Model Study in Gas-Stirred Ladles

“…Several techniques are available in the literature to solve the PBEs numerically. Some popular methods are sectional methods, − method of moments, − semianalytical homotopy methods, and the Monte Carlo method. , A detailed review of these solution methods is available in ref . The Monte Carlo method is a probabilistic method that generates random numbers to obtain statistical approximations of the entities under consideration.…”

Challenges and opportunities concerning numerical solutions for population balances: a critical review

“…It was demonstrated that the CAT is highly accurate numerical approach to solve problems related to multidimensional PBEs, however, computationally expensive due to its notion of distributing particles properties to the neighbouring nodes. In contrast to the sectional methods, the FVSs [117,118,127,141] are highly accurate and efficient as no distribution of particles properties to the neighbouring nodes is required, that is, weights are added to the formulations for conserving the integral properties. It was shown that the FVSs are easy to extend for approximating problems in higher dimensions due to their simpler mathematical formulation and are robust to apply on any kind of uniform and non-uniform grids.…”

Biomineral Flocculation of Kaolinite and Microalgae: Laboratory Experiments and Stochastic Modeling

“…The second analytical solution used in this study is based on a size-dependent additive kernel 𝐴𝐴 𝐴𝐴𝑖𝑖𝑖𝑖 = 𝑉𝑉𝑥𝑥𝑥𝑖𝑖 + 𝑉𝑉𝑦𝑦𝑥𝑖𝑖 + 𝑉𝑉𝑥𝑥𝑥𝑖𝑖 + 𝑉𝑉𝑦𝑦𝑥𝑖𝑖 (Fernandez-Diaz & Gomez-Garcia, 2007;Singh, 2021): (A4)…”

“…where 𝐴𝐴 𝐴𝐴2 = 1 − exp(𝑀𝑀𝑀𝑀) is the dimensionless time and M is the total mass of the particles in the system. In this case, we take N 0 = 1 and 𝐴𝐴 𝐴𝐴𝑥𝑥0 = 𝐴𝐴𝑦𝑦0 = 10 −3 , which are 10 times greater than the values in Singh (2021) to rescale the time close to the first case. It is shown that there is good agreement between the predicted median volumes, the FSDs of V x and that the FSDs along the diagonal with the corresponding theoretical results (Figure A3) at t = 0, 1, 10, 50, and 60 s. Note that at t = 25 s, the modeled FSD along the diagonal is overestimated in larger size classes compared to the analytical FSD, whereas the results at other moments are in good agreement.…”