Spectral solution of population balance equations using barycentric Lagrange polynomials: Application to stirred tank reactor hydrodynamics
Bibliographic record
Abstract
Abstract The population balance equation (PBE) is a fundamental tool for modelling the evolution of particle size distributions in dispersed multiphase systems, such as those involving droplets, bubbles, or solid particles. An accurate and efficient numerical solution of the PBE is essential for understanding and optimizing a wide range of processes in chemical and process engineering. In this study, a spectral method based on barycentric Lagrange polynomial interpolation is developed for solving the PBE. The method is applied to several representative cases, including pure growth, pure breakage, pure coalescence, and combined breakage–coalescence. In each case, numerical results are compared against analytical solutions and against the quadrature method of moments (QMOM) by evaluating the first four moments, demonstrating excellent agreement. The method is further validated using experimental data from a liquid–liquid extraction batch reactor, where simultaneous droplet breakage and coalescence occur. A computational cost study shows that while both the barycentric and standard formulations yield the same high level of accuracy, the barycentric formulation significantly reduces computational time, especially as the number of collocation points increases. These results underscore the effectiveness of the barycentric spectral method as a robust, accurate, and computationally efficient framework for modelling particulate processes in chemical and process engineering applications.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".