Solution of the dynamic population balance equation describing breakage–coalescence systems in agitated vessels: The least‐squares method
Bibliographic record
Abstract
A variety of processes used across, for example the cosmetics, pharmaceutical and chemical industries involve two‐phase liquid–liquid interactions. The quality of liquid–liquid emulsification systems may be importantly related to the droplet size distribution. The population balance equation (PBE) can be used to describe complex processes where the accurate prediction of the dispersed phase plays a major role for the overall behaviour of the system. In recent years, the high‐order least‐squares method has been applied to approximate the solution to population balance (PB) problems. From the chemical engineering point of view, the least‐squares method is associated with complex algebra. Moreover, in previous chemical engineering publications the method has been outlined using rather compact mathematical notations. For this reason, in this study, details of the least‐squares algebra and implementation issues are revealed. The solution strategy is illustratively applied to a test problem: a liquid–liquid emulsification system with breakage and coalescence events in a stirred tank.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".