Largest Lyapunov exponent and Shannon entropy: Two indices to analyze mixing in fluidized beds
Bibliographic record
Abstract
The quality of mixing in fluidized beds greatly influences performance in many applications. Assessing quality of mixing involves measuring the mixing rate and evaluating the bed mixedness. Quantifying the bed mixedness is typically done using mixing indices. However, the application of existing mixing indices to fluidized beds can be problematic due to aeration and complications from the particle phase in the Two-Fluid Model (TFM). The objectives of this study are twofold. First, the largest Lyapunov exponent is proposed to quantify mixing in fluidized beds. Its effectiveness is shown on a mono-disperse bed with varying gas velocities. The increase in mixing rate with higher gas velocity is accurately represented by the largest Lyapunov exponent. Second, the Shannon entropy mixing index is adopted to quantify the bed mixedness for TFM results. This index is tested on a bi-disperse bed to predict segregation and evaluate the effect of bed composition and superficial gas velocity on this process. The effect on segregation is reflected in the entropy components: distributional entropy showed minimal variation, whereas conditional entropy was significantly affected. Evaluating bed mixedness at different length scales showed that increasing bin spatial resolution slightly reduced conditional entropy. The results are validated against experimental data. • The largest Lyapunov exponent quantifies chaos and mixing in a fluidized bed. • The Shannon entropy mixing index effectively measures bed mixedness. • It characterizes overall bed distribution and change in bed composition. • Conditional entropy is key for tracking segregation. • The length scale of assessment impacts the observed mixing level in the bed.
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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".