Modified orthogonal collocation for accurate flux-based material balance calculations in slab, cylindrical, and spherical geometries
Bibliographic record
Abstract
This study addresses a challenge in the application of the weighted Orthogonal Collocation method for solving flux-based material balances in diffusion and reaction systems where Fick’s law is not applicable. The conventional approach encounters difficulties due to the non-zero gradient boundary condition of flux at x = 0, which leads to increased errors and inaccuracies in the solution. To resolve this problem, a modification to the Orthogonal Collocation method is proposed, adjusted specifically to handle the complexities of flux-based material balances. The modified method adapts the traditional collocation approach, ensuring it can accommodate the boundary condition peculiarities inherent in these systems. Testing and comparison demonstrated that the modified method achieves accuracies comparable to the original Orthogonal Collocation method when applied to concentration-based material balances, whereas incorrect use of the original scheme leads to errors that are orders of magnitude greater. • Incorrect flux boundary condition in orthogonal collocation leads to severe errors. • Collocation matrices are recalculated to correct the flux boundary condition error. • Correction of boundary condition restores the accuracy of the collocation method. • The correction was tested successfully with diffusion and reaction problems.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".