Exact closed-form unique and dual solutions to the longitudinal fin with temperature-dependent properties and internal heat generation
Bibliographic record
Abstract
In this study, heat transfer in a longitudinal rectangular fin with temperature-dependent thermal properties and internal heat generation is revisited. The advanced heat transfer models have been used to study the effects of thermo-geometric parameters, coefficient of heat transfer, and thermal conductivity parameters on the temperature distribution, heat transfer, and thermal performance of the longitudinal rectangular fin. It is shown that its governing nonlinear differential equation with proper boundary conditions is exactly solvable. With this aim, we reduce the order of the differential equation to first and then convert it into a total differential equation by multiplying by a convenient integrating factor. A full discussion and exact analytical solution in the implicit form is given for further physical interpretation and it is proved that three possible cases may occur: there is no solution to the problem, the solution is unique, or the solutions are dual, depending on the values of the parameters of the model.
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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".