Transient Heat Conduction in Functionally Graded Hollow Cylinders and Spheres
Bibliographic record
Abstract
In the present work, transient heat conduction in functionally graded (FG) hollow cylinders and spheres is investigated based on the non-Fourier heat conduction theories. Since the heat transmission has been observed to propagate at a finite speed for applications with very low temperature, short-pulse thermal-heating, and micro temporal and spatial scales, dual phase lag (DPL) and hyperbolic heat conduction theories are considered in current study instead of the conventional Fourier heat conduction theory. Except the phase lags which are assumed to be constant, all the other material properties of the hollow cylinders and spheres are taken to change continuously along the radial direction according to a power-law formulation with different non-homogeneity indices. The heat conduction equations are written based on the dual phase lag theory which includes the hyperbolic heat conduction theory as well. These equations are applied for axisymmetric hollow cylinders of infinite lengths and spherically symmetric hollow spheres. Using the Laplace transform and Bessel functions, the analytical solutions for temperature and heat flux are obtained in the Laplace domain. The solutions are then converted into the time domain by employing the fast Laplace inversion technique. The exact expression is obtained for the speed of thermal wave in FG cylinders and spheres based on the DPL and hyperbolic heat conduction theories. Finally, the current results are verified with those reported in the literature based on the hyperbolic heat conduction theory.
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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 | 0.000 | 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 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".