Transient Heat Conduction in a Planar Slab With Convection and Radiation Effects
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Bibliographic record
Abstract
Abstract This article presents new semi-analytical solutions for transient heat conduction in a slab, which exchanges heat at the surface with its surroundings via convection, radiation, or simultaneous convection–radiation mechanisms. The concept of thermal penetration depth together with the integral method of heat balances form the basis of the formulation. Explicit expressions are derived that eliminate the need for numerical integration of the ordinary differential equations (ODEs) obtained from space integration of the heat equation. Verification of the temperature relations with a convection boundary condition is performed using an exact solution. In the case of radiation and combined convection–radiation boundary conditions, the accuracy of the explicit solutions is assessed against a numerical model. In all three cases, the new relations predict the surface temperature with high precision. For the radiation boundary condition, the computed midpoint temperature is also in excellent agreement with the numerical solution. In the case of convection heat exchange at the surface, the temperature at internal locations has close agreement with the exact solution for Biot numbers up to Bi = 1. The model prediction deviates from the exact solution in the interior positions for higher values of Bi with the highest deviation occurring at the center. For example, at Bi = 3, the highest relative error is 7.5%. Likewise, for the case of combined convection–radiation boundary conditions, the predicted midpoint temperature is found to closely follow the numerical solution at early stages of the process only. The predictability of the explicit solution of the convection-only boundary condition is more accurate than other approximate solutions at a Fourier number less than 0.2. Additional relations are presented for determination of the total heat transfer as a function of the Fourier and Biot numbers.
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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 it