# fin_thermal

Family 10: a straight rectangular fin, base held hot, every other face cooled by convection.

Exact anchor: the one-dimensional fin with a convecting tip, q = sqrt(h P k A) theta_b (sinh mL + (h/mk) cosh mL) /
(cosh mL + (h/mk) sinh mL), m^2 = h P / (k A). It holds while the fin is thin to the heat (small Biot number h t / 2k);
a stubby or strongly cooled fin falls below it, which is what the data shows. Answers: the heat flow over the 1D answer,
and the fin efficiency q / (h A_surface theta_b). Both depend only on the shape and on mL.

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `mL` | 0.2 - 3.0 |
| `t_over_L` | 0.05 - 0.25 |
| `w_over_t` | 1.0 - 5.0 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `q_star` | 0.0099 % / 0.056 % / 0.066 % | max 4.4e-06 % | max - |
| `eta` | 0.0099 % / 0.056 % / 0.066 % | max 4.4e-06 % | max - |

## Checks

- points: 240 solved, 0 failed
- three-mesh studies: 48 (every 5th point); the band of each point is 1.25 x the largest standard-mesh error among its 3 nearest studies
- symmetry check (max): 0 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: eta vs mL ok (rho -0.999)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `biot` | 0.007227 | 0.05385 | 0.09472 |
| `eta_over_1d` | 0.998 | 0.9999 | 1 |

## Columns of points.csv

parameters, the answers, `<answer>_band` (relative), the hand-theory ratios, `mirror_err`, `equilibrium_err` (where a single reaction exists), `<answer>_loo_err` / `_neighbour_flag` (leave-one-out prediction from all other points), `<answer>_recheck_diff` / `_flag_resolved` (finer re-solve of a flagged point), `has_gci` / `has_similarity` / `has_calculix`, `nodes`, `solve_s`, `studio` (version).

Solved with NovaSolver Studio through the same commands an AI client uses over MCP; the first section says which solver and mesh. The claim is verification (the equations are solved within the band), not validation against tests.

## License

This data is licensed under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/
Credit: NovaSolver, https://novasolver.jp/studio/verified/
Provided as is, without warranty. Design decisions remain the user's responsibility.