# square_hole_thermal

Family 11: heat flowing from a round hole to the outside of a rectangular block (a pipe in a square of insulation).

The conduction shape factor S* = q / (k D_len theta) of a long block. Hand anchor for a square section (H/W = 1): the
textbook fit S* = 2 pi / ln(1.08 W / d), good by about 1 % for a small hole. A tall block with a small hole approaches
the strip between two walls, S* = 2 pi / ln(4 W / (pi d)). Between them no simple formula holds, which is why the rest of
the space is worth having. A quarter of the section is solved, as a thin slice: the end faces and the two symmetry planes
carry no heat.

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `d_over_W` | 0.1 - 0.8 |
| `H_over_W` | 1.0 - 3.0 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `s_star` | 0.0014 % / 0.012 % / 0.012 % | max 0.00011 % | max - |

## Checks

- points: 150 solved, 0 failed
- three-mesh studies: 30 (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 2, re-solved on a finer mesh 4, unresolved 0
- direction checks: s_star vs d_over_W ok (rho 0.997), s_star vs H_over_W ok (rho -0.073)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `s_over_fit` | 0.8663 | 0.9595 | 0.9932 |
| `s_over_strip` | 1.02 | 1.166 | 1.385 |

## 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.