07 · Cooking Time Is a Biot Problem — tested
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
report.md
# 07 · Cooking Time Is a Biot Problem — tested **Source:** Incoprea Theory, Volume XXI, Theory 11 (cooking-time-biot). **Claim (verbatim):** "Cooking time for a solid food is set by its Biot number, so two pieces of equal mass cook at different rates depending on shape and surface transfer." **Prediction:** "Cooking time should collapse onto Biot and Fourier numbers." **Refutation:** "Time tracking mass would remove shape." ## Why probable Transient heat conduction is exact textbook physics; the food-engineering literature uses Bi/Fo scalings routinely. ## Method Exact series solutions of 1-D transient conduction for slab and sphere (code/biot_test.py): centre temperature as a function of Fourier number Fo = αt/L² at Biot numbers Bi = hL/k = 0.1–100, for the standard case of a piece at 5 °C in boiling water (target centre 70 °C), plus an equal-mass shape comparison. ## Results Fo needed for the centre to reach 70 °C: | Bi | Fo slab | Fo sphere | |---|---|---| | 0.1 | 12.08 | 4.02 | | 1 | 1.71 | 0.57 | | 10 | 0.68 | 0.23 | | 100 | 0.58 | 0.19 | - **Collapse:** at fixed Bi the process is a single curve of centre temperature vs Fo, for any size — time scales as L²/α (Fourier), with the Biot number fixing the family of curves. Doubling thickness (and thus mass, for fixed area) multiplies cooking time by ~4, not 2 — "time tracks mass" fails. - **Equal mass, different shape:** for V = 1000 cm³ (slab half-thickness 5.0 cm vs sphere R 6.2 cm) at Bi = 10, t_slab/t_sphere = (0.68/0.23)·(5.0/6.2)² ≈ **2.0** — two pieces of identical mass and identical heat-transfer conditions cook to the centre at rates differing by a factor ~2. - **Surface transfer matters:** at Bi = 0.1 (poor surface transfer) the required Fo is 12.1 vs 0.58 at Bi = 100 — a 20× difference from the surface term alone. ## Verdict **SUPPORTED.** Cooking time collapses onto Biot/Fourier numbers; equal-mass pieces cook at shape-dependent rates, and time clearly does not track mass. ## Limitations - Constant properties, no phase change/evaporation or internal moisture transport; real foods deviate modestly. - Slab/sphere idealisations of real food shapes. Working report — not peer reviewed. Educational information, not medical advice. AI-assisted test: DeepSeek.