The 3D tunnel treats each item as a single lumped temperature — valid only when internal conduction is fast compared to surface heat loss. This 2D companion drops that assumption and solves real 1D transient heat conduction through the product's own half-thickness, from its center (symmetry plane, no heat flux) to its surface (convective boundary), using a conservative finite-volume scheme:
∂(ρ·c(T)·T)/∂t = ∂/∂x( k(T)·∂T/∂x )
Surface: −k·∂T/∂x|_surface = h·(T_surface − T_air)
Center: ∂T/∂x|_center = 0 (symmetry)
Latent heat is carried by the apparent heat capacity method: c(T) equals the normal specific heat away from the freezing point, but is inflated by L/ΔT inside a narrow band ΔT straddling Tf, so the same energy balance that drives sensible cooling also consumes the latent heat of fusion — no separate front-tracking logic needed. Conductivity k(T) blends from the unfrozen value to the (higher) frozen value across that same band, since ice conducts heat roughly 3× faster than unfrozen tissue. Every face between grid cells uses a single shared harmonic-mean conductivity, which is what makes the scheme exactly energy-conservative even where properties jump sharply at the moving front.
The result: a real internal temperature profile and a genuine freezing front that starts at the surface and works inward — the center can still be warm for a long time after the surface is fully frozen, something no lumped model can show. The engineering cross-check alongside it is Plank's equation (Plank, 1913), the classical food-industry estimate for phase-change freezing time:
t_Plank = ρ·L / (T_f − T_air) · ( P·a/h + R·a²/k_frozen )
P = 1/2, R = 1/8 (infinite-slab shape factors), a = half-thickness
Plank's formula ignores sensible heat and assumes a fixed temperature front, so it is known to differ from a full numerical solution by roughly ±30–100% depending on Biot number — this simulator's own measured phase-change duration (surface-to-center) is displayed next to it so you can see that real-world engineering rule of thumb in action, not just take it on faith.
- Biot number Bi = h·a/k compares surface resistance to internal conduction resistance. Bi ≪ 0.1 means the 3D tunnel's lumped assumption is a good approximation (e.g. thin peas); Bi > 0.5 means internal gradients dominate and only a model like this one captures the real behaviour (e.g. a thick chicken cut).
- Cut thickness scales the half-thickness directly — doubling it roughly quadruples the diffusion-limited freezing time (the R·a² term), exactly the geometry sensitivity real cold-chain engineers design around.