This is the same in-space forging problem as the droplet simulator, but modeled a level deeper: instead of assuming the whole part is one uniform temperature (a "lumped" 0-D model), this solves the real 2D heat-conduction equation across a rod's circular cross-section, cell by cell, with radiative — and on Earth, convective — heat loss applied only at the exposed boundary. That lets the surface cool faster than the core, exactly as it does in a real casting, and produces a genuine moving solidification front creeping inward from the skin (a Stefan problem), instead of a single "solidified: yes/no" flag.
2D heat equation (finite-volume form), per cell:
ρ·C_p,eff(T)·∂T/∂t = k·∇²T (interior conduction)
boundary cells also lose:
q_loss = ε·σ·(T⁴ − T_env⁴) + h·(T − T_env) [h=0 in vacuum]
Latent heat via apparent heat capacity (Stefan problem):
C_p,eff(T) = C_p + L /(2·ΔT_melt) for |T − T_melt| < ΔT_melt
(releases exactly the alloy's latent heat of fusion L as each
cell crosses the melting window — this is what makes the
solidification front physically real instead of a hard cutoff)
- Orbital Vacuum / Earth Atmosphere — toggles the convective boundary term; vacuum keeps only T⁴ radiation at the rim.
- Emissivity ε — scales radiative loss at every exposed boundary cell directly.
- Rod radius — larger rods have a smaller perimeter-to-area ratio, so the core stays hot far longer after the surface has already solidified — the core/surface temperature gap you see growing is that effect.
- Watch the dashed ring: it marks the solidification front (where local temperature has dropped through the melting window). In vacuum it can advance unevenly and leave a large hot core sealed under a solid skin — a real defect risk (shrinkage porosity, uneven grain) that Earth's more uniform convective cooling avoids.
Real-world relevance: continuous casting and in-space wire-forming both need to know not just when a part is "done" but where the liquid core still is while a solid shell has already formed — get it wrong and the shell can rupture or trap voids as the interior finally shrinks.