Coal left in a stockpile slowly oxidizes in the presence of air, and that oxidation releases heat — a real, well-documented industrial hazard (spontaneous combustion of stored coal). The rate of heat release follows the Arrhenius law:
q_gen(T) = A · exp(−Eₐ / R·T) [W/m³]
while heat simultaneously spreads through the pile and escapes at its surface, governed by the 2D heat-diffusion equation on the pile's triangular cross-section:
∂T/∂t = α·∇²T + q_gen(T)/(ρc) − loss at exposed faces
Both terms are solved together, cell by cell, on a grid shaped like the pile's actual cross-section (angle of repose ≈ 35°). Cells fully buried in coal only diffuse heat to their neighbours — coal is a poor conductor (k ≈ 0.25 W/m·K), so a large core is thermally insulated from the surface. Cells on the pile's skin additionally lose heat by convection to the ambient air at rate h·(T−T_amb).
This sets up a genuine positive-feedback instability: because q_gen(T) grows exponentially with T while conductive/convective loss only grows linearly, there is a critical point (the Frank–Kamenetskii thermal-explosion criterion) beyond which generation permanently outpaces loss — the core temperature accelerates without bound toward the pile's smoldering-ignition threshold (≈180 °C, marked on the history graph) instead of settling at a steady elevated temperature. Below that point the same equations settle into a stable, slightly-warmed steady state.
- Bigger pile → lower surface-to-volume ratio → core stays insulated → higher risk.
- Hotter ambient → the whole Arrhenius curve starts from a higher baseline rate → higher risk.
- Lower Eₐ (more reactive coal, e.g. lignite) → exponentially faster reaction at the same temperature → higher risk.
- Better ventilation (higher h) → faster convective heat removal at the surface → lower risk. (Real piles also gain oxygen from airflow, which this model simplifies away — see the article.)