Fourier heat conduction equation
The 1D transient heat equation ∂T/∂t = α ∂²T/∂x² governs temperature evolution in time and space. The thermal diffusivity α = k/(ρ·cp) determines how quickly heat penetrates the material.
- High α (copper, aluminium): heat spreads quickly, reaches steady state fast.
- Low α (concrete, wood, insulation): heat penetrates slowly — effective for thermal mass.
- Typical values: steel 14×10⁻⁶ m²/s, concrete 0.7×10⁻⁶ m²/s, air 19×10⁻⁶ m²/s.
Explicit finite difference method
Space is discretised into nx nodes separated by dx = L/(nx−1). Each time step advances using T_i^{n+1} = T_i^n + r(T_{i+1}^n − 2T_i^n + T_{i−1}^n) where r = α·dt/dx².
- Stability requires r ≤ 0.5 — the simulator enforces this by limiting the time step.
- Boundary nodes are fixed to T_left and T_right (Dirichlet conditions).
- The simulation runs to t_end seconds and returns the final temperature profile across the slab.
Engineering applications
This model applies to insulation design, heat treatment of metals, building envelope energy calculations, and fire resistance analysis of structural elements.
- Insulation design: verify no surface temperature exceeds dew point.
- Steel heat treatment: ensure core reaches austenitising temperature uniformly.
- Fire resistance: check time for unexposed face to reach 140°C above ambient (ISO 834).