Physics · Simulator 013
Hold one end of a rod in a flame and the other in ice. Choose the material, the end temperatures and the starting temperature, then watch heat flow along the rod until the temperature profile settles into a straight line.
In a thin rod with insulated sides, temperature T(x, t) follows the one-dimensional heat equation:
∂T/∂t = α · ∂²T/∂x², α = k / (ρ c)
The thermal diffusivity α (m²/s) combines conductivity k, density ρ and specific heat c. Heat flows from hot to cold, and the curvature of the profile drives change. With fixed end temperatures, the steady state is a straight line between them.
The time to approach equilibrium scales as L²/α, so a rod twice as long takes four times as long. The slowest-decaying mode has time constant τ = L² / (π² α).
Numerical method: explicit finite differences with 60 segments. The time step stays below the stability limit Δt ≤ Δx²/(2α).
Choose insulated for the right end to make it a no-flux boundary. The rod then warms up to the left-end temperature everywhere.
Idealised: insulated sides, constant properties, perfect contact at the ends. Material values are typical room-temperature figures.
A 20 cm aluminium rod (α ≈ 9.7×10⁻⁵ m²/s) starts at 20 °C, and both ends are suddenly held at 100 °C. Roughly how long until the centre is within 1 °C of 100 °C?
The slowest mode decays as e^(−t/τ), with τ = L²/(π²α) ≈ 0.04/(9.87 × 9.7×10⁻⁵) ≈ 42 s. The centre starts 80 °C away, with a first-mode amplitude of about (4/π) × 80 ≈ 102 °C, so it needs t ≈ τ ln(102) ≈ 4.6τ ≈ 195 s, about 3¼ minutes. Set both ends to 100 °C using the “both hot” preset in the controls and check.