Heat Equation (Heat Diffusion) formula
| Symbol | Quantity | SI unit | Meaning |
|---|---|---|---|
| α | Thermal diffusivity | m²/s | Thermal diffusivity — how fast a temperature change spreads |
| k | Thermal conductivity | W/(m·K) | Thermal conductivity |
| ρ | Density | kg/m³ | Density |
| cₚ | Specific heat capacity | J/(kg·K) | Specific heat capacity at constant pressure |
What is Heat Equation (Heat Diffusion)?
Heat Equation (Heat Diffusion): Temperature diffuses just like concentration: ∂T/∂t = α∇²T. The thermal diffusivity α plays the role of Fick's D. — Joseph Fourier, 1822
Heat conduction obeys the same mathematics as mass diffusion. Fourier's heat equation is identical in form to Fick's second law, with thermal diffusivity α taking the place of the diffusion coefficient. α has the same units as D (m²/s), and the diffusion-length rule L ≈ √(αt) tells you how far a temperature change penetrates in time t.
High-α materials (copper, aluminium, diamond) equalise temperature quickly. Low-α materials (wood, polymers, insulation) respond slowly, which is why a wooden spoon stays cool in hot soup.
How to use this calculator
- Choose the quantity you want to find in the Calculate menu. It is highlighted as the result field.
- Enter the known values. Scientific notation such as
1.6e-11is accepted. - Select a unit for each value. Units are converted to SI automatically.
- The answer updates as you type. Open Show calculation steps to see the working, or use Copy result to share a link with your inputs.
Worked example
Copper: k = 401 W/(m·K), ρ = 8960 kg/m³, cₚ = 385 J/(kg·K).
- ρcₚ = 8960 × 385 = 3.45 × 10⁶ J/(m³·K).
- α = 401 / 3.45 × 10⁶ = 1.16 × 10⁻⁴ m²/s = 116 mm²/s.
For comparison, water is about 0.143 mm²/s and air about 22 mm²/s.
Frequently asked questions
What is the difference between thermal conductivity and thermal diffusivity?
Conductivity k measures how much heat flows under a steady temperature gradient. Diffusivity α = k/(ρcₚ) measures how quickly temperature changes during transient heating, because it also accounts for how much heat the material stores.
How long does heat take to penetrate a slab?
A rough estimate is t ≈ L²/α. For a 1 cm steak (α ≈ 0.14 mm²/s) heated from one side, t ≈ (10 mm)² / 0.14 mm²/s ≈ 700 s, or about 12 minutes.