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How It Works
The Fourier number indicates how far heat has penetrated into a body during a transient process. It compares the rate of heat conduction through the material to its capacity to store thermal energy. A large Fo means the body has nearly reached thermal equilibrium; a small Fo means temperatures are still changing rapidly.
It appears in solutions to the heat equation and is essential for deciding whether a lumped-capacitance simplification is valid or whether you must account for internal temperature gradients.
Example Problem
A steel rod (thermal diffusivity α = 1.2 × 10⁻⁵ m²/s) with a 0.05 m radius is heated for 120 seconds. What is the Fourier number?
- Fo = αt / L² = (1.2 × 10⁻⁵ × 120) / 0.05²
- Fo = 0.00144 / 0.0025 = 0.576
With Fo ≈ 0.58, heat has penetrated a meaningful fraction of the rod's radius but the interior has not yet reached the surface temperature.
Frequently Asked Questions
What does a Fourier number greater than 0.2 mean?
In many transient conduction solutions (Heisler charts, one-term series approximations), Fo > 0.2 means the first-term approximation of the infinite series is accurate to within about 2%. Below 0.2, additional terms are needed for an accurate result.
How is the Fourier number used in food processing?
Food engineers use it to estimate cooking, sterilization, and cooling times. For example, determining how long to hold a canned food at 121 °C so the center reaches a safe temperature requires calculating Fo from the can's dimensions and the food's thermal diffusivity.
What is thermal diffusivity and how does it differ from conductivity?
Thermal diffusivity (α = k / ρcₚ) measures how quickly temperature changes propagate through a material. Thermal conductivity (k) measures steady-state heat flow. A material can have high conductivity but low diffusivity if it also has a large heat capacity. Copper has α ≈ 1.1 × 10⁻⁴ m²/s while water is only about 1.4 × 10⁻⁷ m²/s.
Related Calculators
- Nusselt Number Calculator — ratio of convective to conductive heat transfer at a surface.
- Peclet Number Calculator — advective vs. diffusive heat transport in a flow.
- Prandtl Number Calculator — momentum diffusivity relative to thermal diffusivity.
- Biot Number Calculator — ratio of internal to surface thermal resistance, often paired with Fourier number.
- Thermal Diffusivity Calculator — compute the diffusivity value used in the Fourier number formula.
- Thermal Conductivity Calculator — find the conductivity needed for diffusivity and Biot calculations.