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Fourier Number

Also known as Fo · dimensionless time

Fo=k tρ c L2\mathrm{Fo} = \frac{k \, t}{\rho \, c \, L^{2}}

Worked example: Steel, 50 mm, 600 s → Fo 2.866 (alpha = 1.19e-5 m2/s) — press Try an example to run it live, then adjust anything.

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Fourier Number explained

kρcptLFo

The Fourier number is dimensionless time: how far a thermal disturbance has diffused compared with the size of the object. It is normally written Fo = αt/L² with the thermal diffusivity α = k/(ρc), but diffusivity has no entry in this calculator's unit picker, so the group is spelled out as kt/(ρcL²) — identical physics, and it has the pleasant side effect of showing where the diffusivity comes from. High-k, low-ρc materials diffuse heat fast: copper's α is 1.1 × 10⁻⁴ m²/s, steel's 1.2 × 10⁻⁵, brick's 5 × 10⁻⁷, and that thousand-fold spread is why a copper pan responds instantly and a masonry wall takes half a day.

Fo ≈ 1 is the rough marker for "the disturbance has crossed the body". Below Fo = 0.2 the one-term approximations in the textbook charts are not valid and you need the full series; above about 1 the transient is essentially over. Worked example: a 100 mm steel plate (L = 0.05 m half-thickness, k = 45, ρ = 7850, c = 480) after 10 minutes has Fo = 45 × 600/(7850 × 480 × 0.0025) = 2.87, thoroughly soaked through. The same plate in firebrick would need most of a day. This is the number behind cooking times, heat-treat soak schedules and the thermal-mass lag that lets a stone building coast through an afternoon.

Fourier Number formula

Fo=k tρ c L2\mathrm{Fo} = \frac{k \, t}{\rho \, c \, L^{2}}
Where
  • Fo\mathrm{Fo}= Fourier number
  • kk= Thermal conductivity (W/(m·K))
  • tt= Elapsed time (s)
  • ρ\rho= Density (kg/m³)
  • cpc_p= Specific heat (J/(kg·K))
  • LL= Characteristic length (m)

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