Dimensionless heat transfer check for any geometry. Flexible units, optional properties, clear result guidance included. Use it to size experiments, models, and designs fast.
Fourier number compares diffusion time inside a body to the time available. It is widely used in transient conduction checks and similarity analysis.
The Fourier number is defined as: Fo = α t / L²
If you compute diffusivity from properties, this calculator uses: α = k / (ρ cp)
These sample cases illustrate typical ranges and how the dimensionless value changes with time and size.
| Material | α (m²/s) | t | L | Fo | Comment |
|---|---|---|---|---|---|
| Aluminum (typical) | 9.7×10-5 | 60 s | 10 mm | 0.582 | Strong internal gradients likely. |
| Stainless steel (typical) | 4.0×10-6 | 300 s | 20 mm | 0.030 | Early transient development. |
| Glass (typical) | 6.0×10-7 | 1 hr | 5 mm | 86.4 | Later-time response for thin parts. |
| Polymer (typical) | 1.2×10-7 | 2 hr | 25 mm | 1.38 | Moderate-to-late transient range. |
The Fourier number (Fo) is a dimensionless time for transient conduction. It compares diffusion inside a solid to its characteristic size using Fo = αt/L². Small values indicate surface-limited change, while larger values indicate deeper penetration and a more developed temperature field.
Heat diffusion distance scales with √(αt). Because √Fo = √(αt)/L, Fo links “how far heat travels” to “how big the part is.” If Fo = 0.25, the diffusion distance is roughly half of L, which is useful for rapid engineering estimates.
Thermal diffusivity spans orders of magnitude: many polymers are about 0.05–0.2×10-6 m²/s, glass about 0.5–1.0×10-6 m²/s, stainless steel about 3–5×10-6 m²/s, and aluminum alloys about 8–10×10-5 m²/s. Water near room temperature is about 1.4×10-7 m²/s.
L should represent the main conduction path. Common choices are half-thickness for a plane wall, radius for a cylinder, and radius for a sphere. For irregular shapes, an effective scale like V/A can be practical. Use the same definition consistently across comparisons.
Rearrange to estimate time: t = Fo·L²/α. With α = 4×10-6 m²/s and L = 0.02 m, Fo = 0.1 corresponds to about 10 s, Fo = 1 to about 100 s, and Fo = 5 to about 500 s. For thin parts, L is small, so the same Fo is reached quickly.
Fo tracks internal diffusion, but convection at the surface can dominate. The Biot number (Bi = hL/k) indicates whether internal gradients are small, where h is a convection coefficient and k is conductivity. A common guideline is Bi < 0.1 for lumped-capacitance use; otherwise, spatial temperature profiles matter even at later Fo.
Fo appears in transient solutions, charts, and similarity analysis. Engineers use it to compare tests across materials, judge early transient behavior (Fo < 0.1), and anticipate smoother fields at later times (Fo > 1) for many simple geometries. It is also used with Heisler charts and finite-difference models.
Most mistakes come from unit mix-ups and inconsistent L. Convert time to seconds and length to meters before computing. If you compute α from properties, use k in W/(m·K), ρ in kg/m³, and cp in J/(kg·K). Fo must be nonnegative. If α changes with temperature, use a representative average.
No. Fourier number is a heat-transfer similarity parameter for transient conduction. Fourier series are mathematical expansions of functions. They share a name but describe different concepts.
It often indicates early transient behavior, where temperature changes are mainly near the surface and the interior has not responded much. It is a useful screening threshold, not a strict rule.
Use the main conduction path: half-thickness for a wall, radius for a cylinder or sphere, or an effective scale like V/A for complex parts. Keep the same definition when comparing cases.
Yes. Choose the material-property option. The calculator uses α = k/(ρcp) and then computes Fo using your selected time and length inputs.
Large Fo can occur for very small L, long times, or high diffusivity. Also check units, especially mm versus m. A length entered as 10 mm but treated as 10 m will change Fo by 106.
No. Surface resistance matters. If Bi is large, internal gradients can persist even at later times. Use Fo together with Bi and the correct boundary condition for reliable conclusions.
SI units are standard: α in m²/s, t in seconds, and L in meters. This calculator converts common units, but consistent SI reporting makes comparisons and documentation clearer.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.