Fourier Number Calculator

What is Fourier Number?

The Fourier number (Fo) is a dimensionless quantity used in heat transfer analysis to characterize transient heat conduction. It represents the ratio of heat conducted through a body to the heat stored within it. Named after the French mathematician and physicist Jean-Baptiste Joseph Fourier, this number quantifies how deeply heat has penetrated into a body over a given time period relative to its size.

The Fourier number appears in all solutions to the transient heat conduction equation. It helps engineers determine whether a lumped-capacitance simplification is valid or whether internal temperature gradients must be accounted for. A large Fourier number (greater than 0.2) indicates that heat has diffused deeply into the body and the one-term series approximation for temperature distribution is accurate.

Fourier Number Formula

The Fourier number is calculated using the following formula:

Fo = αt / L²

Where:

• Fo — Fourier number (dimensionless)
• α — Thermal diffusivity (m²/s)
• t — Time (s)
• L — Characteristic length (m)

How to Use the Fourier Number Calculator

Select the variable you want to solve for from the dropdown menu. Enter the known values in the appropriate fields and the calculator will compute the result instantly. You can solve for:

• Fourier Number (Fo) — When you know thermal diffusivity, time, and characteristic length
• Thermal Diffusivity (α) — When you know Fo, time, and characteristic length
• Time (t) — When you know Fo, thermal diffusivity, and characteristic length
• Characteristic Length (L) — When you know Fo, thermal diffusivity, and time

The characteristic length is typically the body's volume divided by its surface area (V/A). For a sphere this equals r/3, for a long cylinder r/2, and for a flat plate half the thickness. Multiple unit systems are supported for each variable.

Interpreting the Fourier Number

The value of the Fourier number tells you how deeply heat has penetrated into a body:

• Fo > 0.2 — The temperature profile has nearly equilibrated. One-term series approximations are accurate to within about 2%.
• 0.05 < Fo ≤ 0.2 — Moderate heat penetration. One-term series may have limited accuracy; additional terms may be needed for precise results.
• Fo ≤ 0.05 — Steep internal temperature gradients persist. The thermal wave has only penetrated the outer layers. Full series solutions are required.

Example Calculation

Example: A steel rod with thermal diffusivity α = 1.2 × 10−&sup5; m²/s and characteristic length L = 0.05 m is heated for 120 seconds. What is the Fourier number?

Fo = αt / L² = (1.2 × 10−&sup5; × 120) / (0.05)² = 0.00144 / 0.0025 = 0.576

Since Fo = 0.576 > 0.2, heat has penetrated a meaningful fraction of the rod's radius but the interior has not yet reached the surface temperature. The one-term series approximation is accurate for predicting temperature distribution.

Applications of Fourier Number

• Food Processing — Determining sterilization, pasteurization, and cooking times for canned and packaged foods
• Metallurgy — Predicting quench times and internal temperature gradients during heat treatment of steel parts
• Building Science — Estimating how quickly temperature changes propagate through walls and insulation
• Biomedical Engineering — Modeling tissue heating during laser surgery or cryoablation procedures
• Electronics Cooling — Analyzing transient thermal response of semiconductor devices and heat sinks

Common Mistakes

• Wrong characteristic length — For a sphere use the radius, for a plane wall use the half-thickness, not the full dimension.
• Confusing thermal diffusivity with thermal conductivity — Diffusivity includes density and heat capacity, conductivity does not. Thermal diffusivity α = k / ρc_p.
• Applying the one-term approximation when Fo < 0.2 — The first-term series solution is only accurate to ~2% when Fo exceeds 0.2.