Fir Coefficient Calculator: How to Use It

This calculator works out the Fir coefficient, a dimensionless number used in heat transfer analysis to compare the rate of heat conduction to the rate of heat storage in a solid object. You enter the thermal diffusivity, a characteristic length, and the time of interest, and you get the Fir number, which tells you how far heat has penetrated relative to the object’s size.

Formula

The Fir coefficient (often written as the Fourier number, Fo) is defined as:

Fo = α · t / L²

• α (alpha) = thermal diffusivity of the material, in m²/s (or ft²/s)
• t = time since the start of the heating or cooling process, in seconds (or hours, if consistent)
• L = characteristic length, in meters (or feet). For a slab, L is usually half the thickness; for a cylinder or sphere, it is the radius.

The result is dimensionless – it has no units. It compares the rate of heat conduction (α/L²) to the rate of heat storage (1/t). A high Fir number means heat has had time to penetrate deeply; a low number means the surface temperature has changed but the interior has not yet responded.

Worked Example

Using the default values from the calculator:

• Thermal diffusivity α = 1.0 × 10â�»â�µ m²/s (typical for steel or rock)
• Characteristic length L = 0.1 m (10 cm)
• Time t = 60 seconds (1 minute)

Step 1: Square the characteristic length: L² = 0.1 × 0.1 = 0.01 m²

Step 2: Multiply diffusivity by time: α · t = 1.0 × 10�� × 60 = 6.0 × 10�� m²

Step 3: Divide: Fo = 6.0 × 10�� / 0.01 = 0.06

The calculator shows 0.060. This low value tells you that after 1 minute, heat has only penetrated a small fraction of the object’s thickness – the interior is still nearly at its initial temperature.

When to Use the Fir Coefficient

The Fir coefficient is used in transient heat conduction problems – situations where temperature changes with time. It appears in the solution charts for heating or cooling of simple shapes (slabs, cylinders, spheres) that are suddenly exposed to a different temperature. Engineers use it to estimate how long it takes to heat a steel billet, cool a glass bottle, or thaw a frozen food block.

It is also the time variable in the Heisler charts and the Gurney-Lurie charts, which give the temperature at the center of an object after a given time. By calculating Fo, you can enter those charts and find the temperature ratio.

In practical terms, the Fir coefficient answers: “Has the heat reached the center yet?� If Fo is much less than 0.1, the center temperature has not moved noticeably. If Fo is greater than about 1, the object is nearly at the surrounding temperature throughout.

How to Choose the Characteristic Length

The characteristic length is not always the full thickness. It depends on the shape and the boundary conditions:

• Slab (flat plate) heated from both sides: L is half the thickness, because heat comes from both faces and meets in the middle.
• Slab heated from one side only: L is the full thickness.
• Cylinder (long rod, pipe): L is the radius, not the diameter.
• Sphere: L is the radius.

Using the full thickness when you should use half will give a Fir number four times too small, because L is squared. Always check the geometry of the problem you are solving.

Common Mistakes and Misconceptions

A frequent error is mixing units. If you enter diffusivity in m²/s but length in cm and time in minutes, the result is meaningless. Convert everything to a consistent system first – usually SI (meters, seconds) or imperial (feet, hours).

Another mistake is using the wrong thermal diffusivity. Diffusivity is not the same as thermal conductivity. It is defined as α = k / (� · c_p), where k is thermal conductivity, � is density, and c_p is specific heat. For example, copper has a high diffusivity (~1.1 × 10�� m²/s), while wood has a low one (~1.5 × 10�� m²/s). Using conductivity instead of diffusivity is a common error.

Some people think the Fir coefficient is the same as the Biot number. They are related but different: the Biot number compares internal conduction resistance to external convection resistance, while the Fir number compares conduction to heat storage over time. The Biot number uses a heat transfer coefficient; the Fir number does not.

Limits of the Formula

The simple formula Fo = αt/L² assumes constant thermal properties and a uniform initial temperature. It does not account for:

• Temperature-dependent diffusivity (common in materials that change phase or have wide temperature swings)
• Internal heat generation (e.g., chemical reactions, electrical current)
• Non-uniform initial temperature distributions
• Contact resistance between layers

For most engineering estimates, these effects are small enough to ignore, but for precise work you may need numerical simulation. Also, the Fir number is only a scaling parameter – it does not give the actual temperature. You still need the solution charts or an analytical solution to convert Fo into a temperature.

Frequently Asked Questions

What is the difference between Fir coefficient and Fourier number?

They are the same thing. The Fourier number is the formal name; “Fir� is a common misspelling or shorthand used in some engineering contexts. The symbol is Fo, and the formula is identical.

Can I use this calculator for irregular shapes?

Not directly. The formula assumes a simple shape with a well-defined characteristic length. For an irregular object, you can use an equivalent radius (the radius of a sphere with the same volume-to-surface-area ratio), but the result is an approximation. For complex geometries, use finite element software.

What is a “good� Fir coefficient value?

There is no universal good or bad value – it depends on your goal. If you want the center of an object to reach the surface temperature, you need Fo greater than about 1. If you want to keep the center cold while the surface warms, you want Fo much less than 0.1. The value tells you the stage of the transient process, not whether it is good.

How do I convert Fir coefficient to actual temperature?

The Fir number alone does not give temperature. You need the initial temperature, the surrounding fluid temperature, and the Biot number (if convection is involved). Then you use the Heisler or Gurney-Lurie charts, which plot temperature ratio against Fo for different Biot numbers. Alternatively, use the analytical series solutions for slabs, cylinders, or spheres.

Why does the characteristic length get squared?

Because heat diffusion is a random-walk process: the distance heat travels grows with the square root of time. Squaring the length in the denominator makes the Fir number proportional to time, which matches the diffusion equation. Mathematically, it comes from the dimensionless form of the heat equation.

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in a | Thermal diffusivity | m²/s | 1.0e-5 | 0
in L | Characteristic length | m | 0.1 | 0
in t | Time | s | 60 | 0
out Fo | Fir coefficient | - | a * t / (L^2) | 3