Thermal Diffusivity Calculator

Background.

Hold one end of a steel rod and one end of an identical copper rod in a flame, and the far end of the copper warms first. Copper conducts heat better, but conduction is only half of the story. How fast a temperature change travels through a material also depends on how much heat each slice of it soaks up along the way. Thermal diffusivity combines the two: conductivity divided by the heat needed to warm one cubic metre by one kelvin.

In symbols, α = k ÷ (ρc_p). Thermal conductivity k says how readily heat flows under a temperature gradient. Density times specific heat is the volumetric heat capacity, the energy stored per cubic metre per degree. A material with high diffusivity passes a change along quickly and settles at its new temperature fast; one with low diffusivity holds a temperature difference for a long time.

The unit is square metres per second, and the clearest way to read it is through time. By conduction, heat penetrates a distance L in a time of the order of L² ÷ α, so doubling the thickness of a slab roughly quadruples how long it takes to warm through. For steel with the default properties, about 1.23 × 10⁻⁵ m²/s, that characteristic time is around 8 seconds for a centimetre; for still water, by conduction alone, it is over 11 minutes.

Nothing is looked up here: the page takes the three properties exactly as you enter them. They must describe the same material at the same temperature, phase and direction. Conductivity and specific heat both change with temperature, and in wood, composites and other layered materials conductivity differs along and across the grain, so a single diffusivity holds only for the state and direction your inputs came from.

What is thermal diffusivity calculator?

Thermal diffusivity, α, measures how quickly a temperature change spreads through a material by conduction. It is thermal conductivity divided by volumetric heat capacity, the product of density and specific heat at constant pressure, and it is the coefficient in the heat-diffusion equation ∂T/∂t = α∇²T for a uniform material with no internal heat source. Its SI unit is m²/s; tables often list it in mm²/s, where 1 mm²/s equals 10⁻⁶ m²/s.

How to use this calculator.

1. 01Enter the Thermal conductivity in W/(m·K) from a data sheet or handbook, taken at the temperature you are interested in.
2. 02Enter the Mass density in kg/m³. A figure in g/cm³ needs multiplying by 1,000, so 7.8 g/cm³ becomes 7,800 kg/m³.
3. 03Enter the Specific heat capacity in J/(kg·K). Tables in kJ/(kg·K) need multiplying by 1,000, so 0.47 becomes 470. For gases use the constant-pressure value.
4. 04Read α in m²/s and multiply by 1,000,000 for mm²/s. To estimate how long heat takes to work through a thickness of L metres, divide L² by α.

The formula.

α = k ÷ (ρ × c_p), with k in W/(m·K), ρ in kg/m³ and c_p in J/(kg·K). The units reduce to m²/s: a watt is a joule per second, so W/(m·K) divided by J/(m³·K) leaves m²/s. Physically, k sets how much heat flows for a given temperature gradient, and ρc_p sets how far the temperature rises for each joule stored in a cubic metre; their ratio is the rate at which the temperature profile itself spreads out. Diffusivity rises in direct proportion to conductivity and falls in proportion to both density and specific heat, so a light material with little heat capacity can spread temperature changes quickly despite conducting heat poorly. In transient problems α enters through the Fourier number, αt ÷ L², which compares elapsed time with the diffusion time across a length L.

A worked example.

Frequently asked questions.

References& sources.

1. [1]ISO 22007-2:2022, Plastics — Determination of thermal conductivity and thermal diffusivity, Part 2 (PAYWALL/BIBLIOGRAPHIC).
2. [2]Incropera et al., Fundamentals of Heat and Mass Transfer, 8th ed. (PRINT), transient conduction.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.