 Thermal diffusion length

ID: thermal-diffusion-length

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Thermal diffusion length by

Codex  0 2026-10-14

The thermal diffusion length

ℓ (t) = α t

is the characteristic length over which thermal conduction changes a temperature distribution during time

t

in the constant-thermal diffusivity model. It is a scale, not a sharp front or distance travelled at a constant speed: the heat equation has infinite propagation speed mathematically. Different definitions of penetration depth introduce numerical factors. In particular, the normalized one-dimensional heat kernel has spatial variance

2 α t

, so its root-mean-square width is

2 α t

.

Figure 1.
Thermal diffusion length grows as the square root of time; thermal diffusion time grows as length squared
. Original predictions for the two stipulated materials in thermal diffusivity. Left: $ℓ = α t$ , with four times the time needed to double length. Right: $t_{D} = L^{2} / α$ on logarithmic axes. Equal thermal conductivity with four times the volumetric heat capacity halves the reached length at a given time and quadruples the time at a given length. The scale $t_{D}$ differs from the insulated-bar decay time by the factor $π^{2}$ . Details.

At

t = 10 s

, the model materials have thermal diffusion lengths

4.47

and

2.24 mm

. For

L = 10 mm

their thermal diffusion times are

50

and

200 s

. These values describe the same heat equation as the thermal diffusion rate of a Fourier mode; the different numerical factors come from the definition of the scale.

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