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Heat Transfer

Thermal diffusivity and Fourier number

Dimensionless time for heat to diffuse through a body.

Ref: HTR-05
Rev: v1
Field: Heat Transfer
Result: Fo

α = \frac{k}{ρ c_{p}}, Fo = \frac{α t}{{L_{c}}^{2}}

The Fourier number measures diffusion time relative to the length scale L_c. Values around 0.2 or more mean heat has penetrated the whole body.

Fo says how many times heat has had time to cross the body: small values mean the inside has not yet felt the surface temperature change.

Calculator

Solve for

FoFourier numberComputed

kConductivity

W/(m·K)

0.05400

ρDensity

kg/m³

10020,000

c_pSpecific heat

J/(kg·K)

1005,000

tTime

13,600

L_cLength scale

0.0011

Thermal diffusivity α13.35 10⁻⁶ m²/s
Diffusion time L_c²/α187.2 s

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Fourier number () current value 0.3205

L_cLength scaleKey driver

8.01e-4801

ρDensity

0.12525

tTime

0.0053419.2

kConductivity

3.21e-42.56

c_pSpecific heat

0.03081.54

Steel has α about 13×10⁻⁶ m²/s; heat takes about three minutes to cross a 5 cm length scale.
Doubling the length scale needs four times the time to reach the same Fo.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
L_c Length scale	0.001 to 1 m	8.013e-4	801.3	801.3	+10%	−17.4 %
ρ Density	100 to 20,000 kg/m³	0.125	25	24.87	+10%	−9.09 %
t Time	1 to 3,600 s	0.005342	19.23	19.23	+10%	+10 %
k Conductivity	0.05 to 400 W/(m·K)	3.205e-4	2.564	2.564	+10%	+10 %
c_p Specific heat	100 to 5,000 J/(kg·K)	0.03077	1.538	1.508	+10%	−9.09 %

Sweep of Length scale

Length scale (m)	Fourier number
0.001	801.3
0.002371	142.5
0.005623	25.34
0.01334	4.506
0.03162	0.8013
0.07499	0.1425
0.1778	0.02534
0.4217	0.004506
1	8.013e-4

Sweep one variable

Vary

Fourier number ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Fourier number

Result ·

What the equation gives you. Everything else feeds it.

Conductivity

W/(m·K) · typical 0.05 to 400

Thermal conductivity. Raises Fo.

Density

kg/m³ · typical 100 to 20,000

Density. Lowers Fo.

c_p

Specific heat

J/(kg·K) · typical 100 to 5,000

Specific heat. Lowers Fo.

Time

s · typical 1 to 3,600

Elapsed time. Raises Fo.

L_c

Length scale

m · typical 0.001 to 1

Characteristic length (half-thickness). Squared: the strongest driver.

Worked example with the values above

α = k/(ρc_p) = 50 / (7,800 × 480) = 1.335×10⁻⁵ m²/s.
α t = 8.013×10⁻⁴ m²; L_c² = 0.0025 m².
Fo = 0.3205.

Assumptions and limits

Constant properties.
A representative characteristic length.

Common mistakes

Using the full thickness where half-thickness is needed.
Using minutes for time.

Statistics and uncertainty

Properties vary by 5–10% with temperature, but the length scale enters squared and often dominates.

Where it is used

Transient heating and cooling estimates.
Choosing between lumped and detailed models.

Maths you need

Thermal conductivity: k
Specific heat: cp

Related formulas

Practise this formula