Library / Heat transfer and thermal systems / HTS-25
Heat transfer and thermal systems
Penetration depth of a periodic thermal wave
Depth at which the amplitude of a sinusoidal surface temperature has fallen to 1/e of its surface value in a semi-infinite solid, after transients decay.
- Ref
- HTS-25
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- δ [m]
A sinusoidal surface temperature of angular frequency ω penetrates a semi-infinite solid with an amplitude that decays as exp(−x/δ), where δ = √(2α/ω). At x = δ the amplitude is 1/e of the surface amplitude, and the wave also lags by one radian. It describes daily and seasonal temperature waves after initial transients decay.
Depth at which the amplitude of a sinusoidal surface temperature has fallen to 1/e of its surface value in a semi-infinite solid, after transients decay. ω is the angular frequency.
Calculator
- Period of the oscillation, 2π/ω314.2 s
- Depth where the amplitude is 1 % of the surface value, 4.605δ0.04605 m
- Thermal wavelength, 2πδ0.06283 m
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.
- The wave of period 314.2 s dies to 1/e of its surface amplitude at 0.01 m and to 1 % at 0.04605 m.
- Halving the frequency increases the depth by only about 41 %, since δ varies as 1/√ω.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ω Angular frequency | 1e-6 to 100 rad/s | 1.414e-4 | 1.414 | 1.414 | +10% | −4.65 % |
| α Thermal diffusivity | 1e-8 to 0.001 m²/s | 0.001 | 0.3162 | 0.3152 | +10% | +4.88 % |
Sweep of Angular frequency
| Angular frequency (rad/s) | Penetration depth (m) |
|---|---|
| 1e-6 | 1.414 |
| 1e-5 | 0.4472 |
| 1e-4 | 0.1414 |
| 0.001 | 0.04472 |
| 0.01 | 0.01414 |
| 0.1 | 0.004472 |
| 1 | 0.001414 |
| 10 | 4.472e-4 |
| 100 | 1.414e-4 |
Sweep one variable
Variables and how each one contributes
Penetration depth
What the equation gives you. Everything else feeds it.
Thermal diffusivity
The thermal diffusivity, in m²/s. A higher diffusivity lets the wave penetrate deeper, as the square root. Raising it by 10 % (to 1.1×10−6) moves the result by +4.88 %, from 0.01 to 0.01049 m.
Angular frequency
The angular frequency, in rad/s. A faster oscillation has a shallower penetration, in inverse proportion to the square root. Raising it by 10 % (to 0.022) moves the result by −4.65 %, from 0.01 to 0.009535 m.
Worked example with the values above
- Ratio: 2α/ω = 2 × 1×10−6 / 0.02 = 1×10−4 m².
- Square root.
- δ = 0.01 m.
Assumptions and limits
- Constant properties, a uniform initial temperature and convection to a fixed ambient on both sides for the wall solutions.
- For lumped models, a body with small internal temperature differences and constant conductances.
Common mistakes
- Using the full thickness instead of the half-thickness in Bi and Fo.
- Using the first-term form at small Fo, where more terms are needed.
- Applying a lumped model when the Biot number is not small.
Statistics and uncertainty
First-term wall solutions are accurate to within about 2 % for Fo above 0.2, and lumped formulas need a small Biot number, below about 0.1, to stay within a few per cent.
Where it is used
- Cooling and heating of slabs, plates and electronics enclosures.
- Estimating warm-up and settling times.
Maths you need
- Biot number: Bi = hL/k
- Fourier number: Fo = αt/L²
- Time constant: τ = C/G, capacitance over conductance