Periodic Heating Problem
Theory: a temperature wave that decays and lags
Overview
Cellars are said to be cool in summer and warm in winter. Can heat conduction explain that?
Yes. Surface temperature rises and falls daily and yearly, and those changes travel into the ground as temperature waves. The waves shrink with depth and arrive late, so a little deeper the variation is small, and at some depths the seasons are shifted. The same behaviour applies wherever heat enters and leaves periodically: wall insulation and thermal storage, engine-wall temperature swings, ground-source heat. Let's calculate how far and how late the wave reaches in soil.
Periodic solution for a semi-infinite body
$A$ is the surface amplitude, $d$ the penetration depth, $P$ the period. Every $d$ the amplitude falls by $1/e$ and the phase lags one radian (about 16% of the period). Longer periods and higher diffusivity reach deeper.
Surface heat flux
The amplitude of heat entering and leaving the surface is proportional to effusivity $\sqrt{k\rho c}$ and peaks one-eighth of a period (45°) before temperature. High-effusivity materials exchange more heat for the same temperature swing.
Fourier and underground temperature
In his 1822 Analytical Theory of Heat, Fourier discussed how surface temperature changes propagate underground, showing the annual wave weakens and lags with depth; he hoped to learn about the Earth's interior heat from underground measurements. Lord Kelvin later estimated the age of the Earth with the same conduction ideas. Today this periodic solution is the basis for estimating ground temperature in designing ground-source heat pumps.
Worked examples
Example 1: daily and annual waves in soil
Diffusivity 5×10⁻⁷ m²/s, conductivity 1.0 W/mK.
| Item | Daily | Annual |
|---|---|---|
| Penetration depth d | 0.117 m | 2.24 m |
| Depth to 1% amplitude | 0.54 m | 10.3 m |
| Depth of half-period lag | 0.37 m | 7.0 m |
| Amplitude ratio at 10 cm | 0.43 | 0.96 |
| Lag at 10 cm | 3.3 h | 62 h |
For a 10 K daily surface amplitude, the surface heat-flux amplitude is 121 W/m².
Example 2: comparison with a numerical solution
3 m of soil in 600 cells, driven by a 10 K daily surface cycle for 10 days; the last day was compared with the exact solution.
| Depth | Amplitude (numerical) | Amplitude (exact) | Lag (numerical) | Lag (exact) |
|---|---|---|---|---|
| 0.1 m | 4.258 K | 4.262 K | 3.25 h | 3.26 h |
| 0.3 m | 0.772 K | 0.774 K | 9.77 h | 9.77 h |
Example 3: engine combustion-chamber wall
| Wall material | Penetration depth (2000 rpm, four-stroke) | Depth of 99% decay |
|---|---|---|
| Aluminium | 1.15 mm | 5.3 mm |
| Steel | 0.48 mm | 2.2 mm |
Combustion swings temperature violently, yet it barely penetrates the wall.
Combustion cycles run about 17 times a second, so the penetration depth is only about 1 mm in aluminium and 0.5 mm in steel. Only the surface layer swings; inside, the wall sits at a near-constant mean. Overall thermal stress and cooling can therefore be designed on time-averaged heat flux, but the outer few millimetres see repeated thermal strain, a cause of thermal fatigue. With a one-year period in soil, by contrast, the wave penetrates metres and seasons reverse at 7 m. The same formula gives depths differing by orders of magnitude with period and material.
Analysis tips
- Compute the penetration depth first and make the domain at least about five times deeper.
- Place ten or more grid points within one penetration depth.
- Run several periods until periodic steady state and evaluate the last one.
- Use a time step of about 1/100 of the period or less.
- With several periods (daily and annual), superpose them (for linear problems).
“The frost-free depth”
In cold regions, the burial depth of water pipes and foundations depends on how far winter cold reaches underground. Estimating penetration depth from soil properties and the length of winter shows the annual wave falls to about a third of its amplitude at a little over 2 m. Actual frost depth is often shallower still, thanks to latent heat as soil water freezes and insulation by snow. Regional frost depths are set from such conduction reasoning together with observation.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Domain too shallow | Bottom boundary interferes | At least 5d |
| Evaluating during start-up | Wrong amplitude and lag | Run several periods |
| Coarse grid | Near-surface gradient wrong | Ten points per d |
| Confusing period and angular frequency | Wrong d | d = √(αP/π) |
| Ignoring latent heat (freezing) | Frost depth wrong | Include phase change |
I'd like to learn related topics.
Try the transient conduction tool and ground-source heat pump tool. Related pages include semi-infinite solids, lumped capacitance, Heisler charts and NAFEMS T3.