A daily temperature cycle at a surface penetrates a solid to a characteristic depth d = sqrt(a x P / pi), where a is the thermal diffusivity (conductivity divided by density times specific heat) and P is the period of 24 hours. For adobe or earth that depth is about 0.11 m, for concrete 0.135 m and for granite 0.19 m, which is why useful daily mass is about 10-15 cm thick.
Part of Passive Solar Building Design
The conduction arithmetic that explains how fast heat enters and leaves building mass, how far a daily cycle reaches, and how long the delay is.
A Periodic Surface Temperature Decays Exponentially With Depth
When a surface temperature varies sinusoidally with period P, the temperature wave in a uniform solid decays as exp(-x/d) with depth x and lags the surface by a phase of x/d radians. The characteristic depth is
d = sqrt(a x P / pi), with a = k / (rho x c)
where k is thermal conductivity (W/m K), rho density, c specific heat (so rho x c is the volumetric heat capacity from Mass Materials: Stone, Earth, Water) and P the period in seconds (24 hours is 86,400 s). This is the standard periodic-heat-flow result for a semi-infinite solid (see Carslaw and Jaeger, Conduction of Heat in Solids). The time lag at depth x is (x/d) x (P / 2 pi). For a daily cycle P = 86,400 s and P / 2 pi corresponds to 3.82 hours.
The model assumes a uniform solid, constant properties, a sinusoidal surface and good heat transfer at the surface. Real walls have moisture, layers, finishes and surface films, and real sun is not a sine wave. Property values below are typical handbook figures; conductivity especially varies by 30% or more with moisture and compaction.
Worked Example: Adobe
Take k = 0.7 W/m K, rho x c = 1.6 MJ/m3 K (density 1,700 kg/m3 at about 940 J/kg K).
- a = 0.7 / 1,600,000 = 4.4 x 10^-7 m2/s.
- d = sqrt(4.4 x 10^-7 x 86,400 / pi) = sqrt(0.0121) = 0.11 m.
- Amplitude remaining at depth x is exp(-x/d): at 0.05 m, 63%; at 0.10 m, 40%; at 0.15 m, 25%; at 0.22 m (two d), 13%; at 0.33 m (three d), 5%.
- Lag at depth x is (x/d) x 3.82 hours: 1.7 hours at 0.05 m, 3.5 hours at 0.10 m, 5.2 hours at 0.15 m, 7.7 hours at 0.22 m.
Reading it: the wall layer within about 0.22 m does the daily work, and heat taken in at midday comes back to the surface 5 to 8 hours later, in the evening and night, when it is wanted. Beyond about 0.33 m the daily wave is below 5%: that material is not part of the daily cycle. It still acts as a store across several cloudy days and as a stabilizer of the average temperature, but it does not shift much heat each day.
Other Materials, Same Method
| Material | k (W/m K) | rho x c (MJ/m3 K) | d (m) | Amplitude at 0.15 m |
|---|---|---|---|---|
| Adobe / earth | 0.7 | 1.6 | 0.11 | 25% |
| Brick | 0.8 | 1.5 | 0.12 | 29% |
| Concrete | 1.4 | 2.1 | 0.135 | 33% |
| Granite | 2.8 | 2.16 | 0.19 | 45% |
Sensitivity: for adobe with rho x c fixed at 1.6, k = 0.5 gives d = 0.093 m, k = 1.0 gives d = 0.131 m, so the daily depth is about 9-13 cm over that conductivity range. The carrier’s guidance of 10-15 cm for dense materials matches 1 to 1.4 times d for earth, brick and concrete. For granite the same logic suggests somewhat thicker, 15-20 cm.
The carrier’s reason, that thicker mass responds too slowly and is still absorbing when you need it releasing, is true of the deep part but incomplete: the lag at 0.22 m, 7.7 hours, can put the return of midday heat into the evening and night. The limiting fact is the amplitude: below about 0.22 m the surface sees less than 13% of the cycle, so extra thickness adds storage for cloudy spells but little to the daily swing.
Water Is Different
Water in a container circulates by convection, so its temperature is nearly uniform and the diffusion depth does not apply. Its limits are the area of the container surface that exchanges heat with the room and the sun, and the speed of that exchange. Spread water out in thin, wide, dark containers in the sun.
Measurement
You can check this for your own wall: bury or press a thermometer 0.10 m deep and another at the surface, record both every hour on a sunny day, and see the delay and the amplitude ratio. Compare with exp(-0.10/d) and 3.5 hours for adobe.
Common Mistakes
- Using one thickness for every material — granite needs more than adobe.
- Treating the model as a specification — it explains the order of magnitude; measure the building.
- Thinking deeper mass is wasted — it stores heat across cloudy days, just not within a day.
Safety
Sunlit and stove-heated masonry can burn
Do not judge the temperature of a mass heated by a stove or direct sun with a bare hand: dark surfaces in strong sun can reach scalding temperatures. Use a thermometer, and keep children and the sick away from hot mass surfaces.
What’s Next
- Learn next: Where to Insulate vs Where to Mass — wrapping the layer that does the work
- Combine with: Earth Tubes & Ground Coupling — the same equation at the scale of a year