Earthen Wall Thermal Lag

What is modelled

A wall of thickness L, conductivity k, density ρ and specific heat c sees a sinusoidal outdoor temperature of period 24 h. Heat diffuses through it with diffusivity α = k/(ρc). Because of its heat capacity the inside surface swings less (the decrement factor f) and later (the time lag φ) than the outside. The wall is solved with an implicit finite-difference scheme and compared with the exact periodic solution.

Equations

• ∂T/∂t = α ∂²T/∂x², with convective surfaces q = h (Tair − Tsurf).
• Thermal wave penetration depth δ = √(α P / π); amplitude falls as e−x/δ and lags by x/δ radians.
• Steady U-value U = 1 / (1/ho + L/k + 1/hi).
• Periodic solution from the slab transfer matrix with γ = √(iω/α).

Checks run in the page

• Steady heat flux of the finite-difference solver against U ΔT.
• Decrement factor and time lag of the solver against the complex transfer matrix.
• Amplitude decay inside a very thick wall against e^−x/δ.
• Energy balance: heat stored equals heat in minus heat out.

Honest simplifications

One-dimensional conduction, constant properties, no moisture transport, no solar radiation on the wall other than through the outdoor swing, and a fixed room air temperature. Real mud walls have moisture buffering and cracks.

• Choose Thick mud wall and see a long lag and a strongly reduced swing.
• Choose Thin wall and watch the inside follow the outside.
• Raise the conductivity to see how a denser, more conductive earth loses the benefit.