Fourier's Law of Conduction

The U-value (thermal transmittance) is U = k/Δx for a simple wall, or U = 1/R_total for a composite one. It measures heat loss per unit area per kelvin of temperature difference, so lower is better. Benchmarks: single-glazed windows U ≈ 5–6 W/(m²·K); old uninsulated cavity walls U ≈ 1.5; standard modern insulated walls U ≈ 0.3 (UK Part L and US IECC requirements); Passivhaus walls U ≈ 0.15; vacuum insulation panels U ≈ 0.005–0.008. To convert a U-value to heat flux, multiply by ΔT: a wall with U = 0.3 exposed to a 20 K indoor-outdoor temperature difference loses 6 W/m². Multiply by wall area to get total watts.

In steady state, energy conservation requires that the heat flux be the same at every cross-section of the wall — no heat can accumulate anywhere. Fourier's law says q = −k·dT/dx. If k is constant through the wall and q is constant, then dT/dx must also be constant, which means the temperature varies linearly with position. This is the reason the temperature profile through a homogeneous slab is a straight line. In a composite wall (e.g. brick + insulation + plaster), each layer is a straight line with a different slope: the high-k layer has a shallow slope and the low-k layer has a steep slope. The overall profile is piecewise linear, with the flatter layer containing less temperature drop and the steeper layer containing more.

Yes, but the change is modest for most solids over moderate temperature ranges — typically a few percent per 100 K — so using a single room-temperature k value is acceptable for design work. Gases rise with temperature (air's k goes from 0.024 at 300 K to 0.045 at 600 K), while most metals fall slightly (copper's k drops from 401 at 300 K to 380 at 600 K). The exception is very-high-temperature applications where radiation becomes a significant heat-transfer mechanism inside porous materials (insulations above 500 °C), and where k must be treated as an effective value including radiation across pores. For precise work, use k(T) from a materials database and integrate.

For layers stacked in series — the usual case for a wall with multiple materials — the thermal resistances add: R_total = R₁ + R₂ + R₃ + …, and the U-value is U = 1/R_total. The heat flux is the same through every layer, so q = ΔT_total/R_total. For layers arranged in parallel — such as insulation between timber studs, where the studs are much more conductive than the insulated cavities — the U-values add weighted by area fraction: U_parallel = f₁·U₁ + f₂·U₂. In practice, walls often have both: solid layers in series, and within one layer, insulation and studs in parallel. The combined U is computed by summing resistances for the series parts and averaging by area fraction for the parallel parts.

Fourier's law is the constitutive equation that describes heat flux from a temperature gradient, at any instant. Combining it with energy conservation gives the transient heat conduction equation: ∂T/∂t = α·∇²T, where α = k/(ρ·c_p) is the thermal diffusivity. This partial differential equation describes how temperature evolves over time — during warm-up, cool-down, or a changing boundary condition. Fourier's law alone only describes the steady-state case where the temperature profile is fixed and heat flux is constant. For transient problems, engineers solve the heat equation analytically (for simple geometries) or numerically by finite differences or finite elements. The thermal diffusivity α controls how quickly temperature changes propagate: high-α materials like copper (α ≈ 1.1×10⁻⁴ m²/s) equilibrate in milliseconds, while low-α materials like wood (α ≈ 10⁻⁷ m²/s) take hours.

Conduction is heat transfer through a material by molecular vibrations and (in metals) free-electron motion — it requires no bulk motion of the material, and it is what Fourier's law describes. Convection is heat transfer from a surface to a moving fluid (or vice versa), governed by Newton's law of cooling q = h·ΔT, where h is the convective heat-transfer coefficient (typically 5–25 W/(m²·K) for natural convection in air, up to 1,000 for forced water convection). Radiation is heat transfer by electromagnetic waves, governed by the Stefan–Boltzmann law q = ε·σ·(T₁⁴ − T₂⁴), and it dominates at high temperatures. Real heat-transfer problems usually combine all three: for example, a hot pipe loses heat by conduction through its insulation, by convection to the surrounding air, and by radiation to nearby surfaces. This calculator handles the conduction part only.