Thermodynamics & Heat Transfer · Fourier's law
The law under every warm wall
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The law under every warm wall

Heat does one thing in a solid: it walks from hot to cold, and it walks faster through some materials than others. Fourier's law prices that walk. P=kAΔTdP = \dfrac{k A \Delta T}{d} — read aloud P equals k A delta-T over d, with Δ\Delta the Greek letter delta, said “delta” and meaning “the change in”.

Name every letter before it goes to work. PP is the rate of heat flow in watts — joules per second, a power, not an amount. kk is the material's thermal conductivity in W/(m·K): how many watts cross one metre of it per kelvin of difference. AA is the area the heat crosses, in square metres. ΔT\Delta T is the temperature difference between the hot face and the cold face, in kelvin — and because it is a DIFFERENCE, a kelvin and a degree Celsius are the same size, so no conversion is ever needed here. dd is the thickness along the heat's path, in metres.

Read the shape of it. Three things on top all push heat through: a leakier material, a bigger wall, a colder night. One thing underneath fights back, and it is the thickness. That single fact is the entire insulation industry.

Two habits. First, kk is quoted per METRE, so a thickness written in millimetres must be divided by a thousand before it enters — 140 mm is 0.140 m, and forgetting that makes a wall look a thousand times better than it is. Second, push the units through the rearrangement every time: WmK×m2×K÷m\mathrm{\frac{W}{m\cdot K} \times m^2 \times K \div m} leaves watts, so the arrangement survives. Units guide, they never confess — if they refuse to cancel into the answer's units the rearrangement is wrong, no appeal, but units that do work out never prove you right.