Heat Conduction Rate

Fourier's law says heat flows down a temperature gradient at a rate proportional to how steep that gradient is. Written for a flat slab, the gradient is ΔT/d and the flow is $P = kA\Delta T/d$. Each term earns its place: more area gives the heat more parallel paths, a larger temperature difference drives harder, and greater thickness spreads the same difference over a longer distance and so flattens the gradient. The conductivity $k$ is the material's own willingness to pass heat along, and it spans four orders of magnitude — copper near 400 W/(m·K), glass 0.96, mineral wool 0.04, still air 0.026.

Run the numbers on a single window pane: 1.5 m², 3 mm thick, 20 K from inside to outside. $P = 0.96 \times 1.5 \times 20 / 0.003 = 9600\ \text{W}$. Nine and a half kilowatts through one window. That is obviously wrong, and being clear about why it is wrong is the most useful thing this page can teach.

The equation is fine; the model is incomplete. Heat has to reach the glass from the room air and leave it into the outdoor air, and both of those handoffs are slow. The still-air film clinging to the inside surface has a thermal resistance of about 0.12 m²·K/W and the wind-scoured outside film about 0.03, while the glass itself contributes only $d/k = 0.003/0.96 = 0.0031$. Resistances in series add, so the total is roughly 0.155 m²·K/W, giving $U = 1/R = 6.5\ \text{W/(m}^2\text{·K)}$ and a real heat flow of about 195 W — fiftyfold less than the bare slab calculation. The glass was never the bottleneck. This series-resistance picture is exactly the electrical analogy: ΔT is voltage, $P$ is current, and $d/kA$ is resistance, which is why building science speaks in R-values ($d/k$ per unit area) and U-values ($1/R_{total}$) rather than in this equation directly.

So the standard error is treating one layer as the whole assembly. Add layers by summing their resistances, never by averaging their conductivities, and never forget the two air films — in a well-insulated wall they are negligible, and in a window they dominate. It is also why a double-glazed unit works: the gain comes almost entirely from the trapped gas layer and the two extra surface films, not from doubling the glass.

Three further limits. This is a steady-state relation: it tells you the flow once the temperatures have settled and says nothing whatever about how long a wall takes to get there, which is a question of thermal mass and belongs to the diffusion equation. It is one-dimensional, which means it silently assumes heat goes straight through — a steel stud bridging an insulated cavity carries far more than its share of the area, and a thermal bridge can add a third to an assembly's real loss while the calculated R-value notices nothing. And ΔT is a difference, identical in kelvin and Celsius, but imperial conductivities quoted in BTU·in/(hr·ft²·°F) are a different quantity with a different thickness convention buried in them, so convert deliberately rather than by feel.