How this instrument works
Conduction is heat travelling through matter without that matter travelling with it — vibration handed along atomic lattices, and in metals a wash of free electrons ferrying energy much as they ferry current. Joseph Fourier reduced all of it to one proportionality: flow rises with temperature gradient and with area crossed. He put his memoir to the Paris Academy in 1807, was rebuffed by Lagrange over a trigonometric series he had invented to solve it, won an Academy prize anyway in 1812, and published his finished argument as Théorie analytique de la chaleur in 1822. Those series carry his name now, and they proved to matter rather more widely than one slab of iron ever could.
Material constant k spans four orders of magnitude, which is why picking it correctly dominates every answer this sheet gives. Silica aerogel sits near 0.013 W/m·K, still air at 0.026, mineral wool and rigid foams between 0.03 and 0.04, softwood 0.13, water 0.6, glass 1.0, concrete 1.4, stainless steel 15, aluminium 237, copper 400. Laboratories pin these down on the guarded hot plate: a measured electrical input driven through one sandwich of specimen, with guard heaters killing sideways leakage so that watts entering the sample are exactly watts crossing it. Building practice repackages identical physics as R = d/k for one layer and U = 1/ΣR for a finished assembly.
Four assumptions sit underneath Q̇ = k·A·ΔT ⁄ d, and three of them fail regularly. It is a steady-state result: temperatures have settled and no energy is being stored, so a wall still warming after sunrise wants the transient diffusion equation instead. Flow is taken as one-dimensional through parallel faces, which any stud, joist or steel fixing ruins by bridging that layer with something far more conductive. ΔT belongs to two surfaces, not to the air on either side, since boundary films add resistance of their own — around 0.13 m²·K/W indoors. And k is treated as fixed, though it climbs with temperature in most insulants and rises sharply once moisture gets in.
- Type your k figure into Thermal conductivity (W/m·K): 0.04 for mineral wool, 0.13 for softwood, 1.0 for glass, 400 for copper.
- Enter whichever face the heat crosses into Area, switching between square centimetres, square metres and square feet as your drawing demands.
- Give Temperature difference (K or °C) as a gap between both faces — warm side minus cold — never an absolute thermometer reading.
- Set Material thickness to how far heat travels between those faces; millimetres and centimetres are both on that field's menu.
- Read Heat flow rate in watts, kilowatts or BTU/h. It is a rate, not an amount: multiply by hours to reach an energy bill.
Worked example — ten square metres of loft insulation
Ten square metres of ceiling carries 100 mm of mineral wool. Loft above holds at 2 °C, room below at 22 °C. Thermal conductivity (W/m·K) = 0.04, Area = 10, Temperature difference (K or °C) = 20, Material thickness = 0.1. Multiply across a numerator: 0.04 × 10 × 20 = 8. Divide by the thickness: 8 ⁄ 0.1 gives a Heat flow rate of 80 W.
Eighty watts is one old filament bulb held permanently against plasterboard — cheap enough to run, and precisely why the wool is up there. Strip it out and 12.5 mm of board alone, k near 0.17, would pass about 2.7 kW across an identical 20 K gap. Go in reverse and lay 200 mm rather than 100 mm: thickness doubles, flow halves to 40 W. That inverse relationship is an entire argument for deeper insulation, written on one line.
Questions
Should there be a minus sign in Fourier's law?
In its proper vector form, yes: q = −k ∇T, where the minus sign records that heat runs down a gradient, from hot toward cold. This sheet ducks that bookkeeping by asking for one difference between two faces rather than two separate temperatures, so a positive entry returns positive wattage and you supply direction yourself. Enter a negative Temperature difference and your answer flips sign, meaning flow has simply reversed.
How do k, R-value and U-value relate to each other?
k belongs to a material, R to a particular thickness of it, U to a whole assembly. R = d/k, so 100 mm of k = 0.04 wool gives R = 2.5 m²·K/W. Resistances in series add, and U = 1/ΣR gives watts per square metre per kelvin passed by a finished construction. Mind the units on imported figures: North American R-values are quoted in ft²·°F·h/BTU and run about 5.68 times their metric equivalent, so an R-13 batt is roughly 2.3 in SI.
Why is a real wall always worse than this calculation?
Because real walls are rarely one clean slab. Studs, joists, mortar beds and steel fixings bridge insulation with far more conductive material, and a bridge covering a few percent of area can add tens of percent to the total loss. Gaps, compressed batts and damp fabric all raise effective conductivity too — wet mineral wool performs dramatically worse than dry. Measured assembly figures commonly land 10 to 30 percent above any paper prediction.
Do I subtract air temperatures or surface temperatures?
Surface temperatures, strictly, because Fourier's law describes what happens between the two faces of a material itself. Room air and outside air sit behind stagnant boundary films carrying resistance of roughly 0.13 and 0.04 m²·K/W. For a thick insulating layer those films are a rounding error you can ignore. For a single sheet of glass or metal they dominate completely, which is why feeding bare glazing into this formula returns an absurd figure.
Can I use it for pipe or tank lagging?
Only as an approximation. Heat crossing a cylinder meets a steadily growing area as it moves outward, so a flat-slab form understates what thick lagging on a narrow pipe actually passes. An exact cylindrical result uses a logarithm of outer-to-inner radius ratio in place of plain thickness. Where lagging is thin compared with pipe radius, entering the mean of inner and outer surface areas keeps this sheet within a few percent.
Why do metals conduct heat so much better than foams?
Metals hand energy around using those same free electrons that carry electric current, which is why good electrical conductors are almost always good thermal ones — the Wiedemann–Franz law ties both ratios together. Insulants take an opposite approach: they trap gas in cells too small for convection currents to form, so solid material acts as scaffolding while trapped air does actual insulating. That is why crushing a batt, or soaking it, wrecks its performance.