Heat Conduction (Fourier)

The engineering

One-dimensional, steady-state conduction through a flat slab with constant conductivity — the plane-wall idealization. The k values are room-temperature handbook figures; real conductivity drifts with temperature (carbon steel loses a third of its k by 500 °C) and the fiberglass value assumes dry, uncompressed batts. Curved geometry (pipe insulation) and multiple layers need the log-radius and series-resistance versions.

The four-decade spread in the material menu is the whole story of thermal design: copper conducts ten thousand times better than fiberglass, so in any layered wall the insulation utterly dominates and the metal might as well not be there. The K/W row composes in series like electrical resistors — same algebra, same intuition.

Where this math comes from

Joseph Fourier began studying heat flow while Napoleon's prefect in Grenoble, and his 1822 Théorie analytique de la chaleur delivered both the law (flux proportional to temperature gradient) and the mathematical machinery — Fourier series — invented specifically to solve it. Lord Kelvin called the book 'a great mathematical poem.'

Jean-Baptiste Biot had measured bar conduction around 1804, but it was Fourier's framing that lasted: the same equation now prices building insulation (the US R-value is just L/k in customary units), sizes heat sinks, and — as Fourier himself anticipated in an 1824 essay — underlies the physics of the greenhouse effect.

1. 1804Jean-Baptiste BiotEarly experiments on conduction in bars (circa).
2. 1822Joseph FourierThéorie analytique de la chaleur — the law and the series.
3. 1945Building industryR-value insulation rating popularizes L/k commerce (circa).

See the full timeline of the math behind every calculator →

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