Thermal Linear Expansion

Nearly every solid grows when heated, and it grows by a fixed fraction of whatever length it already had. That is the whole content of $\Delta L = \alpha L_0 \Delta T$: the coefficient α is the fractional growth per degree, so a long member moves more than a short one made of the same stuff. The reason solids expand at all is a detail of the interatomic bond. The potential well an atom sits in is not symmetric — pushing two atoms together costs more energy than pulling them apart by the same distance — so as the atoms vibrate harder, their average separation drifts outward. Expansion is that asymmetry, summed over every bond in the piece.

Take a 100 m copper riser in a hydronic building, filled at 20 °C and run at 82 °C. With α = 16.5 × 10⁻⁶ per K, $\Delta L = 16.5\times10^{-6} \times 100 \times 62 = 0.102\ \text{m}$. The pipe wants to be 102 mm longer than it was when it was installed — a hand's width, in a shaft where the anchors are bolted to concrete. This is why risers get expansion loops, offsets, or bellows, and why the guides that keep the pipe pointing straight matter as much as the loops themselves.

The same coefficient handles areas and volumes to good approximation: an area grows at about 2α and a volume at about 3α, because the fractional growth applies in each dimension independently. Bond two metals with different α back to back and the strip curves as it warms, which was the thermostat for most of the twentieth century. Go the other way and you get Invar, a nickel–iron alloy with α near 1.2 × 10⁻⁶ — a tenth of steel's — for which Charles Édouard Guillaume took the 1920 Nobel Prize in Physics, an award for a material rather than a discovery, given because surveying and clockmaking needed lengths that did not care about the weather.

The largest mistake is applying this equation to a member that is not free to move. A pipe anchored at both ends does not get longer; it develops stress instead. The stress is $\sigma = E\alpha\Delta T$, and for steel a 50 K rise gives $210\times10^9 \times 12\times10^{-6} \times 50 = 126\ \text{MPa}$ — a serious fraction of the yield strength, and completely independent of length. That last point catches people: a short restrained member is under exactly the same stress as a long one, so shortening a run does not relieve anything. Continuous welded rail is laid under deliberate pre-tension for this reason, and a bridge without working expansion joints does not stretch, it buckles.

Two unit traps and one geometric one. ΔT is a difference, so it carries the same number in kelvin and in Celsius — but not in Fahrenheit, where a 50 °F change is 27.8 K, and where a coefficient tabulated "per °F" is five-ninths of the per-kelvin value. Never substitute an absolute temperature for ΔT. And α itself is not constant over a wide range; handbook values are averages over a stated interval, and they drift at cryogenic and high temperatures. The geometric one: a hole in a heated plate gets larger, not smaller. Every dimension scales by the same fraction, including the empty ones, which is why warming a jar lid loosens it.