Chvorinov's Rule

Nicolas Chvorinov published in Giesserei in 1940 the observation that a casting's solidification time is proportional to the square of the ratio of its volume to its cooling surface area. That ratio is called the modulus, it has units of length, and it reduces every shape a foundry can pour to a single number.

The reasoning behind the square is heat conduction into a semi-infinite mould. Heat flows out through the mould interface, and for a mould whose thermal properties dominate — sand, chiefly — the accumulated heat extracted per unit area grows as the square root of time. The casting must give up its superheat plus its latent heat, which is proportional to its volume; the mould can take heat at a rate proportional to its area times 1/√t; equate the two and integrate and the time comes out proportional to (V/A)². The constant B gathers the superheat, the latent heat, the metal's density, and the mould's thermal diffusivity into one fitted number.

The exponent is pinned at 2 on this site, and pinning it is what makes B mean anything. B has dimensions of time divided by length to the power n, so at n = 2 it is s/mm² and at any other exponent it is a numerically unrelated quantity carrying different units. Sources do fit other exponents — values between about 1.5 and 2.0 appear for castings that lose significant heat by radiation from an exposed top surface, and for moulds whose properties do not dominate — and a B taken from such a source cannot be used with n = 2. Check the exponent before the number. There is a time-unit trap on top of that: older foundry references quote B in min/cm², which is 6000 times the s/mm² figure, so 2.5 s/mm² is 0.000417 min/cm².

B is not a property of the alloy alone either. Green sand, dry sand, investment shell and a metal die give very different values for the same metal, because the mould's diffusivity is half of what B contains. Pouring temperature changes it, since the superheat has to be removed before freezing starts. A chill inserted in the mould changes B locally, which is the entire point of using one. The right way to get a B is to pour a simple plate or cube in your own sand with your own alloy, put a thermocouple in the section, and fit it — a constant that belongs to your foundry beats any published figure.

What the rule is actually FOR is riser design, and there the modulus does all the work. Two sections with the same modulus freeze in the same time no matter how differently shaped they are, so a riser needs a modulus about 1.2 times that of the casting section it feeds, and that ratio — not any volume calculation — is how risers are sized in practice. Shape then decides how much metal the riser wastes: a sphere has the best modulus for its volume and is impossible to mould, so a cylinder of height about 1.5 times its diameter is the usual compromise. An insulating or exothermic sleeve raises the effective modulus without any extra metal at all, which is why sleeves are worth their cost on all but the cheapest work.

Count the cooling area honestly, because this is where the rule is most often misapplied. Only surfaces that genuinely conduct heat into the mould belong in A. A face against a core that is already hot, a face against another casting section, and the neck between a riser and the casting are all excluded — and excluding the neck is exactly what lets a riser of modest size out-last the part it feeds. Include a surface that is not cooling and the section looks safer than it is, which is how isolated hot spots and the shrinkage porosity inside them get designed in.

Two limits to keep in sight. The rule assumes a mould that stays intact and a metal with a narrow freezing range; a long-freezing-range alloy develops a mushy zone that feeds badly regardless of what the modulus says, and needs a directional-solidification approach instead. And it says nothing about feeding distance — the length along a section that a single riser can supply — which is a separate set of empirical rules. Chvorinov tells you which part freezes last; it does not tell you whether the liquid can get there.