Pour two castings from the same ladle of the same metal into the same kind of sand mould. One is a thin plate, the other a chunky block. The plate is solid before you have walked back to the furnace; the block is still mushy in the middle minutes later. The metal did not change. The mould did not change. What changed is the geometry — specifically, how much surface the casting offers for its volume of heat to escape through.
That single geometric idea is the heart of foundry practice. Get it right and you can predict which section freezes last, place a riser to feed it, and produce a sound casting. Get it wrong and the last region to solidify shrinks against fully frozen metal around it, leaving a void. This article explains Chvorinov's rule, the casting modulus it depends on, and how to turn both into a solidification time.
Why this calculation matters
A casting does not solidify all at once. Thin sections freeze first, thick sections last, and liquid metal shrinks as it turns solid. Unless that shrinkage is fed with extra liquid from a reservoir, the last region to freeze ends up with a shrinkage cavity — porosity that can scrap the part or seed a fatigue crack in service.
The whole craft of riser design rests on one principle: the riser must stay liquid longer than the casting section it feeds. To apply that principle you need to know, quantitatively, how long different sections take to solidify. Chvorinov's rule gives you that number from geometry alone, before any metal is poured. It tells you which section is the thermal bottleneck, how big a riser must be to outlast it, and where to chill or pad the casting to steer the freezing front. It also explains, in one tidy equation, why a heavy boss or a thick flange is so often the source of trouble.
The core method
Heat leaves a solidifying casting through its surface. The more volume a casting holds relative to its cooling surface area, the longer that heat takes to escape. Chvorinov captured this in a compact rule: solidification time is proportional to the square of the volume-to-surface-area ratio.
t = B * (V / A)^2
Here t is the solidification time, V is the casting volume, A is the surface area through which heat is lost, and B is the mould constant. The ratio V/A appears so often in foundry work that it has its own name — the casting modulus, usually written M:
M = V / A
The modulus has units of length and is, loosely, a measure of section thickness: a thin wall has a small modulus, a chunky block a large one. Because solidification time grows with the square of the modulus, doubling the modulus quadruples the freezing time. That square law is why thick sections cool so disproportionately slowly.
The mould constant B bundles together everything about the mould and the metal that is not geometry — the mould material's ability to absorb heat, the metal's latent heat of fusion, the pouring superheat, and the freezing temperature. It is found by experiment for a given metal-and-mould combination and has units of time per length squared, for example min/cm^2. The practical power of the rule is this: compute the modulus of the casting section and the modulus of a candidate riser, and require the riser's modulus to be the larger of the two. A riser with a higher modulus freezes later and can feed the casting until the casting itself is solid.
A worked example
Take a casting with a volume V = 1000 cm^3 cooling through a surface area A = 600 cm^2, poured into a mould whose constant is B = 2 min/cm^2.
Step 1 — compute the casting modulus. The modulus is the volume-to-area ratio:
M = V / A = 1000 / 600 = 1.67 cm
Step 2 — square the modulus. Chvorinov's rule needs the square of the modulus:
M^2 = (1.67)^2 = 2.78 cm^2
Step 3 — apply the mould constant. Multiply by B to get the solidification time:
t = B * M^2
t = 2 * 2.78
t = 5.6 min
So this casting takes about 5.6 minutes to solidify. The value of the number is comparative. Work out the modulus of every distinct section, and the one with the largest modulus freezes last — that is where shrinkage will concentrate and where a riser must be placed. A riser feeding this section must be sized so its own modulus exceeds 1.67 cm; otherwise it freezes first and feeds nothing. A thin-walled section of the same casting, with a much smaller modulus, would solidify in a small fraction of this time and need no feeding at all.
Common mistakes
Counting non-cooling surfaces in the area. Only surfaces through which heat actually escapes belong in A. Where two casting sections join, or where a riser meets the casting, no heat crosses — those interfaces are not cooling area. Including them inflates A, shrinks the modulus, and underestimates the freezing time.
Treating solidification time as linear in size. It is not. Time scales with the square of the modulus, so a section twice as thick freezes roughly four times more slowly. Linear intuition badly underestimates how stubborn a heavy section is.
Forgetting that the mould constant is specific. B is tied to one particular metal and one particular mould material. A value measured for sand-cast aluminum does not transfer to a steel casting in a different mould. Using a borrowed B can throw the time off by a wide margin.
Sizing a riser by volume instead of modulus. A riser that merely holds enough metal can still freeze before the casting does. The governing requirement is that the riser's modulus exceed the casting section's modulus, so the riser stays liquid longest.
Ignoring chills and section transitions. A chill placed against a thick section raises the local heat loss and shortens its freezing time, deliberately reshaping the modulus map. Sharp transitions between thick and thin sections create hot spots that a uniform-wall assumption will miss.
Try the interactive NovaSolver calculator
Working the square of the modulus by hand is fine once, but seeing how volume and cooling area trade off against freezing time is far more instructive. The Casting Solidification Chvorinov Simulator on NovaSolver lets you set the casting volume, the cooling surface area, the mould constant, and the exponent, and it returns the solidification time, the casting modulus, a cooling index, and a riser-margin guide, with linked geometry, timeline, and sensitivity views so you can watch how a chunkier casting freezes more slowly.
Related calculators
- CNC Machining — Cutting Force & Tool Life Calculator — for the downstream step, machining a solidified casting to its finished dimensions.
- Weld Heat Input Calculator — for another process governed by how fast a molten pool loses heat to the surrounding metal.
- Welding Residual Stress Calculator — to see how uneven cooling, the same physics behind casting shrinkage, locks stress into a part.
You can browse the full set in the manufacturing tools hub.
Closing note
Chvorinov's rule is one of those results that pays back its simplicity many times over. Solidification time depends on the square of the volume-to-surface-area ratio, and that one square law explains why heavy sections are slow to freeze, why they collect shrinkage porosity, and why a riser must always have the larger modulus. Compute the modulus of every section, find the one that freezes last, and feed it deliberately — and a casting that might have hidden a void in its thickest region comes out sound instead.
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