Injection Moulding Cooling Time

An injection moulding cycle is close, fill, pack, cool, eject. On almost every part, the cooling is the longest of the five, and this equation is how long it takes. It comes from Ballman and Shusman, who published it in Modern Plastics in 1959, and it has not needed changing since — because it is not really a moulding equation at all. It is the one-term solution to transient conduction in a flat slab cooled from both faces, which is a piece of nineteenth-century heat transfer that happens to describe a plastic part sitting in a steel tool remarkably well.

The whole answer hangs on one thing: $h$ is squared. Nothing else on the page is. Double the wall and the cooling time goes up fourfold; halve it and the time falls to a quarter. That single fact is the most valuable arithmetic in injection moulding economics, because cooling is the phase that decides the cycle, the cycle decides the parts per hour, and the parts per hour decide whether the job pays. A designer who takes 3 mm down to 2 mm has cut the cooling clock by more than half, and has done more for the part cost than any negotiation over resin price ever will. There is no other lever on this page with that kind of leverage — the diffusivity is a material property you do not control, and the temperatures move the answer only through a logarithm, which is a gentle function.

Now the honesty, and the first item is the one that costs money on the shop floor. The equation ignores the latent heat of crystallisation. Polypropylene, polyethylene, polyamide, acetal — the semi-crystalline family — do not simply cool. They solidify, and solidification releases a great deal of heat that has to be conducted out on top of the sensible heat this equation accounts for. The model knows nothing about it, so on those materials the real cycle runs longer than the prediction, commonly by a third and sometimes by more. On amorphous resins — polystyrene, ABS, polycarbonate, acrylic — there is no crystallisation heat to shed and the prediction is much closer to reality. If you take one thing from this page besides the square on $h$, take that the equation is optimistic on exactly the materials most commonly moulded.

The second trap is the constant, and it is genuine. You will find this same equation printed with $4/\pi$ and printed with $8/\pi^{2}$, and sources swap between them without saying which they mean. They are not typographical variants of each other. The $4/\pi$ form solves for the temperature at the centreline of the wall — the last place to freeze, and the right criterion if you are asking whether the ejectors will push through a soft core. The $8/\pi^{2}$ form solves for the average temperature across the wall. The two constants differ by a factor of $(4/\pi)/(8/\pi^{2}) = \pi/2 \approx 1.571$ inside the logarithm, which works out to roughly 23 % on the answer for typical temperatures. This page uses the centreline form and says so. If a source you are comparing against does not say, you do not know what you have.

The third is that this is only the first term of an infinite series. The full solution is a Fourier series, and dropping everything after the first term is valid once the Fourier number $Fo = \alpha t / (h/2)^{2}$ is above roughly 0.2. Below that the higher terms are still doing real work and the answer is indicative at best. The site's Fourier number page carries the same idea from the heat-transfer side. The solution also assumes an isothermal mould wall — a cavity surface that stays exactly at $T_w$ while the part dumps heat into it — and perfect thermal contact between plastic and steel. Neither is true. The cavity surface warms during the shot, and the contact resistance at the part-mould interface is not zero and grows as the part shrinks away from the steel. Both push the real time up.

And the classic shop mistake: using the nominal wall. The drawing says 2.5 mm and the moulder puts 2.5 mm into the equation, and the tool runs long and nobody knows why. The part cannot be ejected until the thickest section is stiff, and thickest sections are everywhere — a boss, the junction where a rib meets the wall, the pad under a snap fit, the region around the gate. Take a section view and measure the fattest place, including the diagonal across a rib intersection, which is always thicker than either the rib or the wall it joins. That dimension governs. It is also, not coincidentally, where the sink marks are.