Fused Filament Extrusion Flow Balance

What goes into the hot end must come out of the nozzle. A cylinder of filament of diameter $d_f$ pushed in at $v_f$ delivers $(\pi/4)d_f^{2}v_f$ of material per second; a bead of width $w$ and height $h$ laid down at head speed $v_p$ takes $w h v_p$ away. Set them equal and you have the relation behind every extrusion multiplier, every E-step calibration and every under-extruded print.

The number to watch is the product $w h v_p$ on its own — the volumetric throughput in cubic millimetres per second — because that is what the hot end has to melt. A standard brass nozzle on a conventional heater block manages perhaps 8 to 12 mm³/s of PLA before the filament stops absorbing heat fast enough to keep up; a high-flow or volcano-style block with a longer melt zone roughly doubles it. Ask for more than the hot end can melt and the extruder skips or the drive gear grinds a flat on the filament, which shows up as under-extrusion that gets steadily worse as the print goes on. This is why the useful direction of the equation is often solving for $v_p$: fix the feed at what the hot end can actually melt, and read off the fastest the head can move while still filling a bead of that section.

The filament diameter is squared, which is why it deserves a caliper. Filament sold as 1.75 mm commonly measures anywhere from 1.70 to 1.80, and that three per cent spread becomes six per cent in the flow. It is the single most common reason a printer perfectly calibrated on one spool over- or under-extrudes on the next, and measuring the spool and entering the real figure fixes more print-quality complaints than any other single adjustment.

On calling this a mass balance. Strictly it is a volume balance, and it becomes a mass balance because the polymer's density is nearly the same in the solid filament as in the cooled bead. "Nearly" is doing some work. The melt is less dense than the solid; semi-crystalline polymers such as PLA and PETG shrink further as they crystallise; and the deposited beads leave small voids between one another where their rounded edges meet. A printed part is therefore a few per cent lighter than the arithmetic says, and that gap is exactly what the slicer's extrusion multiplier — flow rate, flow percentage, whatever your slicer calls it — is trimmed to close. It is a fudge factor, and it is a physically motivated one.

Two geometric rules bound the answer and neither is in the equation. The extrusion width should sit between about 100 and 150 per cent of the nozzle bore: narrower than the bore cannot be laid reliably, much wider is being squashed out sideways rather than placed. And the layer height should stay under roughly 75 to 80 per cent of the bore, because above that the bead is not pressed hard enough against the layer below to fuse to it — the print looks perfect and delaminates under load. The bead is not really a rectangle either; it is a rectangle with rounded ends, squashed against the previous layer, and the small difference between that stadium cross-section and the rectangle the equation assumes is one more thing the extrusion multiplier quietly absorbs.