Single-Screw Pressure Flow

Bolt a die on the end of an extruder and the melt has to be pushed through it, which means pressure builds at the screw tip. That pressure does not only act forward into the die; it also acts backward down the screw channel, driving melt back up toward the hopper. This page is that back flow, and it is the same Poiseuille pressure-driven flow that runs a pipe, written for a shallow rectangular channel.

The channel depth is cubed here and only to the first power in drag flow, and that single asymmetry is the whole of screw design. Double the metering depth: the drag flow doubles, but the back flow goes up eightfold. A deep-channel screw is a high-output machine whose throughput swings with anything that changes head pressure — a new screen pack, a worn die, a change in melt temperature, a different grade. A shallow-channel screw gives up output in exchange for output that stays where you put it, and it also works the melt harder, which helps mixing and colour dispersion. The ratio between the feed depth and the metering depth is the compression ratio, and it is the single number a screw is specified by after its diameter and L/D.

The second asymmetry is viscosity. Pressure flow is the only term with $\eta$ in it. Everything that changes melt viscosity — a hotter barrel, a higher melt index, the shear thinning that comes with a speed change — moves the back flow and leaves the drag flow untouched. That is why extruder output drifts when the melt temperature drifts, and why a screw designed around one grade can behave quite differently on another with the same nominal melt index but a different molecular weight distribution.

The honest caveat is the same for both extruder pages: this is the isothermal Newtonian analysis. It asks for one viscosity, and one viscosity does not exist in a real screw channel. The melt is strongly shear-thinning, so the effective viscosity varies across the channel depth; it is not isothermal, so it varies down the length as well; and the shear rate itself varies with position. Treat these equations as a design starting point and as a way of understanding which direction a change will push the machine — never as an output prediction. The extruder's own measured output-against-pressure curve beats them every time.

Which points at the most useful direction on the page: run it backwards. Measure output against head pressure on the real machine, take the slope, and back out an effective melt viscosity. That number already contains the shear thinning, the temperature profile and the flight leakage that the model omits. It will not match a rheometer reading at any single shear rate, and it should not — it is the one viscosity that makes the isothermal Newtonian model reproduce your machine, which is a far more useful thing to have.

One warning about the extreme case. Set the pressure flow equal to the drag flow and you have the pressure at which the screw delivers nothing at all — the closed-discharge or dead-head condition, where the melt in the channel circulates instead of advancing. Real lines run nowhere near it, but a plugged screen pack or a frozen die can take a machine there in seconds, and the melt in a dead-headed extruder heats rapidly because all the drive power is going into shearing material that is not leaving. Rupture discs and pressure transducers on the head exist for exactly this.