The Single-Screw Extruder Because molten polymers are usually very viscous, they often need very high pressures to push them through dies. One such “pump” for achieving this is the screw extruder, shown in Fig. E6.5.1. The polymer enters the feed hopper as pellets, falls into the screw channels in

Question

The Single-Screw Extruder

Because molten polymers are usually very viscous, they often need very high pressures to push them through dies. One such “pump” for achieving this is the screw extruder, shown in Fig. E6.5.1. The polymer enters the feed hopper as pellets, falls into the screw channels in the feed section, and is pushed forward by the screw, which rotates at an angular velocity [latex] \omega[/latex], clockwise as seen by an observer looking along the axis from the inlet to the exit (notice that the screw is lefthanded). The heated barrel, together with shear effects, melts the pellets, which then become fluid prior to entering the metering section.

There are three zones in the extruder:

In the feed zone, the granules are transported into the barrel, where they melt just before entering the compression zone. The large constant depth of the channel in the feed zone means that there will be negligible pressure exerted on the downstream melt in the metering zone.
In the compression or “transition” zone, the channel depth decreases from the feed zone to the metering zone. Therefore, the melt will gradually increase in pressure as it travels along the compression zone, reaching a maximum pressure at the beginning of the metering zone.
In the metering zone, length [latex]L_{0}[/latex], the screw channel has a constant shallow depth [latex]h[/latex] (with [latex]h \ll r[/latex] ) and the melt in the channel should now be homogeneous or uniform. Thus, the metering zone of the screw acts like a constant-delivery pump, since the screw is rotating at a constant speed. The melt pressure uniformly increases as it passes along the metering zone. Therefore, calculations of extruder output are based on the metering zone of the screw. Extruder output calculations are relatively simple due to the uniformity of conditions existing in the metering zone of the screw.

The preliminary analysis given here neglects any heat-transfer effects in the metering section and also assumes that the polymer has a constant Newtonian viscosity [latex]\mu[/latex].

The investigation is facilitated by taking the viewpoint of a hypothetical observer located on the screw, in which case the screw surface and the flights appear to be stationary, with the barrel moving with velocity [latex]V=r \omega[/latex] at a helix angle [latex] heta[/latex] to the flight axis, as shown in Fig. E6.5.2. The alternative viewpoint of an observer located on the inside surface of the barrel is not very fruitful, because not only are the flights seen as moving boundaries, but the observations would be periodically blocked as the flights passed over the observer! The width of the screw channel measured perpendicularly to the vertical sides of the flight flanks is designated W.

Accepted Answer

Motion in two principal directions is considered:

Flow parallel to the flight axis, caused by a barrel velocity of [latex]V_{y}=V \cos heta=[/latex] [latex]r \omega \cos heta[/latex] relative to the (now effectively stationary) flights and screw.
Flow normal to the flight axis, caused by a barrel velocity of [latex]V_{x}=-V \sin heta=[/latex] [latex]-r \omega \sin heta[/latex] relative to the (stationary) flights and screw.

In each case, the flow is considered one-dimensional, with "end effects" caused by the presence of the flights being unimportant. A glance at Fig. E6.5.3(b) will give the general idea. Although the flow in the [latex]x[/latex]-direction must reverse itself as it nears the flights, it is reasonable to assume for [latex]h \ll W[/latex] that there is a substantial central region in which the flow is essentially in the positive or negative [latex]x[/latex] direction.

Motion parallel to the flight axis. The reader may wish to investigate the additional simplifying assumptions that give the [latex]y[/latex] momentum balance as:

[latex]\frac{\partial p}{\partial y}=\mu \frac{d^{2} v_{y}}{d z^{2}} .[/latex] (E6.5.1)

Integration twice yields the velocity profile as:

[latex]v_{y}=\frac{1}{2 \mu}\left(\frac{\partial p}{\partial y} ight) z^{2}+c_{1} z+c_{2}=\underbrace{\frac{1}{2 \mu}\left(-\frac{\partial p}{\partial y} ight)\left(h z-z^{2} ight)}_{ ext {Poiseuille flow }}+\underbrace{\frac{z}{h} r \omega \cos heta}_{ ext {Couette flow }} .[/latex] (E6.5.2)

Here, the integration constants [latex]c_{1}[/latex] and [latex]c_{2}[/latex] have been determined in the usual way by applying the boundary conditions:

[latex]z=0: \quad v_{y}=0 ; \quad z=h: \quad v_{y}=V_{y}=V \cos heta=r \omega \cos heta.[/latex] (E6.5.3)

Note that the negative of the pressure gradient is given in terms of the inlet pressure [latex]p_{1}[/latex], the exit pressure [latex]p_{2}[/latex], and the total length [latex]L\left(=L_{0} / \sin heta ight)[/latex] measured along the screw flight axis by:

[latex]-\frac{\partial p}{\partial y}=-\frac{p_{2}-p_{1}}{L}=\frac{p_{1}-p_{2}}{L},[/latex] (E6.5.4)

and is a negative quantity, since the screw action builds up pressure and [latex]p_{2}>p_{1}[/latex]. Thus, Eqn. (E6.5.2) predicts a Poiseuille-type backflow (caused by the adverse pressure gradient) and a Couette-type forward flow (caused by the relative motion of the barrel to the screw). The combination is shown in Fig. E6.5.3(a).

The total flow rate [latex]Q_{y}[/latex] of polymer melt in the direction of the flight axis is obtained by integrating the velocity between the screw and barrel, and recognizing that the width between flights is [latex]W[/latex] :

[latex]Q_{y}=W \int_{0}^{h} v_{y} d z=\underbrace{\frac{W h^{3}}{12 \mu}\left(\frac{p_{1}-p_{2}}{L} ight)}_{ ext {Poiseuille }}+\underbrace{\frac{1}{2} W h r \omega \cos heta}_{ ext {Couette }} .[/latex] (E6.5.5)

The actual value of [latex]Q_{y}[/latex] will depend on the resistance of the die located at the extruder exit. In a hypothetical case, in which the die offers no resistance, there would be no pressure increase in the extruder [latex]\left(p_{2}=p_{1} ight)[/latex], leaving only the Couette term in Eqn. (E6.5.5). For the practical situation in which the die offers significant resistance, the Poiseuille term would serve to diminish the flow rate given by the Couette term.

Motion normal to the flight axis. By a development very similar to that for flow parallel to the flight axis, we obtain:

[latex]\frac{\partial p}{\partial x}=\mu \frac{d^{2} v_{x}}{d z^{2}},[/latex] (E6.5.6)

[latex]v_{x}=\underbrace{\frac{1}{2 \mu}\left(-\frac{\partial p}{\partial x} ight)\left(h z-z^{2} ight)}_{ ext {Poiseuille flow }}-\underbrace{\frac{z}{h} r \omega \sin heta}_{ ext {Couette flow }}[/latex], (E6.5.7)

[latex]Q_{x}=\int_{0}^{h} v_{x} d z=\underbrace{\frac{h^{3}}{12 \mu}\left(-\frac{\partial p}{\partial x} ight)}_{ ext {Poiseuille }}-\underbrace{\frac{1}{2} h r \omega \sin heta}_{ ext {Couette }}=0 .[/latex] (E6.5.8)

Here, [latex]Q_{x}[/latex] is the flow rate in the [latex]x[/latex] direction, per unit depth along the flight axis, and must equal zero, because the flights at either end of the path act as barriers. The negative of the pressure gradient is therefore:

[latex]-\frac{\partial p}{\partial x}=\frac{6 \mu r \omega \sin heta}{h^{2}},[/latex] (E6.5.9)

so that the velocity profile is given by:

[latex]v_{x}=\frac{z}{h} r \omega\left(2-3 \frac{z}{h} ight) \sin heta ext {. }[/latex] (E6.5.10)

Note from Eqn. (E6.5.10) that [latex]v_{x}[/latex] is zero when either [latex]z=0[/latex] (on the screw surface) or [latex]z / h=2 / 3[/latex]. The reader may wish to sketch the general appearance of [latex]v_{z}(z)[/latex].

Question 6.5: The Single-Screw Extruder Because molten polymers are usuall...

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