**Chapter 4 Spinning**

As mentioned previously, fibers are formed by the extrusion of the polymer
melt or spin dope through tiny holes in a spinneret plate. Such a plate
may contain 1,000 holes or more. Textile fibers are relatively fine, so
the diameter of the hole may be only a few mils, where one mil is 0.001
inches (25.4 µm). The thickness of the filament is generally
not given in linear dimensions, but rather in terms of mass per length.
For some reason the fiber industry has adopted the terms denier and denier
per filament, dpf, to express the filament size. One dpf corresponds to
a mass of 1 g in a length of 9000 meters! If the density of the polymer
is 1 g/cm3, this would correspond to a diameter of 1.2 x 10^{-3}
cm, or about half a mil. Typically, textile fibers are in the range of
3 to 15 dpf. Recall that one g is roughly 1/30 of an ounce.

In melt spinning, the filaments are normally drawn down, or stretched, just downstream of the spinneret holes. The stretch is of the order of 2 to 3x, so the spinneret hole may be 50 to 75% larger than the filament diameter when it is first cooled. Additional post-formation stretching may also be used, however, so that the final filament diameter may be one-half or less than the diameter of the spinneret hole.

The spinneret hole is usually only slightly longer than its width, in part to minimize pressure drop at the plate. But the plate still has to be strong enough to withstand the upstream pressure. For this reason, the melt passes through a conical section before reaching the final spinneret hole, so that the plate can be relative thick (see Fig. 4.1). Pressure drop through this converging section is very difficult to calculate for these polymeric materials, because the extensional flow rheology is usually not well characterized. One can readily visualize the alignment of the polymer molecules in the converging section, where the polymer undergoes a severe stretching step. Ignoring this pressure loss, let us focus on the spinneret hole itself.

An important question here is whether one can use the Hagen-Poiseuille
equation to compute the pressure drop in such a short tube. To help answer
this question, we first compute the entrance length, which is approximately
equal to the axial distance downstream from a tube entrance at which the
momentum boundary layers merge at the center axis, where a fully parabolic
profile is established. This distance, from Bird, Stewart, and Lightfoot,
Transport Phenomena, p. 47, is

Le/D = 0.035 Re (4-1)

where Le is the entrance length, D is the tube diameter and Re is the Reynolds number.

For a representative calculation, we consider nylon melt, with a viscosity of 200 Poises, being spun from a hole 10 mils in diameter at a final spinning speed of 2,000 ypm and a stretch of 5x between the spinneret and the final take-up. This means that the bulk velocity, ub, in the spinneret hole is 400 ypm. The Reynolds number, for a specific gravity of about one, is:

Re = ub D/n = 0.077 (4-2)

so the entrance length calculated from Eqn. 4.1 is less than 3 thousandths
of the diameter of the hole. Therefore, we can safely use the Hagen-Poiseuille
equation to calculate the pressure drop. The equation is:

(4-3)

For the nylon example we just explored, the pressure drop is predicted, for a length of 3.0 mils, to be 2200 psi. This pressure drop might be a bit excessive in practice, but the method of calculation remains illustrative. One part of the calculation which was not taken into account is the power-law behavior of most polymer melts and polymer solutions. Such behavior usually is revealed by a shear-thinning response, in which the apparent viscosity decreases as shear rates increase. This would lead to significantly lower pressure drops for the spinneret plate.

As the polymer exits the spinneret hole, it tends to swell and this swell is especially noticeable at low filament tensions. Apparently, the polymer molecules must coil under the shearing action within the hole and, as it exits, the polymer molecules are free to uncoil, as seen by an expansion of the polymer stream jetting out of the hole. This phenomenon is referred to as "die swell," and can even amount to a doubling or more of stream diameter. Newtonian fluids can also be shown to exhibit a swelling at the exits of tubes, even at very low Re; the predicted extent of swell is about 14% for Newtonian fluids. Since, in fiber spinning, the filaments are under tension, the extent of die swell is considerably reduced. Furthermore, the extent of swell appears to have no influence of final filament properties. In order for the filaments to undergo stretching, some power must go into the stretching motion immediately downstream of the spinneret plate, but the amount of this power is negligible, as described in the next section.