**Chapter 5 Fiber Treatment**

In order to start up a fiber spinning line, the filament bundle must be "strung up" along the process path. The first stage of the spinning process is vertically downwards, so gravity will tend to pull the line downwards. The line is caught with a large vacuum nozzle and, almost simultaneously, the line beneath the nozzle is cut with scissors. The line now is sucked into the nozzle, which functions exactly like a home cannister style vacuum cleaners, with the fiber piling up in a large drum. The suction at the nozzle keeps the spinning line under tension. The line is then passed over pairs of drive rolls, described below, and then over certain stretching/orientation steps, if needed. Finally, the line is taken to a take-up device which winds the fiber bundle onto bobbins. In order to maintain a uniform takeup speed, the rotational speed of the bobbins must decrease as additional fiber is laid on it and the average diameter increases. This decrease in rotational speed is controlled by keeping the tension constant in the takeup line. The tension is sensed by deflection of an idler wheel located between the last drive roll and the take-up machine. The idler wheel is located on a type of cantilever beam, so that the deflection is a measure of the tension in the line. The position of the wheel, and thus the line tension, is maintained constant by adjustment of the take-up speed.

The drive rolls are always characterized by having non-parallel axes. This permits the threadline to advance in the axial direction on the rolls as the line is strung multiple times around the rolls. The rolls, which are usually highly polished, permit no slip on the surface on account of the multiple wraps on each pair of rolls and the tension in the line. This is somewhat like the calculation of frictional force for a rope wound around a post, a calculation the student probably performed in a statics course.

Because the drive rolls permit good speed control for each pair of rolls, it is possible, by having the second pair of drive rolls moving faster than the first, to control the degree of stretch the fiber undergoes between the two pairs of rolls. This stretch, which may range from a few percent to 20 or 30%, helps to impart desired molecular orientation, and thus desired mechanical properties, to the fiber bundle. Sometimes it is necessary to heat the bundle to a temperature above Tg, the glass transition temperature, but below the melt temperature, in order to facilitate significant stretches without imparting excessive line tension. Usually, the strain gives the fiber a higher modulus (stress over strain), or stiffness, in the axial direction and higher tenacity, or strength to break per unit cross sectional area.

Table 5.1 summarizes several areas which would be convenient for post-presentation
elaboration, or, perhaps, illustrative asides. For example:

__Analytical solution of Partial Differential Equations__ The diffusion
equation is seen frequently in the physical sciences, and presents a nice
starting point for mathematical topics, including techniques of separation
of variables, inhomogeneous boundary conditions, etc.

__ __

__ Gurney-Lurie & Heissler Charts__ These charts depict
very nicely several physical results of unsteady-state conduction and diffusion.
These include the "nearly exponential" decline of unaccomplished
change with time and, similarly, the dependence on the square of characteristic
dimension, via the Fourier number. Similarly, the relative resistance effect
can be seen clearly, along with quick calculations to indicate quantitatively,
when a lumped-parameter model, where, for example, internal resistance could
be neglected, would be appropriate.

**Application to Other Cylindrical Geometries**** **Direct applications
could include glass fiber-optic cable and extrusion of rod-shaped polymer
material, although one could extend the calculations to food processing,
such as sterilization times for cans of food placed in a steam oven, or
cooling of cans of soda, etc.

__ Complication of Accounting for Declining Radius or Heat of Fusion__
The student can recall that these complications were removed during our
treatment so that we could proceed with the analysis. One could account
for the declining radius by patching together successive solutions of uniform
diameter and uniform temperature to approximate the smoothly varying geometry
with one varying in steps. This would require knowledge of the extensional
rheology as a function of temperature, however. Similarly, one would have
to know the heat of fusion in order to account for its effect. If the total
amount of heat to be removed by fusion is of the same order as that of the
sensible heat, then the Fourier time calculation is likely to be too short
by a factor of two or so.

**Extension to other Geometries**** **These would include infinite
slabs, for example in the manufacture of float glass or cooling of metal
slabs or ribbons of steel. Extension of the infinite slabs to Newman's rule
so that three-dimensional bricks can be treated by using three separate
one-dimensional solutions. Spheres are also very important, for example,
in the removal of unreacted monomer, VCM (vinyl chloride monomer), from
PVC (poly-vinylchloride) spheres made by suspension polymerization or drying
of porous spheres, and cooling of molten metal pellets, etc.

**Convection Currents Inside an Object**** **The time to cool a
soda can will be less than that calculated from the charts because convection
currents will become established with the can that will hasten heat transfer
from the can. This is somewhat like the familiar example from heat transfer
courses which states that ice cubes can be made more quickly from tepid
water than from cold water on account of the establishment of such convection
currents, which will more than compensate for the greater amount of heat
to be transferred. The author does not recall seeing any confirming experimental
results, however.