In the spinning of molten polymers, such as nylon, polyester, and polypropylene, melt spinning begins with a cooling of the molten filament after it leaves the spinneret. At the same time, the filament is pulled downwards towards the take-up section and this resulting tension in the molten filament provides a stretching action in the molten filament itself. In most melt spinning operations the degree of stretch is of the order of 3x, which means that the velocity of the initially cooled, or solid, fiber is about three times the average velocity of the melt coming out of the spinneret. For some filaments, this initial stretch is very important in helping to establish properties in the polymer which depend on whether one deals with the properties in the fiber axis direction or in the fiber radius direction. This directional dependence of properties is called anisotropy and the usual example is that of a slab of wood, in which strength and fracture properties along the grain are quite different from properties across the grain. (With many fibers, however, these properties are controlled downstream, where the fibers are reheated, stretched further, and cooled again.)

In any case, the polymer melt, once it comes out of the spinneret hole, starts to cool down and also starts to stretch out. Because the "apparent viscosity" of the melt increases rapidly as the melt cools, most of the stretching takes place in a region relatively close to the spinneret hole whereas "most" of the cooling takes place well away from this hole. But these terms and descriptions are not exact and are not easily quantified. The real advantage in using these descriptions is that it permits us to make a simplifying assumption as we analyze the melt spinning process. The assumption is: We can separate the stretching and cooling operations into two separate distinct regions, with the first occurring relative close to the spinneret (say, within 10% of the distance to the first take-up, or speed-control roll), and the second over the remaining distance to the first take-up roll. If need be, we could return later to this assumption to determine its degree of accuracy, but let us accept it for the moment.

The stretching region, within which the relatively long polymer molecules become aligned along the filament axis, might be characterized by very complex rheology. Within the field of polymer processing, rheology deals with the relationship between stress and the history of strain; for Newtonian fluids, you can recall that the fluid stress is proportional to the instantaneous rate of strain (the shear rate). We are not really too concerned with the polymer melt rheology here, however, since it will not likely be important in determining the power required for the first drive or take-up roll. Frictional and interfacial stresses are likely to be far more important. Therefore, in terms of design considerations, we can probably ignore that part of the melt-spinning process in which the initial, post-spinneret stretching of the polymer melt occurs and focus instead on the cooling step of the melt spinning operation.

Perhaps the most important design consideration in the melt spinning
process is the cooling of the filaments, Fig. 4-2. In order to simplify
our analysis, we restrict our focus to the point where the filaments have
reached a uniform diameter (recall that we previously asserted that this
is a relatively short distance) and that they are at some initial temperature
which will be somewhat cooler (by about 20 C°) than the melt temperature
at the spinneret exit. At this point, the temperature within the filament
will depend on radial position, with the maximum occurring at the center,
on the filament axis. We shall invoke an approximation of a flat temperature
profile, in which the temperature does not vary with r at this intial position,
in order to utilize existing mathematical solutions. An important part
of learning engineering is to learn how to take "appropriate shortcuts"
which save time with little sacrifice in accuracy. This is one example.
The melt spinning process is steady: viewing the spinning threadlines at
a fixed position (the socalled Eulerian perspective) shows that nothing
appears to change with time. If one situates oneself on the moving threadline
(figuratively, of course) there does certainly appear to be a time dependence
to the temperature of the filament. This viewpoint of moving with the material
is called the Lagrangian perspective. Whereas the Eulerian perspective
requires you to measure the threadline temperature as a function of r,
radial position within the filament, and z, axial position along the filament,
in order to follow the cooling, the Lagrangian perspective allows you to
follow the cooling as a function of r and t, where t is time. Zero time
should be some convenient reference-here it would correspond to locating
yourself on the filament at the end of the stretching region and at the
"beginning" of the cooling region. This is equivalent to the
cooling of an infinite rod which is fixed in space. The governing differential
equation, which can be derived easily using shell balance techniques is:

(4-4)

where is
time and is
the thermal diffusivity of the polymer. The student will readily recognize
this as a partial differential equation, since the temperature T depends
on both r and .
In order to solve the equation quantitatively, one must specify initial
and boundary conditions. The boundary is naturally R the outside radius
of the filament. So the initial condition ( =
0) is simply:

T (r,) = T

_{o }for r< R (4-5)

where T_{o} is a constant. We need two boundary conditions,
corresponding to r = 0 and r = R. At r = 0,

T remains finite; although some prefer to say:

(4-6)

based on symmetry arguments. At r = R, the heat arriving at the surface
by conduction from within must match the heat leaving by convection:

(4-7)

at r = R and all >
0. k is the thermal conductivity of the filament (we shall assume that
this conductivity does not change as the polymer solidifies. h is the heat
transfer coefficient governing the heat transfer from the surface to the
surrounding air. h can be estimated from various correlations if information
about the velocity and direction of the cooling air is given. An example
of such a correlation for heat transfer from a cylinder in crossflow is
given by Churchill and Bernstein (J. Heat Transfer, **99**, 300 (1977)):

(4-8)

where NuD is the Nusselt number, hD/k, (k is the thermal conductivity of the fluid in crossflow and D is the cylinder diameter) and ReD is the Reynolds number based on the fluid in crossflow. Pr is the Prandtl number, v/,and is also based on the crossflow fluid. The student will recall that an important advantage of presenting correlations in terms of dimensionless variables like Nu (dimensionless heat transfer coefficient) and Re (inertial stresses divided by viscous stresses) is that the resulting expression is often simpler, revealing more clearly the relationships among such variables.

We shall also assume that the heat of fusion is negligible, primarily because we want to simplify the calculation. This assumption, especially for crystalline polymers, could be very poor, however, and could lead an underestimate of the cooling time by a factor of two.

As engineers, or even as normal, sane people, we would not want to solve
the differential equation for every single set of geometries, thermal diffusivities
and initial and boundary conditions. We can avoid this needless energy
expenditure if we express the differential equation in dimensionless form:

(4-9)

where:

(4-10)

Y is called the unaccomplished temperature change, since it starts at
unity at time zero and declines from there. n is the normalized radial
position, and X is dimensionless time, sometimes called the Fourier number.
One final dimensionless group, m, (also equal to 1/Bi) expresses the relative
resistance outside the filament to that within the filament:

(4-11)

Finally, the initial and boundary conditions become:

Y = 1 at X = 0 and 0 < n < 1; (4-12)

at n = 0 and X > 0; (4-13)

at n = 1 and X > 0; (4-14)

The solution, Y(n, X) is then valid for any case of unsteady-state heat
conduction within a cylindrical geometry with a uniform initial temperature
and convective heat transfer from the surface to a surrounding fluid at
a uniform temperature To . The solution is shown in
graphical form on the slide and is available in almost all transport textbooks.
The resulting charts are known variously as "Gurney-Lurie Charts"
or "Heissler Charts," depending on which reference or form of
charts you use. An analytical solution for a slightly less general case
is given below. Note that, by use of dimensionless variables, we have successfully
created a result which is applicable to a broad range of geometries and
material properties. For the special case in which heat transfer resistance
from the surface of the fluid to the surrounding fluid is negligible, one
can set h, the heat transfer coefficient, to infinty (this, of course,
is equivalent to setting the temperature of that surface to that of the
surrounding fluid for all >
0) and the analytical solution is:

(4-15)

where J0 is the zero order Bessel function of the first kind and ai is the ith root of J0(ai) = 0. This solution is presented in the transport text Momentum, Heat, and Mass Transfer, by Bennett and Myers, 3rd ed., p. 286.

A solution for nonzero m, or finite heat transfer resistance to the crossflow fluid, is

(4-16)

where J0 and J1 are Bessel functions of the first kind (zero and first
order, respectively), and ai is the ith root of aiJ1(ai)/J0(ai) = Bi.

**Key elements that you have learned in this section include**:

Difference and relationship between Lagrangian and Eulerian perspective,

Existence of unsteady-state heat conduction charts,

Heat transfer resistance within objects relative to resistance outside these objects,

Value of dedimensionalization as a means to obtain more general solutions to complex equations,

A typical heat transfer correlation for heat transfer coefficient, h

Spinning of synthetic fibers with a melt spinning process is conceptually simple, involving little more that the extrusion of molten polymer through fine holes and solidification of the resulting filaments by cooling.