SPINNING A MOLTEN THREADLINE Steady-Slate Isothermal Viscous Flows M. A .
MATOVICH’ AND J.
R. A . P E A R S O N
Department of Chemical Enginem’ng, University of Cambridge, Pembroke Street, Cambridge, England The steady, nonuniform, axially symmetric, extensional flow characteristic of a molten threadline was investigated. Most attention was given to incompressible isothermal cases. Approximation procedures based on the gradient of the thread line diameter being small were developed: These make the problem one-dimensional. Specific solutions were obtained for Newtonian and Coleman and NOH second-order fluids. The relevance of a generalized Trouton viscosity model is analyzed. Possible extensions to nonisothermal flows are discussed.
E SEEK here to investigate the mechanics of the flow of ‘fluid-like materials when they are drawn into long thin fibers. In particular, we consider the industrially important process of continuous filament spinning (Lodge, 1953) : In this, molten polymer-e.g., nylon, terylene/Dacron, polypropylene-is pumped steadily through a set of small circular holes in a metal plate (spinneret) and drawn down by axial tension; the material extends, cools, and solidifies, and is finally mound on driven spools or bobbins which provide the axial tension required for the operation. The process can, rather arbitrarily, be separated into three regimes, corresponding to three successive regions (Figure 1). Region 1. The “die-swell” or extrusion region, close to the spinneret, in which the material moves relatively slowly and lateral axial velocity gradients caused by flow through the spinneret decay away. Region 2. h molten iidraw-down’J region, which succeeds Region 1, in which the material accelerates in the axial direction, forming a thin filament; lateral axial velocity gradients are assumed small in this region. Region 3. h cold “draw-down” region, which succeeds Region 2, in which the material behaves more like a solid; this region may be physically the longest, and is the one in which air drag is most significant.
R e are concerned mainly with Region 2, which for our purposes is characterized by high rates of axial extension. In practice, there is no clear-cut boundary between the various regions, each merging into the next, and so in what follows artificial-i.e., virtual-start and end points are assumed for Region 2. The analysis given is based on an approximation, itself depending for its validity on the requirement that da/dx, the space derivative of the fiber radius in the axial direction, be small compared with unity. Having obtained a “solution” based on this approximation, we show a posteriori that da/dx may be made arbitrarily small independently by suitable choice of the end boundary conditions, and that terms neglected are of suitably small order. Hence the approach is consistent, even if it may not always be physically relevant. The approximation scheme also allows axial temperature variations to be accommodated in the solution scheme, without making further assumptions about radial temperature gradients. However, the rheological information necessary to specify the material is not significantly simplified by our 1
Present address, Shell Development Co., Emeryville, Calif.
94608. 512
l&EC
FUNDAMENTALS
molten polymer
i+i spinneret die -swell
molten draw-down
(ii)
cold draw-down
(iii) II
Y
to take-up device Figure 1.
Diagram of continuous filament spinning
primary assumption on da/ak. A suitable rheological equation of state must be chosen independently. Elementary solutions for a Newtonian (linear viscous) fluid are reviewed, account being taken of viscous, inertia, gravity, and surface tension forces, while an “exact” solution for second-order Coleman and No11 fluids is presented; this latter may be regarded as a perturbation about a purely viscous solution, and displays interesting singular properties. The only previous detailed analytical study of this problem was reported by Ziabicki and coworkers (Ziabicki, 1961,1964; Ziabicki et al., 1965; Ziabicki and Kedzierska, 1961; Ziabicki and Takserman-Kroeer, 1964). However, the precise mechanical structure of the problem is not made explicit in their work. Consequently their derivation of the basic integrated momentum balance is obscure and their discussion of the error involved in the model is incomplete. The present work attempts to define the structure of the problem while presenting several analytical solutions for the radius profile of the threadline not seen heretofore. The singular nature of one of the expansion procedures employed is examined. A more detailed account of much of this work is given by Matovich (1966), while general remarks and a simplified presentation are given by Pearson (1966). The problem of
stability---Le., of unsteadiness--of threadline flows will be discussed in a subsequent communication. Flow Equations a n d Boundary Conditions
We adopt the cylindrical polar coordinate system (r, x) shown in Figure 2 with velocity components (u, v). The radius of the axisymmetric threadline is given by a (2). The total tension in the threadline is F t and the mass flow axially Q (constant for a steady flow). The ambient pressure, pa, outside the threadline is taken as zero. The density, p , and the surface tension, c,are taken as constant, and gravity g is taken to act in the x-direction (a usual arrangement for the industrial process). We consider only steady flows. Assuming for the moment that the flow is isothermal, the local differential equations governing the flow are: Continuity
assumption that the material outside the threadline (usually air in the relevant industrial processes) exerts no forces on the fluid in the threadline. Boundary (or initial) conditions at the beginning (x = 0, say, since we may choose our origin argitrarily) and end (z = 1, say, where 1 is as yet undetermined) of the flow field of interest are more difficult to define unambiguously. Although Equations 1 to 3 and Boundary Conditions 5 to 6 apply to the flow in Regions 1 and 3 as well as in Region 2, we want specifically to restrict attention to a region in which a’ 0 = y > 0, because this makes a and a' -+ 0 with e, and a fortiori to the generation terms. To keep Equation 76 invariant under the transformation, we find that, neglecting the generation term,
L,(2) --L (xe-7) a,( 2 ) a (xe--y )
9
I K k 1 L, m N n,, nz
S
O(x) nould be an approximate solution corresponding to do),do),etc., while +(
