Flow in a Wire Coating Die

To further illustrate the use of the force balance on a differential element of fluid to obtain an equation for the shear stress distribution, the design of a wire coating die is considered next. The problem is to design a wire coating die to coat a 

655 X 10 m diameter wire with a 0.330 x 10 m thick layer of HDPE at 200 °C at the highest extrusion rate possible. (Assume again that the extrusion rate limit is the onset of melt fracture at r = 1.4 x 10 Pa.) 

The rheological properties of the fluid are described by the power-law model and elastic effects (i.e., die swell) can be neglected. 

We observe that although the pressure drop is only slightly lower than for the Newtonian case, the extrusion rate is more than twice as large. 

FIGURE 2.11 Wire coating die with an imposed pressure gradient 

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The flow in a wire coating die can be modeled using a combination of drag and pressure flow. Derive an expression for the velocity field inside the die. 

Figure 4.32 Schematic representation of flow in a wire-coating die [88].

Figure 4.33 Simulated flow patterns of (a) Newtonian and (b) shear-thinning polymer melts (LDPE) in a wire-coating die. The Newtonian fluid exhibits a big vortex, while in the same geometry the LDPE melt flows without recirculation [88].

Example 1. Consider a wire-coating die (Problem 16.10) in which a wire of radius Ri is drawn with a velocity V through a concentric die of inner radius R, and length L by a force P. The annulus between wire and die is filled with a fluid. Here, however, the die is rotating about its (and the wire s) axis with an angular velocity o). There is no pressure drop across the die, and we can neglect entrance and exit effects. This is known as helical flow, from the path followed by a fluid element. 

A wire crosshead die is used to manufacture wire coatings, which is illustrated in Fig. 11.10. This specialized die turns the melt flow 90° before it leaves the die. At this turn, the wire to be coated enters the melt stream and exits the die co-axially with the polymer. This process yields a seamless polymer coating around the wire. 

Annular flow is encountered in pipe extrusion dies, wire coating dies and film blowing dies. In the problem under consideration, a Power law fluid is flowing through an annular gap between two coaxial cylinders of radii kR and R, with k < 1 as schematically depicted in Fig. 5.15. The maximum in the velocity profile is located at r = j3R, where / is a constant to be determined. Due to the geometrical characteristics and ignoring entrance effects, the flow is unidirectional, i.e., u = (ur,ue,uz) = (0,0, wz(r)). 

A general equation for the isothermal pressure flow of an incompressible Newtonian fluid in a die (without moving parts like, e.g., wire coating dies), can be written as... 

It was determined that the shear rate in PVC passing through a copper wire coating die was in excess of 4000000 Is. This resulted in volatilisation of the dioctyl phthalate plasticiser and changes in the polymer molecular structure, as determined by infrared spectroscopy. The flow was studied using capillary rheometry with a die of a 0.15 mm diameter. It was concluded that the change in... 

Next, the posttorpedo section may be analyzed. The solution to the generalized annular Couette flow problem may be profitably used for wire-coating dies with appropriate modifications to account for the taper in the different sections (i.e., the variation of k with axial distance z). On eliminating A from Eqs. (7) and (10), the following equation for the determination of n, x) is obtained ... 

Figure4.34 Simulated flow patterns of an LDPEmelt in an industrial wire-coating die. The polymer melt flows smoothly in a streamlined flow both (a) inside and (b) outside the die, verifying the good die design [37].

A typical wire coating die was shown in Figure 1.4. The design that is shown there is somewhat beyond our capabilities at this point. For this reason we consider only the annular flow region as shown in Figure 2.10. We make the following assumptions ... 

The melt flows from the extmder into the die where it flows around the bend and around the core tube. On the far side of the core tube, it forms a weld. Melt sticks to and is pulled by the moving wire. Details of the sizes and shapes of the die parts in contact with the melt are important in obtaining a smooth coating at high rates. The die exit usually is the same diameter as that of the coated wire and there is litde drawdown. Die openings are small and pressures inside the die are high at ca 35 MPa (5000 psi). Wire takeup systems operate as high as 2000 m /min. 

One important aspect of wire-coating is the thickness distribution of the polymer on the surface of the wire as well as the velocity distribution within the die. A simplified wire coating process is presented in the Fig. 6.37, where the wire radius is defined by R and the annulus radius by kR. This type of flow is often referred to as an axial annular Couette flow. 

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