Thin filament equations

We have said little here about the proper selection of initial conditions for the thin filament equations. This issue has received some attention through the use of numerical methods like those described in the next chapter, which make it possible to link the shear and exit flow in the spinneret to the spinline. A related but more difficult problem that has received very little attention is the proper modeUng of heat transfer at the top of the spinline, where crossflow air is likely to be impeded by the geometry, and heat transfer from the metal spinneret plate might be important. 

The examples we have studied thus far have had rather simple kinematics flow parallel or nearly parallel to a wall and ideal or nearly ideal extension. Thus, we have been able to obtain exact solutions for the flow or to obtain approximate solutions based on the small difference between the actual flow and an ideal case for which an exact solution is available. Even for the case of fiber spinning, where an analytical solution to the thin filament equations cannot be obtained under conditions relevant to industrial practice, we simply need to obtain a numerical solution to a pair of ordinary differential equations, which is a task that can be accomplished using elementary and readily available commercial software. 

The solution to the thin filament equations for an isothermal Maxwell fluid in the absence of inertia, air drag, and gravity is shown in Figure 10.2 iox Dr = 20. The calculation is for the initial value of the ratio Xrrhzz set to 0, but the stress ratio approaches 0 on the spinline for all choices in the permissible range from 0... 

The blown film process was briefly described in Section 1.2.6. The process is shown schematically in Figure 10.7. There are many similarities between the blown film and the fiber spinline because of the free surface and the very small transverse dimension relative to the distance between melt extrusion and solidification, and thin sheet equations analogous to the thin filament equations are typically used, although the hoop stress must now be taken into account. The equations for a Newtonian fluid were first published by Pearson and Petrie in 1970, and their approach has been used by nearly all investigators since. There are two steady-state momentum equations because variations in both thickness and width in the stretching direction are important. The mechanics of the solid region above the ill-defined freezeline are... 

We now turn to the transient thin filament equations for a spinline, as given in Appendix 7B. For simplicity, we will restrict ourselves to isothermal spinning of a Newtonian fluid in the absence of inertia, air drag, or gravity, in which case the relevant equations are... 

Surface tension-driven breakup into droplets is rarely important in melt spinning, where the large viscous and elastic forces overwhelm the surface tension forces, ft is an important mechanism in the formation of the dispersed phase in polymer blends, and it is important in solution processing. The surface tension-driven breakup of a viscoelastic filament has been analyzed using both thin filament equations and a transient finite element analysis, but we will not pursue the topic here because it is not relevant to our present discussion. 

Fiber spinning, discussed in Chapter 7, is a process in which the residence time is short hence, transient viscoelastic effects are likely to be important. Our starting point for a steady-state thin filament analysis is Equation 7.26 ... 

Draw resonance is a particularly good example of a processing instability because it has a clear signature and a sharp onset. Furthermore, the onset is amenable to rigorous analysis because of the simplicity of the thin filament spinning equations developed in Chapter 7. The approach we use here is linear stability theory, which asks a very specific question ... 

The significance of fhe second assumption is that the deformation field in the film is essentially elongational. Shear stresses are not present, if the film thickness is very small. The first assumption makes the final equations simpler without losing any significant information. Finally, overall these assumptions resemble those used in Section 9.1.2 in the thin-filament theory for the fiber-spinning process. 

The first examples of CVD thin film experiments involved the deposition of W onto carbon lamp filaments by reduction of WCle with H2, as reported in a patent at the end of the nineteenth century. Afterwards Ti, Ni, Zr, Ta and other pure metallic films were obtained by CVD processes (Equations 3.1.1 and3.1.2).2... 

We assume that the filament is drawn down uniformly that is, the radius must be independent of z. In that case, the radial velocity must be independent of z, since the filament will remain uniform only if the rate of thinning (the radial velocity) is the same everywhere. It therefore follows from Equation 7.3 that dvjdz must be independent of z we denote dv jdz by Ye, the rate of extension. Ye may be a function of time. We have assumed that = 0 at z = 0, so it follows that the velocity field is of the form... 

The method of resonance vibrations (Section C above) has also been used, in the form of standing longitudinal wave measurements and flexural vibrations of short fiber segments. - In the latter case, the fiber cross-section shape and dimensions must be known with high accuracy the equations for calculating E and E" are similar to equations 12 and 13 (for circular cross-section) but with different numerical coefficients for one end clamped and one free. Despite the small flexural stiffness of thin fibers, this method has been employed on filaments as thin as 0.03 mm . 

Thermal ionization Thermal ionization sources produce ions by heating sample-coated metallic filaments. Samples in solution are first deposited as microlitre drops on the filaments and dried at low temperatures forming thin salt or oxide layers (Figure 4). The prepared filament is transferred into the mass spectrometer source, the source is evacuated, and the filament is then heated to temperatures ranging from 800 to 2000"C. The probability of forming an ion in thermal equilibrium with the surface of the hot metal filament depends on the temperature (T), work function of the filament (W), and the ionization potential associated with the production of a given ion species (IP). This is described by the Langmuir- Saha equation as the relative number of ionized particles (N+) to neutral species (N ) evaporated from the filament (Eqn [4])... 

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