The dynamics of melt spinning was analyzed by deriving a set of simultaneous partial differential equations.Equation of heat balance: Equation of force balance: Equation of material balance: In the above equations, the distance x from the spinneret and time τ are the independent variables; temperature t, local velocity v, and cross‐sectional area A are the dependent variables. It is assumed that tensile viscosity β is a function of temperature t alone, and the spinning tension F is a pure time function. Steady‐state (∂/∂τ = 0) solutions of the above equations showed a fairly good agreement with the experimentally measured values of temperature t(x) and thickness A(x).

SynopsisIn the part installment of the present paper, the authors formulated the dynamics of melt spinning by introducing a set of fundamental equations that consist of the equations of heat, force, and material balances. Some steady-state solutions were also given. Additional steady-state solutions corresponding to many different spinning conditions for polyester and polypropylene filament yarns consistently show good agreement with experimental results. These steady-state solutions that give filament crosssection A ( z ) and filament temperature t ( z ) as functions of position z are correlated with yarn qualities: yarn density and birefringence, crystallinity and molecular orientation, are correlated respectively with the speed of polymer cooling at 100°C. and the maximum tensile stress ( F / A ) , acting on the filament. A transient solution of the fundamental equations computed on an IBM 1401 machine shows that the filament cross-section A at the take-up roll forms a large transient peak after a st.epwise increase in the speed v, of cooling air.The fundamental equations, therefore, clarify the dynamic relations between cooling air speed and yarn weight variations.This agrees with experiments fairly well.

SynopsisThe stability of melt spinning has been studied theoretically by solving for transients the perturbed form of the simultaneous partial differential equations of melt spinning introduced by the author in a previous study. Computed stability limits summarized in the form of maps in the (t*--St) plane with the cooling air speed serving as the third parameter showed that the thread must be in a molten state at the take-up before an instability can develop and that the cooling of the thread by air plays a predominant role in stabilizing the melt spinning. Here, t* is air temperature and St is the Stanton number. Draw resonances were observed experimentally in a water-quenched melt spinning of PET fiber and in the casting of PP film. Experimental results agreed well with theoretical simulations with respect to oscillation periods and stability. Draw resonance observed by Bergonzoni et al. was closely simulated by means of the present theories.

SynopsisDraw resonance, a periodic variation of spin line diameter in unstable melt spinning, was measured for its wave form under 34 different spinning conditions for PET and PP. In an attempt to simulate the measured wave form, the equations of continuity and momentum for the isothermal melt spinning of power law fluids were solved for their limit cycle solutions expressed in the time variations in the cross-sectional area at the take-up. Power law exponent p and draw down ratio $ , uniquely define the solution. Theoretical curves were superposed on the experimental amplitude-versus-$, diagram and oscillation period-versus-$w diagram to assign p value to each experimental point. Excellent agreement between theory and experiment was obtained with PET in that p values were nearly independent of $w and of the diagram used in the determination of the p value, amplitude diagram, or oscillation period diagram. Motion pictures (16 mm) of the side profiles of the pulsing spinline showed good agreement with the theoretical side profiles constructed from the corresponding limit cycle solution. It was proposed that the stability of melt spinning has no direct equivalence to the spinnability of fluids.

SynopsisThe exact wave form of draw resonance in isothermal spinning of Newtonian liquids was sought by solving numerically the simultaneous partial differential equations' of melt spinning in their original nonlinear form without recourse to perturbation. When the drawdown ratio of spinning exceeded 20, solution of the equations became a limit cycle, a sustained oscillation having amplitude and period independent of initial conditions. As the draw down ratio was further increased, the amplitude of the limit cycle grew very rapidly, and the wave form became close to a pulse train predicting an extreme thinning of the thread at regular intervals along the thread. The above solution for Newtonian liquids agreed well with experiment with respect t o oscillation period. Agreement, however, was poor in amplitude, indicating that possibly the amplitude of draw resonance is affected by deviations of polymer viscosity from Newtonian.