Ind. Eng. Chem. Res. 1994.33, 2384-2390
2384
Frequency Response Analysis of Polymer Melt Spinning? B r i a n M. Devereuxt and Morton M. Denn' Department of Chemical Engineering, University of California at Berkeley, Berkeley, California 94720
T h e dynamics of a viscoelastic polymer melt spinline are analyzed by solving the linearized perturbation equations in the frequency domain using a realistic constitutive equation for the melt rheology. T h e computed frequency response is in reasonably good agreement with experiments on the spinning of poly(ethy1ene terephthalate), but the agreement is poor for polypropylene, which is highly elastic and very shear-thinning. T h e implication is that a better understanding is needed of the process of solidification under stress, which is the major omission in the spinline model.
Introduction The process for the manufacture of synthetic fibers is shown schematically in Figure 1. Molten polymer is extruded through small holes (spinnerets)into a quench environment where the extruded filaments are drawn down in diameter and cooled to below the solidification point, after which further solid-phase processing occurs. Orientation is induced during the melt draw, and many of the propertiesof the commercial fiber are dependent on details of the fluid mechanics, heat transfer, and structure development during the melt drawing. Variations in the properties of the final product are often the result of phenomena which occur on the melt spinline, particularly interactions between the cooling fiber and the quench air. We are concerned in this work with the sensitivity of the process to small input disturbances. The dynamics of polymer melt spinning were first studied independently by Kase and Matsuo (1967) and Pearson and Matovich (1969). treating the polymer melt as an inelastic liquid and the process as isothermal. Subsequent work has been concerned with accounting properly for heat transfer from the drawn filament to the ambient quench air and the proper rheological characterization of the melt; see, for example, the chapter by Denn (1983). Most of the published literature on process dynamics has focused on the onset of a self-sustained oscillatory instability known as 'draw resonance", but the more important industrial problem is the propagation of processing disturbances, primarily those associated with the quench, which affect the properties of the final drawn filament. Young and Denn (1989)reported on the effect of forced quench air disturbances on the drawn-filament diameter in laboratory experimentson tbespinning of poly(ethy1ene terephthalate) (PET)and polypropylene(PP)fibers. They utilized a dynamic simulation code for the time-domain response written by one of us (B.M.D.), based on a Newtonian fluidmodel,tocompute the frequencyresponse for the PET experiments, and they found relatively g d agreement. The simulation was not applicable to the polypropylene data, however. We describe here an analysis of the frequency response of a fiber spinline which utilizes a description of fluid rheology adequate to describe both small-amplitude oscillatory and finite-strain steady shear data. Hence, it
DIE SWELL REGION
+ QUENCH AIR SlREAM
+
+
DRAW DOWN REGION
SOL!S FIBER
t?
@
Figure I. Schematic of the melt spinning p-
should be applicable to a highly shear-thinning and viscoelastic polymer like polypropylene. Traditional "thin filament" equations are employed; the dynamic equations are linearized about the steady state and then solved in the frequencydomain. In this way the solution technique deals only with a linear boundary value problem for a set of complex ordinary differential equations, which can be solved at each forcing frequency without iteration. Spinline Model The thmfdamentequationsarewrittenintermsof croensectional averaged variables. The field equations for mass, momentum, and energy, respectively, are as follows:
* DedicatedtoArthurB.Metzner,whofirstintrodudMorton Denn to the problem of vismlastic spinline dynamica over 20 years ago. 8 Present address: UOP, 777 Old Sawmill River Road, Tarrytown, NY 10591-6700. To whom correspondence should be addressed. E-mail:
denn @nolet.berkeley.edu.
088&5885/94/2633-2384$04.50/0
I A Y
0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2385 Phan-Thien model, which has the following form:
(
-Ti+ A -
Here, u is the average velocity, p the melt density, A the cross-sectional area, T the melt temperature, and T, the temperature of the quench medium. T~~ and Trr are the axial and radial components, respectively, of the extra stress. A discussion of the approximations inherent in these equations can be found in Vassilatos and co-workers (1992). The drag and heat transfer coefficients are given by correlations due to Matsui (1976) and Kase and Matauo (19671, respectively:
cf = 0.56Re4*"
(4)
6* T i ) =2AiD
' 6 * t Gi
Gi
where 6*/6*t is the Gordon-Schowalter non-affine convected derivative defined as
-a*() - D( --
L.( ) - ( ).LT
Dt
6*t
with L=VV-XD
(10)
and the relaxation time is related to the linear viscoelastic value Ai,0 by
h = 0.42Re'/3[1 + 6 4 ( ~ ~ / u ) ~ l ' ~ ~( 5 )
kaD
Ua is the cross-flow quench velocity. The coefficient of 0.56 in eq 4 represents the midrange of available data. The temperature dependence of the polymer physical properties was estimated from the Polymer Handbook (Brandrup and Immergut, 1975). The temperature dependence of the physical properties of dry air and water was fit to tabulated values in Welty et al. (1976). The possible influence of the different Prandtl numbers of air and water is not included in the correlations. The boundary conditions for this set of equations are split. It is assumed that the initial velocity, area, and temperature are known:
D is the rate of deformation tensor. Xi,o and Gi are obtained from linear viscoelastic measurements. The two parameters x and e are obtained from steady shear data. The model reduces to the upper-convected Maxwell model when these two parameters are set to zero. The temperature dependence of constitutive parameters is contained in the relaxation times through a time-temperature shift factor. Details of the determination of rheological parameters are contained in Devereux (1993) and in the supplementary material (see paragraph at end of paper regarding availability of supplementary material); relevant rheological functions are given in Appendix A. The Phan-Thien constitutive equation reduces to the following form for spinning kinematics:
-+ u a ~ z iz- 2Gi a u - 2(1 - x)T,,,~ at
A(z=O) = A,
0%)
T(z=O) = To
(7)
Spinneret values are usually taken for these variables, with probably only a small error associated with the profile rearrangement at the spinneret exit (cf. Keunings et al., 1983). Since the flow rate will be specified, only one of the pair uo and A0 is independent. The takeup point zfis given implicitly by the condition that the velocity and temperature reach the specified takeup rate and solidification temperature, respectively, at the same position. (We assume there is no further stretching of the filament after solidification.)
Rheological Model
az
az
az
aTrr,i a u KiTrr,i + u a7 + Gi a u + (1- x).smi+ -= 0 at
az
(12b)
4.0
with
Only one degree of freedom is available for the stress boundary conditions at z = 0. We assume that the spinneret flow has the same distribution among the partial stresses as fully-developed capillary flow and the radial partial stresses are negligible, giving T,,,~(Z=~)
A?Gi
I -
We considered three differential-type rheological models for the melt, all generalizations of the Maxwell model: those of Phan-Thien and Tanner, as modified by PhanThien (1978);Marrucci (Acierno et al., 1976);and Larson (1984). All three models require as input the linear viscoelastic spectrum. The models were all adequate in fitting steady shear data for the polymers studied here, but only the Phan-Thien model predicted reasonable behavior for spinline simulations under a wide range of steady spinning conditions. A detailed analysis of the response of all three models is given in Devereux (1993); we restrict ourselves here to results obtained using the
au +
(13)
The remaining initial stress is fixed by matching the downstream boundary condition. The relaxation times for PET are very small, making the steady-state equations "stiff", and it was impractical to use more than two relaxation modes. Up to four modes were used in the dynamic simulations for polypropylene. The values of the parameters x and e and the linear viscoelastic parameters for the two polymers used by Young and Denn (19871, a Celanese 0.65 IV PET (M,of order
2386 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994
Table 1. Phan-Thien Parameter Values Darameter PET X 0.15 f 0.20
PP
fixed spatial domain by using the takeup boundary condition
0.12 0.035
O(z=zf,,,t)+ CT(Z=Zf,s,t)= 0,
~~
Table 2. Discrete Relaxation Spectra for PET at 280 "C and PP at 200 OC
where Of is an imposed perturbation in takeup velocity and
Gi (Pa)
Xi.0 ( 8 )
PET (T= 280 "C) 0.282 X
le2
0.120x 0.673 X
1v
1 Mode
0.952 X lo6
The mathematical character of the set of equations is determined by the eigenvalues of the matrix B. The system is fully hyperbolic if the eigenvalues are all real and a complete set of linearly-independent eigenvectors can be found. Beris and Liu (1988) showed that the set of four nonlinear equations governing the isothermal spinning of an upper-convected Maxwell fluid is fully hyperbolic. The complexity of the B matrix for the equations studied here makes proof of full hyperbolicity infeasible, but the eigenvalues were always found to be real for the simulations in this work. The linearized perturbation equations were Fourier transformed, leading to the set of linear ordinary differential equations
2 Modes 0.132 X lo7 0.164 X lo6
le2 PP (T= 200 "C) 1 Mode
0.402 X 1@
0.555 X loo 2 Modes
0.425 X lo5 0.115 X 10"
0.247 X 10-' 0.102 x 10' 4 Modes
0.183 X le2 0.317 X 10-' 0.431 X 100 0.387 X 10'
0.134 X 0.253 X 0.227 X 0.527X
los
106 l@ lo2
40 000) and a Himont polypropylene with M , = 3.0 X los and polydispersity indexM,IM, = 2.8, are given in Tables 1and 2. Most investigators have taken 6 equal to 0.015, the value reported by Phan-Thien (1978) for a low-density polyethylene, for calculations with all polymer melts. The values used here are consistent with steady shear data and the expected degree of extension thickening. The steady-state simulation was checked against PET spinning profiles reported by George (1982). Agreement was similar to that found by Gagon and Denn (1981) using the Phan-Thien equation to simulate the same experiments, but with a different set of parameter values.
Frequency Response The partial differential equations for the field and constitutive variables are of the general form
where yi is the vector of dependent variables and U k is the vector of quench variables (Ta,ua). These equations were linearizedabout the steady-state profiles with perturbation variables 9i and ai defined as Yi = Yi,&l
+ 9J
(15a)
ui = Ui,Jl
+ fii)
(15b)
where the subscript "s" denotes the steady state. The general form of these equations in matrix notation is
a9
-+ at
a9
B -+ Cg + D Q =0 az
(16)
The detailed coefficients are given in Devereux (1993) and are given in Appendix B, which is available as supplementary material. The perturbations in initial conditions are taken to be zero in all results reported here, since the major sensitivity is to the quench. The solidification point z f will vary dynamically, but, as shown by Chang et al. (1982),the linearized equations can be formulated over a
dP dz
-= -B-'(C
+ i d ) ? - B-~DO
(18)
Here, the upper-case symbolsfor the perturbation variables denote the Fourier transforms, which have both real and imaginary parts. The boundary conditions were also transformed. The boundary-value problem in the frequency domain can then be solved by integrating the set of linear ordinary differential equations twice, each time with a linearly-independent set of initial conditions, and then interpolating to match the transformed downstream condition. This approach eliminates the need to solve a difficult set of hyperbolic equations in the time domain. The special case of a Newtonian fluid and isothermal flow (no quench) is worked out in Denn (1983);complete details are in Devereux (1993). The frequency response,which is the dynamical response to sinusoidal forcing at all frequencies, is a complex function of frequency formed from the ratio of the transform of an output variable to the transform of an input. All information about the linear dynamical behavior is contained in the frequency response, which is usually represented as a graph of the logarithm of the magnitude of the complex number (the amplitude ratio, or gain) and the value of the phase angle as functions of the logarithm of frequency (Bode plots). The amplitude ratio contains most of the information about sensitivity, but the phase angle must be examined to identify a transition from stable to unstable conditions.
Results
All simulated frequency responses exhibited a resonant peak structure characteristic of a hyperbolic system with split boundary conditions. (Such peaks occur in the Bode diagrams for countercurrent heat exchangers, for example; cf. Friedly (1972).) The physical basis of the resonance is best understood by consideringa disturbance originating at the spinneret. The information content of the disturbance propagates downstream at characteristic velocities corresponding to the positive eigenvalues of matrix B. When these primary waves reach the downstream boundary, which is the solidification point, a reflected wave
Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2387 Table 3. Conditions for Youna's Svinnina Esmriments (1987) run no.
~~~
UY (mls)
mass flow rate (&Is) To ("2)
3 (PET) 0.70 1.016 X 10-3 1.20 x 1 w 7.08 1.11 x l(r 295.0
2 (PET) 1.32 1.016 X 1V 1.20 x ll3' 5.43 1.11 x 1W 295.0 ~
~
carrying information about the tension adjustment required to maintain a fixed takeup speed propagates upstream at a different characteristic velocity, the sole negative eigenvalue. Resonance results when the period of the sinusoidal disturbance is an integral multiple of the sum of the residence times of the primary and reflected waves. Hence, the location of the resonant peaks is primarily determined by the longest characteristic residence time, or the slowest characteristic velocity of the hyperbolic system. For the processing conditions considered here, the slowest characteristic is always equal to the filament velocity; hence, the first resonant peak is located roughly a t Zrrlt,, where t , is the material residence time on the spinline. Rheology affects the resonant frequencies only indirectly through changes in the residence time. A detailed analysis of the effects of changes in process and material parameters on sensitivity is contained in Devereux (1993). Among the major observations are that the interplay between cooling and rheology is complex for an elastic polymer like polypropylene, and that while cooling is usually stabilizing an increasing cross-flow velocity can increase sensitivity and even bring about a transition to the draw resonance instability. Similarly, lowering the die temperature or decreasing the spinline length increases sensitivity and can be destabilizing. In all these cases the increased sensitivity seems to be associated with increasing the viscoelastic relaxation time. Transitions to draw resonance like those described here have been reported in polypropylene spinning by Gupta and Drechsel (1981), Yo0 (1987). and Lee (1987). Wewill focus hereontheexperimentsofYoung(Young, 1987;Young and Denn, 1989),which is the only published experimental study of the frequency response of a melt spinline to a controlled disturbance. A schematic of the experimental system is shown in Figure 2. Molten polymer was drawn down in ambient air for approximately 1 m into a shallow water quench. The water quench was necessary to ensure that the polymer solidified before contacting the guide wheels. An approximately squarewave disturbance in cross-flow air velocity was provided by an air jet modulated by an on-off solenoid valve, and the magnitude of the velocity was measured with a hotwire anemometer. A laser micrometer provided measurements of the final filament diameter at approximately millisecond intervals. The magnitude of the transfer function relating the final filament area to the cross-flow air velocity was calculated from the discrete Fourier transforms of the measured time series as (19) For the simulations, the disturbance was idealized as a perfect square wave,so thatthe amplitude of thesinusoidal component a t the fundamental frequency was taken as half the amplitude of the square wave. The spatial variation of the disturbance was neglected and assumed
~~~
~~
6 (PP) 1.32 l.fi18 x 1 w
9 (PP) 0.525
ih4x iG
m i f i x iw 10-1
6.99 1.17 X lW 200.0
6.88 1.17 x lW 203.0
;:sox
pulsed air stream hot wire
anemometer solenoid valve
FL U
c -
_quench w _ a__f e r _ _
laser micrometer
FignreZ. Schematicof the experimentalapparatusof Young (1987).
to be equivalent to a disturbance of constant amplitude over a range of 5 cm. The amplitude was also assumed to be independent of frequency, although hot-wire anemometer measurements indicated that the cross-flow air velocity varied by as much as 20% over the frequency range studied. Simulated frequency responses for 10of Young's (1987) experiments, five each on PET and polypropylene, are reported in Devereux (1993); we have selected two from each series for discussion here. Process conditions are summarizedinTable3,where therunnumberscorrespnd to Young's designation. In these four cases the cross-flow air was applied just below the spinneret a t 30 O C and an average speed of 1m/s; quench perturbations were taken to be uniform over the first 5 em of the spinline. Takeup speeds were about half those typically used for the manufactureoftirecordandabout 20% ofspeedstypically used for textile fiber manufacture. The steady-state simulations using two relaxation modes for PET and four for polypropylene indicated little or no drawdown in the water bath for the PET, but about 15% of the draw was calculated to occur in the bath for polypropylene. The Maxwell and Phan-Thien fluids have nearly identical frequency characteristics for PET spinning conditions, and the dynamical (but not steady-state) results are essentially the same for one or two relaxation modes. Figures 3 and 4 probably represent the worst and best agreement between the PET experiments and simulations. Frequency response calculations for a Newtonian liquid are included for comparison. (Run 2 was simulated by Young and Denn; the Bode diagrams differ slightly because of differences in the representation of the forcing
2388 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 1on
PET: run2 =experimental Newtonian ........ UC Maxwell (I) ............... UC Maxwell (2) Phan-Thien ( I ) Phan-Thien (2)
-- -
10.‘
1on
i
oexperimental
- - _. _ _Phan-Thien _. ............... Phan-Thien
10‘
10’
frequency (rad/s)
Figure 3. Experimental and computed gain between quench flow and takeup area, PET, run 2.
1on
(2) (4)
I
10’
10 ‘
frequency (rad/s)
Figure 5. Experimental and computed gain between quench. flow and takeup area, PP, run 6. 1on
lo-’
PET: run3 0 experimental Newtonian UC Maxwell (I) ............... UC Maxwell (2) Phan-Thien ( I ) Phan-Thien (2)
.’-
‘1
U
U : I .I
.c
d)
,.’
--- -----10.~
I
1oo
I
PP
:
runs
0 experimental
/‘
1
I
10’
10‘
frequency (rad/s)
10.’
1on
--- -...............
Newtonian Phan-Thien (2) Phan-Thien (4)
10’
frequency (rad/s)
Id
Figure 4. Experimental and computed gain between quench flow and takeup area, PET, run 3.
Figure 6. Experimental and computed gain between quench flow and takeup area, PP, run 9.
function which are unimportant in the present context.) The location of the first resonant peak is captured, and the maximum gain is approximately correct; in all cases the viscoelastic simulations overestimate the maximum sensitivity. The sharp minima, which are characteristic of hyperbolic systems of this type, do not appear in the data, however. Young and Denn noted the presence of a large amount of frequency cascading with PET,a nonlinear phenomenon in which power is transferred to higher harmonics; “smearing” of the output fundamental into nearby frequencies also occurred, but was more common with polypropylene. Nonlinear effects might contribute to the absence of the sharp minima. The agreement between the computed and experimental frequency response for polypropylene, shown in Figures 5 and 6, is much less satisfactory. The simulation
underestimates the magnitude of the gain by a considerable amount and fails badly in predicting the location of the first resonant peak. The computed peak locations are the same for the viscoelastic and Newtonian fluid models because the computed residence times are nearly the same; since the location of the first resonant peak more or less depends only on the material residence time, it appears that a part of the problem lies with the steady-state simulations. (One reviewer has suggested the first experimental peak may be indicative of a subharmonic in a nonlinear system. We have no reason to believe there are subharmonics present.) In two of the five cases the frequency response calculations indicated that the spinning should be unstable and exhibit draw resonance, while there was no indication of draw resonance in the experiments.
Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2389 Polypropylene is known to crystallize on a spinline, and the absence in the model of crystallization kinetics and of any influence of the degree of crystallinity on melt rheology might be a critical factor in the poor agreement with experiment. The results of Magill (1962) for quiescent crystallization rates of isotactic polypropylene would rule out any significant crystallization prior to the water bath, but an orienting stress field could increase the crystallization rate by orders of magnitude.
to ensure that the zero-shear viscosity
Conclusion The results of this study are disappointing. It has long been established that PET spinning can be simulated with elementary rheological models, and simulations are used routinely for process analysis. The calculations done here for polypropylene utilize the same spinline model, with a rheological characterization that is as complete as available data can justify, and they are clearly inadequate to describe the experimental observations of PP spinline dynamics (and possibly the steady state as well). Hence, simulations for spinning highly-elastic shear-thinning polymers like polypropylene must be used with great caution. The most obvious factor missing in the model is an adequate description of the solidification process under stress, which is an area that has received little attention to date.
with parameters 70 = 2.72 X lo2Pass, Ao = 2.5 X 1Vs, and b = 0.8 chosen to satisfy the correlations for linear PET of Gregory (1973) for the zero-shear viscosity
Acknowledgment This work was supported by the National Science Foundation under Grant MSM-8216381. Appendix A. Rheological Functions Linear viscoelastic behavior is based on the continuous spectrum, H(X). The moduli and relaxation times for a discrete spectrum, Xi,o and Gi,are obtained by averaging the continuous spectrum over N intervals,
was consistent with the experimental value. For PET the spectrum was assumed to follow the form suggested by Wissbrun (19861, H(h) = b(~ldX~)(X/X,)”’/[l+ ( X / Q b I 2
(A41
+
qo(Pa.s) = 1.13 X 10-14M,3.5expF11.9755 6802.1/T (K)) (A5a) and mean relaxation time X(s) = 1.17 x 10-9Mw3.5 exp(-11.9755
+
6802.1/T (K)) (A5b) The definition of mean relaxation time as used by Gregory is the reciprocal of the shear rate at the onset of shear thinning for the steady shear viscosity function. Discrete spectra were calculated over equally-spaced logarithmic intervals of time for both polymers for N ranging from 1to 10. Five relaxation modes gave good fits to the continuous spectra. For the dynamic simulations N = 1and 2 was used for PET, and N = 2 and 4 for PP. These values are given in Table 2. Both polymers were taken to be thermorheologically simple, in which case the temperature dependence of the spectrum is accounted for by a time-temperature shift factor which relates relaxation times at different temperatures to that at an arbitrary reference temperature TO, = X(T)/X(To)
(A61
Experimental data on PP were well-represented by a Vogel form for the shift factor, and
aT = X(T)/X(T0=200“C) = exp[4.529 X 103/T + 0.0176/(T - 343.15)] (A7)
For polypropylene the approximation methods of Williams and Ferry (1953) were used to estimate the continuous spectrum from both experimental storage (G’)and loss (G”)data at 200 “C, as follows:
where the final term is included to guarantee solidlike behavior at 70 “C. For PET the shift factor was taken directly from the Gregory correlation, eq A5b, with a term added to cause solidlike behavior. Both PET and linear polyolefins are known (White et al., 1985, 1987) to follow the empirical Cox-Merz (1958) rule, 7(4 = Iv*(w=+)l
(-48)
where 7 is the shear viscosity and )7*1is the magnitude of the complex viscosity, q*(o)= w-l(Gt2
where m =-- d l n H d In X
H was determined as the weighted average of the two values which provided the best fit to the experimental moduli when the continuous spectrum was discretized. The spectrum was extrapolated beyond the range of the data
+ G”2)’/2
(A9)
The nonlinear parameters in the Phan-Thien equation were fit to the viscosity function as determined by the Cox-Merz rule, using dynamic data for PP and the Wissbrun spectrum for PET. The shear viscosity is most sensitive to x, while e is very sensitive to extensional behavior. Reliable extensional data are not available on these polymers. Linear PET is expected to be extension thinning (White et al., 1985), but a value of e giving a
2390 Ind. Eng. Chem. Res., Vol. 33,No. 10, 1994
small amount of extension thickening was needed in order to obtain an adequate fit with x to the shear viscosity. By analogy to data on linear high-density polyethylene with polydispersity near 3 (White et al., 1987),the value of c was selected to give amaximum extensional viscosity equal to 3 times the limiting value at low extension rate. The resulting values are contained in Table 1. Comparisons of the constitutive equation with PP data and PET correlations are given by Devereux (1993) and are included in Appendix C, available as supplementary material.
Supplementary Material Available: Text comprising Appendix B, Coefficients of Linearized Equations; Figures C.l-C.12 in Appendix C, Comparisons of Constitutive Equations with Rheological Data (22 pages). Ordering information is given on any current masthead page. Literature Cited Acierno, D.; LaMantia, F. P.; Marrucci, G.; Rizzo, G.; Titomanlio, G. A Non-Linear Viscoelastic Model with Structure Dependent RelaxationTimes 11.Comparisonwith L.D. Polyethylene Transient Stress Results. J. Non-Newtonian Fluid Mech. 1976,1,147-157. Beris, A. N.; Liu, B. Time-Dependent Fiber Spinning Equations. 1. Analysisof the Mathematical Behaviour.J. Non- Newtonian Fluid Mech. 1988,26,341-361. Brandrup, J., Immergut, E. H., Eds. Polymer Handbook, 2nd ed.; Wiley: New York, 1975. Chang, J. C.; Denn, M. M.; Kase, S. Dynamic Simulation of LowSpeed Melt Spinning. Ind. Eng. Chem. Fundam. 1982,21,13-17. Cox, W. P.; Merz, E. H. Correlation of Dynamic and Steady Flow Viscosities. J. Polym. Sci. 1958,28, 619-622. Denn, M. M. In Computational Analysis of Polymer Processing; Pearson, J. R. A,, Richardson, S. M., Eds.; Applied Science Publishers: London, 1983;Chapter 6. Devereux, B. M. Computer Simulation of the Melt Spinning Process. Ph.D. Thesis, University of California, Berkeley, 1993. Friedly, J. C. Dynamic Behauior of Processes; Prentice-Hall: Englewood Cliffs, NJ, 1972. Gagon, D. K.; Denn, M. M. Computer Simulation of Polymer Melt Spinning. Polym. Eng. Sci. 1981,21,844-853. George, H. H. Model of Steady-State Melt Spinning at Intermediate Take-up Speeds. Polym. Eng. Sci. 1982,22,292-299. Gregory, D. R. Departure from the Newtonian Behaviour of Molten Poly(ethy1ene terephthalate). Trans. SOC.Rheol. 1973,17,191195. Gupta, R.K.; Drechsel, P. D. Draw Resonance in the Non-isothermal Melt Spinning of Polypropylene. Proceedings 2nd World Congress of Chemical Engineering; The Canadian Institute of Chemical Engineering: Ottawa, 1981;Vol. 6,pp 364-371.
Kase, S.; Matsuo, T. Studies on Melt Spinning. II.Steady-State and Transient Solutions of Fundamental Equations Compared With Experimental Results. J. Appl. Polym. Sci. 1967,11,251-287. Keunings, R.; Crochet, M. J.; Denn, M. M. Profile Development in Continuous Drawing of Viscoelastic Liquids. Ind. Eng. Chem. Fundam. 1983,22,347-355. Larson, R. G. A Constitutive Equation for Polymer Melts Based on Partially Extending Strand Convection. J. Rheol. 1984,28,545571. Lee, W. K. A Correlation of the Onset of Draw Resonance Spinline Instability. Chem. Eng. Commun. 1987,53,117-130. Magill, J. H. A New Technique for Following Rapid Rates of Crystallization. 11. Isotactic Polypropylene. Polymer 1962,3,3542. Matovich, M. A.; Pearson, J. R. A. Spinning a Molten ThreadlineSteady State Isothermal ViscousFlows. Ind. Eng. Chem. Fundam. 1969,8,512-520. Matsui, M. Air Drag on a Continuous Filament in Melt Spinning. Tram. SOC.Rheol. 1976,20,465-473. Phan-Thien, N. A Nonlinear Network Viscoelastic Model. J. Rheol. 1978,22,259-283. Vassilatos, G.; Schmelzer, E. R.; Denn, M. M. Issues Concerning the Rate of Heat Transfer from a Spinline. Int. Polym. Process. 1992, 8,144-150. Welty,J.R.; Wicks, C. E.; Wilson,R. E. Fundamentals OfMomentum, Heat, and Mass Transfer; Wiley: New York, 1976. White, J. L.; Yamane, H.; et al. A Collaborative Study of the RheologicalProperties and Unstable Melt Spinning Characteristics of Linear and Branched Polyethylene Terephtalates. Pure Appl. Chem. 1985,57,1443-1452. White, J. L.; Yamane, H.; et al. ACollaborative Study of the Stability of Extrusion, Melt Spinning and Tubular Film Extrusion of Some High-, Low- and Linear Low-DensityPolyethylene Samples. Pure Appl. Chem. 1987,59,194-216. Williams, M. L.;Ferry, J. D. Second Approximation Calculations of Mechanical and Electrical Relaxation and Retardation Distributions. J. Polym. Sci. 1953,11, 169-175. Wissbrun, K. F. Numerical Comparison of Empirical Rules for Prediction of Nonlinear Rheology from Linear Viscoelasticity.J. Rheol. 1986,30,1143-1164. Yoo, H. J. Draw Resonance in Polypropylene Melt Spinning. Polym. Eng. Sci. 1987,27,192-201. Young, D. G. Melt Spinning Dynamics. Ph.D. Thesis, University of California, Berkeley, 1987. Young, D. G.; Denn, M. M. Disturbance Propagation in Melt Spinning. Chem. Eng. Sci. 1989,44,1807-1818. Received for review January 11, 1994 Accepted June 13,1994 @
0 Abstract published in Advance ACS Abstracts, September 1, 1994.
