Stability of Isothermal Fiber Spinning of a Newtonian Fluid Gerard J. Donnelly and Charles B. Weinberger’ Department of Chemical Engineering, Drexel University. Philadelphia, Pennsylvania 19 104
A Newtonian silicone oil was “spun” at increasing take-up speeds until an oscillatory instability commonly known as draw resonance occurred. The speeds at onset of the instability corresponded to ratios of take-up speed to jet speed (draw ratio) near 20, in good agreement with the theoretical prediction of Pearson and Mate vich (1969). The speed ratio at onset did not appear to depend upon the calculated tensile force, although the severity of oscillation, at correspondingdraw ratios, was greater when the tensile force was higher in the spinning zone. Periods of oscillation could be correlated with residence time of a fluid element in the spinning zone.
Introduction In the commercial spinning of synthetic fibers, process speeds are frequently limited by an instability known as draw resonance, manifest as an oscillatory variation in fiber diameter along the length of the solidified fiber. During spinning, a process in which an extruded fluid is stretched by a take-up system, draw resonance can be seen as an oscillation in diameter of the fluid stream at some fixed point in the draw-down zone. Experimental work by Bergonzoni and Dicresce (1966) and Miller (1963) with polymeric materials shows that the most significant variable controlling the onset of draw resonance is E , the ratio of take-up velocity to jet velocity. E , sometimes called the “draw ratio,” is directly related to the total extensional strain experienced by a fluid element as it passes through the spinning zone. Linearized perturbation analyses by Pearson and Matovich (1969), Kase (1974), and Gelder (1971) have confirmed the importance of E in controlling the onset of draw resonance (for both Newtonian and non-Newtonian fluids). Moreover their theoretical results for spinning a Newtonian fluid under isothermal, constant-force conditions show that the process i s unstable for a draw ratio, E , greater than 20.21. The theoretical analyses also yield a predicted value for the period of resonance, T, in terms of the filament length, L , the jet velocity, UO, and E . Shah and Pearson (1972) extended the earlier work to account for effects of inertia, surface tension, and gravity. The same type of linearized perturbation analysis has been applied by Pearson and Shah (1974) to the more complicated, but perhaps more relevant, cases of cooling fluid threadline and spinning of power-law fluids. However, before these more recent analyses are used to interpret experimental results of draw resonance with rheologically complex materials, it is appropriate to test first the theoretical predictions for a Newtonian fluid. Experimental measurement of first, the critical value of the draw ratio, E , and second, the period of resonance, T, permit such a test of the predicted values. We also measured the severity of draw resonance as a function of draw ratio, E , for E > 20; this dependence was not predicted explicitly by the theoretical analysis.
Theory In this section we are concerned mainly with specifying the equations necessary (a) to check the assumption of constant-force spinning and (b) to define the residence time of a fluid particle in the draw-down zone. In addition, since Newtonian fluids do exhibit a die swell a t low Reynolds numbers, the appropriate definition of the initial velocity, 334
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
UO, is discussed. The stability analyses of the spinning process have been reviewed recently by Pearson and Shah (1972) and will not be repeated. The appropriate dynamical equation for spinning (including the usual assumption of a flat velocity profile) is given by Matovich and Pearson (1969)
Here, u is velocity in the axial x direction, Q is volumetric flow rate, u is surface tension, and the tensile stress T is the axial-radial principal stress difference
F is the tensile force and A is the local cross-sectional area. For a Newtonian fluid du r=3u-
(3) . ,
dx
’
a result obtained by Trouton (1906). Clarke (1968) solved eq 1 for the case of a Newtonian fluid with negligible surface tension; Trouton (1906) obtained a simpler expression for u ( x ) for the case where inertia could also be neglected (where only the second and third terms of eq 1are nonzero). The Trouton solution is u(x) =2pg sinh2 3WC l2
;[
(x
+ Q)]
(4)
where c 1 and cg are constants to be determined from the specified velocities. The Trouton solution is particularly appropriate in our work since our experimental conditions include those of negligible surface tension and inertia (Re/ In this case, then We < 0.23 and Re < 0.68 X
Equation 5 permits us to determine how closely constantforce conditions were maintained in the fluid threadline. The residence time, 8, of a fluid particle in the spinning zone is simply
where L is the length of the zone. For the case of negligible inertia and surface tension, 0 is obtained from eq 4 and 6 8=
12
- acrcl [~ 0 t h Pg
with c 1 and c2 constants of eq 4.
(X
+ c ~ ) 0) ] ~
(7)
I 5D
I
I
I
-
fi fi
SYRINGE
PUMP
4
9
.5
D
64.1
20
0
44.4 30.8
20
0
1
1
I
I
17.0 17.3 17.2
40
4.0
L
\
5
5 FI LAMENl LENGTH
TAKE UP ROLL
3
-
50
0:
d Id
20
-
f:.ol1-l.-0
8-7
Figure 1. Sketch of apparatus. 10
For constant-force spinning, the residence time is
I
I
I
I
15
PO
25
30
I 35
I
I
40
45
DRAW RATIO. E =,(V /%I
Figure 2. Detection of critical draw ratio.
where E is the draw ratio, ~ 1 1 ~ In 0 . eq 8 the dependence of 8,f upon E is rather weak;the initial velocity, U O , and the filament length, L , dominate the residence time. Recall that the period of resonance predicted by Matovich and Pearson (1969) and Gelder (1971), 1.35 L T=--
In E uo
possesses similar dependence on L and uo as does Ocf. Their results therefore suggest a close relationship between residence time and period of resonance; this was confirmed by our experimental findings, as described below. Since the draw ratio, E , is simply the ratio of the take-up speed, u l , to the initial speed, U O , accurate determination of E requires accurate measurement of these two speeds. Of these two, uo presents the greater problem experimentally because of the tendency of the fluid jet to “swell” as it emerges from the die. The magnitude of this die swell, which has been predicted by Nickel1 et al. (1974) and measured by Batchelor et al. (1973), corresponds to a diameter increase of 13.5%. This translates into a speed at the point of maximum die swell approximately 29% below the average speed at the die exit. The speed a t maximum die swell would appear the more appropriate one with which to define E, since it corresponds to (a) the lowest speed and (b) the speed where the velocity profile is more nearly flat (the flat velocity profile assumption is embodied in the analyses of Matovich and Pearson (1969) and Gelder (1971)). This basis for E does add experimental complexity because it requires that the die swell be measured each time, since the tensile force in the filament tends to suppress the extent of the die swell from that of the predictable tension-free value. In any case, the present study includes die swell measurements and thus should permit one to determine experimentally which definition for E is more suitable, since the resulting two values for the critical draw ratio can be compared to the theoretical value of 20.2. Experimental Section
The fluid used was a silicone oil (a dimethylpolysiloxane, #200, made by Dow Corning Co.) whose shear viscosity, determined with a Brookfield viscometer, is 1000 P. Previous stress relaxation measurements by Weinberger and Goddard (1974) with the fluid confirmed the fluid’s Newtonian character. The extensional-flow apparatus consisted of a syringe pump mounted in a vertical position on an adjustable jack and a take-up roll with a variable speed motor; the apparatus is similar to one used by Weinberger (1970). The sili-
cone oil was extruded downward through a circular nozzle (LID = 2 and i.d. = 2.8 mm) and continuously wound up tangentially onto a rotating cylinder with a diameter of 50.4 mm (see Figure 1).The silicone oil wasscraped off the cylinder with a doctor blade positioned approximately half a revolution around from the contact point. The principal experimental variables and their ranges were: length of fluid filament, 20 or 40 mm; take-up speed, 15 to 80 rpm; volumetric flow rate, 30.8, 44.4 or 64.1 mm3/ sec. For a given volumetric flowrate and filament length, the take-up speed was increased until oscillatory variation in the fiber diameter was observed through a cathetometer. At this point, the cylinder speed and period of resonance were measured and photographs were taken of the resonating filament a t moments of maximum and minimum filament diameters. The take-up speed was then increased and the above procedure repeated. Diameter measurements were taken a t 10 mm from the die exit for all runs a t a filament length of 20 mm and a t 10 and 20 mm for the run a t a filament length of 40 mm. Filament diameters were measured from the photographs to obtain the diameter ratio, DR, as a function of E , the draw ratio. DR was correlated with E by a linear least-squares fit and the intersection of this fit with a line of DR = 1, that is, nonresonance, determined the criticaI draw ratio, Ecrit, or draw ratio a t which onset of resonance occurs. This technique for experimental determination of Ecrit is much less ambiguous than the subjective technique of visual observation. No elaborate precautions were taken to control the temperature of the spinning filament, but the ambient air temperature was nominally controlled a t 22(40.3)’C. Calculations by Weinberger (1970) showed that the temperature rise from viscous dissipation of energy in the spinning threadline could be neglected. Results and Discussion Experimental values for the severity of draw resonance, in terms of filament diameter ratio, as a function of draw ratio, are shown in Figure 2. The various slopes correspond to runs with different experimental conditions. Although the conditions were different, all of the runs yield a value for the critical draw ratio, Ecrit,of roughly 17.2. These values for Ecritare based upon a jet velocity, UO, calculated as velocity at the die. Recall that this initial velocity yields lower values for Ecritthan does the measured die swell velocity; see Table I. These two sets of values for Ecritbracket the predicted value of 20.2, thereby corroborating the theoretical analysis. Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
335
Table I. Variation in Tensile Force and Critical Draw Ratio Flowrate,
Filament Force Force at mm3/ length, at die, wind-up, (E),,,, sec L , mm dyn dyn ul/uo Q,
Run no. 1 2 3 4
64.1 44.4 30.8 30.8
20 20 20 40
300 217 158 112
262 177 122 48
17.0 17.3 17.2 17.2
(E),it,
Run no. 1 2 3 4
64.1 44.4 30.8 30.8
20 20 20 40
0.6 0.9 1.3 2.2
21.9 22.3 22.2 22.2
Resonance times, T , sec Exptl
Pred
1.2 1.8 2.4 3.4
0.9 1.3 1.9 3.8
Comparison of the experimental conditions of Table I with the slopes of Figure 2 reveals an apparent relation between tensile force and severity of draw resonance. For the run with the lowest tensile force, a doubling of the draw ratio past Ecrit resulted in a diameter ratio less than 1.75, whereas for the run with the highest tensile force, this same diameter ratio was reached when the draw ratio was increased by less than 10%. Presumably, the important variable here is average tensile stress (engineering), but an answer to whether it is the fundamental causative variable requires further analytical or experimental work. As was noted earlier, the stability criterion of Matovich and Pearson (1969) was developed for the spinning condition where viscous forces dominate. This constant-force spinning condition was checked for all the experimental conditions used and the results are presented in Table I (the tensile forces were computed from eq 4 and 5 , using the measured die and wind-up velocities). We felt that a 13% change in force (run no. 1) was sufficiently small to justify that run as a test of the theory; the other data are provided for the sake of completeness. Interestingly, the onset of resonance does not appear to depend upon the constancy of force. According to Shah and Pearson (1972), the effect of gravity is a stabilizing one, since, as Re/Fr increases beyond about 2, the predicted Ecritrises. For the first three runs, Re/Fr was less than 2.5, but for run no. 4, the ratio was 10.2, leading to a predicted Ecritof roughly 50. For this run, however, the experimental value for Ecrit was still about 17.2. Although this single data point certainly does not provide a good test of this portion of the theoretical predictions, it does provide incentive for further experimental tests with fluids spun under nonconstant force conditions. The data do not allow us to choose between die velocity and “swell” velocity for the definition of Ecrit,since each definition gives a value that deviates by approximately the same amount, but in opposing directions, from the theoretical value. Incidentally, experimental measurements of die swell for E slightly less than Ecritrevealed a 14% increase in jet diameter, in good agreement with a figure of 13.5% obtained experimentally by Batchelor et al. (1973). The close relationship between residence time, 8, and period of resonance, Texp,is illustrated in Table 11. Texpis 336
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Velocity ratio, E
DR at x = 1 0 mm
DR at x = 20 mm
% increase in DR
19.3 25.3 32.2 36.9
1.07 1.26 1.54 1.68
1.09 1.31 1.61 1.75
2 4 5 4
u,/uo’
Table 11. Periods of Resonance and Residence Times Flowrate, Filament Residence length, times, Q, 8 , sec mm3/sec L , mm
Table 111. Growth of Disturbance with Axial Position (L = 40 mm, DR = Diameter Ratio)
roughly twice 8, for a variety of experimental conditions. In addition, the calculated values for 8 do not include the die swell volume and are certainly low. In any case, the significant result here is the apparent correlation between 8 and Texp, a result expected from eq 8 and 9. Furthermore, the experimental and predicted periods of resonance agree within 25%, thereby providing additional corroboration of the Pearson analysis. One question that arose during the experimental phase of our work was whether or not the instability occurs as a radius disturbance near the die which grows as the material moves from the die to the take-up system. To answer this question, diameter measurements were taken a t both 10 and 20 mm from the die for the 40-mm filament length test, Table 111. For draw ratios ranging from 20 to 40, the diameter ratios for both distances (at a given E ) agreed within 5%, approximately equal to the uncertainty of the measurement. Thus, the physical picture appears to be closer to one of an oscillatory instability occurring over the full filament length rather than one of a growing instability. Acknowledgment The technical assistance of Mr. L. Haas is gratefully acknowledged. Nomenclature A = cross-sectional area of fluid filament, mm2 CI,C2 = constants of eq 4 DR = filament diameter ratio E = draw ratio, U ~ / V O Ecrit = draw ratio a t onset of draw resonance F = tensile force in filament Fr = Froude number, uo2/gL L = filament length, mm Q = volumetric flowrate, mm3/sec Re = Reynolds number, pLuo/3p T = period of resonance, sec T e x p= observed period of resonance, sec Tpred= predicted period of resonance, 1.35L/uo,ln E , sec u = axial velocity, mm/sec uo = average axial velocity a t die, mm/sec uo’ = average axial velocity a t point of die swell, mm/sec u1 = take-up velocity, mm/sec We = Weber number, 2 p / a ( Q / T ) ~ / ~ u o ~ / ~ x = axial distance, mm Greek Symbols 8 = residence time of fluid element in draw-down zone, sec OCf = residence time defined by eq 8 p = viscosity of fluid, P p = fluid density, gm/cm3 u = surface tension T = total tensile stress, F/A, dyn/cm2 L i t e r a t u r e Cited Batchelor, J.. Perry, J. P., Horsfall. F., Po/ymer, 14, 297 (1973). Bergonzoni, A,, DiCresce, A. J.. J. Po/ym. Sci., 6, 45, 50 (1966). Clarke. N. S., J. Nuid Mech., 31, 481 (1968).
GeMer, D.. I d . Eng. Chem., Fundam., 10, 534 (1971). Kase, S..J. Appl. Polym. Sci., 18, 3279 (1974). Matovich. M. A,, Pearson, J. R. A,. Ind. Eng. Chem., Fundam., 8, 512 (1969). Miller, J. C., S.P.E. Trans., 3, 134 (1963). Nickell, R . E., Tanner, R. I., Caswell. E., J. FIMMech., 65, 189 (1974). Pearson. J. R. A,, Matovich. M. A,. Ind. Eng. Chem., Fundam., 8, 605 (1969). Pearson, J. R. A,, Shah, Y. T., Trans. SOC.Rheol., 16, 519 (1972). Pearson, J. R. A., Shah, Y. T.. Ind. Eng. Chem., Fundam., 13, 134 (1974).
Shah, Y. T., Pearson, J. R. A.. Ind. Eng. Chem., Fundam.. 11, 150 (1972). Trouton, F. T.. Proc. Roy. SOC.Ser. A, 77, 426 (1906). Weinberger, C. B., Ph.D. Dissertation, University of Michigan, 1970 Weinberger, C. B., Goddard, J. D., Int. J. Multiphase Now, 1, 465 (1974).
Received for reuiew January 20, 1975 Accepted June 30,1975
A Model for Gas-Liquid Slug Flow in Horizontal and Near Horizontal Tubes Abraham E. Dukler’ and Martin G. Hubbard Chemical Engineering Department, Universify of Houston, Houston, Texas 77004
A model is presented which permits the prediction in detail of the unsteady hydrodynamic behavior of gas-liquid slug flow. The model is based on the observation that a fast moving slug overruns a slow moving liquid film accelerating it to full slug velocity in a mixing eddy located at the front of the slug. A new film is shed behind the slug which decelerates with time. Mixing in the slug takes place first due to the mixing eddy and then due to the usual diffusion due to turbulence. The model predicts slug fluid velocity, velocity of propagation of the nose of the slug, film velocity as a function of time and distance, length of the slug, film region behind the slug, and mixing eddy and shape of the surface of the film region. Agreement with experimental data is good.
Introduction Gas-liquid flow in conduits is a more complex phenomenon than single-phase flow primarily because the spacial distribution of the two phases is unknown and difficult to specify quantitatively. A variety of such distributions have been qualitatively described by numerous investigators (Hewitt and Hall Taylor, 1970; Hoogendoorn and Welling, 1965; Kosterin, 1949). Hubbard and Dukler (1966) suggested that these many observed patterns really represented the superposition of only three basic distributions: separated, intermittent or slug, and distributed flows. A still unresolved problem is the prediction of the particular combination to be expected given the flow rates, fluid properties, conduit size, and inclination. A useful empirical correlation has been proposed (Baker, 1954). Intermittent or slug flow exists over a wide range of flow rates for moderate pipe sizes in a horizontal configuration. Such a flow pattern is inherently unsteady with large time variation of the mass flow rate, pressure, and velocity at any cross section normal to the tube axis. This is so even when the gas and liquid flow to the system is steady. As a result, processes of heat and mass transfer are also unsteady with substantial fluctuations in temperature and concentration. This poses special and difficult problems for the designer. I t is the purpose of this paper to present a systematic model for the hydrodynamics of slug flow from which the time varying behavior can be predicted. The model presented here is based partly on the work of Hubbard (1965) which was presented by Hubbard and Dukler (1968) but unpublished. That version has been substantially revised to eliminate certain empirical aspects and certain simplifications made in the original work which have now been found to be unnecessary. Some Related Research A number of early investigations (Govier and Omer, 1962; Hoogendoorn, 1959; Martinelli and Nelson, 1948)
measured pressure drop and in some cases average holdup under conditions where slug flow was observed to exist. A careful study of these data and the experimental techniques reveal serious limitations in the results. In many instances a portion of the pressure measuring system was not in fully developed slug flow. In some cases the slug length (estimated by methods to be reported here) exceeded the distance between pressure taps. In most of these studies only air space existed between pressure taps at times. In all cases pressure drop was measured using highly damped manometers to smooth out the fluctuations. Furthermore, these studies provide no information on the characteristics of slug flow such as frequencies, spacing, slug length velocities, etc. The earliest attempt to study the details of slug flow were by Kordyban (1961). He proposed a simple model where slugs move at the velocity of the gas and “skate” or slide over the top of a substrate film without interaction or mixing between the slug and film. Experimental data were taken in 6 ft long tubes of 0.315 and 0.420-in. i.d. with pressure taps separated by 1 ft. Thus the validity of the data is seriously in question. Based on their concept, the authors develop an expression for pressure drop but this is in poor agreement with their data. In fact the classical Martinelli correlation is shown to be a better predictor of the data than their own correlation. In 1963 Kordyban and Ranov (1963) reported new experimental data from a 10-ft long pipe of 1.25-in. diameter. They generate slugs by forcing the two-phase mixture through a riser before entering the test section. If this riser had not been used it is doubtful if they would have been able to observe slug flow, especially a t the higher flow rates in their work. In this instance the pressure measuring technique produced data that were more meaningful. However, comparison with the earlier Kordyban model leads the authors themselves to suggest that the model is not adequate. The beginnings of an attempt to understand the details Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
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