Ind. Eng. Chem. Fundam. 1981, 20, 147-149 Dickinson, F. M.; Monger, 0. P. Blochem. J . 1973, 131, 261. Eberson, L. E.; Weinberg, N. L. Chem. Eng. News Jan 25, 1971, 40, 40. Friday, D. K. M.S. Thesis, University of Vkglnla, Charlottesville, VA, 1980. Janik, B.; Eking, P. J. Chem. Rev. lSS8, 68, 295. Kelly, R. M.; Kirwan, D. J. Biotech. Bbeng. 1977, 19, 1215. Mahler, H. R.; Cordes, F. 0. “Biological Chemistry”, 2nd ed., Herper and Row: New York, 1971.
147
Olander, D. R., AIChE J., 19S0, 6,233. Prescott, J. H. Chem. Eng. Nov 8, 1965. 238. Searby, P. E. M. S. Thesis, University of Virginia, Charlottesville, VA, 1977.
Received for review May 27, 1980 Accepted February 6,1981
Effects of Inertia, Surface Tension, and Gravity on the Stability of Isothermal Drawing of Newtonian Fluids J. C. Chang and
M. M. Denn*
Department of Chemical Engineering, Universe of Delaware, Newark, Delaware 1971 1
F. 1.Geyllng Bell Laboratories, Murray Hill, New Jersey 07974
The stabili of continuous isothermal drawing of Newtonian fluids has been analyzed, taking inertial, surface tension, and gravitational effects into account. The results agree qualitatively with an earlier analysis, but there are important quantitative differences. Agreement with experiment is good.
Introduction An instability known as draw resonance is sometimes observed in processes involving the continuous drawing of liquid filaments. The instability is characterized by oscillations in tension and drawn filament diameter with a well defined period and amplitude. Some illustrative data and general reviews of theory and experiment are contained in the works of Petrie and Denn (1976), Kase and Denn (1978), and Denn (1980). Most experimental studies have focused on polymeric liquids, because of applications in processes such as fiber spinning and extrusion coating. The first theoretical analyses, however, were carried out for Newtonian liquids, and the Newtonian fluid limiting case continues to be of interest because of possible applications in glass fiber drawing, as well as because it provides an analytical framework in which to explore certain mechanisms without the addition of rheological complexity. One set of experiments on draw resonance has been reported on a truly Newtonian fluid (Chang and Denn, 1979), and Donnelly and Weinberger (1975) and D’Andrea and Weinberger (1976) have reported draw resonance experiments on a silicone fluid that is so slightly viscoelastic over the deformation rate range employed that a Newtonian theory might be adequate. These experiments were carried out isothermally, so they provide some insight into underlying mechanisms, but they are not directly applicable to practical processing. Glass and polymer fiber drawing always occur under nonisothermal conditions, where the effect of heat transfer on temperature-dependent physical properties is very important. The isothermal, Newtonian theory of draw resonance predicts instability when the area reduction (drawdown) ratio exceeds 20.2 for conditions in which gravity, surface tension, inertia, and air drag are unimportant, and the experiments of Donnelly and Weinberger seem generally to codirm this value. The experiments of Chang and Denn 0 196-431 3/81/1020-0 147$0 1.2510
and DAndrea and Weinberger were done under conditions where the first three of these effects could not be ignored, however. Here, agreement with thoretical values obtained by Shah and Pearson (1972) is not good. We report here new calculations for the onset of draw resonance in the isothermal drawing of a Newtonian liquid, using two computational procedures that differ from the method of Shah and Pearson. The general qualitative effects of gravity, surface tension, and inertia reported by Shah and Pearson are confirmed, but there are important quantitative differences that bring the theory and experiments closer. The work reported here combines the results of independent studies done at the University of Delaware and at Bell Laboratories. Spinning Equations The asymptotic dimensionless stress, momentum, and continuity equations for isothermal Newtonian spinning are (Kase and Matsuo, 1965; Matovich and Pearson, 1969)
(3)
Here T, u, and a are dimensionless total axial stress, axial velocity, and area, respectively (note that some authors use a to denote radius), and 0 and 5 are dimensionless time and axial position. The dimensionless groups, Reynolds, Froude, and Weber numbers are, respectively Re = pLV~/3q (44
Fr = vQ2/gL We = 2a01/zvQ2p/a1/2a 0 1981 American Chemical Society
(4b) (44
148
Ind. Eng. Chem. Fundam., Vol. 20, No. 2, 1981
Table 11. Critical Draw Ratio Computed Using Various Numerical Methods at Two Spinning Conditions
Table I. Critical Eigenvalue at D R = 20.2 with R e = ReIFr = ReIWe = 0 11 21 31 41 49
-
extrapolation, N -,
--1.0959 -0.1489 -0.0547 -0.0294 -0.0205 0
-12.594 -13.561 -13.808 -13.895 -13.930 -14.012
These equations neglect air drag. The origin is taken at the point of maximum extrudate swell, where the area and velocity are presumed known. The validity of these boundary conditions at steady state is discussed by Fisher et al. (1980) in a numerical study of the jet exit region and the approach to the asymptotic equations. The velocity is fixed at the takeup. The dimensionless boundary conditions are then u = a = 1 at l = 0 (54 u = DR at [ = 1 (5b) D R is the draw ratio. The steady-state solution to eq 1through 5 is obtained by setting ala6 = 0 in eq 2 and 3, specifying the initial stress, To,and solving the resulting initial value problem iteratively by adjusting To until eq 5b is satisfied. Stability Equations The response of the system to infinitesimal perturbations is obtained by substituting the following eigenfunction expansions into eq 1through 3 and 5 and neglecting nonlinear terms (Denn, 1975)
a(4,O) = a,([)
+ ?eAks4k(S)+ 2eAk*%k*(t) ( 6 4
u(f,O) = u,(O
+ k2 =l
k=l
k=l
2
+ k = l eXk*'+*&) (6b)
Subscript s denotes the steady-state solution and * denotes the complex conjugate. We thus obtain the following linear eigenvalue problem
( 4 k ) and (+k} are complex functions, and the eigenvalues (hk) are, in general, complex numbers. The system is unstable to infinitesimal disturbances if the real part of any Ak is positive. Note that the term Re/Fr does not enter explicitly into the eigenvalue equations because it appears as a nonhomogeneous term in eq 2; the grativational effect is contained in u s ( [ ) . Solution Two computational methods have been used to solve eq 7. In one, following Gelder (1971), finite-difference ap-
critical draw ratio finite difference conditions with Richardson direct Shah and R e RelWe RelFr extrapolation integration Pearson 0 0
0 1
10 0
32.55 5.5
32.43 5.7
52 9
Table 111. Effect of Interactions and Sensitivity of Critical Draw Ratio to Changes in h.and We. The Last Entry Corresponds to the Experiments of Chang and Denn (1979)
Re
Fr
We
AI
DR
0.0244 0.0244 0.0244 0.0244
m
m
0.003
-
0.003
0.0979 0.0979
14.13 19.19 12.68 17.99
25.0 41.8 17.2 29.9
-
R@orkorB h We
Figure 1. DR112as a function of Re, Re/Fr, and Re/ We for Re/Fr = Re/ We = 0, Re = Re/ We = 0, and Re = Re/Fr = 0, respectively. The solid lines are the present calculations, and the dashed lines are the calculations of Shah and Pearson (1972).
proximations were used to replace the derivatives, leading to a matrix eigenvalue problem, and the eigenvalues were obtained for increasingly more discretization points, N . Richardson extrapolation was then used to obtain the limiting value of the critical eigenvalue for N tending to infinity. A sequence of such calculations is shown in Table I for DR = 20.2 and Re = Re/Fr = Re/ W e = 0. To within the accuracy of the computational scheme, this is the critical draw ratio for these conditions, since the real part of the eigenvalue, XR, passes from negative to positive. The critical eigenvalue is always found to be the one with the smallest modulus. The second method, following Fisher and Denn (1976), is the direct numerical integration of eq 7, normalizing d+k(0)/d[ to unity and varying hk until the boundary condition at takeup is satisfied. For Re = Re/Fr = Re/ W e = 0, this method gives a critical draw ratio of 20.21 with hR = 0, XI = 13.989, which agrees well with the calculation in Table I. Two further comparisons of the two methods are shown in Table 11, together with values read from the graphs of Shah and Pearson (1972). Shah and Pearson integrated the linearized initial-boundary value problem numerically for forced responses to harmonic disturbances and used the divergence or decay of the time-dependent solutions to determine the neutral stability criterion. Agreement between the two methods used here is good, but the results differ from those reported by Shah and Pearson. The computed critical draw ratio is shown in Figure 1 as a function of Re, Ref Fr, and Ref W e for the three cases Re/Fr = R e / W e = 0, Re = R e / W e = 0, and Re = Re/Fr = 0, respectively. These calculations, and all others reported here subsequently, were obtained using the direct integration method. The curves reported by Shah and Pearson are also shown in Figure 1. The qualitative features are the same, but there are quantitative differences.
Ind. Eng. Chem. Fundam., Vol. 20, No. 2, 1981
149
Table IV. Comparison of Theoretical Onset of Instability t o Experiments of Donnelly and Weinberger (1975) exptl computed
run 1 2 3
4
u,,mm/s 10.41 7.21 5.00 5.00
L,mm Re 20 20 20 40
X lo4 Fr X lo4 5.53 6.94 4.80 2.54 3.33 1.28 6.67 0.64
We X lo4 135.5 65.0 31.3 31.3
Table V. Comparison of Theoretical Onset of Instability t o Experiments of D’Andrea and Weinberger (1976) comRe X Fr X We X Re/ R e / exptl puted run lo4 lo4 lo4 Fr We DR DR 25.60 1.27 103.7 20 0.25 26.0 35.0 31.50 1.03 103.7 30 0.30 37.1 41.8 35.70 0.91 103.7 39 0.35 46.2 47.2 18.30 0.79 46.8 23 0.39 26.9 32.4 19.20 0.55 49.3 34 0.39 43.0 41.1 40.70 0.80 103.7 50 0.39 52.0 54.0 10.4 33 0.81 17.8 27.1 8.39 0.25 5.3 48 0.81 31.9 33.6 4.32 0.09 34.30 0.50 42.6 69 0.81 46.9 44.6
Shah and Pearson’s critical draw ratio consistently exceeds ours for all finite Re/ W e and RelFr, and for Re < 0.02. This could be a consequence of numerical damping, introduced into their numerical integration scheme to supress numerical instability, and the inability to discern long-term divergence of transient responses from a limited number of cycles. It is not clear why their critical draw ratio is below ours for Re greater than about 0.04, where the stability boundary curve rises steeply. Comparison with Experiment Chang and Denn (1979) observed draw resonance in the isothermal spinning of a Newtonian corn syrup. Spinning conditions corresponded to the last row in Table 111, where the effect of interactions and the sensitivity to changes in Fr and W e is illustrated. The experimental onset of draw resonance was at DR = 26, and at D R = 24.5 in a similar run but with a 5 9% reduction in spinline length. The period of oscillation computed from the linear theory (tP = 2?rL/uoXI)was 0.4 s, while the observed period was 2.3 s. While there is no a priori reason to expect close agreement between the period predicted by linear theory and the period of the finite-amplitude limit cycle that follows instability (compare Denn, 1975), a difference of this magnitude is surprising. Limit cycle analyses for Re = Re/Fr = Re/ W e = 0 by Fisher and Denn (1975) and Ishihara and Kase (1975) predict a period that is only slightly larger than that given by linear theory at draw ratios just beyond the onset of draw resonance. Weinberger and co-workers have carried out isothermal draw resonance studies on a silicone oil that is only slightly viscoelastic and might be expected to behave approximately like a Newtonian fluid. The experimental data of Donnelly and Weinberger (1975) are shown in Table IV, together with the calculated critical draw ratio and the period of oscillations computed from the linear theory. The experimental draw ratio is based on the velocity in the die; the critical draw ratio in all cases taking extrudate swell into account was approximately 22. The calculated
DR
tp, s
DR
AI
tp, s
17.0 17.3 17.2 17.2
1.2 1.8 2.4 3.4
20.6 20.9 20.9 26.6
14.63 14.92 15.30 19.36
0.8 1.2 1.6 2.6
period is roughly two-thirds of the experimentally observed period, which is much better agreement than in the experiments of Chang and Denn. The experiments of D’Andrea and Weinberger (1976) are compared with theory in Table V. (It should be noted that the jet diameters reported by D’Andrea and Weinberger are in fact radii.) The experimental draw ratio is based on the velocity in the die. Neither extrudate swell nor the period of oscillations was reported. The agreement between computed and experimental draw ratios is quite close in nearly all cases. If extrudate swell were accounted for, then the experimental onset of draw resonance in most runs would probably be at a slightly larger critical draw ratio than that computed from theory. (This conclusion differs from that originally reached by D’Andrea and Weinberger, who compared their experiments to Shah and Pearson’s calculations for Re = Re/ W e = 0). Conclusions Linear stability theory predicts the onset of draw resonance for isothermal spinning of Newtonian fluids, including the effects of gravity and surface tension; experiments in a range that would test the Reynolds number dependence have not been carried out. The experimentally observed period does not agree with that predicted from the linear theory, however. The general qualitative features found by Shah and Pearson (1972) are correct, but their results differ quantitatively from direct numerical solutions of the linear eigenvalue problem. Acknowledgment J. C. Chang and M. M. Denn received support from the National Science Foundation under Grant ENG-76-15880. Manuscript preparation was done while M. M. Denn was on leave in the Chemical Engineering Department of the California Institute of Technology. Literature Cited Chang, J. C.; Denn, M. M. J. Non-Newtonian F/uM Mech. 1979, 5 , 369. D’Andrea, R. G.; Welnberger, C. B. AICh€J. 1976, 22, 923. Denn, M. M. “Stability of Reaction and Transport Processes”, Rentice-Hall: Engiewood CWs, NJ, 1975. Denn, M. M. Ann. Rev. FIuMMech. W60, 12, 365. Donneliy, G. J.; Weinberger, C. B. Ind. Eng. Chem. Fundam. 1975, 14, 334. Fisher, R. J.; Denn, M. M. Chem. Eng. Scl. 1975, 30, 1129. Fisher, R. J.; Denn, M. M. AIChE J . 1972, 22, 236. Fisher, R. J.; Denn, M. M.; Tanner, R. I. Ind. Eng. Chem. Fundam. lS60, 19, 195. GeMer, D. Ind. Eng. Chem. Fundam. 1971, 10, 534. Ishlhara, H.; Kase, S. J. Appl. Polym. S d . 1975, 19, 557. Kase, S.; Denn, M. M. Proceedings, 1978 Joint Automatic Control Conference”, 1978, p 11-71, Kase, S.; Matsuo, T. J . folym. Sei. WeS, A-3, 2591. Matovich, M. A.; Pearson, J. R. A. Ind. Eng. Chem. Fundam. W6g, 8 , 512. Petrle, C. J. S.; Denn, M. M. AICh€J. 1976, 22, 209. Shah, Y. T.; Pearson, J. R. A. Ind. Eng. Chem. Fundam. 1972, 1 7 , 150.
Received for review June 5, 1980 Accepted December 22, 1980
