Ind. Eng. Chem. Res. 1989,28, 1910-1912
1910
7664-38-2;HCl, 7647-01-0.
ed.; CRC Press: Boca Raton, FL, 1982-1983; p D-173.
Jian-Zhong Zhou,* Li-Li Jiao, Yuan-Fu Su
Literature Cited Makin, A. B. Isothermal Solubility of the Ternary System Na2HPO,-NaCI-H,O at 25 OC. Zh. Neorg Khim. 1957,2, 2794-7. Onoda Cement Co. Br. Patent 1,036,207 1966. Weast, R. C., Ed. CRC Handbook of Chemistry and Physics, 63rd
Chemical Engineering Research Centre East China Institute of Chemical Technology Shanghai 200237, China Received for review March 23, 1989 Accepted August 20, 1989
Theory for the Extension of a Newtonian Filament by Gravity and/or a Take-up Force The extension of a Newtonian filament is a problem of classical interest. For the case in which viscous and gravity forces are dominant, Petrie lays out three possible solutions, one involving the hyperbolic sine, another involving the sine, and a degenerate case of both (a hyperbola). Starting with Trouton, the hyperbolic sine solution has always been selected for analyzing isothermal spinning experiments under all conditions of take-up force. That is not always correct. For the gravity-only spinning or for gravity augmented by a small take-up force, the solution must be of the sine form. A dimensionless group $ (involving the ratio of the gravity force to the viscous force, and the draw ratio) is derived that leads to the hyperbolic sine if $ 1, to the hyperbola if IF. = 1, and to the sine if $ > 1, thus giving a criterion for selecting the governing solution. The extension of an isothermal Newtonian filament with or without a take-up force is a classical problem in Newtonian fluid mechanics. Theory for low-speed isothermal Newtonian spinning provides the basis for understanding and analyzing complicated isothermal viscoelastic spinning experiments. Neglecting inertial, drag, and surface tension forces, Trouton (1906) presented the governing nonlinear differential equation that describes slow-speed isothermal Newtonian spinning. He noted that a hyperbolic sine function satisfies the governing equation, approaching a hyperbola in the limit of long filaments. The first detailed discussion was by Marshall and Pigford (1947), who showed two changes of variables by which one reduces the nonlinear equation to a linear one. A succinct review was presented by Petrie (1979), in which he identified and sketched three solutions. One involved the hyperbolic sine, another the ordinary sne, and the third, a degenerate form of the first two (a hyperbola). Petrie, however, did not suggest any criterion for choosing the form of the solution that would depend upon the imposed boundary conditions. In the literature, the hyperbolic sine solution has always been selected for analyzing the isothermal spinning experiments under all conditions of take-up force. This choice, starting with Trouton (1906), carried through the work of Matovich and Pearson (1969), Donnelly and Weinberger (1975), and Prilutski (1984). Our concern is with the analytical solutions. The choice of boundary conditions is, therefore, an important part of the problem. The initial velocity is the first condition; however, there are several choices for the second. In the most plausible cases, one imagines either an imposed take-up force or a gravity-only flow in which the force of gravity ceases to act at some point. Ziabicki (1976) identified another case by imagining that the filament “piles up”, producing a zero velocity on a stationary receiving surface. In the related experimental work, Pate1 and Bogue (1989) very briefly showed that the hyperbolic sine form cannot, in fact, accommodate the boundary conditions for problems in which the dominant force is gravity. They stated a criterion to choose the governing form of the solution. The present work expands on that analysis. Also,
the criterion to choose the governing form of the solution is derived here based solely on the operating conditions. The mathematical proof of the criterion is presented in the Appendix. Governing Equations and Solutions The governing dimensionless differential equation, neglecting the inertial, drag, and surface tension terms, is
where u* = u/uo, x* = x / L , uo is the velocity at x = 0, and L is the length of the filament. We will often use the symbol NG = (pgL2/3vuo),which is a Stokes number. It is the ratio of the gravity force to the viscous force. Marshall and Pigford (1947) described the changes of the variables that are necessary to reduce this nonlinear equation to a linear one. (One first lets p = du*/dx* and p(dp/du*) = d2u*/d(r*)2,leading to the Bernoulli equation, which is a nonlinear first-order differential equation, and the substitution z = p2, which transforms it to a linear equation.) An intermediate equation is instructive: (du*/dx*)2 = 2N&* C(U*)~ (2)
+
The character of the solution depends on the sign of C. The three possible solutions were summarized by Petrie (1979): u* =
(2)
sinh2 (A(x* + B ) )
(3) (4)
u* = (NG/2)(x*
+ B)2
(5) where A and B are constants of integration. Eauation 5 follows as a degenerate case from echer eq 3 0; eq 4 by letting A approach zero. If we restate eq 5 in terms of the dimensionless radius R *, it will be of the form R*(x* + B ) = K , leading to the designation “hyperbola solution”. I. Cases Involving Falling Filaments, with or without an Imposed Take-up Force. The physically
0888-5885/89/2628-1910$01.50/0 0 1989 American Chemical Society
sine solution
hyperbolic sine solution
hyperbola solution
\Ir> 1
9< 1
+=1
20
1
.... . . .-pi
FL=0 or small DOS
sine solution
hyperbola solution
hyperbolic sine solution
q =1
\y< 1
\t >1
(x/L)
I
F, =neg
I
F,
I
neg
FL=neg
I
Figure 1. Nature of experiments involving gravity and/or imposed end force. (A, top) Experiments involving falling filaments or spin lines. (B, bottom) Imagined experiments involving a receiving surface to impart a zero velocity (probably not physically attainable).
important cases are sketched in Figure 1A. For pure gravity spinning, the boundary condition is FL= 0 (leading to du*/dx* = 0 at x* = l ) , which can be approximately achieved by passing a filament through a neutrally buoyant bath or by filament breakage at the end, as in Trouton’s experiments (1906). In either case, it is possible to approximately achieve the condition of zero stress at the end, leading to du*/dx* = 0 at x* = 1. The other boundary condition is u = uo (or u* = 1) at x * = 0. One cannot impose the condition du*/dx* = 0 at x * = 1in the case of the hyperbolic sine solution. The derivative of eq 3 leads to (sinh y)(cosh y) on the right-hand side, with y = A(x* + B ) . This can be zero only by making sinh y equal to zero at the end of the filament. But when we return to eq 3, it gives a zero velocity at the end of the filament. More possibilities are available in the case of the sine solution (eq 4). We can make the derivative zero either by making the argument of sin y zero (not acceptable for the reasons noted above) or by setting the argument of cos y equal to s / 2 . The latter choice is the correct one and leads us to B = [(a/(2A)) - 11,where the constant A can be obtained by using the initial condition: 1=
(2)
cos2 A
By use of eq 6, it can be shown for the case of gravityonly spinning that a maximum in the elongation rate profile occurs for NG > ( ~ / 2 =) ~2.47. For values of NG < 2.47, the elongation rate will monotonically decrease to zero at the end. It can also be shown that only the sine solution will permit a maximum in the elongation rate profile. It is interesting to note that Trouton (1906) did not report the sine solution to the differential equation, which would have been the correct solution for his gravity-induced extension. He was, however, able to approximately fit his diameter data by using the degenerate hyperbola solution, which can be arrived at from both the hyperbolic sine and the ordinary sine solution in the case of a long filament. As discussed below, it is possible to apply a small take-up force (small externally imposed draw ratio) and still remain within the framework of a sine solution. In such cases, the solution is given by the general solution (eq
Figure 2. Comparison of different solutions for the velocity of the falling filament a t constant Stokes number and various imposed = 7.11, Q = 3.60 (sine). (2) Gravity draw ratios. (1) Gravity only: plus small take-up force: p = 11.42, Q = 1.77 (sine). (3) Gravity plus medium take-up force: p = 17.33, Q = 1.0 (hyperbola). (4) Gravity plus large take-up force: @ = 23.6, Q = 0.67 (hyperbolic sine).
4), with constants A and B determined by the two implicit equations (7)
Under other conditions (high draw ratios), the hyperbolic sine solution applies. An important special case of the hyperbolic sine solution occurs in the case of a constant force spin line. This case can be arrived at by setting A = (1/2)(1n p) and letting B be large enough such that sinh (AB) = cosh (AB) = A / (2/NG)l12, a large B corresponding to a small NG. Under these conditions, eq 3 reduces to the required solution u* = exp(x* In p). A final case comes from the hyperbola solution (eq 5). This degenerate case requires only one boundary condition, namely, u* = 1at x* = 0, leading to B = (2/NG)l12 as the constant to be inserted in eq 5. The three typical solutions are plotted in Figure 2 as a function of the gravity number (NG) and the draw ratio (p). The criterion for deciding among the cases will be discussed later. 11. Cases Involving Falling Columns Impinging on a Stationary Plate. These cases were motivated by a discussion in Ziabicki’s book (1976) and are sketched in Figure 1B. It is unlikely that any of them can be achieved in the idealized way shown, but the qualitative concept of piling up on a stationary plate is certainly plausible. The boundary conditions are u* = 1 at x * = 0 and u* = 0 at x* = 1. In all cases, the second boundary condition leads to B = -1. The constant A comes from the first boundary condition and involves an implicit solution of the equation 1 = (NG/2A2)(sinh2 A) or 1 = (NG/2A2)(sin2A). Criterion for Determining the Form of the Solution Because the three solutions are quite different, one needs a criterion to decide which one to select for a particular set of operating conditions (i.e., boundary conditions). The key to defining such a criterion comes from considering the behavior of the hyperbola. Combining the result B =
1912 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989
(2/NG)lI2with eq 5 and noting that the draw ratio p occurs at x * = 1, one finally obtains the result \k = 1, where \k is defined by
By definition, \k = 1 leads to the hyperbola solution. As shown in the Appendix, \k > 1 leads to the sine solution and \k < 1 to the hyperbolic sine solution. The criterion developed above works also for the cases involving impingement of a fluid on a stationary plate. For such cases, = 0 and the definition of \k reduces to \k = (NG/2). It can be shown that the criterion for choosing among the cases is as before: \k < 1, hyperbolic sine; 9 = 1, hyperbola; \k > 1, sine. For pure gravity spinning, the draw ratio is determined "by nature" and is not an independent variable to be imposed. We designated this draw ratio by pG, and it is of interest to consider which limiting values constrain it. One limit is physically clear: as NG is made small (by using a very short spin line or a very high viscosity fluid), little m. In another draw-down occurs and PG 1,while \k limit, Pc increases as the gravity number (NG) increases but the dimensionless number (\k) tends toward a limit (2.4674...). The limit involves such long filaments and/or small viscosities (for example, at NG = 1O00,9 = 2.59, still short of a limit) that it is probably of little practical use, due to breakage of the filament by surface tension or the dominance of inertial effects. It is, however, of theoretical interest and can be derived by considering eq 6, from which one can determine that A and PG are connected by the relation cos A = 1/PG1I2,with the limit that A 7r/2 as NG m. Inserting these relations into the definition of \k (eq 9), one is finally led to the result that \k (x/2I2 as NG m. These results were confirmed by using computer simulations. These results are the basis for the qualitative analysis developed previously. Namely, for pure gravity spinning (where \k > 2.467) or for spinning with a small imposed draw ratio, P (such that \k > 11, the sine solution is valid. Conversely, for large draw ratios, one must use the hyperbolic sine result. Much of the experiment work in the past has, in fact, been done a t draw ratios and Stokes numbers such that \k < 1 (Donnelly and Weinberger, 1975). The selection of the hyperbolic sine solution is correct in those cases, but this cannot be assumed in general and cannot be assumed for gravity-only spinning. Patel and Bogue (1989) successfullyused a generalized sine function to fit the diameter profiles obtained by gravityonly spinning of low molecular weight polypropylenes as suggested by the above proposed criterion. The generalized hyperbolic sine function could not fit the data a t all because the hyperbolic sine solution cannot permit a maximum in elongation rate profile.
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Acknowledgment The author acknowledges with thanks helpful discussions with Prof. Donald C. Bogue during the course of this work. The author also acknowledges the financial support
of both the Center for Materials Processing a t the University of Tennessee and the Exxon Chemical Company. Appendix: Proof That \k > 1 (or \k < 1) Leads to the Sine (or Hyperbolic Sine) Forming the square root of the ratio of the two boundary condition equations for the sine solution (eq 8 divided by eq 7), substracting 1, inverting, and combining with eq 7 leads to
Invoking the identity sin a - sin y = 2 cos ((1/2)(a + y)} sin ((1/2)(a - y)) and noting that the left-hand side of eq A-1 is just \k, one obtains finally
where C = cos ((1/2)(A+ 2AB)) and D = (sin E ) / & with E = A/2. Now the absolute values of C and D are always 5 1. (We only need to consider the absolute values because of the square in eq A-2.) We can thus conclude that the bracket in eq A-2 is always I 1, leading to the result that \k I 1. This, then, is the criterion for the sine solution. By an analogous derivation, one can show that \k I1 is the criterion for the hyperbolic sine solution. In the present context, we can write simply 3 > 1 for the sine solution and \k < 1 for the hyperbolic sine solution, since both degenerate to the hyperbola for \k = 1. Literature Cited Donnelly, G. J.; Weinberger, C. B. Stability of Isothermal Fiber Spinning of a Newtonian Fluid. Ind. Eng. Chem. Fundam. 1975, 14, 334-337. Marshall, W. R., Jr.; Pigford, R. L. Bessel Functions. In The A p -
plication of Differential Equations to Chemical Engineering Problems; University of Delaware Press: Newark, 1947; p 73. Matovich, M. A.; Pearson, J. R. A. Spinning a Molten Threadline. Ind. Eng. Chem. Fundam. 1969,8, 512-520. Patel, R. M.; Bogue, D. C. Measurement of the Elongational Properties of Polymer Melts by Gravity Spinning. J. Rheol. 1989, 33(4), 607-627. Petrie, C. J. S. Theory of Spinning. In Elongation Flows; Pitman: London, 1979. Prilutski, G. M. The Rheology of Polymeric Liquid Crystals. Ph.D. Dissertation, University of Delaware, Newark, 1984. Trouton, F. T. On the Coefficient of Viscous Traction and Its Relation to that of Viscosity. Proc. R. Soc. London 1906, A-77, 426-440. Ziabicki, A. Melt-Spinning. In Fundamentals of Fibre Formation; Wiley: New York, 1976.
Rajen M. Patel Department of Materials Science and Engineering University of Tennessee Knoxville, Tennessee 37996 Received for review February 21, 1989 Revised manuscript received August 13, 1989 Accepted September 26, 1989
