Literature Cited
Aiba, S., Huang, S.Y., Chem. Eng. Sci., 24, 1149 (1969). Bandyopadhay, B., Humphrey, A. E., Taguchi, H., Biotechnol. Bioeng., 9, 533 (1967). Bates, R. L., Fondy, P. L., Fenic, J. G., in “Mixing,” V. W. Uhl, J. B. Gray, Eds., Vol. 1, Academic Press, New York, N. Y., 1966, Chapt. 3, p 129. Calderbank, P. H., Trans. Inst. Chem. Eng., 37, 173 (1959). Cooper, C. M., Fernstrom, G. A., Miller, S. A., Ind. Eng. Chem., 36, 504 (1944). Heineken, F. G., “On the Use of Fast-Response Dissolved Oxygen Probes for Oxygen Transfer Studies,” presented at the Third International Fermentation Symposium, Rutgers-The State University, New Brunswick, N. J., 1967.
Oldshue, J. Y., Biotechnol. Bioeng., 8, 3 (1966). Resnick, W., Gal-Or, B., Aduan. Chem. Eng., 7, 296 (1968). Richards, J. W., Progr. Ind. Microbiol., 3, 143 (1961). Rushton, J. H., Costich, E. W., Everett, H. J., Chem. Eng. Progr., 46, 395, 367 (1950). Yoshida, F., Yamaguchi, T., Hattori, K., J . Ferment. Technol., 36, 1019 (1968). Westerterp, K. R., Chem. Eng. Sci., 18, 495 (1963).
RECEIVED for review February 20, 1970 ACCEPTED January 27, 1971 Presented at the Division of Microbial Chemistry and Technology, 158th Meeting, ACS, New York, N. Y., September 1969.
Wet Spinning of Purely Viscous Fluids Mechanical Aspects Joseph Yerushalmi and Reuel Shinnarl Department of Chemical Engineering, The City College of the City University of New York, New York, N . Y . 10031
The mechanical aspects of the steady spinning of purely viscous fluids in a typical wet spinning process are examined. Approximate solutions are obtained for Newtonian fluids in situations in which the axial velocity may be regarded as independent of the radial direction. The salient result of the analysis i s the existence of a maximum filament length for any prescribed set of spinning conditions. With the fluid inertia neglected, the maximum filament length is directly proportional to the initial radius of the filament and the square root of the ratio of the viscosities of the filament fluid and the surrounding liquid.
I n a typical process for the production of synthetic fibers (Figure I ) , a molten polymer, or a polymer solution, is continuously extruded through a symmetric array of narrow orifices (the spinnerette). The resulting filaments are then drawn by a wind-up device a t a speed greater than the extrusion velocity at the spinnerette holes. Molten polymers are generally extruded into an atmosphere (air) of temperature lower than the melting range of the polymer (melt spinning), while polymer solutions are commonly drawn either through a gas phase into which the solvent evaporates (dry spinning), or through a bath of liquid into which the solvent diffuses (wet spinning). The exchange of heat (or mass) with the surrounding medium results in a considerable hardening (viscosity increase of the filament fluid). A rigorous mathematical description of the spinning process requires the simultaneous application of energy, mass, and momentum balances, together with a suitable rheological equation of state. The resulting set of equations cannot be solved at present. I n this study, we shall, therefore, confine ourselves to an examination of the steady spinning of purely viscous fluids in a typical wet spinning process. Further, we shall consider only the mechanical aspects of the flow, disregarding the coagulation mechan-
’ T o whom correspondence should be addressed. 196
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
ism and the consequent hardening of the filament. This specialized approach is adopted in view of the complexity of the general problem and in the interest of obtaining some valid insight into that problem. I t should not, of course, imply the absence of the hardening phenomenon, or that the filament fluid is necessarily purely viscous. Even considered alone, the mechanical aspects of the spinning problem presents a formidable mathematical task. In addition to the nonlinearity of the momentum equations, the main difficulty is associated with the free boundary of the filament. The free streamlines are not known in advance and must be determined as part of the solution. Further, the boundary conditions a t the free surface are themselves dependent upon the unknown orientation of the free streamlines. The problem can be simplified considerably by recognizing that, except for a relatively short region near the spinnerette hole, the transport of momentum at any section of the filament occurs predominantly in the axial direction. That is, the mechanics of the flow are governed largely by the balance between the axial inertia caused by the action of the wind-up roll, the resultant tensile stress, and the external drag exerted on the fluid by the surrounding medium. Under these conditions, the axial component of the velocity can be regarded as independent of the radial direction. This is the so-called uniform veloc-
z-direction
(a)
g J = , : Extrusion Chamber S inncrette
wind-u
fd
Filaments
u,av, v,av, &qx+-) dr
=--
-
a ar
- (rT,)
a T, +az
(2)
continuity :
l a
- -
r ar
-L-
Z
i + az r
dVI dz
(ru,) + - = 0
(3)
In the above, u, and u, are the radial and axial components of the velocity, respectively. p is the density of the filament fluid. T,,, T,, etc., are the designated physical components of the extra stress. The boundary conditions specify the value of the axial velocity at both ends of the spinning way, and the stress distribution on the free surface of the filament. The former conditions are simply,
m
u,(z = 0,r) = V , v,(z = L , r ) = VL
Figure 1. Schematic diagram of the spinning process a. General view
b. Single filament
ity approximation. For a purely viscous fluid the momentum equations can then be reduced to an ordinary differential equation which relates the axial velocity t o the spinning distance. Matovitch and Pearson (1969), who studied the steady isothermal spinning of purely viscous fluids in a typical melt spinning process, have shown that the flow problem arising from the uniform velocity approximation represents a suitable approximation to the general Droblem. The mathematical formulation of the flow problem generally parallels that of Matovitch and Pearson. However. there is an essential difference between the flow problems that arise in melt and wet spinning. The difference lies in the importance of the external drag exerted on the filament by the surrounding liquid in the latter case. I n melt spinning, the external drag is of secondary magnitude (and was, in fact, neglected in the analysis of Matovitch and Pearson), while in wet spinning, the interaction of the filament and the surrounding fluids represents a primary constraint on the spinning operation. The formulation of Matovitch and Pearson also contains slight errors in the treatment of the boundary conditions a t the free surface of the filament. We shall, accordingly, develop the governing equations of the flow problem in wet spinning in some detail, thereafter reducing them to the onedimensional approximate problem. The solutions for a Newtonian fluid reveal the existence of a maximum filament length that, for any prescribed set of spinning conditions, is primarily dependent on the square root of the ratio of the viscosities of the filament and the surrounding fluids.
(4) (5)
The conditions prevailing a t the free surface are determined from the interaction of the filament and its surrounding. Accordingly, the normal component of the stress is balanced against the stress induced by the surface tension force, and the tangential component of the stress is balanced by the external drag caused by the surrounding medium. Denoting the variable radius of the filament by R = R ( z ) and using R’ = dR/dz and R” = d2R’ dz’, the above conditions can be written as
[ T , - Rr(-p + TLL)Ir = R = -Fd - 2 HuR’ [-p + T,, - T,R’], R = -FdR’ + 2 Ha
(6) (7) is the coefficient of surface tension, and 2 H =
where 0 is the mean curvature of the free surface, given by
-1
2 H = R [ 1+ ( R T +
R” [ l + (R’I2l3’
(8)
Fd is the external drag. A suitable expression for Fd will be incorporated into the reduced problem which is considered in the next section. At present, we leave it unspecified. In addition to the four boundary conditions expressed above, it follows from the fact that the filament surface is a streamline that a t the free surface the following kinematic relation holds, (9 1
vr(z,r = R ) = DR/Dt = v,(z,r = R)R’ where DIDt is the material-time derivative. The Reduced Problem
The following conditions are assumed:
(i) vL >> vr (ii) dv,/dr = 0
(1Oi) (1Oii)
Using condition (i), it follows from Equation 9 that Problem Formulation
R’ K 1
(11)
The flow is considered steady and axisymmetric and fixed cylindrical coordinates (r,o,z) are adopted with the z-axis aligned with the axis of symmetry of the filament. Omitting the gravitational force, the equations of motion and continuity are:
Using condition (ii), it follows from the continuity equation that
r-direction:
and, further, that
p j ru,av, + - ) =v,au, - g + - - ( r T r )i + a- - 2 az ar r ar
a T,, az
(1) r
p I R 0
v,rdrd8
=
pav,R2= W
0
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
197
Equation 13 thus becomes the continuity equation for the reduced problem with W being the mass flowrate across any section of the filament. The value of W can be obtained from
W = pxVoR: (14) where R, is the radius of the spinnerette hole. The reduction of the momentum equations (in accordance with the above simplifications) can be facilitated by an order of magnitude estimate. Accordingly, let 6 and L correspond, respectively, to some characteristic lengths in the radial and axial directions. From Equation 11 we conclude that
S/L
