Ind. Eng. Chem. Fundam. 1983, 22, 347-355
347
Profile Development in Continuous Drawing of Viscoelastic Liquids Roland Keunlngs and Marcel J. Crochet Unit6 de Mcanlque Appiiqde, Unlversl Cathollque de Louvain, B- 1348 Louvain-ia-Neuve, Belgium
Morton M. Denn' Department of Chemical Engineering, University of California, Berkeley, California 94720
Stress, velocity, and diameter profiles are computed for continuous drawing (fiber spinning) of viscoelastic liquids from the fully developed flow in the splnneret to the region of uniform extension where the asymptotic thin filament equations apply. The calculations extend to a capillary recoverable shear of 1.O for Maxwell and Phan-Thien-Tanner fluids, and to 4.0 for an Oldroyd fluid B; spinline force levels are reached at which significant viscoelastic effects are observed. The thin filament equations generally become valid within two diameters of the spinneret exit, but downstream of the point of maximum extrudate swell. The force at whlch viscoelastic effects in the spinline become important is adequately estimated from asymptotic solutions to the thin filament equations. The finite element calculations generally support the practice of taking the initial ratio of transverse-to-axial extra stresses to be zero for integration of the thin filament equations.
1. Introduction
The process of continuous drawing of polymeric liquids to form fibers (melt spinning) is shown schematically in Figure 1. A filament is extruded from a small hole in the spinneret plate into an ambient atmosphere that is below the solidification temperature. The molten filament is drawn by a takeup device that imposes a velocity that is greater than the extrusion velocity. The mechanics of spinning are discussed in the reviews by Denn (1980) and White (1982) and in the monographs by Petrie (1979) and Ziabicki (1976). In the commercial process a large number of filaments will usually be extruded from a single spinneret and be taken up together in a yarn. Commercial spinning is carried out at takeup speeds in excess of 1000 m/min, where air drag and inertial effects must be taken into account. Laboratory experiments are usually carried out at lower speeds, and frequently isothermally; in an isothermal experiment the environment is maintained at the spinneret temperature, and solidification is effected at a fixed point by a rapid chill, usually with a water bath. Analyses of spinning are carried out with a set of thin filament equations that are based on the assumption of no shear stresses and small rate-of-change of curvature; these equations are not valid close to the spinneret. The thin filament equations are usually assumed to apply downstream from a point close to the position of maximum extrudate swell. This point will be within about one filament diameter of the spinneret, so the uncertainty in the location of the origin should not be important on a long commercial spinline, although it could be important in laboratory experiments. The thin filament equations require initial data for filament area and velocity if the fluid is inelastic; if the fluid is viscoelastic then it is also necessary to specify the ratio of axial to transverse extrastresses and the fraction of the axial stress that is contained in each mode of the viscoelastic spectrum. The selection of the appropriate initial conditions for the thin f i e n t equations is a major outstanding problem in the analysis and simulation of melt spinning. Velocity rearrangement and the approach to the asymptotic equations has been studied by finite element methods for low-speed isothermal spinning of a Newtonian liquid by
Fisher et al. (1980), who found that the thin filament equations became adequate in all cases within one filament diameter of the spinneret. White and Roman (1976,1977) have developed an approximate theory of extrudate swell under tension, following the unconstrained extrudate swell analysis of Tanner (1970); this theory has not generally been applied in spinline simulations. The transverse-toaxial extra-stress ratio for most viscoelastic fluid models can be shown to go asymptotically to zero far from the spinneret in the thin filament approximation, and Denn et al. (1975) have found that the thin filament equations for isothermal low-speed spinning of a Maxwell fluid are relatively insensitive to this ratio when it varies between zero and the inelastic fluid limit of -0.5; Petrie (1979) has used an approximate analysis of stress decay beyond the spinneret to show that a value of zero for the extra-stress ratio is consistent with neglecting shear stresses in the thin filament equations. We present here the results of a computational study of spinning of a class of viscoelastic fluids that includes the Maxwell fluid and generalizations used in recent simulation studies of melt spinning. Two finite element algorithms are employed, giving similar results, and some results are obtained at high levels of fluid elasticity. The calculations show that the asymptotic equations become valid for the viscoelastic fluids within two filament diameters of the spinneret, and they generally support setting the transverse extra-stress to zero as an initial condition. The White-Roman solution for extrudate swell, modified to include an inelastic contribution, is an adequate approximation up to a recoverable shear of about 2, and at low imposed axial force. The point of maximum extrudate swell occurs as much as one diameter upstream of the point where the asymptotic equations become valid, so the maximum area is not an appropriate initial condition; the spinneret velocity and area are probably adequate initial conditions for long spinlines, since this simply introduces a small error in the location of the origin. All calculations reported here are valid for low speed, isothermal spinning. The axial distance scale over which computations are carried out is only five spinneret diameters downstream of the spinneret exit, however, corresponding to only a few millimeters for typical commercial
0 1983 American Chemical Society Q196-4313/83/1Q22-Q347801.5Q/Q
348
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983
Quench Air
Ta
v
PT1,eq.(2.6a) -
. .
1 -
/
3!A _ _ _ _ _ _ _ _ _ _ _ _ _ _--~ -_NeWt_Oll'S? . _ ____ --_ c _
0
1 w = Drw,
Fr=O
--
- Fz*O 2
Figure 2. Geometry and boundary conditions.
conditions, and temperature changes w i l l be small over this distance. Furthermore, inertial and air drag contributions are not important in the region close to the spinneret. Thus, the conclusions regarding initial conditions for the thin filament equations are aplicable to simulations of nonisothermal, high speed spinlines as well. 2. Problem Formulation The geometry for continuous drawing of a single round filament is shown in Figure 2. The spinneret is a cylindrical tube of diameter do, in which fluid flows with an average axial velocity wo and flow rate Q = *dO2wo/4.A takeup device at some distance downstream from the point of extrusion imposes a force F; we are assuming low-speed spinning (negligible inertia and air drag),in which case this force is transmitted axially a t a constant value. This enables us to carry out the computation over an axial distance that is less than the entire liquid filament length by imposing the force as a boundary condition. Body (gravitational) forces can usually be neglected in melt spinning. In the absence of inertia the steady state equations for conservation of momentum and mass can then be written u = -PI T (2.1)
+
+ V.T = 0 v * v= 0
2
3
4
hT,
5
Figure 3. Steady-state extensional stress for Newtonian, Maxwell, and Phan-Thien-Tannerfluids.
Figure 1. Schematic of the melt spinning process.
-Vp
1
(2.2)
(2.3)
u is the total (Cauchy) stress, T is the extra-stress, p is the isotropic pressure, and I is the identity tensor. As noted above, eq 2.2 will be applicable to the spinneret region even for high takeup speeds. Boundary conditions are as shown in Figure 2. It is assumed that the spinneret is sufficiently long to ensure fully developed flow a t the plane where the computation is begun. AU of the constitutive equations to be considered subsequently have an essentially constant shear viscosity, so the upstream axial velocity in the spinneret is parabolic. The extra-stresses must also be specified at the entry plane for a viscoelastic fluid, and these are taken to be the stresses in fully developed flow. The fluid adheres to the wall of the spinneret ( v = O),while normal and tangential contact forces vanish on the free surface following extrusion; vanishing of the tangential contact force follows from the assumption of negligible air drag. The radial velocity
and the axial contact stress vanish on the axis of symmetry. The downstream plane is placed sufficiently far from the spinneret exit to assume complete readjustment of the velocity to a uniform profile; in that case the total axial is uniformly distributed and equal to the imstress, oEZ, posed takeup force divided by the area, while the shear stress is small, and hence the tangential contact force may be neglected with respect to the axial force. Simulations of spinning experiments with the thin filament equations have generally been carried out for the Maxwell fluid and the generalization by Phan-Thien and Tanner (PTT) (1977; Phan-Thien, 1978); the only published viscoelastic fluid simulation of pilot scale spinning experiments at commercial speeds (Gagon and Denn, 1981) is for the PTT fluid. The PTT constitutive equation for a fluid with a single relaxation time is
Y(q,T) T
+ X(T" + C;(DT+ TD)) = 2pD
(2.4)
D is the deformation rate tensor and V denotes the 01droyd upper-convected derivative, which for a time-dependent flow is T" = vVT - LT - TLT
(2.5)
A, p, q, and C; are material parameters, and L = Vv is the velocity gradient. X and p are the relaxation time and viscosity measured in linear viscoelastic experiments. $. is a measure of shear thinning and would typically be of order 0.2 for a polymer such as low-density polyethylene. When C; does not vanish, the shear stress in simple shear flow decreases when the shear rate increases beyond a critical value; the problem may be remedied by adding a purely viscous stress to the extra-stress defined by (2.4). The function Y(q,T) is important only in extensional flows; it is given by Phan-Thien (1978) as
Y(q,T) = ex.(
q i tr T )
(2.6a)
Phan-Thien and Tanner (1977) used a linear form that is equivalent to eq (2.6a) for small values of the argument
x
Y(q,T) = 1 + q- tr T P
(2.6b)
The upper-convected Maxwell equation is recovered for q = = 0, and the Newtonian fluid follows from the additional condition A = 0. All calculations reported here were carried out with C; = 0, in which case the shear viscosity is constant (except for a negligible contribution associated with nonzero q). The behavior of the several rheological models in the idealized extensional flow of uniform uniaxial extension at a constant stretch rate re, following attainment of a steady-state stress, is shown in
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983 349
Figure 3, with q = 0.015 for the PTT fluid. The Maxwell fluid predicts unbounded stresses at XI’, = 0.5, whereas the stresses are always bounded for the PTT fluid. For numerical reasons to be discussed subsequently, it is sometimes helpful to include a retardation time in the stress constitutive equation. The equation that generalizes the Maxwell fluid is the Oldroyd fluid B T XITv = 2p(D X2Dv) (2.7)
+
+
An equivalent formulation is T = T1
+ T,
(2.8a)
+ XITvl = 2p1D
TI
(2.8b)
T, = 2p2D 11 = 111 + 112; 112 = 1 1 ( X 2 / U
(2.84 (2.8d)
The behavior of the Oldroyd fluid B in steady laminar shear flow and in steady extension is the same as that of the Maxwell fluid. Equation 2.8b is easily replaced by a PTT-type partial stress of the form of eq 2.4; in that case a shear stress that is a monotonically increasing function of shear rate is possible only for p 2 / p = X2/X1 > if 5 # 0. 3. Dimensional Analysis The dimensionless axial velocity w/wo obtained for low-speed isothermal spinning without gravitational effects from the thin filament equations must be a unique function of z/do and a single dimensionless group for a Newtonian fluid B
4F/3rpdow0
(3.1)
The analytical solution to the thin filament equations for a Newtonian fluid, referred to an arbitrary origin, is W
- = exp(Bz/do)
(3.2)
WO
Bz/do is the ratio of the axial stress at the origin, F / (ad2/4), to the product of the steady extensional viscosity, 3p, and a rough estimate of the stretch rate, wo/z. Viscoelasticity imposes a second dimensionless group. A convenient choice is the recoverable shear in the fully developed viscometric flow
SR= (Tzz - Trr)/2Trz
(3.3)
For the Maxwell liquid, and (to a good approximation) for the PTT liquid with 4 = 0, we have SR
= 8X~o/do
(3.4)
while for the Oldroyd fluid B and the equivalent PTT-like generalization with 4 = 0, we have
SR= (8Xwo/do)(~i/~) (3.5) S R is a measure of the level of fluid elasticity in the spinneret. The solution to the thin filament equations is a smooth function of S R , with the origin taken at an arbitrary location where the thin filament equations are valid; for large values of the product BSR and small values of T,(0)/Tz,(O) the solution for a Maxwell fluid approaches the following function (Denn et al., 1975)
_W -- 1 + - - - 8- 1 n2 WO
SRdO
4 BSR
(
I+-R ;O :)
-
The product BSR has an important physical interpretation, as shown in eq 3.7 for a Maxwell fluid BSR =
8F
(3.7)
3 ~ / is h the tensile (Young’s)modulus for a linearly elastic (Hookean) solid, which is the response exhibited by a Maxwell material at short times. Thus, within a factor of eight, BSR is the ratio of the imposed tensile stress to the tensile modulus of the material. In the limit BSR a the velocity for a Maxwell fluid becomes linear in the axial coordinate z with a slope 8/SR The asymptotic behavior of the thin filament equations for the Oldroyd fluid B has not been studied as extensively, but the properties seem to be like those of the Maxwell fluid (Petrie, 1979). It is conventional to describe some spinning results in terms of the area reduction ratio, or draw ratio, Dr. The draw ratio is equal to the dimensionless velocity, w/wo, at the point where the force is imposed. It follows from eq 3.2 that the draw ratio increases without bound with increasing force (increasing B ) for a Newtonian fluid, while, except for a small contribution from the Trr/Tz, term, eq 3.6 imposes the following bound for a Maxwell liquid
-
8 L DrIl+-SR
(3.8)
L is the distance from the origin of the thin filament equations to the point of imposition of the force. This bound is a consequence of the fact that a Maxwell fluid can reach infinite stresses at finite extension rates (Figure 3), and a similar bound does not exist for the PTT fluid with q > 0. The analysis of the thin filament equations by Denn and co-workers (1975) employed dimensionless groups CY and e that arise naturally for the thin filament equations, but are less useful when taking the spinneret flow into account. These groups are related to B and SRas follows e = - do (3.9) 8L ’ 3BL In all of the work reported here the coordinate origin is at the exit of the spinneret, so comparisons with solutions of the thin filament equations must take the different origins into account in defining both Dr and L. 4. Range of Parameters The dimensionless groups SRand B are independent. The former characterizes the viscoelastic flow in the spinneret, while both groups arise in the spinline portion. A numerical experiment in which B is varied at constant S R corresponds to the physical experiment of changing the force with a constant throughput. Changing SRat constant B corresponds to fixing the throughput and imposed force and then examining the response of a sequence of fluids. Numerical solutions of free extrudate flow (B = 0) of Maxwell and PTT fluids from a cylinder have not been obtained for values of S R that are greater than approximately unity. Use of a retardation time, as in the Oldroyd fluid B, permits calculations to S R of 4, but Maxwell and PTT fluid calculations will be limited to SR5 1 because of limitations on the extrusion flow. The asymptotic solution for highly elastic liquids in eq 3.6 is valid for large values of the product BSR. Numerical solutions of the thin filament equations by Denn et al. (1975) for Dr = 20 show that the limiting solution for BSR m is approached closely for BSR around 10, with sig(y
-
=
SRdO
-*
350
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983
SDiLi{Y(q,T*)T*+ X[Tv* + E(D*T* + T*D*)] 2pD*) da) = 0 (5.2) where Zl is the domain of integration. The equation for mass conservation is
SD#iV.v*d2J = 0
(5.3)
In a first method, which we call MIX1, the system is closed with the Galerkin form of the momentum equation (5.4) where t is the imposed contact stress and e is the domain boundary. In a second method, MIX2, which we have used only for the Maxwell fluid, we solve eq 5.2 and 5.3 (with Y = 1, E = 0) and write the Galerkin form of the momentum equation as
Figure 4. Meshes used for calculation.
nificant elastic effects for BSR around 1.3. The finite element numerical solution of fiber spinning of a Newtonian fluid by Fisher et al. (1980) showed that extrudate swell essentially vanished for B = 1. An estimate of the force required to produce significant viscoelastic effects can be obtained by computing the value of B from eq 3.2 for a Newtonian fluid required to obtain the limiting draw ratio for a Maxwell fluid in eq 3.8; this calculation gives
B
N
doL In (1 +
&)
The problem scaling is such that only SRand B need be specified. For numerical purposes we take a spinneret with unit radius, do = 2, a unit mean velocity, w o = 1,and a unit shear viscosity, p. SRand B are then fixed by choosing X and F, respectively. Most calculations were done with the force imposed at L / d o = 5; a few calculations were done with L / d a = 7 for verification purposes. With L / d o = 5 it follows from eq 3.9 that CY = 0.025S~,and the limiting draw ratio for a Maxwell fluid from eq 3.8 is 1 + 4o/sR. For SRof order unity we expect significant elastic effects from eq 4.1 for B about 0.75. 5. Numerical Technique In the fiiite element method, the domain of integration is partitioned into a set of triangles and quadrilaterals. Figure 4 shows two meshes used for the calculation; the second mesh served as a test for verifying the validity of the boundary conditions in the take-up section. The meshes are shown in a typical distorted configuration; wide elements are needed downstream, since the radius of the filament becomes quite small for large values of B. Figure 4c shows the distorted configuration at the maximum draw ratio for which a numerical solution has been obtained. The last elements of the fiber indeed become quite thin. It is assumed that the velocity vector v , the extra-stress tensor T, and the pressure p are approximated by finite sums of the form v* = T* = ET'+,; p* = E@+J (5.1)
cvJ+,;
The shape functions are complete second-order polynomials over triangles and biquadratic polynomials over the parent quadrilateral, while the shape functions are complete fit-order polynomials over triangles and bilinear polynomials over the parent quadrilateral. In order to determine the nodal values vJ,T J ,and p J ,it is necessary to solve a nonlinear system composed of the Galerkin form of the governing equations. The PTT constitutive equation is written +J
+J
JD(V$i)'(-pI
+ 2pD* -ATv*) da) = S,$;t d@
(5.5)
More details on MIX1 and MIX2 may be found in Crochet (1982) and Crochet and Keunings (1982a). (For the sake of simplicity, the numerical method has been presented here in terms of Cartesian tensors in an orthogonal coordinate system. In the present paper, we will make use of curvilinear cylindrical coordinates. The use of cylindrical coordinates leads of course to a system of partial differential equations which differs from its Cartesian counterpart; moreover, the integration must be performed over toroidal volume elements. Details may be found in Crochet et al. (1983), section 10.6.) When X vanishes, method MIX2 is identical with the classical velocity-pressure formulation developed by Nickell et al. (1974);when X does not vanish, the algorithm resembles a perturbation of the Newtonian scheme. For small values of A, and in particular for the Newtonian case, the solutions obtained with MIX2 are smoother than those obtained with MIX1. When X increases, however, it is found that method MIXl has better convergence properties. It has been shown by Crochet and Keunings (1982a) that extrudate swell calculations, within the range of SR reached with present techniques, are sensitive within a few percent to the choice of mesh and of method MIXl or MIX2; this explains why, in later sections, slightly different numbers will be found when both techniques are used for solving the same problem. The Galerkin equations for MIXl are changed only slightly for the Oldroyd fluid B or the P'I"I'-like equivalent. Equations 5.2 and 5.4 are replaced by
ID+JY(q, TI*)T1* + X1[Tvl* + [@*TI* + T,*D*)I 2p1D*)dD = 0 (5.6) IB(V$JT(-p*I + Tl* +2pZD*) da) = &+it
d@
(5.7)
This gives an algebraic system in the nodal values of v*, p * , and T1*; the viscous component T2* is an internal variable of the system, which can be calculated a posteriori from eq 2 . 8 ~ . Details can be found in Crochet and Keunings (1982b). The shape of the free surface is found by successive approximations, with the technique described in Crochet and Keunings (1980). In the Newtonian case, the initial profile is taken from the asymptotic solution calculated from the edge of the spinneret. In the non-Newtonian case, we start from a profile obtained with the previous value of B.
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983 351 5.-
Table I. Characteristic Dimensions for a Newtonian Fluid as a Function of B
8 . 1
\
B 0.50
0.25 error, MIX1, % error, MIX& % S w ,MIXl S w ,MIX2 z s , MIXl z s , MIX2 Dr, MIX1 D r , MIX2 z,, MIXl z, MIX2
0.9 0.9 1.062 1.064 0.35 0.35 2. I 2. I 0.9 0.6
0.75
asymptotic solution
1.0
-
6. Newtonian Fluid SRis zero for a Newtonian fluid. We have been able to reach a value of B = 1 with both techniques MIXl and MIX2; a value of B = 1.5 has been reached with MIX2, but the results are less satisfactory because of the very thin final element. It is useful to examine several characteristic quantities associated with Newtonian fiber drawing as B varies from 0.25 to 1.5. The Galerkin form (5.3) of the continuity equation does not allow for local incompressibility, which is only satisfied in the mean. Since conservation of mass is an important factor in fiber spinning, we evaluate the mean velocity of the liquid in each section with two methods: (i) the imposed flow rate is divided by the local cross section, and (ii) the mean axial velocity is calculated on the basis of the nodal values. These two quantities should be identical; the relative error is always found to be less than 1%, except in the last two elements, which are too thin for a precise calculation. The maximum error in the last two elements is shown in Table I; as we might expect, the error becomes large when B grows, since the draw ratio increases exponentially. The swelling ratio is defined by
(6.1)
where d, is the maximum diameter on the fiber. The dimensionless location (z/do)of the maximum diameter is denoted by z,. Table I shows the values of S w and z, obtained with methods MIXl and MIX2; they are in excellent agreement. There is less than 1%swell when B = 1,and z, is very close to the exit of the spinneret for B = 0.75. These results are consistent with the calculations of Fisher et al. (1980). An essential quantity for comparing numerical and asymptotic solutions is the axial position at which the velocity profile may be considered to be uniform. We have calculated the relative difference between each nodal velocity and the average velocity within each section. We take the velocity profile as uniform where the maximum departure from the mean is less than 5%; the corresponding dimensionless position is denoted z., It is found that the value of z, is larger with MIXl than with MIX2; it should be noted, however, that the sections where uniformity is evaluated are located at discrete values of z, and 0.9 simply corresponds to the section which follows 0.6. The computed mean velocities using MIX2 are shown in Figure 5, together with the asymptotic solution given by eq 3.2. The agreement is excellent as long as the origin for the asymptotic solution is taken at z,; the draw ratio would be overestimated in all cases if z, were taken as the origin. Two important facts follow for use of the thin
A
\ oy \ i .
1.5
1.9 4.9 0. I 15.2 2.0 4.9 0.9 1.005 1.026 1.013 1.0 1.011 1.003 1.026 0.05 0.05 0.2 0 0.05 0.05 0.2 33 120 9.5 1440 9.4 33 120 0.9 1.8 0.9 0.9 0.6 0.6 0.6
SW= dm,/dO
,
numerical solution
3-I
Figure 5. Computed mean axial velocity, Newtonian fluid.
-L T'
D
8=025
x 6.1
Trr
-_____
. -
2------
4
~
1-
i
I* o
{
v
7
4
7
j
z
o
3
t Y
-1 -
Figure 6. Computed ratio of mean extra-stresses, Newtonian fluid.
filament equations. First, while the profile will be shifted, use of the spinneret area and velocity as initial conditions for the thin filament equations results in a maximum error in the computed force of 4% for B I 0.5. Second, it can be seen from Figure 5 that the asymptotic solutions nearly intersect at z/do = 0, In (a/w0) = -0.24. Thus, the asymptotic solution will give nearly the correct B-Dr dependence for z/do > z,, and B from zero to 1.5 if the spinneret exit is takken as the origin, with a mean inital velocity of 0 . 7 9 and ~ ~ a diameter of 1.12d,,. The solutions reported by Fisher et al. (1980) show this property up to B = 0.33, but there is some deviation at B = 1.0. The thin filament equations for a Newtonain fluid require that the ratio of average extra-stresses Tzz/TPr be -2.0. The computed ratio is shown in Figure 6 for B = 0.25 and 1.0; there is considerable deviation in the region of velocity rearrangement, but a ratio of -2.0 is reached and maintained at zun. 7. Maxwell and PTT Fluids The maximum recoverable shear SRfor which a smooth solution can be obtained with the present numerical techniques for an undrawn jet for Maxwell and PTT fluids is approximately unity. We have been able to obtain solutions with S R = 1 for a Maxwell fluid t o a maximum value of B = 2 with method MIX1, and B =0.85 with MIX2. The solution with MIXl degrades seriously for B I 1.5, and results must be used cautiously; the algorithm does not converge beyond B = 2. These upper values of B correspond to the parameter range (BSR 1.3) beyond which Denn et al. (1975) found significant elastic effects in the solution of the thin filament equations, and they are beyond the force computed from eq 4.1. The maximum draw ratio permitted by the thin filament equations in the limit of infinite force is 41. The characteristic dimensions computed for the Maxwell fluid with SR= 1are shown in Table 11. The relative error in the average velocity is somewhat larger at the smaller values of B than that calculated for the Newtonian fluid,
-
352
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983
Table IT. Characteristic Dimensions as a Function of B for SR = 1 B
error. MIX1. % error; MIX2; 5% S w ,MIXl S w ,MIX2 z s , MIXl z s , MIX2 Dr, MIXl Dr, MIX2 z, MIX1 z, MIX2
0.25
0.50
0.75
2.6 0.9 1.149 1.178 0.35 0.6 2.2 2.1 0.9 0.9
2.4 0.9 1.099 1.116 0.35 0.35 7.0 6.6 0.9 0.9
2.4 1.5 1.068 1.074 0.2 0.2 14.3 14.1 0.9 0.9
1.0
1.25
1.50
1.75
2.0
2.5
3.0
3.7 -
6.1 -
-
1.042
1.028
1.017
1.008
0.1 -
0.1
0.05
0.05
20.4
24.5
27.5
29.5
-
31.4
0.9
0.9 -
0.9
0.9
-
-
-
-
-
-
8.6 1.0
-
-
0
-
-
-
-
-
0.9
-
Table 111. Value of -FzZ/Frras a Function of B for SR = 1 B MIXl,z, MIX2,z, MIXl,zT MIX2, ZT
0.25
0.5
0.75
1.0
1.1 2.0 1.9 2.2
1.8 2.4 2.2 2.5
2.4 2.8 2.8 3
3.2
1.25 1.50 1.75 2.0 4.1 5.3 6.6 8
-
-
3.7
4.8
6.3
7.8
9.2
-
-
-
-
-
-
-
-
8. In Or
t
o *q.(2.64
&+
x
eq.(2.6b)
Figure 8. Computed mean axial velocity, Maxwell fluid. i
607
a
4.t
-
0
.
-L
5
8.2.
br
50-
~
3.1
2.t
1.t
B
6-
0
PTT. Sa.2. eq.(Z.Ba)
05
,/
175
/
'5
/
125
40-
-
30 :
1
I ,
E 15
2
25
3
/'
201
Figure 7. Computed draw ratio at z j d o = 5 as a function of B for various fluid models and values of recoverable shear.
but it remains within acceptable bounds. Here, however, the maximum error occurs within one diameter of the spinneret. The swelling ratio is larger for the Maxwell fluid than for the Newtonian fluid, as expected. The values obtained for the swelling ratio with MIX2 are somewhat higher than those obtained with MIX1, as found by Crochet and Keunings (1982a) for free jets. The location of maximum swell occurs further from the spinneret than in the Newtonian case. The position at which the velocity is uniform within 5% of the mean is at 0.9do from the spinneret exit in all cases. The draw ratio, which is shown graphically in Figure 7, exhibits the major qualitative difference from Newtonian behavior. For B = 1 the draw ratio is an order of magnitude smaller than for the Newtonian fluid under the same force, and the limiting value of 4 1 seems to be approached asymptotically. The ratio of extra-stress components is an essential initial condition for the thin filament equations. -Tzz/Trr is shown in Table I11 at 2,; there are differences between the results from MIXl and MIX2. -Tzz/Trris shown in Table I11 at z;, there are differences between the results from MIXl and MIX2 The stresses become uniform to within 10% only further downstream, at a dimensionless position denoted zT;for SR= 1, zT = 1.3 for all values of B. The stress ratio is also shown at ZT in Table 111. Tzz/Tr always lies between -2.0 and -a, which is the range in
Figure 9. Computed ratio of mean extra-stresses, Maxwell fluid.
which Denn et al. (1975) found the solution to the isothermal thin filament equations to be insensitive to the initial stress ratio. The mean velocity and stress ratios computed using MIXl are plotted against z / d o in Figures 8 and 9, respectively. Trrquickly becomes negligible relative to Tzz, in accordance with the asymptotic solutions (Denn et al., 1975). A linear scale is used for velocity, and the profiles at the larger values of B do seem to be approaching a straight line with slope 8 / S R , in accordance with eq 3.6. The changes in slope for B 2 1.5 are an indication of degeneration of the solution, however. This is more clearly illustrated in Figure 10, where contours of constant axial velocity are plotted; substantial oscillations can be seen in Figure 1Oc in the contour lines for B = 1.5, just beyond the spinneret exit. The contours are unchanged in Figure 10d, where the force was applied at Lid, = 7 in order to obtain a longer downstream region. The remaining calculations have been carried out using MIX1, since higher values of B are achievable for the viscoelastic fluid and the two methods give similar results where they overlap. Convergence of the numerical al-
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983 353
Table IV. Characteristic Dimensions as a Function of B for SR = 0.5'
B 0.25
0.50
0.75
error, M error, PTT b error, PTT a
2.8 2.8 2.8
2.8 2.7 2.7
2.7 2.6 2.6
S w ,M S w ,PTT b S w ,PTT a z,, M
1.079 1.076 1.076 0.35 0.35 0.35 2.6 2.6 2.6 0.9 0.9 0.9
1.037 1.034 1.034 0.2 0.2 0.2 8.7 8.8 8.8 0.9 0.9 0.9
z,, PTT b z,, PTT a
Dr, M Dr, PTT b Dr, PTT a zun, M zun b zun,PTT a 9
a
1.0 2.8 2.7 2.7
1.017 1.015 1.015 0.1 0.1 0.1 21.2 21.7 21.7 0.9 0.9 0.9
1.007 1.005 1.005 0.05 0.05 0.05 34.0 35.5 35.6 0.9 0.9 0.9
1.25
1.5
1.75
2.0
2.25
2.50
2.75
3.0 3.2 3.1
3.3 3.4 3.3
3.5 3.7 3.6
4.0 4.2 4.1
4.5 4.6 4.6
5.2 5.2 5.1
6.1 6.0 5.7
1.0 1.0 1.0 0.0 0.0 0.0 43.5 46.7 47.0 0.9 0.9 0.9
1.0 1.0 1.0 0.0 0.0 0.0 50.4 55.8 56.4 0.9 0.9 0.9
1.0 1.0 1.0 0.0 0.0 0.0 55.6 63.4 64.5 0.9 0.9 0.9
1.0 1.0 1.0
1.0 1.0 1.0
0.0 0.0 0.0
0.0 0.0 0.0
1.0 1.0 1.0 0.0 0.0
1.0 1.0 1.0 0.0 0.0
59.6 70.2 72.0 0.9 0.9 0.9
62.8 76.5 79.2 0.9 0.9 0.9
0.0
0.0
65.3 82.4 86.6 0.9 0.9 0.9
67.4 88.1 94.5 0.6 0.9 0.9
M: Maxwell; PTT b: linear Y,eq 2.6b; PTTa: exponential Y ,eq 2.6a.
Table V. Value of -Fzz/Fr,. as a Function of B for SR = 0.5 0.25 Maxwell PTT b PTT a
1.8 1.8 1.8
0.50 2.2 2.2 2.2
0.75
1.0
2.8 2.8 2.8
3.7 3.7 3.7
1.25 5.1 5.2 5.2
B 1.50
1.75 9.5 9.7 9.7
7.1 7.2 7.2
2.0
2.25
2.50
2.75
12.3 12.7 12.7
15.3 16.0 16.0
18.5 19.5 19.6
21.8 23.3 23.5
Table VI. Comparison of Sw and Dr for the Maxwell and Oldroyd B Fluids; SR = 1
Sw. Maxwell S w ;Oldroyd B Dr, Maxwell Dr, Oldroyd B
0.25
0.5
0.75
B 1.0
1.149 1.135 2.2 2.3
1.099 1.086 7 7.1
1.068 1.055 14.3 14.2
1.042 1.036 20.4 19.7
gorithm was obtained up to a value of B = 2.75 for SR = 0.5 for the Maxwell and PTT fluids; 5 was set equal to zero in the latter, giving an essentially constant shear viscosity, with q set equal to 0.015. The characteristic dimensions and stress ratios at zT = 1.8 are shown in Tables IV and V, respectively. There is negligible difference between Maxwell and PTT fluids in all respects except in the attainable draw ratio at a fixed force. The Maxwell fluid shows lower draw ratios and always remains below the limiting value of Dr = 81 given by the asymptotic solution in eq 3.8. This limit is not observed by either version of the PTT fluid, for which infiiite stresses at a finite stretch rate are not possible. The draw ratios are lower for the linear stress coefficient, eq 2.6b, than for the exponential coefficient, eq 2.6a; it can be seen in Figure 3 that the former admits larger stresses in steady extension, so this behavior is to be expected. The computed draw ratios at L / d o = 5 for a given force are shown in Figure 7 for the Newtonian fluid and Maxwell and PTT fluids at SR = 0.5 and 1. One important conclusion that can be reached from these calculations is that the onset of significant viscoelastic effects in the spinline depends on B and SR in essentially the manner estimated from the asympototic solutions in eq 4.1. The draw ratio for the Maxwell fluid falls to half of that for the Newtonian fluid at B = 2 / 3 and 5/6 for SR = 1 and 0.5, respectively, while the values of B computed from eq 4.1 are 0.74 and 0.88. Comparison of Tables I11 and V indicates that the initial extra-stress ratio is only weakly dependent on SRbut is a strong function of B. For B greater than about 1.25 the initial ratio -Trr/Tzzis 0.2 or less, and the value of zero normally used
1.25
1.5
1.75
1.028 ' 1.027 24.5 23.3
1.017 1.018 27.5 25.7
1.008 1.01 29.5 27.3
8. 0.5
0. 0.5 1. 1,.5 H
" 1.
0. 4.5 1. 1,5
1!5
1. 4 4.
I 5.
i.'
8-1.5
4
1.5 '2. 4. 6.
!
1,
10.
t.
I
L
I
+2-r,=-
20
Figure 10. Contours of constant axial velocity, Maxwell fluid.
for this ratio as an initial condition for the thin filament equations is appropriate. 8. Retardation Time The presence of a retardation time in the constitutive equation, together with the use of large entry lengths, enables calculation to significantly higher levels of fluid
354
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983
Table VII. Characteristic Dimensions as a Function of B for SR = 2; Comparison between the Oldroyd B Fluid and the PTT Fluid with a Viscous Component
B 0.25 2.5 2.3 1.263 1.23 2 0.6 0.6 1.8 1.9 0.6 0.6
error, Oldroyd B error, PTT a Sw,Oldroyd B S w , PTT a z s , Oldroyd B z s , PTT a D r , Oldroyd B D r , PTT a z u n , Oldroyd B z U n ,PTT a
0.5 2.3 2.1 1.199 1.173 0.4 0.4 5.1 5.3 0.6 0.6
0.75
1.0
1.25
1.5
2.7 2.5 1.152 1.127 0.4 0.25 8.7 9.2 0.6 0.6
3.2 2.9 1.119 1.098 0.25 0.25 11.1 12.1 0.6 0.6
4.1 3.6 1.091 1.070 0.25 0.25 12.5 14.3 0.6 0.6
-
5.2 -
1.043 -
0.15
16.3 -
0.6
Table VIII. Characteristic Dimensions for an Oldroyd B Fluid as a Function of SR,for B = 0.25 SR
sw
0 0.5 1 1.5 2 2.5 2.75 3 3.25 3.5 3.75 4
1.060 1.077 1.135 1.194 1.263 1.342 1.387 1.440 1.508 1.572 1.637 1.723
Dr 2.70 2.60 2.31 2.05 1.81 1.58 1.47 1.35 1.21 1.11
1.01 0.91
error, % 1.7 1.2 1 1.6 2.5 3.3 3.6 3.7 3.5 3.3 3.2 3.3
-
z,,
0.6 0.6 0.6 0.6 0.6 0.85 0.85 1.25 1.25 1.25 1.25 1.75
*S
0.4 0.4 0.4
0.6 0.6 0.6 0.6 0.6 0.6 0.85 0.85 0.85
Figure 12. Final mesh and axial velocity contours, Oldroyd fluid B; B = 0.25, S R = 4. sw 1.84
6.0.25
0
Figure 11. Mesh used for calculations with fluid models having a retardation time.
elasticity. Crochet and Keunings (1982b) have obtained free extrudate swell calculations up to S R = 4 by this means, where SRis defined by eq 3.5 for the Oldroyd fluid B (eq 2.7) and the PTT-type generalization of eq 2.8b. All calculations reported here are for p l = 8p2 (XI = 9X2)and, for the PTT-type generalization, 5 = 0 and q = 0.015. We have used the mesh shown in typical deformed shape in Figure 11,with a tube length equal to 8do and the extrudate length maintained at 5d0. Computed swell ratios and draw ratios for the Maxwell and Oldroyd fluids are shown in Table VI for S R = 1,which is the maximum recoverable shear obtainable for the Maxwell fluid. The results are very close; the differences may be a consequence of the particular choice of the ratio X2/X, and of the fact that two different meshes were used. A similar comparison is made in Table VI1 between the Oldroyd fluid B and the PTT generalization at SR= 2. There is little difference between the two models, although the Oldroyd fluid B shows slightly more extrudate swell and a smaller attainable draw ratio at the same imposed force. The draw ratio at SR= 2 is plotted in Figure 7 as a function of B; the viscoelastic behavior is consistent with that expected from the asymptotic solutions. Values up to a recoverable shear of 4 have been obtained for B = 0.25, with results and limitations similar to those found by Crochet and Keunings (1982b) for free extrudate swell. The characteristic dimensions are shown in Table VIII. There is considerable extrudate swell, and very little net drawdown over five diameters; indeed, Dr is less than unity for SR= 4. The position of maximum diameter moves only from 0.4do to 0.85do as S R is changed from 0 to 4,but the velocity does not become uniform until 1.75d0
0
numerical
1.8.
\.
O
141
WHITE and ROMAN sR
lJ/ 1.
0.
1
2
3
,
4
Figure 13. Computed swell ratios compared to White-Roman theory, Oldroyd fluid B; B = 0.25. s,.2
h
13t\,
,
'\ e..'
12t
WHITE and ROMAN
'.
b-__-.P---_ numerlcal
11 +
1 *--
0
+-025
-
__t
05
075
B A .
1
125
Figure 14. Computed swell ratios compared to White-Roman theory, Oldroyd fluid B; S R = 2.
at the highest recoverable shear. The axial velocity contours are shown in Figure 12 for S R = 4;there is a velocity overshoot near the exit section. White and Roman (1976,1977) have developed an approximate theory of extrudate swell in spinning using constrained elastic recovery arguments that generalize the free swell theory of Tanner (1970). In our notation their result can be written 3 Sw6 = 1 + 1/SR2 - -BSRSw2 (8.1) 8
Tanner's theory is recovered for B = 0. This type of theory cannot predict inelastic swell, and a slight modification, following Tanner (1970), is introduced
Ind. Eng. Chem. Fundam. 1983, 22, 355-357
C(B) is the value of S w less one for a Newtonian fluid at the same value of B; C(B)is obtained from Table I. The computed swell ratios for an Oldroyd fluid B are compared to eq 8.2 in Figure 13 for B = 0.25. Agreement is good up to SR of about 2, after which the analytical solution underestimates the actual swell. The deviation is quite large at S R = 4. The comparison is similar to that between the Tanner (1970) theory and the free extrudate swell calculations of Crochet and Keunings (1982b). Results for S R = 2 are shown in Figure 14 up to B = 1.25. There is a small difference, and the computed swell drops off more rapidly with force than eq 8.2, but the general trend is consistent. In view of the difference between the position of maximum swell and the position of velocity uniformity, and the inability to make an a priori estimate of the location of maximum swell, the use of eq 8.2 seems to offer no advantage over spinneret conditions as initial conditions for the thin filament equations. 9. Conclusion The elasticity levels that have been reached in these calculations are in a range that is relevant to polymer processing practice. The velocity and stress arrangement from spinneret to extensional flow always takes place in a small region, and the uncertainty about the location of the origin of the thin filament equations will not be im-
355
portant on long spinlines. The maximum swell is adequately predicted by the White-Roman equation up to a recoverable shear of 2, but it is not evident that this equation provides any advantage over use of the spinneret area and velocity as initial conditions for the thin filament equations. The calculations generally support use of a zero initial condition for the ratio of transverse to axial extrastresses in the thin filament equations. Literature Cited Crochet, M. J. "The Flow of a Maxwell Fluid around a Sphere"; I n Gallagher, R. H., ed.; "Finlte Elements in Fluids IV"; Wlley: New York, 1982. Crochet, M. J.; Davles, A. R.; Walters, K. "Numerical Simulation of Non-Newtonlan Flow"; Elsevier: London, 1983. Crochet, M. J.; Keunings, R. J. Non-Newtonian F/u/d Mech. 1980, 7 , 199. Crochet, M. J.; Keunlngs, R. J . Non-Newtonkn Nu/d Mech. 1982a, 10, 85. Crochet, M. J.; Keunings, R. J . Non-NewtonianFluidMech. 1982b, 10, 339. Denn, M. M. Ann. Rev. F/u/dMech. 1980, 12, 365. Denn, M. M.; Petrie, C. J. S.; Avenas, P. AIChE J . 1975, 21, 791. Fisher, R. J.; Denn, M. M.; Tanner, R. I. Ind. Eng. Chem. Fundam. 1980, 19, 195. Gagon, D. K.; Denn, M. M. Polym. Eng. Sci. 1981, 27, 844. Nickeil. R. E.; Tanner, R. I.; Caswell, B. J. F/u/d Mech. 1974, 65, 189. Petrie, C. J. S. "Elongational Flows"; Pitman: London, 1979. Phan-Thien, N. J . Rheol. 1978, 22, 259. Phan-Thlen, N.; Tanner, R. I. J . Non-Newtonlan F/u/dMech. 1977, 2,353. Tanner, R. I. J . Polym. Sci. 1970, 8 , 2067. Whlte, J. L. Po/ym. Eng. Rev. 1962, 1, 297. Whlte, J. L.; Roman, J. F. J. Appl. Polym. Sci. 1976, 20, 1005. White, J. L.; Roman, J. F. J . Appl. Po/ym. Sci. 1977, 21, 869. Ziabicki, A. "Fundamentals of Fibre Formation": Wiley: New York, 1976.
Received for review August 18, 1982 Revised manuscript received March 16, 1983 Accepted March 28, 1983
COMMUNICATIONS Effect of Vapor Efflux from a Spherical Particle on Heat Transfer from a Hot Gas
The efflux of vapor from a particle reduces the heat transfer to the particle. This effect was evaluated for spherical and elongated (cylindrical) particles. One example of this phenomenon is the devolatilization of small coal particles, for which previous investigators have used the Ackermann correction which was derived for flat-plate geometry. Computation shows that for the case of 74-pm coal particles in an entrained-flow gasifier, the flat-plate solution yields significantly greater values for this effect, and hence significantly lower rates of heat transfer, than the spherical-geometry solution.
Introduction The heat transfer rate to a surface is reduced by the presence of a mass flux from that surface. When the mass flux is independent of the heat transfer, this effect can be incorporated into the boundary conditions as the Ackermann correction to the heat transfer coefficient, that is
h = Ah-ho
(1)
where
0196-4313/83/1022-0355$01.50/0
For derivation of this solution, the reader is referred to Sherwood et al. (1975). One practical problem which exhibits this physics is the thermal decomposition of solids with the generation of gaseous products. Coal devolatilization is an example of such a decomposition. A number of investigators have employed the factor C,, or a factor derived from it, in their studies of the devolatilization of powdered coal in pyrolysis, gasification, or combustion processes. For example, it was used by James and Mills (1976) and Sprouse (1979) in its original form, by Ubhayakar et al. (1977) after assumption of the small-particle limit (Nu = 21, and by Coates and Glassett (1974) after further assumption of uniform first-order reaction within the particle. However, all the above authors employed eq 2 and 3, which were derived for a flat-plate geometry, for devolatilizing particles which were in all other respects considered spherical. 0 1983 American Chemical Society
