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Slip effects in tapered dies.

INTRODUCTION

Tapered dies are very important in polymer processing, including profile extrusion, molding, and film blowing (1), (2). The calculation of pressure drop for polymer melt flow through such dies is of considerable practical importance to the design engineer (1), The importance of converging dies in experimental rheology to assess the extensional rheological properties of polymers was stressed and examined by Cogswell (3), (4). Vlachopoulos and Scott (1) summarize all references studying viscous flow phenomena under no-slip through tapered dies using analytical techniques up to 1985. Numerical solutions to the converging flow problem of viscoelastic fluids were also provided to address the effect of extensional viscosity, and in general of viscoelasticity, in converging dies (5-7). These effects become significant at high entrance angles where the lubrication approximation is not a valid assumption, and extensional rheology plays a crucial role (7).

All the aforementioned studies use the assumption of no-slip at the solid boundary. However, it is known that highly viscous fluids (specifically polymer melts) slip at solid boundaries when the wall shear stress exceeds a critical value (8-10). Therefore, the aforementioned studies are of no use when slip occurs at the wall. This study presents analytical solutions for viscous power-law fluids flowing through tapered capillary (truncated cone) and slit (wedge) dies under slip conditions, ranging from no-slip to severe slip. The results are compared with two-dimensional (2D) finite element simulations to identify the limits of their applicability.

Another motivation for this work was to explain experimental results of poly-tetra-fluoro-ethylene (PTFE) paste flow through tapered dies of various contraction angles (11), (12). The authors have reported that the pressure drop versus contraction angle relationship goes through a definite minimum at an angle of around 30 to 40 degrees. Corresponding data for molten polymers show an initial decrease of the pressure drop, which saturates for angles greater than 30 to 40 degrees in accord with viscous and viscoelastic finite element results (7), (13). The increase in the pressure drop of paste flow at higher angles was interpreted as a manifestation of solid-like behavior also experimentally observed by Horrobin and Nedderman (14). However, solid like pastes exhibit significant slip at solid boundaries. As will be explained here through finite element simulations, viscous fluids in a certain range of severe wall slip conditions may exhibit such an increase of pressure drop at high entrance angles.

ANALYTICAL EXPRESSIONS FOR PRESSURE DROP

Tapered Capillary Dies

We start the analysis for incompressible flow of a power-law fluid through a tapered die under wall slip. No tensorial formalisms will be followed as the analytical solutions refer to 1-D problems. The constitutive equation for a power-law fluid is written as:

[tau] = K[[gamma].sup.n] (1)

where K is defined as the consistency index and n is the power-law exponent. A Newtonian fluid is obtained by setting K = [mu] (Newtonian viscosity) and n = 1.

For the slip analysis, a nonlinear slip velocity law is usually assumed that approximates the actual slip behavior of several fluids, including molten polymers and polymer solutions (15) as well as pastes (16), (17):

[u.sub.S] = -[beta][[tau].sub.W.sup.b] (2)

where [beta] is the slip coefficient, b is the slip-law exponent, and [[tau].sub.w] is the wall shear stress. For example, for molten polymers, quadratic and cubic slip laws have been reported [[u.sub.s] = -[beta][[tau].sub.w.sup.2] in Ref. 8 or [u.sub.s] = -[beta][[tau].sub.w.sup.3] in Ref. 10]. However, experimental data over a short range of shear stress values (0.1-0.2 MPa) can be approximated by a linear slip law to a first approximation. These linear approximations make the problem tractable analytically, and thus they can be used for cases where analytical solutions are to be employed. Obviously, for the numerical results, nonlinear laws can be used as discussed earlier.

The velocity profile for fully developed capillary flow under slip conditions described by Eq. 2 is given by:

[u.sub.z] = [[R.sup.[[1/n]+1]]/[1 + [1/n]]][( - [1/2K][dp/dz]).sup.[1/n]][1 - [(r/R).sup.[[1/n]+1]]] + [u.sub.S] (3)

where R is the capillary radius. Assuming linear slip (b = 1 in Eq. 2) and replacing

[u.sub.S] = [[[beta]R]/2]( -dp/dz) (4)

into Eq. 3, the following form of the velocity profile can be produced,

[u.sub.z] = [[R.sup.[[1/n]+1]]/[1 + [1/n]]][[( -[1/2K][dp/dz])].sup.[1/n]][1 - [(r/R).sup.[1+[1/n]]]] + [[[beta]R]/2](-dp/dz) (5)

Integrating the aforementioned Eq. 5 over the cross sectional area of the capillary die to obtain the volumetric flow rate, yields:

Q = [[pi]/[3 + [1/n]]][[[[R.sup.[3n+1]]/2K](-dp/dz)].sup.[1/n]] + [[[pi][beta][R.sup.3]]/2](-dp/dz) (6)

The tapered capillary die depicted in Fig. 1 has a half contraction angle of [phi] and its radius changes with axial location, z, from [R.sub.0] at location z = 0 to [R.sub.L] at location z = L according to the following expression:

[FIGURE 1 OMITTED]

R = [R.sub.0] - ([R.sub.0] - [R.sub.L])[z/L] (7)

Using Eq. 7, Eq. 6 can be transformed into:

Q = [[pi]/[[1/n] + 3]] [[[[[R.sup.[3n+1]]tan[phi]]/2K][dp/dR]].sup.[1/n]] + [[[pi][beta][R.sup.3]]/2]tan[phi][dp/dR] (8)

Equation 8 will be used in the next section to derive analytical expressions for various flows through tapered capillary dies.

The final analytical equations as well as the numerical results will be presented in a non-dimensional form. For this, all lengths are scaled by the exit die radius [R.sub.L] (or half gap [H.sub.L]), all velocities by the average velocity U at the die exit (U = [Q/([pi][R.sub.L.sup.2])] for capillary flow and U = [Q/(2[WH.sub.L]) for slit slow where W is the width of the slit), all pressures and stresses by K[(U/[R.sub.L]).sup.n] (for capillary flow) or K[(U/[R.sub.L]).sup.n] (for slit flow). Then the dimensionless numbers which control the flow phenomena are:

(a) the dimensionless lengths R* [equivalent to] [R/[R.sub.L]] or H* [equivalent to] [H/[H.sub.L]] (for slit die)

(b) power-law index, n. The power-law index n ranges from 0 (extremely shear-thinning fluids) to 2 (extremely shear-thickening fluids). For n = 1, we have the Newtonian fluid case.

(c) for slip at the wall, the dimensionless slip parameter, [B.sub.s1], is given by:

[B.sub.sl] = [[beta]/U][([eta][bar.[gamma]]).sup.b] = [[beta]/U][(K[(U/[R.sub.L]).sup.n]).sup.b] * (capillary flow)

or

[B.sub.sl] = [[beta]/U][([eta][bar.[gamma]]).sup.b] = [[beta]/U][(K[(U/[H.sub.L]).sup.n]).sup.b]. (for slit flow)

Newtonian Case with Linear Slip. Substitute in Eq. 8, n = 1 and K = [mu] (Newtonian viscosity). Then using the lubrication approximation (neglecting the r-component of the velocity), Eq. 8 can be integrated with respect to pressure drop to obtain the total pressure drop, for a Newtonian fluid flowing through a tapered angle of half angle, [phi], as depicted in Fig. 1:

[DELTA]p = [Q/A][[[A.sup.3]/[B.sup.3]] ln [[R.sub.0]/[R.sub.L]] + [[A.sup.2]/[B.sup.2]]([1/[R.sub.0]] - [1/[R.sub.L]]) - [A/2B]([1/[R.sub.0.sup.2]] - [1/[R.sub.L.sup.0]]) - [[A.sup.3]/[B.sup.3]] ln [[[R.sub.0] + [B/A]]/[[R.sub.L] + [B/A]]]] (9)

where

A = [[[pi]tan[phi]]/[8[micro]]] and B = [[[pi][beta]tan[phi]]/2]

In dimensionless form, Eq. 9 becomes:

[DELTA]p * [equivalent to] [[[DELTA]p]/[U[mu]/[R.sub.L]]] = [1/[tan[phi]]][1/[8[B.sub.sl.sup.3]]] ln[R*.sub.0] + [1/[2[B.sub.sl.sup.2]]]([1/[R*.sub.0]] - 1) - [1/[B.sub.sl]]([1/[R*.sub.0.sup.2]] - 1) - [1/[8[B.sub.sl.sup.3]]] ln [[R* + 4[B.sub.sl]]/[1 + 4[B.sub.sl]]] (10)

Power-law Case (n = 1/2) with Linear Slip. Substitute in Eq. 8, n = 1/2. Integration results into:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where

A = [[pi]/[20[K.sup.2]]][(tan[phi]).sup.2] and B = [[[pi][beta]tan[phi]]/2]

In dimensionless form, Eq. 11 becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Power-law Case with Nonlinear Slip, 1/n = b. The general power-law case can be solved in conjunction with a specific nonlinear slip law, namely:

[u.sub.s] = -[beta][[absolute value of [[tau].sub.w]].sup.[b-1]][[tau].sub.w] or [u.sub.s] = [beta][([R/2] tan[phi][dp/dR]).sup.[1/n]] (13)

As discussed earlier, the slip velocity data can be approximated with either a linear law or a nonlinear law, since most of the time, the experimental data are available over a short range of shear stresses. Therefore, to comply with the true viscosity of the material, its slip behavior can be approximated using a power, b (b = 1/n) in order to obtain an analytical solution. For example, for a typical power-law exponent of 1/3 for a molten polymer, a slip law with an exponent of 3 can be used as determined by Hatzikiriakos and Dealy (10) for a series of HDPEs. In the general case, the slip law exponent can be adjusted and approximated by b = 1/n.

Introducing the slip law given by Eq. 13, Eq. 6 can be written as:

Q = [[pi]/[3 + [1/n]]][[[[[R.sup.[3n+1]]tan[phi]]/2K][dp/dR]].sup.[1/n]] + [pi][beta][([[R.sup.[1+[2/b]]]/2]tan[phi][dp/dR]).sup.b]

or replacing b = [1/n]

Q = [[pi]/[3 + [1/n]]][[[[[R.sup.[3n+1]]tan[phi]]/2K][dp/dR]].sup.[1/n]] + [pi][beta][([[R.sup.[1+2n]]/2]tan[phi][dp/dR]).sup.[1/n]] (14)

Integrating using the lubrication approximation, the following expression is obtained for the total pressure drop, for a power-law fluid flowing through a tapered die of half angle, [phi], as depicted in Fig. 1 under a nonlinear slip law with an exponent of b = [1/n]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where

A = [[pi]/[[1/n] + 3]][[[tan[phi]]/2K].sup.[1/n]] and B = [pi][beta][[[tan[phi]]/2].sup.[1/n]]

In dimensionless form, Eq. 15 becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Tapered Slit Dies

The x-velocity component for fully developed flow between two infinite parallel plates under slip conditions for a fluid with viscosity described by Eq. 2 is given by:

[u.sub.x] = [[H.sup.[1+[1/n]]]/[[1/n] + 1]][(-[1/K][dp/dx]).sup.[1/n]][1 - [(y/H).sup.[1+[1/n]]]] + [u.sub.S] (17)

where x is the direction of flow, y is normal to the direction of flow, and H is the half distance between the plates. Using the linear slip law given by Eq. 2 for the flow between two flat plates can be written as:

[u.sub.s] = [beta]H(-dp/dx) (18)

Introducing Eq. 18 into Eq. 17,

[u.sub.x] = [[H.sup.[1+[1/n]]]/[[1/n] + 1]][(-[1/K][dp/dx]).sup.[1/n]][1 - [(y/H).sup.[1+[1/n]]]] + [beta]H(-dp/dx) (19)

Integrating over the cross sectional area of the slit die to obtain the volumetric flow rate per unit width, W:

[Q/W] = [2/[2 + [1/n]]][[[[H.sup.[1+2n]]/K](-dp/dx)].sup.[1/n]] + 2[beta][H.sup.2](-dp/dx) (20)

The tapered slit die depicted in Fig. 2, has a half contraction angle of [phi] and its half height changes with axial location, x, from [H.sub.0] at location x = 0 to [H.sub.L] at location x = L according to the following expression:

[FIGURE 2 OMITTED]

H = [H.sub.0] - ([H.sub.0] - [K.sub.L])[x/L](21)

Using Eq. 19, Eq. 20 can be transformed into:

[Q/W] = [2n/[2n + 1]][[[[[H.sup.[1+2n]]tan[phi]]/K][dp/dH]].sup.[1/n]] + 2[beta][H.sup.2]tan[phi][dp/dH] (22)

Equation 22 will be used now to derive analytical expressions for various cases.

Newtonian Case with Linear Slip. Substitute into Eq. 22, n = 1 and K = [mu] (Newtonian viscosity). Then using the lubrication approximation (neglecting the y-component of the velocity), Eq. 22 can be integrated with respect to pressure drop to give the total pressure drop for a Newtonian fluid flowing through a tapered slit die of half angle, [phi], as depicted in Fig. 2:

[DELTA]p = [Q/WA][[[A.sup.2]/[B.sup.2]]ln[[H.sub.0]/[H.sub.L]] + [A/B]([1/[H.sub.0]] - [1/[H.sub.L]]) - [[A.sup.2]/[B.sup.2]]ln[[[H.sub.0] - [B/A]]/[[H.sub.L] + [B/A]]]] (23)

where

A = [[2tan[phi]]/[3[micro]]] and B = 2[beta] tan[phi]

In dimensionless form, Eq. 23 becomes:

[DELTA]p* [equivalent to] [[[DELTA]p]/[[mu](U/[H.sub.L])]] = [1/[tan[phi]]][[1/[B.sub.sl.sup.2]]ln[H*.sub.0] + [1/[B.sub.sl]]([1/[H*.sub.0]] - 1) - [1/[B.sub.sl.sup.2]] ln [[[H*.sub.0] + 3[B.sub.sl]]/[1 + 3[B.sub.sl]]]] (24)

Power-law Case (n = 1/2) with Linear Slip. Substitute in Eq. 23, n = 1/2. Integration yields:

[DELTA]p = [-[B/2A] + [square root of ([[B.sup.2]/[4[A.sup.2]]] + [Q/WA])]][[1/[H.sub.L]] - [1/[H.sub.0]]] (25)

where

A = [1/2][[[tan[phi]]/K].sup.2] and B = 2[beta] tan[phi]

In dimensionless form, Eq. 25 becomes:

[DELTA]p* = [[[DELTA]p]/[K[(U/[H.sub.L]).sup.0.5]]] = [2/[tan[phi]]][-2[B.sub.sl] + [square root of ([B.sub.sl.sup.2] + 1)]][1 - [1/[H*.sub.0]]] (26)

Power-law Case with Nonlinear Slip, l/n = b. Introducing the nonlinear slip law and the general power-law fluid as given by Eq. 1, the following expression is obtained for the total pressure drop, for a power-law fluid flowing through a tapered slit die having a half angle of [phi], as depicted in Fig. 2 under a nonlinear slip law with an exponent of b = 1/n:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where

A = [2/[2 + [1/n]]][[[tan[phi]]/K].sup.[1/n]] and B = 2[beta][(tan[phi]).sup.[1/n]]

In dimensionless form, Eq. 27 becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

FINITE ELEMENT CALCULATIONS

Governing Equations and Rheological Modeling

We consider the conservation equations of mass and momentum for incompressible fluids under isothermal, creeping, steady flow conditions. These are written as (18):

[DELTA] * [bar.u] = 0 (29)

0 = - [DELTA]p + [DELTA] * [bar.[tau]] (30)

where [bar.u] is the velocity vector, p is the pressure and [bar.[tau]] is the extra stress tensor.

The viscous stresses are given for inelastic non-Newtonian power-law fluids by the relation (5-7):

[tau] = [eta]([absolute value of [gamma]])[bar.[gamma]] = K[[absolute value of [gamma]].sup.[n-1]][bar.[gamma]] (31)

where [eta]([absolute value of [gamma]]) is the apparent non-Newtonian viscosity given by the power-law model, which is a function of the magnitude [absolute value of [gamma]] of the rate-of-strain tensor [bar.[gamma]] = [gradient][bar.u] + [gradient][[bar.u].sup.T], given by:

[absolute value of [gamma]] = [square root of([1/2][II.sub.[gamma]])] = [([1/2]([bar.[gamma]]:[bar.[gamma]])).sup.[1/2]] (32)

where [II.sub.[gamma]] is the second invariant of [bar.[gamma]]

[II.sub.[gamma]] = ([bar.[gamma]]:[bar.[gamma]]) = [summation over (i)][summation over (j)][[gamma].sub.ij][[gamma].sub.ij] (33)

Furthermore, at the solid wall boundaries, a slip law is assumed (13), (19). This means that the no-slip boundary condition is replaced by the usual Navier slip boundary condition at the solid walls (tangential velocity given by the slip law, normal velocity set to zero), i.e.,

[beta][[absolute value of [bar.t][bar.n]:[bar.[tau]]].sup.[b-1]]([bar.t][bar.n]:[bar.[tau]]) = [bar.t] * [bar.u], [bar.n] * [bar.u] = 0 (34)

where n is the unit outward normal vector to a surface and t is the tangential unit vector in the direction of flow.

The aforementioned rheological model (Eq. 31) is introduced into the conservation of momentum (Eq. 30) and closes the system of equations. Boundary conditions are necessary for the solution of the above system of equations. Figures 1 and 2 show the solution domains for the tapered capillary and slit dies. Because of symmetry only one half of the flow domain is considered. The extra boundary conditions for the 2-D flow are: symmetry along the centerline, zero radial velocity profile at entry and exit, and at entry a velocity profile corresponding to the assumed flow rate.

As discussed earlier, results will be presented as dimensionless. It is noted that when the slip parameter [B.sub.sl] = 0, we have no-slip, whole [B.sub.sl] [approximately equal to] 1 corresponds to macroscopically obvious slip; for [B.sub.sl] > 100 we have an approach to a perfectly plug velocity profile. The form of [B.sub.sl] shows that for different slip-law exponents, b, we may have the same [B.sub.sl] values, so that b is another independent parameter to vary.

Parametric studies for the following discussed parameters will be undertaken in the present numerical study; the power law exponent n, the slip law exponent b, the slip parameter [B.sub.sl] and the taper angle [phi].

Method of Solution

The numerical solution is obtained with the Finite Element Method (FEM), employing as primary variables the two velocities and pressure ([u.sub.r]-[u.sub.z]-p formulation for the cylindrical die or [u.sub.x]-[u.sub.y]-p formulation for the slit die) (18). We use Lagrangian quadrilateral elements with biquadratic interpolation for the velocities and bilinear interpolation for the pressures.

The finite element meshed used in the simulations are shown in Fig. 3 for the different angles ranging from 2[phi] = 8[degrees] to 2[phi] = 90 [degrees]. The reduction ratio for all tapered capillary dies were RR = [[R.sub.0.sup.2]/[R.sub.L.sup.2]] = [R*.sup.2] = [(18.75).sup.2] = 352. We observe that the angle dictates the length of the die according to the formula:

[FIGURE 3 OMITTED]

L = [[[R.sub.0] - [R.sub.L]]/[tan[phi]]] (35)

Thus, as [phi] [right arrow] 0, L [right arrow] [infinity]. For each geometry, two meshes are shown (put together for brevity). The lower half shows the less dense mesh M1, while the upper half shows mesh M2, with which all simulations were done. The characteristics of the meshes are given in Table 1. M2 has 4800 elements with 41 points in the radial direction, and has resulted by subdividing M1 into four sub-elements. The denser mesh has been used for the final runs to guarantee results, which are mesh-independent. The same meshes were used for the tapered slit dies.
TABLE 1. Finite element meshes and their characteristics used in the
FEM simulations for tapered dies.

MESH  No. element  No. radial points  No. elements  No. nodes  No. DOF
        rows x
       columns

M1     10 x 120            21             1200          5061    10,933
M2     20 x 240            41             4800        23,001    51,059


RESULTS AND DISCUSSION

Tapered Capillary Dies

No Slip. The simulations have been carried out for the range of parameters mentioned earlier. We begin by showing in Fig. 4 the results for power-law fluids (both shear-thinning and shear-thickening) as obtained by both the analytical expression (Eq. 15 with [B.sub.sl] = 0, that is no-slip) and numerical simulations (FEM) for different taper angles, 2[phi]. Interesting trends are found, some of which are counterintuitive or at least not anticipated. The minimum in the dimensionless pressure occurs not for n = 1 (Newtonian fluid) but for a slightly shear-thinning fluid with n [approximately equal to] 0.9 for all contraction angles. After the minimum, both shear-thinning and shear-thickening serve to increase the values of [DELTA]p*, with the shear-thinning showing a more abrupt increase. This is not easily anticipated without the aforementioned FEM calculations and the analytical expressions presented above.

[FIGURE 4 OMITTED]

Another finding (more or less anticipated) has to do with the accuracy of the analytical solutions as compared with the FEM results for small angles. We observe that for 2[phi] = 8[degrees] and 15[degrees], and to a certain degree for 2[phi] = 30[degrees], the two methods give very close results (the analytical curves are denoted by dashed lines). Here the lubrication approximation is a valid assumption. For 2[phi] = 30[degrees], the analytical solutions underestimate the numerical 2-D results, and this becomes worse as the taper angle increases. The disagreement increases with increase of shear thinning (decrease of the n-exponent). The present results corroborate the findings of (1) but, of course, the results plotted in Fig. 4 are much more detailed and cover a very wide range of cases studied. As the results are presented in dimensionless form they are independent of the apparent shear rate.

Slip. The simulations have been carried out for the range of slip parameter 0 [less than or equal to] [B.sub.sl] [less than or equal to] 1000 using a linear slip law. A slip parameter, [B.sub.sl], greater than about 10 corresponds to almost a plug flow. The results are presented first in Fig. 5 for the Newtonian fluid case (n = 1) by the two approaches (analytical Eq. 9, referred to as THEORY in the graph, and FEM calculations) for tapered capillary dies having various angles from 2[phi] = 8[degrees] to 2[phi] = 90[degrees]. The semi-log plot of Fig. 5 shows the typical sigmoidal pressure drop curves, [DELTA]P*, found for slip effects (20). Slip serves to decrease the dimensionless pressure drop [DELTA]P* and this becomes most evident for 0.1 < [B.sub.sl] < 10. Again for small angles (less than 30 degrees) the two methods give comparable results, while for 2[phi] = 30[degrees] and higher, the analytical solutions underestimate the numerical 2-D results more and more with increase of 2[phi] and increase of slip coefficient, [B.sub.sl]. This becomes more evident in Fig. 6, where the results plotted in Fig. 5 are replotted in semi-log for only high values of the slip parameter [B.sub.s1] values (1 [less than or equal to] [B.sub.s1] [less than or equal to] 1000). While the analytical results show a continuous drop with increase of slip, the 2-D FEM simulations show a saturation (levelling-off), which is more evident for higher tapered angles.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

When the results are plotted as a function of the tapered angle (see Fig. 7), we observe an interesting phenomenon due to slip. Firstly for no-slip, the pressure drop decreases monotonically with increasing taper angle, and this is the case also for all analytical solutions with slip. However, the 2-D FEM results show that the monotonic decrease for no-slip becomes flatter as slip increases, then the curve passes through a minimum (for [B.sub.sl] = 1 this occurs for 2[phi] between 40[degrees] and 60[degrees]; for [B.sub.sl] = 10 this occurs around 2[phi] = 15[degrees]), and finally the curves show a monotonic increase with increasing angle. This nonmonotonic behavior is present not only for Newtonian but for all power-law fluids studied (shown later). Again the FEM calculations agree with the analytical results for contraction angles less than 30[degrees] degrees, however, this time only for small values of slip ([B.sub.s1] < 0.1). It is worthwhile to stress that the FEM calculations disagree totally with the analytical expressions when slip increases to high levels (see the cases corresponding to [B.sub.s1] = 1 and 10).

[FIGURE 7 OMITTED]

In what follows we present only results from FEM calculations for different tapered angles. Figure 8 shows the dimensionless pressure drop, [DELTA]p* as a function of 2[phi], for Newtonian fluids (n = 1) and for shear-thinning power-law fluids having n = 0.5 and n = 0.2. We observe that the minimum is present for all cases depending on the degree of slip. However, the Newtonian results give a more enhanced minimum than the shear-thinning cases. Similar results were also produced for the shear-thickening cases (n > 1) with minima but are not shown here.

[FIGURE 8 OMITTED]

Nonlinear Slip. As mentioned earlier, nonlinear slip laws are quite often found in polymer melts, including quadratic and cubic laws. Runs were therefore undertaken for Newtonian and power-law fluids for all three cases (linear, quadratic, cubic) of the slip exponent b = 1, 2, and 3, respectively. Sample results are shown in Fig. 9 for a die angle 2[phi] = 15[degrees] and a power-law fluid with n = 0.5. For low slip numbers and up to [B.sub.s1] [approximately equal to] 0.1, there are small differences caused by the slip exponent. However, in the rapid-decrease range of 0.1 < [B.sub.s1] < 10, the changes become evident, with the linear law giving the highest pressure drops at the same [B.sub.s1] number. This is because the linear law results into higher slip for the same slip coefficient [B.sub.s1]. The differences in the results for the three slip laws persist even for very high [B.sub.sl] numbers, up to 1000, and were found for all fluids and all angles.

[FIGURE 9 OMITTED]

It is then instructive to plot the same results as a function of the die angle, 2[phi]. This is done in Fig. 10, for power-law fluids with n = 0.5, and for different [B.sub.sl] numbers. Figure 10a shows the results for [B.sub.sl] = 1, where a small minimum is observed for all slip cases around 2[phi] = 60[degrees]. Figure 10b shows results for [B.sub.sl] = 10, where now the minima are more pronounced and are shifted to the left as the slip-law exponent decreases. Figure 10c shows results for [B.sub.sl] = 100, where even more dramatic changes occur, and the minimum for the linear slip law has disappeared (occurring apparently at even lower die angles), while the minimum for the quadratic slip law has been shifted to around 2[phi] = 20[degrees]. Thus, it is important to realize that the minima depend on the slip parameter as well as the slip-law exponent, and they change appreciably as the die angles change.

[FIGURE 10 OMITTED]

Tapered Slit Dies

Similar results have been found for the slit dies and are not repeated here. In particular the numerical results in tapered slit dies exbibit the same disagreement with the analytical ones obtained from Eqs. 23, 25, and 27. In particular, the degree of disagreement increases with an increase of the taper angle, an increase of the severity of slip (slip coefficient), and a decrease of the power-law exponent.

CONCLUSIONS

Approximate analytical equations are derived for the calculation of pressure drop for viscous power-law fluids flowing through tapered capillary and slit dies for a wide range of wall-slip conditions. The predicted pressure drop values are compared with 2D finite element calculations to identify contraction angles for which the analytical equations can be used. It is found that the disagreement between the analytical equations and the 2D finite element calculations increases with an increase of the contraction angle and with an increase of the degree of wall slip. At a given flow rate, the pressure drop from the analytical equations is found to decrease continuously with contraction angle, which agrees with the finite element calculations only at small contraction angles. At larger contraction angles, the 2D calculations show that pressure drop increases with contraction angle as opposed to the no-slip case where pressure drop saturates. This implies the existence of a minimum pressure at a specific taper angle. This minimum has been found to depend on the rheological parameters of the fluid, and the degree of slip. Such minima in pressure drop have scientific importance for the die designer.

The observation that pressure drop increases with taper angle under slip conditions, more so with an increase of the degree of slip, explains recent experimental findings of poly-tetra-fluoro-ethylene (PTFE) paste flow through tapered dies (11), (12). The authors have reported that the pressure drop versus contraction angle relationship goes through a minimum at an angle of around 30 to 40 degrees. The increase in the pressure drop of paste flow at higher angles was interpreted as a manifestation of solid-like behavior, also experimentally observed by Horrobin and Nedderman (14). However, solid-like pastes exhibit significant slip at solid boundaries. Numerical calculations under no slip show that pressure drop saturates for taper angles greater than a certain value. Therefore, such pressure increases with taper angle can be used as a method to detect the presence and degree of slip in flows. This is the subject of investigation in a forthcoming article.

REFERENCES

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(11.) A.B. Ariawan, S. Ebnesajjad, and S.G. Hatzikiriakos, Can. J. Chem. Eng., 80, 1153 (2002).

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(13.) E. Mitsoulis, I.B. Kazatchkov, and S.G. Hatzikiriakos, Rheol. Acta., 44, 418 (2005).

(14.) D.J. Horrobin and N.R. Nedderman, Chem. Eng. Sci., 53, 3215 (1998).

(15.) L.A. Archer, "Wall Slip: Measurement and Modeling Issues," in Polymer Processing Instabilities: Control and Understanding, S.G. Hatzikiriakos and K. Migler, Eds., Marcel Dekker, New York (2004).

(16.) P.D. Patil, J. Feng, and S.G. Hatzikiriakos, J. Non-Newtonian Fluid Mech., 139, 44 (2006).

(17.) P.D. Patil, I. Ochoa, J. Feng, and S.G. Hatzikiriakos, J. Non-Newtonian Fluid Mech., 153, 25 (2008).

(18.) K.M. Huebner and E.A. Thornton, The Finite Element Method for Engineers, Wiley, New York (1982).

(19.) W.J. Silliman and L.E. Scriven, .J. Comput. Phys., 34, 287 (1980).

(20.) E. Mitsoulis, J. Fluids Eng., 129, 1384 (2007).

Savvas G. Hatzikiriakos, (1) Evan Mitsoulis (2)

(1) Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, British Columbia, Canada

(2) School of Mining Engineering and Metallurgy, National Technical University of Athens, Athens, Greece

Correspondence to: Savvas G. Hatzikiriakos; e-mail: hatzikir@apsc.ubc.ca

Contract grant sponsors: Natural Sciences and Engineering Research Council (NSERC) of Canada, NTUA (KARATHEODORI) of Greece.

DOI 10.1002/pen.21430
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Author:Hatzikiriakos, Savvas G.; Mitsoulis, Evan
Publication:Polymer Engineering and Science
Article Type:Technical report
Date:Oct 1, 2009
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