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Single screw extrusion of viscoplastic fluids subject to different slip coefficients at screw and barrel surfaces.

INTRODUCTION

The occurrence of wall slip, i.e., a relative velocity between the fluid velocity at the wall and the wall velocity is well documented (1-17), for a variety of fluids including polyethylenes (1-7), aqueous and organic polymer solutions (8-11), and gels and concentrated suspensions (12-16), and rubbers (17). The recent studies of White, et al. (17) on elastomers, and Kalyon and coworkers (7) on polyethylenes, have revealed that wall slip occurs as a function of materials of construction (7, 17) and roughness of the surface (7). The importance of wall slip in processing of wide classes of materials, especially very concentrated suspensions, has been long recognized by the industry which has devised various ways to affect wall slip behavior in continuous processing. Design changes have included the selection of different materials of construction and/or roughness profiles for the barrel and screw surfaces and the machining of special grooves in extrusion processing.

The wall slip behavior of a concentrated suspension consisting of 60 percent by volume glass spheres in an acrylonltrile terminated polybutadiene matrix is demonstrated in Fig. 1. The suspension sample undergoes steady torsional flow in between two parallel disks; the top one is rotating and the bottom is stationary. Before the onset of deformation, a straight line marker is placed at the free surface of the suspension and the edges of the two disks, as shown in Fig. 1a. Upon deformation, discontinuities appear at both suspension/wall interfaces, suggesting wall slip. The wall slip vs. the wall shear stress behavior of fluids which exhibit slip can be characterized by employing viscometric flows (1-3, 6, 9, 12, 13, 15, 16, 18, 19). Kalyon and co-workers (20) have shown that the slip velocity vs. shear stress data can be utilized in the modeling of single screw extruders. The typical wall slip velocity, [U.sub.s], vs. wall shear stress, [[Tau].sub.w], behavior of two concentrated suspensions is shown in Fig. 2 (15, 16). The data can be fitted by:

[U.sub.s] = [Beta][[Tau].sub.w] (1)

where [Beta] is a material constant i.e., Navier's slip coefficient which also depends on the nature of the walls of the rheometer (7, 17). For these two suspensions, the values of [Beta] were determined to be 7.4 X [10.sup.-4] mm/(Pa-s) for the suspension filled with 77 percent by volume solids, and 9.2 X [10.sup.-5] mm/(Pa-s) for the suspension containing 60 percent by volume solids. It is this capability to characterize an interfacial constitutive equation for various fluids and materials of construction, which renders the analysis contained in this paper especially relevant.

The Finite Element Method based formulation of extrusion flow subject to wall slip is available (20), but is too complicated for general adaptation and use. Menning (21) and Meijer and Verbraak (22) have investigated extrusion of Newtonian fluids subject to wall slip. However, the use of the Newtonian fluid is very restrictive and generally, the fluids which exhibit wall slip are viscoplastic. Recently, an analytical model for the extrusion and lubrication flows of viscoplastic fluids, subject to wall slip, was reported by Lawal, et al. (23). However, the possibility of different slip coefficients prevailing on the screw and barrel surfaces was not considered.

In Fig. 3, the wall slip occurring during the drag-induced, steady torsional flow of a concentrated suspension of 60% vol. solid glass spheres in a Newtonian matrix (acrylonitrile terminated polybutadiene) at 90 [degrees] C is shown. The suspension is a viscoplastic fluid. The suspension has an affinity for wall slip at the bottom plate and exhibits a no slip condition at the top plate. This behavior is affected by the roughness intentionally machined onto the surface of the disk located at the top. This suggests a positive Navier's wall slip coefficient at the bottom surface. This simple experiment illustrates that techniques involving selections of surface roughness and materials of construction indeed provide the capability to engineer the wall slip behavior of viscoplastic fluids. Obviously, the presence of different wall slip coefficients at the screw and barrel surfaces should give rise to significant effects during the continuous processing of viscoplastic fluids.

Here we present an analytical model describing the single screw extrusion behavior of viscoplastic fluids in shallow channels (parallel plate or generalized Couette flow) subject to different wall slip coefficients at screw and barrel surfaces.

ANALYSIS

The simplified flow channel is shown in Fig. 4, where the viscoplastic fluid is conveyed and pressurized in between a wall moving with velocity, [v.sub.w], i.e., barrel surface and a stationary screw surface. This is the plane Couette flow for which the z-component of the equation of motion becomes (compressive stresses are positive):

-[Delta][[Tau].sub.yz]/[Delta]y = dP/dz (2)

where P is pressure and [[Tau].sub.yz] is the shear stress. For this one-dimensional flow, the Herschel-Bulkley fluid model describing the shear viscosity of viscoplastic fluids is given by:

[Mathematical Expression Omitted]

d[[Upsilon].sub.z]/dy = 0 [absolute value of] [[Tau].sub.yz] [is less than or equal to] [[Tau].sub.0] (3b)

Here m and n are material parameters and [[Tau].sub.0] is the yield stress, i.e., a critical value of the stress magnitude below which viscoplastic materials do not deform. Minus sign is to be used when [[Tau].sub.yz] [is less than] 0. At stress magnitudes [absolute value of] [[Tau].sub.yz] which are less than the yield stress, only the rigid body motion, i.e., plug flow, is possible.

Defining the following dimensionless variables:

[u.sub.z] = [[Upsilon].sub.z]/[[Upsilon].sub.w]; [Xi] = y/H; [Mathematical Expression Omitted] (4a)

Equation 2 becomes:

d/d[Xi]([[absolute value of] d[u.sub.z]/d[Xi].sup.n - 1] d[u.sub.z]/d[Xi]) = [Lambda] (4b)

Equation 4b is applicable only in the deformation region and is replaced by the rigid body translation requirement, Eq. 3b, in the plug region whenever it exists. The volume flow rate Q is given by:

[Mathematical Expression Omitted]

where the dimensionless volume flow rate, [Omega], is:

[Omega] = [integral of] [u.sub.z] d[Xi] between limits 1 and 0 (5b)

and W is the width of the slit in the x-direction. The solution for a simplified case. i.e., Bingham fluids (n = 1) and without wall slip was provided by Tichy (24). The analytical solution for power-law fluids, again without wall slip, is also available (25, 26).

In two-dimensional flow configurations. Navier's slip at the wall condition is in the form (27):

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the unit tangent vector to the surface, [Mathematical Expression Omitted] the unit outward normal, and [Beta] is the slip parameter. The limits for [Beta] = [infinity] and [Beta] = 0 give perfect slip and no slip conditions, respectively. [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are the velocity vectors of the fluid and the solid surface, respectively. and [Mathematical Expression Omitted] is the total stress tensor. In general, [Beta] may depend on the invariants of the stress tensor, but here it is assumed to be a material constant which for a given fluid depends on the materials of construction and roughness of the solid surfaces. This does not impose any serious limitation on the results as experimental data generally confirm this to be the case (15, 16).

For the geometry under consideration, if we define the slip velocity as the difference between the fluid velocity and the velocity of the solid surface and apply Eq 6 to the surfaces, we obtain in dimensionless form:

[u.sub.st] = ([[Beta].sub.t]/[[Upsilon].sub.w])[[Tau].sub.yz](H) (7a)

[u.sub.sb] = -([[Beta].sub.b]/[[Upsilon].sub.w])[[Tau].sub.yz](0) (7b)

where [u.sub.st] and [u.sub.sb] are the slip velocities at the top and the bottom plates respectively, [[Beta].sub.t] and [[Beta].sub.b] are the corresponding slip parameters at the top and bottom surfaces, and [[Tau].sub.yz](H) and [[Tau].sub.yz](0) are related through:

[[Tau].sub.yz](H) = [[Tau].sub.yz](0) - H dP/dz (8)

Returning to Eq 4, three distinct cases are possible depending on the values of the parameters of the problem. These cases are:

* Case 1: No plug flow region; which occurs when the stress magnitude is greater than the yield stress in the entire flow domain.

* Case 2: Plug flow occurs with the non-deforming plug attached to the bottom surface for positive pressure gradient.

* Case 3: Plug flow occurs, with the rigid core sandwiched in between two deforming zones. This case will be referred to as the "floating plug" case.

Designating the lower interface between the deformation region and the plug flow region by [[Lambda].sub.1], and the upper interface by [[Lambda].sub.2], the analysis for the three cases for [Lambda] [is greater than] 0 are provided in the Appendix.

CASE STUDIES AND DISCUSSION

The analytical model presented in the Appendix can be employed to solve for the velocity distribution and the extrusion flow rate vs. pressurization rate in the extruder, given a fluid, wall slip coefficients at screw and barrel surfaces, geometry, and operating conditions. Here, various typical results will be shown employing a vlscoplastic fluid with the material parameters given in Table 1. The dimensions and extrusion conditions used in the case study are also included in Table 1.

The mass flow rate, vs. the pressurization rate behavior of the viscoplastic fluid under the conditions of Table 1, is shown in Fig. 5. The results are presented as a function of the slip coefficient at the top barrel surface, [[Beta].sub.t], over the slip coefficient at the bottom i.e., screw surface, [[Beta].sub.b], i.e., [Psi] = [[Beta].sub.t]/[[Beta].sub.b]. The highest mass flow rate at any pressurization rate is observed, when there is a no-slip condition at the barrel surface with a finite slip velocity at the screw surface, i.e., [Psi] = 0. As the wall slip coefficient at the barrel surface increases and approaches the value of the slip coefficient at the screw surface (i.e., [Psi] approaches one), the production rate decreases. The increase in the pressurization rate occurs as a result of the decrease in the mass flow rate. Such results can be employed to determine optimum operating ranges for the extrusion process, where the pressurization rate and mass flow rate values are acceptable.

It is interesting to compare the mass flow rate vs. the pressurization rate results presented in Fig. 5 with the case, where there is no wall slip occurring at both the barrel and screw surfaces. The results for no slip occurring at both screw and barrel walls corresponding to the same viscoplastic fluid, geometry, and operating conditions are shown in Fig. 6. For pressurization rates which are less than 44 MPa/m, the mass flow rate values achieved with the no slip condition at both surfaces are considerably smaller than the mass flow rates obtained when there is no slip at the barrel wall but the viscoplastic material slips at the screw surface, i.e., [Psi] = 0 in Fig. 5. For values of pressurization rate in the range of 10 to 30 MPa/m, the mas flow rates obtained with [Psi] = 0.1, and [Psi] = 0.5 are also greater than those corresponding to the no-slip condition at both screw and barrel surfaces. For pressurization rate values greater than 44 MPa/m, the no-slip condition at both screw and the barrel surfaces generates greater mass flow rates than those produced by the finite slip coefficient ratio, [Psi], values.

There are significant ramifications to these findings. This case study clearly demonstrates that greater production rates can be achieved by proper engineering of the surfaces of the barrel and the screw in extrusion process at certain operating regimes.

The typical velocity distributions at three different ratios of the slip coefficients at the top, barrel surface, [[Beta].sub.t] and bottom, screw surface, [[Beta].sub.b], i.e., [Phi] = [[Beta].sub.t]/[[Beta].sub.b], are shown in Fig. 7. These results are obtained at 15 MPa/m. The viscoplastic fluid with the properties given in Table 1 exhibits a plug-flow region attached to the bottom stationary surface for the slip coefficient ratios, [Phi], varying between 0, i.e. zero at barrel surface, and finite slip coefficient at screw surface, to three, i.e., the wall slip coefficient at barrel surface is three times as high as that of the screw surface. As the slip coefficient ratio, [Phi], increases the plug flow region occupies a greater portion of the flow channel. Concomitantly, increasing slip coefficient ratio, [Phi], gives rise to smaller mass flow rates. For example, at a pressurization rate, dP/dz, of 15 MPa/m and at a linear screw speed of 0.3 m/s, the viscoplastic fluid defined in Table 1 generates a dimensionless volume flow rate, [Omega] of 0.45 (mass flow rate of 101 kg/h) at a slip coefficient ratio, [Phi], of zero. The dimensionless volume flow rate decreases to 0.36 (mass flow rate of 83 kg/h) as the slip coefficient ratio increases to 0.5 and to [Omega] = 0.1 (mass flow rate of 24 kg/h) at a slip coefficient ratio of 3.

The velocity distribution for the no slip condition occurring at both the screw and barrel surfaces is also shown in Fig. 7 for the same pressurization rate. Upon the imposition of the no slip condition at both surfaces the plug flow region is eliminated. The volumetric flow rate achieved with the no slip condition occurring at both surfaces is smaller than the volumetric flow rate achieved at zero slip coefficient ratio.

The velocity distributions pertaining to the pressurization rate of 50 MPa/m are shown in Fig. 8. At this higher pressurization rate there is back mixing occurring for the slip coefficient ratios varying between zero and three. The presence of wall slip at the screw surface increases the volumetric flow rate of material with negative velocities. On the other hand, the imposition of the no slip condition at both surfaces eliminates back mixing and gives rise to a higher volumetric flow rate at this pressurization rate.

These results also indicate that the control of the wall slip coefficients in continuous processing of viscoplastic fluids affects their distributive mixing. Under the conditions of Fig. 7, the creation of the no-slip condition at the barrel surface i.e., [Phi] = 0, gives rise to high velocities at the top, barrel surface, with concomitant deformation of the fluid in the same region. As the slip coefficient ratio, [Phi], increases the velocity values, hence the total volumetric flow rate, decreases with a concomitant decrease in the deformation rates introduced into the material. The distributive mixing capability of a continuous processor depends on the total strain introduced as well as the TABULAR DATA OMITTED reorientation of the interface area between components being mixed. Thus, with increasing slip coefficient ratio, [Phi], the distributive mixing ability of the single screw extruder decreases.

The analytical model presented in this study can be utilized also to determine the prevailing slip coefficient ratio in existing extruders running viscoplastic fluids, prone to wall slip. Series of experiments at constant geometry could incorporate changes in the barrel velocity and volumetric flow rate (in starved feeding) followed by the determination of the pressurization rate in the extruder through series of pressure transducers. Such experiments can provide the slip coefficient ratio at the top and bottom surfaces, i.e., barrel and screw surfaces for a given viscoplastic suspension and extruder and could suggest various engineering approaches for better design or for achieving optimum operating conditions in conjunction with the analysis provided here.

CONCLUSIONS

A new analytical model of extrusion flows of viscoplastic fluids subject to different slip coefficients at the barrel and screw surfaces is presented. Many viscoplastic fluids, including those found in plastic, composite, rubber and elastomer, and energetics industries, exhibit wall slip, which can be controlled to some extent by proper choices of materials of construction, surface roughness, and introduction of grooves. Such engineering of surfaces can be carried-out differently for barrel and screw surfaces to generate different slip coefficients at the barrel and screw surfaces. The presented analysis should furnish the capability to develop design expressions for such cases. The results indicate that the pressurization capability of the extruder decreases with the increasing ratio of slip coefficient at barrel surface over screw surface. However, the presented analysis also indicates that over certain operating condition ranges, the presence of wall slip at the screw surface with no slip at the barrel surface can give rise to greater extrusion production rates. The mixing capabilities of the extruder deteriorate with increasing slip ratio. Overall, the analytical model developed here can be used to furnish design expressions for viscoplastic fluids with known properties, including wall slip coefficients. On the other hand, the model presented here can be used to determine experimentally the wall slip coefficient ratios in existing extrusion hardware, for various viscoplastic fluids, which exhibit wall slip. The results can be used not only to understand existing boundary conditions but also to introduce improvements in engineering and design and optimization of extrusion lines.

APPENDIX

Case 1-no plug ragion ([Lambda] [is greater than] 0)

Since [du.sub.z]/d[Xi] [is greater than] 0, Eq 4 becomes:

d/d[Xi][([du.sub.z]/d[Xi]).sup.n] = [Lambda] (9)

which can be integrated once to give:

[Lambda]([Xi] - [[Lambda].sub.2]) = [([du.sub.z]/d[Xi]).sup.n] (10)

having made use of the condition that [du.sub.z]/d[Xi] = 0 at [Xi] = [[Lambda].sub.2]. Further integration of Eq 10 subject to the boundary condition [u.sub.z](0) = [u.sub.sb] gives:

[u.sub.z] = [[Lambda].sup.s]/(s + 1)[([Xi] - [[Lambda].sub.2]).sup.s + 1] - [[Lambda].sup.s]/(s + 1)[(-[[Lambda].sub.2]).sup.s + 1] + [u.sub.sb] (11)

where s = 1/n. If we apply the yield stress condition at [[Lambda].sub.2], [[Tau].sub.yz](0) be given by:

[[Tau].sub.yz](0) = H dP/dz [[Lambda].sub.2] - [[Tau].sub.0] (12)

and [u.sub.sb] is then determined as:

[u.sub.sb] = [[Kappa].sub.1] + [[Alpha].sub.1][[Lambda].sub.2] (13)

where [[Kappa].sub.1] = ([[Beta].sub.b][[Tau].sub.o]/[[Upsilon].sub.w]) and [[Alpha].sub.1] = ([[Beta].sub.b]H/[[Upsilon].sub.w])(-dP/dz). For Bingham plastic materials, i.e., n = 1 and m as the plastic viscosity, the ratio (-[[Kappa].sub.1][Lambda]/[[Alpha].sub.1]) is the Bingham number. If we combine Eqs 8 and 12, the expression for [[Tau].sub.yz](H) becomes:

[[Tau].sub.yz](H) = H dP/dz([[Lambda].sub.2] - 1) - [[Tau].sub.o] (14)

and [u.sub.st] is then given by:

[u.sub.st] = -[[Kappa].sub.2] - [[Alpha].sub.2]([[Lambda].sub.2] - 1) (15)

where [[Kappa].sub.2] = ([[Beta].sub.t][[Tau].sub.o]/[[Upsilon].sub.w]) and [[Alpha].sub.2] = ([[Beta].sub.t]H/[[Upsilon].sub.w])(-dP/dz). Equation 11, upon substitution for [u.sub.sb], becomes:

[u.sub.z] = [[Lambda].sup.s]/(s + 1)[([Xi] - [[Lambda].sub.2]).sup.s + 1] - [[Lambda].sup.s]/(s + 1)[(-[[Lambda].sub.2]).sup.s + 1] + [[Kappa].sub.1] + [[Alpha].sub.1][[Lambda].sub.2] (16)

The boundary condition at the top surface, i.e., [u.sub.z](1) = 1 + [u.sub.st] provides relationship for [[Lambda].sub.2], thus:

[[Lambda].sup.s]/(s + 1)[(1 - [[Lambda].sub.2]).sup.s + 1] - [[Lambda].sup.s]/(s + 1)[(-[[Lambda].sub.2]).sup.s + 1] + [[Kappa].sub.1] + [[Alpha].sub.1][[Lambda].sub.2] + [[Kappa].sub.2] + [[Alpha].sub.2]([[Lambda].sub.2] - 1) - 1 = 0 (17)

Integrating Eq 16, the dimensionless volume flow rate is:

[Omega] = [[Lambda].sup.s]/(s + 1)(s + 2)[[(1 - [[Lambda].sub.2]).sup.s + 2] - [(-[[Lambda].sub.2]).sup.s + 2]] - [[Lambda].sup.s]/(s + 1)[(-[[Lambda].sub.2]).sup.s + 1] + [[Kappa].sub.1] + [[Alpha].sub.1][[Lambda].sub.2] (18)

Case 2-plug attached to the bottom surface ([Lambda] [is greater than] 0)

In the region where the fluid is being deformed, i.e., [[Lambda].sub.2] [is less than or equal to] [Xi] [is less than or equal to] 1, Eq 10 applies. When Eq 10 is integrated subject to the boundary condition at the top surface, the following velocity distribution is obtained:

[u.sub.z] = [[Lambda].sup.s]/(s + 1) [([Xi] - [[Lambda].sub.2]).sup.s + 1] - [[Lambda].sup.s]/(s + 1)[(1 - [[Lambda].sub.2]).sup.s + 1] - [[Kappa].sub.2] - [[Alpha].sub.2]([[Lambda].sub.2] - 1) + 1 [[Lambda].sub.2] [is less than or equal to] [Xi] [is less than or equal to] 1 (19)

In the plug region:

[u.sub.z] = 1 - [[Lambda].sup.s]/(s + 1) [(1 - [[Lambda].sub.2]).sup.s + 1] - [[Kappa].sub.2] - [[Alpha].sub.2]([[Lambda].sub.2] - 1) 0 [is less than or equal to] [Xi] [is less than or equal to] [[Lambda].sub.2] (20)

To obtain the relationship for [[Lambda].sub.2], we use the condition [u.sub.z]([[Lambda].sub.2]) = [u.sub.sb] to give:

[[Lambda].sup.s]/(s + 1)[(1 - [[Lambda].sub.2]).sup.s + 1] + [[Kappa].sub.1] + [[Alpha].sub.1][[Lambda].sub.2] + [[Kappa].sub.2] + [[Alpha].sub.2]([[Lambda].sub.2] - 1) - 1 = 0 (21)

Finally, the dimensionless volume flow rate, [Omega] is:

[[Lambda].sup.s]/(s + 1)(s + 2)[(1 - [[Lambda].sub.2]).sup.s + 2] - [[Lambda].sup.s]/(s + 1)[(1 - [[Lambda].sub.2]).sup.s + 1] - [[Kappa].sub.2] - [[Alpha].sub.2]([[Lambda].sub.2] - 1) + 1 (22)

Case 3-floating plug region ([Lambda] [is greater than] 0)

In the upper section, the solution to the velocity equation is as obtained for Case 2, i.e., Eq 19 and therefore need not be repeated. However, in the lower section, [du.sub.z]/d[Xi] [is less than] 0, and Eq 4 then becomes:

d/d[Xi][(- [du.sub.z]/d[Xi]).sup.n] = -[Lambda] (23)

Integrating Eq 23 once and using the condition that [du.sub.z]/d[Xi] = 0 at [Xi] = [[Lambda].sub.1], we obtain:

[Lambda]([[Lambda].sub.1] - [Xi]) = [(- [du.sub.z]/d[Xi]).sup.n] (24)

Further integration and the application of the bottom surface boundary condition. i.e., [u.sub.z](0) = [u.sub.sb] gives:

[Mathematical Expression Omitted]

In the plug region,

[u.sub.z] = 1 - [[Lambda].sup.s]/(s + 1)[(1 - [[Lambda].sub.2]).sup.s + 1] - [[Kappa].sub.2] - [[Alpha].sub.2]([[Lambda].sub.2] - 1) [[Lambda].sub.1] [is less than or equal to] [Xi] [is less than or equal to] [[Lambda].sub.2] (26)

and since the plug region velocity is constant, the velocity of the fluid in the upper section at [Xi] = [[Lambda].sub.2] should be the same as the fluid velocity in the lower section at [Xi] = [[Lambda].sub.1], hence:

[Mathematical Expression Omitted]

The position of the lower interface of the plug region, [[Lambda].sub.1], is related to [[Lambda].sub.2] through:

[[Lambda].sub.2] - [[Lambda].sub.1] = 2[[Tau].sub.o]/h dp/dz = [Delta] (28)

Here the dimensionless ratio, [Delta] = 2[[Tau].sub.o]/H(dP/dz) is equal to -2[[Kappa].sub.1]/[[Alpha].sub.1] or -2[[Kappa].sub.2]/[[Alpha].sub.2] and it is assumed that the value of this ratio for zero wall slip coefficients is to be obtained using L'Hopital's rule.

Upon substitution, Eq 27 becomes:

1 - [[Lambda].sup.s]/(s + 1)[(1 - [[Lambda].sub.2]).sup.s + 1] - [[Kappa].sub.1] - [[Alpha].sub.1][[Lambda].sub.2] - [[Kappa].sub.2] - [[Alpha].sub.2]([[Lambda].sub.2] - 1) + [[Lambda].sup.s]/(s + 1)[([[Lambda].sub.2] - 2[[Tau].sub.o]/H dP/dz).sup.s + 1] = 0 (29)

This is the required equation for the determination of [[Lambda].sub.2], and [[Lambda].sub.1] can be subsequently recovered from Eq 28. Integration of the velocity profile gives the dimensionless volume flow rate as:

[Mathematical Expression Omitted]

This completes the analysis for dynamic pressurization where dP/dz is positive, i.e., [Lambda] [is greater than] 0. For a negative pressure gradient, the governing equations are different and are summarized in Table 3.

Proper Case Determination

The first step in the procedure for determining the proper case is to determine if a plug flow region exists in the flow domain. Starting with [Lambda] [is greater than] 0, if we set [[Lambda].sub.2] to 0 in Eq 17, it can be shown easily that the condition for a plug region not to exist is given by:

[[Lambda].sup.s]/(s + 1) [is less than or equal to] 1 - [[Kappa].sub.1] - [[Kappa].sub.2] + [[Alpha].sub.2]

for all [[Lambda].sub.2] - [[Lambda].sub.1] [is greater than or equal to] 0 (Case 1) (31)

and that the reverse holds when Eq 31 is not satisfied, i.e.:

[[Lambda].sup.s]/(s + 1) [is greater than or equal to] 1 - [[Kappa].sub.1] - [[Kappa].sub.2] + [[Alpha].sub.2]

for all [[Lambda].sub.2] - [[Lambda].sub.1] [is greater than or equal to] 0 (Cases 2&3) (32)

To distinguish between Case 2 and Case 3, the value of [Lambda]* which is a limit on the thickness of the core region is calculated. The expression for [Lambda]* is obtainable from Eq 27 by replacing [[Lambda].sub.2] by [Lambda]* and setting [[Lambda].sub.1] to 0, hence:

[[Lambda].sup.s]/(s + 1)[(1 - [Lambda]*).sup.s + 1] + [[Kappa].sub.1] + [[Alpha].sub.1][Lambda]* + [[Kappa].sub.2] + [[Alpha].sub.2]([Lambda]* - 1) - 1 = 0 (33)

and Case 2 is obtained either if:

[[Lambda].sub.2] - [[Lambda].sub.1] = (2[[Tau].sub.o]/H dP/dz) [is greater than or equal to] 1 (34a)

or

[[Lambda].sub.2] - [[Lambda].sub.1] [is greater than or equal to] [Lambda]* (34b)

while the condition for Case 3 is:

[[Lambda].sub.2] - [[Lambda].sub.1] [is less than or equal to] [Lambda]* (35)

For [Lambda] [is less than] 0, the corresponding equations are:

Case 1

[(-[Lambda]).sup.s]/(s + 1) [is less than or equal to] 1 - [[Kappa].sub.1] - [[Alpha].sub.1] - [[Kappa].sub.2] for all [[Lambda].sub.2] - [[Lambda].sub.1] [is greater than or equal to] 0 (36)

Case 2

[(-[Lambda]).sup.s]/(s + 1) [is greater than or equal to] 1 - [[Kappa].sub.1] - [[Alpha].sub.1] - [[Kappa].sub.2] (37)

[[Lambda].sub.2] - [[Lambda].sub.1] = (-2[[Tau].sub.0]/H dP/dz) [is greater than or equal to] 1 (38a)

Table 3.

Case 1-no plug region ([Lambda] [is less than] 0)

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where [[Lambda].sub.1] is given by:

[Mathematical Expression Omitted]

Case 2-plug attached to the top surface ([Lambda] [is less than] 0)

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where [[Lambda].sub.1] is given by:

[Mathematical Expression Omitted]

Case 3-floating plug region ([Lambda] [is less than] 0)

[Mathematical Expression Omitted]

[u.sub.z] = 1 + [(-[Lambda]).sup.s]/(s + 1)[(1 - [[Lambda].sub.2]).sup.s + 1] - [[Kappa].sub.2] - [[Alpha].sub.2]([[Lambda].sub.1] - 1) [[Lambda].sub.1] [is less than or equal to] [Xi] [is less than or equal to] [[Lambda].sub.1] (3b)

[u.sub.z] = [(-[Lambda]).sup.s]/(s + 1)[(1 - [[Lambda].sub.2]).sup.s + 1] - [(-[Lambda]).sup.s]/(s + 1)[([Xi] - [[Lambda].sub.2]).sup.s + 1] - [[Kappa].sub.2] - [[Alpha].sub.2]([[Lambda].sub.1] - 1) + 1 [[Lambda].sub.2] [is greater than or equal to] [Xi] [is less than or equal to] 1 (3c)

[Mathematical Expression Omitted]

where [[Lambda].sub.2] and [[Lambda].sub.1] are related through:

[[Lambda].sub.2] - [[Lambda].sub.1] = -2[[Tau].sub.o]/H dP/dz (3e)

and [[Lambda].sub.1] is obtainable from:

[Mathematical Expression Omitted]

or

[[Lambda].sub.2] - [[Lambda].sub.1] [is greater than or equal to] 1 - [Lambda](*) (38b)

where [Lambda](*) is calculated from:

1 - [[Kappa].sub.1] - [[Alpha].sub.1][Lambda](*) - [[Kappa].sub.2] - [[Alpha].sub.2]([Lambda](*) - 1) - [(-[Lambda]).sup.s] / (s + 1) [([Lambda](*)).sup.s + 1] = 0 (39)

Case 3

[(-[Lambda]).sup.s] / (s + 1) [is greater than or equal to] 1 - [[Kappa].sub.1] - [[Alpha].sub.1] - [[Kappa].sub.2] (40)

[[Lambda].sub.2] - [[Lambda].sub.1] [is less than or equal to] 1 - [Lambda](*) (41)

ACKNOWLEDGMENTS

This study has been sponsored by the Department of the Navy, Office of the Chief of Naval Research, and the content of the information does not necessarily reflect the position or the policy of the Government. The experimental work summarized in Figs. 1 to 3 was carried out by Ms. B. Aral of HFMI. We acknowledge helpful discussions with Prof. U. Yilmazer of METU/Turkey.

REFERENCES

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Table 1. The Geometry, Operating Conditions, and Material Parameters Used in the Case Study.

Herschel-Bulkley Fluid

Shear rate sensitivity parameter, n = 0.5 m = 7800 Pa-[s.sup.0.5] [[Tau].sub.0] = 83,800 Pa

Navier's wall slip coefficient for the screw root = 7.4 X [10.sup.-7] m/(Pa - s) Density = 2100 kg/[m.sup.3]

Channel Dimensions Channel depth, H = 0.005 m Channel width, W = 0.02 m

Operating Condition Linear screw velocity = 0.3 m/s
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Author:Lawal, Adeniyi; Kalyon, Dilhan M.
Publication:Polymer Engineering and Science
Date:Oct 15, 1994
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