# Flow/pressure characteristics and modeling of deformation processes of single-screw extruders.

INTRODUCTIONExtrusion machines (worm presses) are among the most widespread equipments to be used as a component part of various production lines, units for the production of polymeric films, inflated and shaped articles, tubes, sheets, and other. Extruders are used in continuously process polymeric materials. The interplay between extruder configuration, operating conditions, and polymer rheology is important in dictating the final properties of the extrudate. The rheological properties of the polymer have a significant impact on fluid flow in an extruder (1-5).

The principal technical characteristic of extruders, named flow/pressure balance, permits to evaluate their practical output in processing various polymers as a function of the hydraulic resistance which depends on the flow of an extruded polymeric material through the channels of the forming tool. The simulation of the flow conditions and properties inside single and twin extruders are encouraging areas in this field (6-9).

The known scientific researches in the field of theoretical determination of flow/pressure characteristic of extruders (worm presses) are not suitable for practical use. Because, calculated data based upon the well-known theoretical relationships do not check with those found by experiment. The basic reason for this situation is that the used theoretical relationships, which can determine flow/pressure characteristic of extruders fail to take proper account of the rheological properties of processed polymeric materials. For example, the theoretical relationship cited in (10) considers only the viscous properties of polymer melts, whereas the majority of these melts are viscoelastic media.

Some unsuccessful attempts account the viscoelasticity of polymers in solving the considered problem (11). However, theoretical and experimental results are in qualitative agreement, there is no sufficient accuracy of quantitative correspondence by use of inadequate rheological model. Investigations that include fluids with complex rheological properties and higher pressure gradients are currently under progress (12), (13).

When the flow/pressure characteristic of an extruder determining, this study specifies inverse motion of the working parts of extruder, by fixed screw and moveable cylinder wall (see Fig. 1).

[FIGURE 1 OMITTED]

The solution of the problem may be simplified with the following assumptions. The flow of polymer melt in the channel of the proportioning zone of the screw is inertia free, isothermal, and steady state. The polymer melt is an incompressible medium, which "adheres" to the walls of the channel. The curvature of the screw channel and the effect of the lateral walls are disregarded. We considered the screw design most frequently used in processing polymers, in which the depth and pitch of the groove in its proportioning zone are constants.

MATHEMATICAL DESCRIPTION

The motion equations of a polymer melt in the channel of the proportioning zone of the screw in a rectangular Cartesian coordinate system (see Fig. 1) with allowance for the assumptions adopted are as follows:

[[partial derivative]P/[partial derivative]x] [equivalent to] A = [[[partial derivative][[tau].sub.xy]]/[partial derivative]y]], [[partial derivative]P/[partial derivative]z] [equivalent to] B = [[[partial derivative][[tau].sub.yz]]/[partial derivative]y] (1)

where [[tau].sub.xy], [[tau].sub.yz] are components of the stress tensor and P = P(x, z) is a function of the pressure distribution in the screw channel, which is governed by the resistance of the forming tool (extrusion head).

Solutions of differential Eq. 1 will be as follows:

[[tau].sub.xy] = Ay + [A.sub.1], [[tau].sub.yz] = By + [B.sub.1] (2)

where [A.sub.1], [B.sub.1] are constants of integration.

For consideration both viscous and elastic properties of polymeric melts, following viscoelastic model is implemented (5), (14).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [bar.[sigma]] denotes (::) stress tensor; P, Lagrange multiplier, determined by boundary condition; [bar.[delta]], identity tensor; [bar.c], Cauchy strain tensor; [[bar.e].sub.f], flow strain rate tensor; [bar.[omega]], vortex tensor; [bar.e], strain rate tensor; [[theta].sub.0](T), relaxation time; [G.sub.0](T), tensile modulus; W, strain energy function W = W ([I.sub.1], [I.sub.2]); [I.sub.1] and [I.sub.2], primary and secondary strain tensor invariants; [psi], dimensionless parameter ([psi] = 0 at [bar.[omega]] = 0 and [psi] = I at [bar.[omega]] [not equal to] 0); [Florin] ([I.sub.1], [I.sub.2]), dimensionless function that defines relaxation time, and 2[W.sup.S] = W([I.sub.1], [I.sub.2]) + W([I.sub.2], [I.sub.1]), symmetric function of W.

The last one can be shown by: [W.sub.1] = [[partial derivative]W/[partial derivative][I.sub.1]]], [W.sub.2] = [[partial derivative]W/[partial derivative][I.sub.2]], [W.sub.1.sup.S] = [[partial derivative][W.sub.s]/[partial derivative][I.sub.1]], [W.sub.2.sup.S] = [[partial derivative][W.sub.s]/[partial derivative][I.sub.2]].

In practice, there is a problem for application of Eq. 3. This problem arises from the choice of strain energy function W = W ([I.sub.1], [I.sub.2]). Most researchers use Mooney-Rivlin potential, but there are differences between experimental and theoretical results for prediction of stress and strain. Results of recent research show that in various kinematical deformations, the following potential can be used (5).

W = 0.25[G.sub.0]([I.sub.1] + [I.sub.2] - 6) (4)

The analysis of the research (15) shows that accounting for expression (4), dimensionless function [Florin] ([I.sub.1], [I.sub.2]) as a component of the rheological model (3) along with polymer shearing ([I.sub.1] = [I.sub.2]), will be as follows:

[Florin]([I.sub.1]) = [1 + [square root of 1 - [([2/[[I.sub.1] - 1]]).sup.2]]].sup.2][1 - [square root of 1 - [([2/[[I.sub.1] - 1]]).sup.2]]] (5)

Within the stated coordinates (see Fig. 1), shearing of the polymeric media which flow along the screw channel is considered to be two-dimensional. Thus, strain rate tensor and vortex which are the parts of the rheological model (3) will be as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [[gamma].sub.zy] [d[V.sub.z](y)/dy], [[gamma].sub.xy] = [d[V.sub.x](y)/dy], V(y) is the velocity vector of the media flowing in the screw channel, and [V.sub.z](y) [V.sub.x](y) are its components in the correspondent axes directions; [[gamma].sub.ij] = [[gamma].sub.ji] at i [not equal to] j.

Generally, accounting for the chosen coordinates, components of elastic strain tensor and its multiplicative inverse tensor will be as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [c.sub.ij] = [c.sub.ji] at i [not equal to] j.

It's easy to prove that, on the conditions of shear stresses, the first and the second invariants of elastic strain tensor will be equal ([I.sub.1] = [I.sub.2]). Therefore, it is sufficient to reduce the considered two-dimensional media shearing to one-dimensional. As the aforementioned invariants are equal, we consider the following kinematical equation:

[C.sub.zz] + [C.sub.yy] + [C.sub.xx] = [C.sub.xx][C.sub.yy] - [C.sub.xy.sup.2] + [C.sub.xx][C.sub.zz] - [C.sub.xz.sup.2] + [C.sub.yy][C.sub.zz] - [C.sub.yz.sup.2] (8)

The adopted assumption that the polymer is an incompressible medium, that is, its volume is constant when straining, leads to the next condition:

[I.sub.3] [equivalent to] det[bar.c] = 1 (9)

In expanded form, the latter condition appears as the following equation:

[C.sub.xx][C.sub.yy][C.sub.zz] - [C.sub.zz][C.sub.xy.sup.2] - [C.sub.xx][C.sub.yz.sup.2] - [C.sub.yy][C.sub.xz.sup.2] + 2[C.sub.xy][C.sub.yz][C.sub.xz] = 1 (10)

RESULTS AND DISCUSSION

The received results allow determining the components of flow strain rate tensor of the viscoelastic polymeric media flow using the second tensor equation of Theological model (3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Applying the third tensor equation of rheological model (3) and taking into account the expressions (4)-(11) for stable media flow ([[bar.dc]/dt] = 0) and ordinary stresses

([[gamma].sub.yz] = const; [[gamma].sub.yx] = const), we get the following determined system of scalar equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [[GAMMA].sub.xy] = [[gamma].sub.xy][[theta].sub.0](T); [[GAMMA].sub.yz] = [[gamma].sub.yz][[theta].sub.0](T).

The solution of the system of Eq. 12 will be as follows:

[C.sub.zz] = [[[c.sub.xy.sup.2] + [c.sub.yz.sup.2] [[1 + [c.sub.yz.sup.2] + [c.sub.xy.sup.2]]/[[square root of 1 - ([c.sub.yz.sup.2] + [c.sub.xy.sup.2])]]]]/[[c.sub.2.sup.yz] + [c.sub.xy.sup.2]]]; [C.sub.xx] = [[[c.sub.yz.sup.2] + [c.sub.xy.sup.2] [[1 + [c.sub.yz.sup.2] + [c.sub.xy.sup.2]]/[[square root of 1 - ([c.sub.yz.sup.2] + [c.sub.xy.sup.2])]]]]; [C.sub.yy] = [square root of 1 - ([c.sub.yz.sup.2] + [C.sub.xy.sup.2])]; [C.sub.xz] = [C.sub.xy][C.sub.yz][[[[1 + [c.sub.yz.sup.2] + [c.sub.xy.sup.2]]]-1/[[square root of 1 - ([c.sub.yz.sup.2] + [c.sub.xy.sup.2])]]]]/[[c.sub.yz.sup.2] + [c.sub.xy.sup.2]]]; (13)

where functions [c.sub.xy], [c.sub.yz] at stated dimensionless shearing rate ([[GAMMA].sub.xy], [[GAMMA].sub.yz]) are determined by the solution of the next, following to system of Eq. 12:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The specified relationships (13), (14) permit to state the following equations to be effectively used in the next analysis:

[C.sub.zz] = 1 + [[C.sub.yz]/[C.sub.xy]][C.sub.xz], [C.sub.xx] = 1 + [[C.sub.xy]/[C.sub.yz]][C.sub.xz] (15)

Taking into account the previous equations, the first tensor equation of rheological model (3) leads to the following expressions, which determine the values of shear components of stress tensor:

[[sigma].sub.yz] [equivalent to] [[tau].sub.yz] = [[G.sub.0](T)/2][[c.sub.yz](1 + [c.sub.xx]) - [c.sub.xy][c.sub.xz]] = [G.sub.0](T)[c.sub.yz] [[sigma].sub.xy] [equivalent to] [[tau].sub.xy] = [[G.sub.0](T)/2] [[c.sub.xy](1 + [C.sub.zz]) - [C.sub.xy][C.sub.yz]] = [G.sub.0](T)[c.sub.xy] (16)

Rheological Eq. 14 with substituted expressions (16) and in the context of received solutions (2) result the following system of equations, the solution of which determines components of velocity vector of the flowing polymeric media in the channel of the proportioning zone of the screw:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

where

F(y) = [[1 + [square root of ([([[~.A].sub.y] + [[~.A].sub.1]).sup.2] + [([[~.B].sub.y] + [[~.B].sub.1]).sup.2])]].sup.2] x [[1 - [square root of ([([[~.A].sub.y] + [[~.A].sub.1]).sup.2] + [([[~.B].sub.y] + [[~.B].sub.1]).sup.2])]].sup.1]; [bar.A] = [A/[G.sub.0](T)], [[~.A].sub.1] = [[A.sub.1]/[G.sub.0](T)], [~.B] = [B/[G.sub.0](T)], [[~.B].sub.1] = [[B.sub.1]/[G.sub.0](T)].

F(y) = [[1 + [square root of ([([[~.A].sub.y] + [[~.A].sub.1]).sup.2] + [([[~.B].sub.y] + [[~.B].sub.1]).sup.2])]].sup.2] x [[1 - [square root of ([([[~.A].sub.y] + [[~.A].sub.1]).sup.2] + [([[~.B].sub.y] + [[~.B].sub.1]).sup.2])]].sup.1]; [bar.A] = [A/[G.sub.0](T)], [[~.A].sub.1] = [[A.sub.1]/[G.sub.0](T)], [~.B] = [B/[G.sub.0](T)], [[~.B].sub.1] = [[B.sub.1]/[G.sub.0](T)].

With the known value of pressure gradient (B = [DELTA]P/[L.sub.zd]) in the channel length of the proportioning zone of the screw [L.sub.zd] which is governed by hydraulic resistance in the channels of the used forming tool, three remaining parameters--A, [A.sub.1], and [B.sub.1] are determined due to the next conditions.

The velocity of media motion on the movable wall of the cylinder of extruder (inverse motion) is equal to the latter on an accepted assumption: the volume flow rate of media when circulating across the channel is zero.

The above conditions correspond to the following system of equations, the solution of which specifies required parameters (A, [A.sub.1], and [B.sub.1]) under given pressure gradient B:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where D denotes (::) the screw diameter in its proportioning zone; H, the depth of the screw channel in proportioning zone; n, the screw rotation frequency; [phi], angle of inclination of the screw channel; [q.sub.x], the volume flow rate of media when circulating across the channel on a unit of length of its proportioning zone.

Now, under specified values of A, [A.sub.1], [B.sub.1] and corresponding pressure gradient B, volume flow rate of the viscoelastic polymer melt in the channel of the screw will be determined as follows:

[Q.sub.z] = [w/[[[theta].sub.0](T)]] x [[integral].sub.0.sup.H] [[integral].sub.0.sup.y] [[[~.By] + [[~.B].sub.1]]/[1 - ([([~.Ay] + [[~.A].sub.1]).sup.2] + [([~.By] + [[~.B].sub.1]).sup.2])]] (19)

where w is the width of the channel.

Figure 2 shows good agreement between the experimental and calculated data received from theoretical relationship (19).

[FIGURE 2 OMITTED]

CONCLUSIONS

The analysis of relationship (19) approves that the output of extruder depends not only on geometrical parameters of screw, their rotation frequency and the resistance of forming tool, but also significantly on rheological characteristics of the processed polymeric articles (relaxation time and elastic shear modulus). This factor should be meant in developing and designing extrusion equipment. For example, when designing extruders for processing polymeric media with high values of elastic shear modulus (raw compound) the length of proportioning zone of the screw may be relatively small (about some few diameters of the screw). For polymers with low values of elastic shear modulus, the relative length of proportioning zone of the screw should preferably be not less than 10 screw diameters. The appliance of relationship (19) with algebraic manipulator easily determines quantitative effect of rheological and other parameters on the performance of extruders. The validity of obtained and presented results permits to recommend them reasonably not only for estimation of extruder output but also for making calculations necessary in extruder equipment design: checking calculation of strength and stability of a screw, calculation of driving power of the equipment being designed, strength calculation of a plasticizing cylinder, etc.

REFERENCES

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Correspondence to: H. Hosseini; e-mail: pedram465@yahoo.com

DOI 10.1002/pen.21722

View this article online at wileyonlinelibrary.com.

[C] 2010 Society of Plastics Engineers

A.A. Borisov, (1) B.V. Berdyshev, (1) H. Hosseini, (2) B. Shirkavand-Hadavand (3)

(1) Department of Polymer Engineering, Moscow State University of Environmental Engineering, Moscow, Russia

(2) Department of Chemical Engineering, Islamic Azad University, Abadan Branch, Iran

(3) Institute for Color Science and Technology, Tehran, 1668814811, Iran

DOI 10.1002/pen.21722

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Author: | Borisov, A.A.; Berdyshev, B.V.; Hosseini, H.; Shirkavand-Hadavand, B. |
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Publication: | Polymer Engineering and Science |

Article Type: | Report |

Geographic Code: | 4EXRU |

Date: | Oct 1, 2010 |

Words: | 2751 |

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