Numerical Simulation of Melting Process in Single Screw Extruder with Vibration Force Field

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1 Nerical Silation of Melting Process in Single Screw Extrder with Vibration Force Field Yanhong Feng Jinping Q National Eng. Research Center of Novel Eqipent for Polyer Processing, Soth China University of Technology, Gangzho 51641, China Abstract Melting capacity is always the bottleneck of perforance iproveent of single-screw extrder, and becoe the nods of the extrsion theory. As a novel extrsion technology, the electroagnetic dynaic plasticating (EMDP) extrder for plastics has acqired great econoic and social benefit progressively. Bt the non-linear viscoelastic behavior of polyer and phase change which is a non-linear proble increase the degree of difficlty of the research on the elting echanis with vibration force field and restrict the exertion of the potential of EMDP extrder in a degree. In this paper, according to the assption of flowing solid, the hydrokinetics is adopted to solve the whole elting proble region, which solves the phase change and the solid-elt interface tracking probles. A 2D elting odel is established. This elting odel fits for the research on the inflence of vibration force field on the elting process. Since polyer has tie-dependent non-linear viscoelastic characteristic with vibration force field, and the elting process is a phase change proble, a ser-defined progra of self-aended non-isotheral Maxwell constittive eqation is developed with the UPFs of ANSYS. This constittive eqation can reflect the relaxation tie spectr of polyer. The responses of elting process to varios vibration paraeters are silated. And the rle of the inflence of freqency and aplitde on elting process is analyzed. Introdction In single screw extrder for plastics, the barrel is iobile, the rotation of the screw pshes polyer conveying toward the extrder head. Dring this corse, the solid bed ade of solid granles elts owing to the condctive heat and heat of viscos dissipation, and this stage is called elting process. In EMDP extrder, sinsoidal velocity in circferencial direction is sperposed on the steady rotation of screw, and additional vibration velocity can also be introdced in the axial direction, ths the screw can rotate plsantly. We call the elting process in the EMDP extrder the dynaic elting process. The research on the dynaic elting process is different fro the traditional elting process by the following three characteristics: 1. the dynaic elting process is periodic, the transient effect shold be considered; 2. Since polyer elt will behave tie-dependent non-linear viscoelastic characteristic with vibration force field, the hypothesis of pre viscos property of elt won t be sitable any ore; 3. the heat generated by dynaic dissipation is another characteristic of dynaic elting. Therefore, the dynaic elting process is a non-linear, non-isotheral, close copled transient proble. To carry ot the nerical silation of dynaic elting process we have to: 1. establish a elting odel which not only represent the actal process bt also is feasible; 2. solve the phase-change silation proble; 3. silate the transient flow of viscoelastic flid.

2 Preparation Dynaic elting odel The elting section can be divided into for sbsections: pper-elt-fil sbsection,elt-pool sbsection, five-zone elting sbsection and solid-bed break-p sbsection. When the screw is not cooled internally, the srface of the screw channel will be covered with thin elt fil very soon which is called side elt fil and lower elt fil, the elting process coes into the fivezone elting sbsection, and this sbsection occpies the ost part of the elting section. The appearance of elt fil changes the convey echanis of solid bed fro the solid friction to the shear effect of elt. In this paper, we focs on the five-zone elting sbsection. See Figre 1. It shows the scheatic diagra of five-zone elting odel with nrolled screw. In this Figre the barrel is stationary, the screw rotates, becase of the helix angle, the rotation velocity can be decoposed into Vscrewz in the down channel direction and Vscrewx in the cross channel direction. The solid polyer conveys in the down channel direction at velocity of Vsz. While in the EMDP extrder periodic velocity coponents, naely Vvibz in the down channel direction and Vvibx in the cross channel direction are sperposed on Vscrewz and Vscrewx respectively. Figre 1. Scheatic diagra of five-zone elting odel Considering the special physical properties of polyer which is different fro the other inorganic aterials, we pt forward an iportant assption: flowing solid phase to describe the conveying process of solid phase in the elt. The following will illstrate the rationality of flowing solid phase fro the physical strctre, properties of polyer and elting behavior dring extrsion. 1) the characteristic of physical strctre of polyer Considering the phase state of polyer, the glass state, rbbery state and elt state are all belong to liqid state, becase the arrangeents between olecle are rando. The ain difference between these states is the deforability, which is difference of echanics state. 2) the characteristic of physical property of polyer

3 Considering the viscoelasticity of aterial, the difference between solid and liqid is jst the difference of relaxation tie. 3) the characteristic of elting process of polyer extrsion Dring the solid conveying process in the EMDP extrder, the copaction effect of solid pellets is enforced by vibration force field, and the solid pellets are fored into a plg-like contin. So the polyer solid phase dring elting process can be considered as continos liqid. Fro the above explanations, the assption that sing the viscoelasticity property of polyer to describe the elting process after the solid bed is srronded by elt is rational. According to the assption of flowing solid phase, hydrokinetics can be applied to solve the whole doain of elting proble which incldes the so-called solid phase zone and elt zone. So this ethod not only is rational, bt also solves the dynaic tracing proble of solid/elt interface sccessflly. Becase of the adoption of viscoelastic constittive eqation, ore dense eshes are needed to captre the special conveying fashion of elt adjacent to the vibration velocity bondary. And the solid/elt interface where the rheology and physical properties of polyer change sharply is varying along the extrsion direction, which also indicates ore dense eshes along the channel depth are needed. Therefore, twodiensional elting odel is sed. Considering that the contribtion of elt pool to elting ability is neglectable, we se the 2D elting odel along the down channel direction. Becase or focs is the effect of vibration paraeters on the dynaic elting process, it ight be as well to frther extract the entrance part of 2D elting odel to represent the initial stage of elting process. The coordinate frae sed in the analysis, which is shown in Figre 2 (a) geoetry odel, has x axis along the down channel direction and y axis along the channel depth direction. Figre 2. 2D elting odel The 2D elting odel is eshed into 4 eleents with FLUID141, as shown in Figre 2 (c) finite eleent eshes. And the larger the velocity gradient or teperatre gradient is, the denser the eshes are.

4 Bondary Conditions Velocity bondary conditions The average velocity in the down channel direction of solid phase at the entrance can be defined as w s & = ρ WH s (1) where ρ s is the density of the copacted solid polyer, W is the width of the channel, H is the depth at the start point of the copression section. The barrel is stationary, while the screw rotates. According to the no-slip assption, the velocity bondary conditions can be given as: v x = v y = H y y = H = (2) where v x y = H and v y y = H are the velocity coponents at the inner srface of barrel in the downstrea direction and transverse direction respectively. The plsative velocity bondary condition at the srface of screw can be defined as v y y = = (3) v x y = v x = v xa = v x const y= + v xa y= = a 2πf sin( 2πft ) (4) (5) (6) where v and v are the velocity coponents at the srface of screw in the downstrea direction x y = y y = and transverse direction, v x is the average velocity (naely, steady velocity), a is aplitde, f is freqency. The plsative velocity bondary condition at the srface of screw is applied sing the fnction bondary condition of ANSYS. Theral bondary conditions 1) Apply constant teperatre T b at the barrel srface. 2) Set the teperatre of screw srface the sae as that of barrel srface. 3) Asse the teperatre of solid phase at the entrance of elting section T s. Pressre bondary conditions Using the pressre difference between inlet and otlet as pressre bondary condition, apply higher pressre p at otlet near extrder head; apply lower pressre p in at the inlet near the feed section. ot See Figre 2 (b) bondary conditions.

5 Using User-Prograable Sbrotines The oleclar otion of polyer is of relaxation. The relaxation spectr is the ost coon fnction to prescribe the relationship of the viscoelasticity of polyer and tie or freqency. Microcosically, it characterize the contribtion of different strctre nits to relaxation tie; acroscopically, it connects those viscoelastic fnction, sch as the relaxation odlar, dynaic viscosity. The elting process is considered to be stable in traditional researches of elting echanis, and nonisotheral pre viscos constittive eqations are ostly sed. While dring the dynaic elting process, the screw rotates plsatively, casing the velocity field in elt change periodically, so do the shear rate in elt, which frther case the apparent viscosity and relaxation tie change periodically, so the flowing behavior dring the dynaic elting process is tie-dependent viscoelastic. While silating the dynaic elting process, constittive eqation that can reflect the tie-dependent viscoelastic characteristic st be sed. In recognition of the fact that the viscosity odels provided by ANSYS can not satisfy the reqests of all sers, a ser-prograable sbrotine (UserVisLaw) is also provided by ANSYS with access to the following variables: position, tie, pressre, teperatre, velocity coponent, velocity gradient coponent. Using the above inforation, a self-aended MAXWELL viscoelastic constittive eqation was prograed with FORTRAN. The MAXWELL constittive eqation (Reference 1) is: τ τ + λ = t ( γ &, T ) 2η ( γ &, T ) d (7) where, ( γ &, T ) ( γ &, T ) η G T λ = is relaxation tie, G is blk odls, d ( L+ L )/ 2 with L = ( υ) T, ( γ&,t ) = is strain rate tensor T & is shear rate, which is the second invariant of strain rate tensor d. γ = 2d : d η is the apparent viscosity which is a fnction of shear rate and teperatre. In this paper the apparent viscosity is also sed to describe the resistibility of polyer to deforation at teperatre lower than elting point. Since the polyer crystal elts within a teperatre range, the teperatre at which the polyer crystal elts copletely is sally called T, and the teperatre range fro the start point of elting to the final point of elting is called elting range described as: η η( T, & γ ) = exp exp ( T T ) [ b ( T T )] s r [ b ( T T )] ηt ηt ηt η T + 1+ c & γ ηt η T + 1+ c & γ T r. Ths the apparent viscosity can be T T g < T < T T T < T T T where T g is glass teperatre, η is viscosity with zero shear rate at teperatre T ( T T r ) Tr, T and η T are the viscosity with zero shear rate and second Newtonian viscosity at teperatre T respectively, c and are characteristic constants of polyer. Since the stress can not be described as the explicit fnction of velocity and gradient of velocity when silating the viscoelastic flow, the stress is a variable to be solved, which case the non-linear characteristics of viscoelstic flows and call ore challenge and ore copter resorces than those T r (8) η

6 nerical silation of Newtonian flows and generalized Newtonian flows. According to weather the stress is an original variable, the finite eleent ethods for viscoelastic flows can be divided into ixed ethod and split ethod. In this work, the split ethod (Reference2,3) is adopted; the velocity and pressre are the two variables to be solved. Ths the Galerkin weak for of the above proble can be stated as: to obtain the n+1 th iterative ( n+ ) ( 1) vale ( 1 n+ υ, p ) V P, the following eqations st be satisfied, Ω ( n+ 1) ( n+ 1) ( 2η d : φ p φ ) ref ( n 1) ( υ ) Ω + ϕ Ω =, ϕ P p d p dω = F φ d Ω + Ω Ω f φ dω Ω τ * : φ dω, φ V (9) (1) where φ and ϕ p are the shape fnctions of velocity and pressre respectively, V and P are fnction spaces of υ, p defined in Ω doain, η ref is the reference viscosity, naely η ( γ&,t ), τ = τ 2η d n ref, the extra stress τ can be solved sing the constittive eqation (7) with the forer n n iterative vale ( υ, p ). According to the above constittive eqation, we prograed the User-Defined Viscosity progra UserVisLaw, and the property type choice of viscosity is USER, the for coefficients NOMI, COF1, COF2, and COF3 are corresponding to η T, η T, c and respectively. This constittive progra not only takes the viscoelastic characteristic of polyer into accont, bt also reflects the relaxation spectr. Specifying Flid Properties for FLOTRAN Theral condctivity and Density The theral condctivity and density, are represented as piecewise continos linear fnctions of teperatre as flowing. ks, T < T T k ks k( T ) = ks + Tr k, T T ρs, T < T T ρ ρs ρ ( T ) = ρs + Tr ρ, T T r ( T T ), T T T < T (11) r r ( T T ), T T T < T (12) where, ks and k represents the theral condctivity of solid phase and elt respectively, represents the density of solid phase and elt respectively. r ρ s and ρ

7 Specific heat The specific heat, on the other hand, is given in sch a way as to inclde the latent heat of fsion for crystal polyer as the peak that occrs in the elting range. C C p( T ) = C C ps ps p, + T, T < T λ T r T T T ( T T ), T T T < T (13) r where, C ps and C p represents the special heat of solid phase and elt respectively, λ is latent heat of fsion. See Figre 3. r Figre 3. Teperatre crve of specific heat The above flid properties are specified with flid property table. Analysis Reslts & Discssion With the copletion of the odel, transient analyses of dynaic elting are perfored, and obtain the velocity and teperatre reslts. See Figre 4. It shows the velocity distribtion at the exit of flow channel. See Figres 5 and 6. They show the teperatre distribtion and density distribtion respectively. These reslts show the clear velocity, teperatre and density difference between solid state and liqid state polyer, which are coincident with those reslts obtained with traditional ethod and indicate the validity of new ethod provided in this paper.

8 Figre 4. Vector plot of velocity at the exit Figre 5. Teperatre contors

9 Figre 6. Density contors See Figre 7. It shows the instantaneos velocity vector plot along lower elt fil at the starting tie of a sinsoidal vibration. After this tie point the velocity of screw srface increases. See Figre 8. It shows the instantaneos velocity vector plot at the iddle tie of a sinsoidal vibration. After this tie point the velocity of screw srface decreases. See Figre 9. It shows the velocity distribtion crves at those above tie points and the velocity distribtion reslt obtained fro the correspondent steady analysis. As seen fro Figre 9, becase of the viscoelastic characteristic of polyer, the velocity crve dring the velocity increasing stage will not sperpose on the velocity crve dring the velocity decreasing stage, even thogh the screw velocities are the sae at these tie points. And both of these crves don t sperpose on the velocity crve of steady state. Figre 7. Velocity vector plot along lower elt fil at the starting tie of a sinsoidal vibration

10 Figre 8. Velocity vector plot along lower elt fil at the iddle tie of a sinsoidal vibration Figre 9. Velocity distribtes along lower elt fil at balance position See Figre 1. It shows the crves of velocity distribtion along lower elt fil at different tie. Velocity distribtion differences between velocity increasing stage and correspondent velocity decreasing stage for loops of velocity distribtion.

Figre 1. Crves of velocity distribtion along lower elt fil at different tie As seen fro Figre 11, the velocity-varying crves lag behind the velocity-varying crve of the point at the screw srface.

11 Figre 1. Crves of velocity distribtion along lower elt fil at different tie As seen fro Figre 11, the velocity-varying crves lag behind the velocity-varying crve of the point at the screw srface. And the velocity-varying crves will always lag behind those velocity-varying crves that are closer to the screw srface. Figre 11. Velocities at different points along lower elt fil varying within a vibration period As shear rate is not a direct reslt available in the postprocessing of ANSYS, a progra of shear rate is prograed sing APDL. See Figre 12. It shows the difference of average shear rate distribtion in the lower elt fil along channel depth between steady elting and dynaic elting with different vibration paraeters. It is jst becase the non-linear viscoelastic response of the velocity distribtion of polyer

12 elt with vibration force field akes the average shear rate distribtion in the elt change. Fro Figre 12, it can be seen that the introdce of vibration force field redces the average shear rate near the screw srface which rotates plsatively, while increase the average shear rate near the solid/elt interface. Coparing with the teperatre reslt, it is fond that the average shear rate increent reaches the axi near 115, which is to say that the introdction of vibration force field into elting process case change of average shear rate distribtion, and ake the average shear rate at the interface of solid/elt increase, which is in favor of the reoval of the newly-generated elt that is of higher viscosity with lower teperatre and accelerate elting process. Fro the above silation reslts, it can be seen that the larger the aplitde or freqency is, the ore obvios the effect of redcing average shear rate near driving srface and enhancing average shear rate near solid/elt interface is. Figre 12. Difference of average shear rate distribtion Conclsion In this effort, the assption of flowing solid phase was sed to describe the conveying behavior of polyer solid that has relaxation characteristic dring elting process. A 2D elting odel that can reflect the inflence of the paraeters of vibration force field on elting process was established. A ser-defined viscosity odel of self-aended MAXWELL viscoelastic constittive eqation was prograed via UPFs. The dynaic elting processes were silated. According to the silation reslts of the elting behavior of polyer with vibration force field, it is fond that the introdction of VFF into elting process can optiize the tie-average shear rate distribtion, naely, redce the average shear rate near the driving interface, enhance the average shear rate near the solid/elt interface and accelerate the dynaic refresh of solid/elt interface, generate a great deal of viscos dissipation heat near the solid/elt interface and proote elting process.

13 References 1) R.Y. Chang, W.L. Yang. J.Non-Newtonian Flid Mech. 1994, 51: ) Henrik Koblitz Tasssen. J.Non-Newtonian Flid Mech. 1999, 84: ) P.T. Frank, Baajijens. J.Non-Newtonian Flid Mech. 1998, 79:

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