BACKMIXING IN SCREW EXTRUDERS

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1 BACKMIXING IN SCREW EXTRUDERS Chris Rauwedaal, Rauwedaal Extrusio Egieerig, Ic. Paul Grama, The Madiso Group Abstract Mixig is a critical fuctio i most extrusio operatios. Oe of the most difficult mixig tasks is backmixig. A extrusio operatio where good backmixig is very importat is whe a low percetage color cocetrate, CC, is added to a virgi polymer. I this case, the iitial distace betwee the CC pellets may be mm or greater. If the fial striatio thickess eeds to be reduced to the micro level, the reductio of the striatio thickess eeds to be at least five orders of magitude this is quite a tough task! This paper will aalyze how the velocity profiles, axial mixig, ad residece time distributio are related. It will be show why simple coveyig screws have poor axial mixig capability. New mixer geometries that are specifically desiged to improve backmixig will be discussed. Results from extrusio experimets will be preseted. Cross Sectioal Mixig ad Axial Mixig Most aalyses of mixig focus o cross sectioal mixig, e.g. (). The cross sectioal mixig is determied mostly by the Couette shear rate betwee the rotatig screw ad statioary barrel. Usig the flat plate approximatio ad assumig pure drag flow, this shear rate ca be approximated by: πdn γ = H & () where D is the barrel diameter, N the rotatioal speed, ad H the chael depth. More accurate expressios for the shear rate have bee developed usig cylidrical coordiates (). Typical values of the Couette shear rate i sigle screw extruders rage from 5 to sec -. With a typical residece time i the melt coveyig zoe of about secods, the resultig total shear strai rages from about, to, uits. This meas that the striatio thickess i cross sectioal mixig is reduced by about three orders of magitude. I may cases, this is ot eough to achieve a level of mixig that appears uiform to the aked eye. plates, the dimesioless velocity φ=v/v max ca be writte as a fuctio of the dimesioless ormal coordiate ξ=y/h as follows: + φ ( ξ ) = ξ () Figure shows the velocity distributio for several values of the power law idex. The velocity profile for a Newtoia fluid (=.) is a parabola. The shear rate i the ceter of the chael is zero. As a result, o mixig will take place there. The ceter regio flattes as the power law idex reduces. I other words, the velocity profile becomes closer to a plug flow profile as the power law idex approaches zero. This meas that the low shear rate regio expads as the fluid becomes more shear thiig. Thus, the regio with poor mixig becomes larger whe the power law idex reduces. From this simple aalysis it becomes clear that the situatio for backmixig is substatially more difficult tha for cross sectioal mixig. For shear thiig fluids there is a cosiderable regio i the ceter of the chael where little or o axial mixig takes place. Residece Time Distributio The RTD ca be determied from the velocity profiles i the chael. The axial flow i a screw extruder is a pressure flow because there are o axial velocity compoets of the screw or barrel. As a first approximatio, the axial pressure flow ca be cosidered a flow betwee parallel plates. The velocity profile for pressure flow of a power law fluid betwee parallel plates is (4): WH H P V = ( + ) ml & (3) Where W is the width of the chael, H the height of the chael, P the pressure drop over axial legth L, the power law idex, ad m the cosistecy idex of the fluid. The velocity profile ca be writte as: Axial mixig or backmixig occurs by pressure flow (3). For pressure flow of a power law fluid betwee parallel

2 H H P v y) = ( + ) ml y H + ( (4) where ormal coordiate y rages from H/ to +H/. The velocity profile ca be writte as: v y) = v max y H + ( (5) The exteral RTD fuctio f(t)dt ca be determied from: dv& f ( t) dt = V& ( + ) y = ( + ) H H + dy Coordiate y ca be expressed as a fuctio of time by the followig substitutio: (6) L t = (7) v(y) With this substitutio, the exteral RTD fuctio ca be writte as: f ( t) dt ( + ) t ( + ) t + = (8) The miimum residece time t ca be determied from: L t = (9) v max The cumulative RTD fuctio F(t) ca be foud by itegratig the exteral RTD fuctio; it ca be expressed as: F t) + = t t t + + t ( () For a Newtoia fluid the RTD fuctio becomes: t = t + F( t) t t t t 3 dt () If we express the RTD as a fuctio of the dimesioless residece time θ, where θ is the actual residece time divided by the mea residece time, we get: F( θ ) = ( + + ) θ + + ( + ) θ () The expressio above for a power law fluid has ot bee published before. With this expressio the RTD ca plotted at several values of the power law idex, see figure. It is clear from figure that the RTD becomes arrower as the value of the power law idex reduces. This meas that backmixig reduces as the fluid becomes more shear thiig (lower power law idex). Figure cofirms what we have already see i the velocity profiles of figure. As the fluid becomes more shear thiig, the velocity profile becomes closer to plug flow ad, cosequetly, the RTD becomes arrower ad backmixig more problematic. RTD i Screw Extruders Pito ad Tadmor (5) developed expressios for the RTD i sigle screw extruders. The cumulative RTD fuctio ca be writte as: ( ξ + ( ξ ) + ξ 3 ) F( t) = F( ξ ) =.5 3 ξ (3) The dimesioless time θ (time divided by mea residece time) ca be expressed as a fuctio of the dimesioless ormal coordiate ξ: θ 3ξ + 3 6ξ ( ξ + + ξ 3ξ = (4) + ξ 3ξ ) The two expressios above were derived for a Newtoia fluid usig the flat plate approximatio cosiderig both dow- ad cross-chael velocity compoets. Figure 3 shows the RTD for a sigle screw extruder as well as for pressure flow of a Newtoia fluid betwee flat plates. The sigle screw RTD is arrower tha the flat plate RTD because of the recirculatio of the fluid i the screw chael. Fluid speds more time i the lower portio of the chael ξ=-/3 tha i the upper portio of the chael ξ=/3-. I order to properly determie the RTD of a extruder we have to cosider ot oly dow- ad cross-chael velocity compoets but also ormal velocity compoets that occur at the flight flaks. This will require a umerical aalysis, either FDA, FEA, or BEA. Eve

3 though the depth of the chael is usually quite small compared to the chael width, the residece time at the flight flaks is substatial because the ormal velocities are quite small. Joo ad Kwo (6) poited out limitatios of the Pito aalysis. As oe would expect, the Pito model uderpredicts the residece times relative to a full three-dimesioal aalysis, particularly with large values of the axial pressure gradiet. The Pito RTD for a sigle screw extruder is arrower tha the RTD for pressure flow betwee flat plate for a Newtoia fluid. The effect of shear thiig is to further arrow the RTD as discussed earlier. These two effects explai why backmixig is such a critical issue i screw extruders. Methods to Improve Backmixig A major cocer i backmixig is the fluid i the ceter regio of the screw chael where the axial shear strai is zero or close to zero. I a simple coveyig screw the fluid i the ier recirculatio regio will stay withi this regio util it reaches the ed of the screw. Whe this happes, the material flowig ito the die will be poorly mixed. Mixig pis ad slots i the screw flights will improve axial mixig because they achieve a short term splittig ad reorietatio of the fluid. The effect of mixig pis o backmixig is illustrated i figure 4. These results were obtaied usig a three-dimesioal BEM flow aalysis. It is clear that oe row of mixig pis has limited effect o axial mixig. Backmixig ca be improved by varyig the spacig betwee the pis, thus, itetioally creatig streams of differet axial velocities. The challege i improvig axial mixig is to efficietly trasfer fluid from the ier recirculatio regio to the outer regio ad vice versa. A simple but effective method of doig this is the iside-out mixer show i figure 5. The flight i this mixer is offset so that the material i the ceter regio is cut by the offset flight ad the pushed to the screw ad barrel surfaces by the ormal pressure gradiets that occur at the flight flak. Results of particle trackig usig BEA are show i figure 6. The redistributio of the material is show i figure 7. It shows how the fluid from the ceter regio is cut by the offset flight ad pushed to screw surface at the pushig side of the flight ad to the barrel surface at the trailig side of the flight. Figure 7 shows the redistributio of the fluid at several axial locatios from the flight offset. It is clear that a cosiderable axial distace is ecessary to brig about the redistributio of the material. Coclusios Backmixig is oe of the most difficult mixig tasks i screw extruders. This is due to the fact that the axial strai rates i the extrusio process are very low, particularly i the ceter regio of the chael. The axial velocity profile is close to plug flow, particularly for strogly shear-thiig fluids. Backmixig problems are especially severe whe addig a small percetage cocetrate to the extruder. Whe both the polymer ad cocetrate are i pellet form, the iitial striatio thickess ca be of the order of mm. If a fial striatio thickess of micro is required, the axial mixig has to achieve a reductio of striatio thickess of at least 5 orders of magitude. Cosiderig that the axial shear rate i melt coveyig is close to zero i the ceter regio of the chael, it is clear that axial mixig will be isufficiet uless efficiet mixig devices are used. The most efficiet way to improve axial mixig is to redistribute material from the ceter of the chael to the outer regio of the chael ad vice versa. Oe mixer that aims to achieve such redistributio is the iside-out mixer. Aother mixer that was desiged specifically to improve axial mixig is the CRD7 mixer, see figure 8. Aother method of reducig mixig problems with color cocetrates is to reduce the iitial striatio thickess. This ca be doe by reducig both the virgi plastic pellet size ad the color cocetrate pellet size. Graules are better tha pellets ad powder is better tha graules from a mixig poit of view. Smaller particle sizes may lead to other problems though, such as coveyig problems ad air etrapmet. Addig the colorat i liquid form ca also reduce the iitial striatio thickess. This is oe of the mai reaso liquid colorats are used. However, they ca create problems by formig a lubricatig layer o the barrel surface ad reducig the coveyig efficiecy of the extruder. Refereces. C. Rauwedaal, Polymer Mixig, A Self-Study Guide, Haser Publishers, Muich (998). C. Rauwedaal, T. A. Osswald, G. Tellez, ad P. J. Grama, Flow Aalysis i Screw Extruders Effect of Kiematic Coditios, Iteratioal Polymer Processig, XIII, 4, (998) 3. Z. Tadmor ad C.G. Gogos, Priciples of Polymer Processig, Joh Wiley ad Sos, New York (979) 4. R.B. Bird, W.E. Stewart, ad E.N. Lightfoot, Trasport Pheomea, Wiley, New York (96) 5. G. Pito ad Z. Tadmor, Polym. Eg. Sci.,, 79 (97) 6. J.W. Joo ad T.H. Kwo, Polym. Eg. Sci., 33, 5, 959 (993)

4 Velocity Distributio Power Law Fluid. Dimesioless Velocity =. =.5 = Dimesioless Normal Coordiate Figure, Velocity profiles for various power law idex values RTD of Power Law Fluid i Flat Plate Pressure Flow.9 Cumulative RTD Fuctio =.5 =. = Dimesioless Time Figure, Residece time distributio curves for several power law idex values RTD Sigle Screw Extruder.9 Cumulative RTD Fuctio Sigle Screw Extruder Flat Plate Newtoia Dimesioless Time Figure 3, RTD of sigle screw extruder ad flat plate

5 Figure 4, Particle trackig results i a mixig sectio with elogatioal mixig pis Figure 5, Solid model of Iside-Out mixer

6 Figure 6, Particle trackig i the Iside-Out mixer Cross sectio Cross sectio Figure 7, Redistributio i the Iside-Out mixer Trasport directio Figure 8, The CRD7 mixer

The axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c.

The axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c. 5.4 Applicatio of Perturbatio Methods to the Dispersio Model for Tubular Reactors The axial dispersio model for tubular reactors at steady state ca be described by the followig equatios: d c Pe dz z =