Fundamentals of Polymer Engineering Part 16 ppt

15 Polymer Processing 15.1 INTRODUCTION Thus far, we have talked about how polymers are synthesized, how they are characterized, and how they behave as solids, melts, or in the form of solutions. Ultimately, however, it is necessary to convert the polymer into useful products. Typically, these may be rods, pipes, films, fibers, or molded articles. Most frequently these materials are made from a single polymer. Increasingly, though, blends, filled polymers, and composite materials are used. It should be noted that even when a single polymer is used, it is rarely a chemically pure material. Almost invariably, it contains additives that act as dyes, plasticizers, antioxidants, and so on. A variety of physical structures can result, depending on the kind of polymer and additives used and also on how these materials are processed. Because the final structure obtained determines the physical properties of the product, the process used has to be chosen with care. For inexpensive, high-volume disposable items such as beverage containers or toys, the most inexpensive process is used. The manufacture of high-value-added sophisticated items such as compact disks or optical lenses, on the other hand, requires a great deal of engineering and an intimate knowledge of the fundamentals of polymer behavior. In this chapter, we discuss just three of the most common polymer processing operations-extrusion, injection molding, and fiber spinning. Because 630 Copyright © 2003 Marcel Dekker, Inc. mostpolymersaresoldintheformofpellets,anextruderisrequiredtomelt, homogenize,andpumpthethermoplasticmaterial.Althougharticlessuchas tubes,rods,andflatsheetscanbemadebyextrusion,anextruderisoftencoupled withotherpolymerprocessingmachinery.Aknowledgeofextrusionistherefore aprerequisiteforstudyingotherpolymerprocessingoperations.Notethat extrusionisacontinuousoperation.Asopposedtothis,injectionmoldingisa cyclicoperationusedtomakeaverywidevarietyoflow-andhigh-technology itemsofeverydayuse.Itisnowalsobeingusedtofabricateceramicheatengine components,whichhavecomplexshapesandareusefulforhigh-temperature applications.Ceramicssuchassiliconcarbideandsiliconnitridearehard, refractorymaterials,andinjectionmoldingisoneoftheveryfewprocesses thatcanbeusedforthepurposeofmassproduction.Fiberspinningisstudiednot onlybecauseitisthemainstayofthesynthetictextilesindustrybutalsobecauseit isusedtomakenovelfibersoutofliquid-crystallinepolymers,graphite,glass, andceramicsforuseincompositematerials.Itisalsoaprocesswherepolymer elasticitybecomesimportantduetotheextensionalnatureoftheflowfield.In contrast,extrusionandinjectionmoldingareshear-dominatedprocesses. Althoughthisisthelastchapterofthebook,itisreallyanintroductionto theverypracticalandfascinatingtopicofmanufacturingitemsmadefrom polymericmaterials.Themajorpurposeofthischapteristodescribethese threeoperationsandalsotoshowhowfirstprinciplesareusedtomathematically simulatetheseoranyotherprocess.Suchanalysesareasinequanonfor improvingproductquality,fordesigningnewproducts,andforprocessoptimiza- tion.Thematerialpresentedhereisnecessarilysimplified,andwehaverestricted ourselveswhereverpossibletosteady-stateandisothermaloperations;extensions tomorerealisticsituationsareconceptuallystraightforward,andwehave providedcitationsoftheappropriatetechnicalliterature.Formoredetailson theprocessofconstructingmathematicalmodels,thereaderisdirectedtothe excellentbookbyDenn[1]. 15.2EXTRUSION Thisisthemostcommonpolymerprocessingoperation.Itisgenerallyusedto meltandpumpthermoplasticpolymersthroughadie,whichgivesadesiredshape totheextrudate.Althoughextrusioncanbecarriedoutusingpressure-drivenand plunger-drivendevices,itisthescrewextruderthatisusedalmostuniversallyin industrialapplications.Extruderscancontainmultiplescrews,butweshall initiallyfocusonthesingle-screwextruder,whichconsistsofahelicalscrew rotatinginsideacylindricalbarrel.ThisisshownschematicallyinFigure15.1[2]. The polymer is generally fed to the extruder in the form of pellets through a hopper, which leads to the channel formed between the screw and the barrel. If Polymer Processing 631 Copyright © 2003 Marcel Dekker, Inc. the polymer being processed is hygroscopic, it is usually dried beforehand and the hopper is blanketed by a dry inert gas such as nitrogen. Although the barrel is heated to a temperature above the melting point of the polymer, the region very close to the base of the hopper is often water cooled to prevent polymer from melting in the hopper and forming a solid plug, which would block additional polymer from entering the extruder. The rotation of the screw forces the polymer to move along the channel, and it does so initially as a solid plug, then as a semisolid, and, finally, as a melt. The channel depth is usually fairly large in the solids-conveying zone and it decreases progressively as the polymer melts and ultimately becomes constant in the melt zone. Although it is convenient to think in terms of these three separate zones and the screw geometry often reflects this thinking, it is probably true that the processes of solids conveying, melting, and melt pressurization occur simultaneously. For the purposes of analysis, though, we shall still treat the three zones separately. Of course, if the extruder is fed directly by a polymerization reactor, as happens during the manufacture of synthetic textiles by melt spinning, the solids-conveying and melting zones are absent. Only the melt zone remains; it is also called the metering zone. The purpose of any mathematical model of steady-state extrusion is to relate quantities such as energy dissipation, the volumetric flow rate, the melting profile, the temperature profile, and the pressure profile to the extruder geometry, to the processing variables such as barrel temperature and screw rpm, and to the material properties of the polymer. In order to accomplish this task, we also need to know the details of the die that is attached to the extruder. Because the process, in general, is nonisothermal and the rheological models are nonlinear, analytical solutions cannot be obtained for realistic cases of interest. Invariably, numerical techniques of solution have to be used. Here, though, we will consider the simplest possible cases with a view toward both elucidating the physics of the problem and illustrating the approach to be taken for problem solving. Analyses FIGURE 15.1 Schematic representation of a single-screw extruder. (From Ref. 2) 632 Chapter 15 Copyright © 2003 Marcel Dekker, Inc. formorerealisticsituationsareavailableintheliterature;here,themathematicsis morecomplicated,butthebasicapproachisthesame.Itshouldberealized, though,thattheprocessofdeterminingthescrewgeometrytoyielddesired extruderperformanceismuchmoredifficultthandeterminingextruderperfor- manceforagivengeometry. 15.2.1ScrewGeometry AsectionofasimplifiedscrewisshowninFigure15.2todefinethevariablesthat characterizethescrewgeometry[3].ThenotationusedisthatofTadmorand Klein[3].TheinsidediameterofthebarrelisD b ,whereasthescrewdiameteris D;bothofthesequantitiescanrangefrom1to12in.Atypicalvalueoftheratio ofthescrewlengthtoitsdiameteris24.ThechanneldepthisH,anditisclear fromFigure15.1thatbothHandDvarywithaxialposition.Theradialclearance between the tip of the flights and the inner surface of the barrel is d f , and L is the axial distance moved by the screw during one full revolution. The width of the screw flight in the axial direction is b, and the width in a direc tion perpendicular to the flight is e. Finally, W is the distance between flights measured perpendi- cular to the flights, and y, the helix angle, is the angle between the flight and the plane perpendicular to the screw axis. In general, y, b, and W vary with radial position; nonetheless, we will take them to be constant. We will also assume that d f is negligible and that there is no leakage of material over the flights. Also, for purposes of analysis, we will assume that D b is approximately the same as D. To begin the analysis, we recognize that flow occurs because friction at the surface of the barrel makes the plastic material slide down the channel and go toward the extruder exit as the screw is rotated. This motion of a material element, resulting from the relative velocity between the barrel and the screw, can be FIGURE 15.2 Line a–a indicates a cut perpendicular to the flight at the barrel surface. (From Ref. 3.) Polymer Processing 633 Copyright © 2003 Marcel Dekker, Inc. studiedmoreeasilybyallowingthebarreltorotateinadirectionoppositetothat ofscrewrotationandholdingthescrewstationary.Further,ifwerealizethatthe curvatureofthescrewishardlyfeltbythepolymer,wecanconsiderthatthe polymerismovingdownalong,rectangularcross-sectionalchannelduetothe movementoftheuppersurface.ThisisshowninFigure15.3.Ineffect,wecan unwindthechannelanduseCartesiancoordinatesfortheanalysis. 15.2.2Solids-ConveyingZone Thisistheregionfromthepointatwhichmaterialentersthehoppertoapointin theextruderchannelwheremeltingbegins.Althoughalargenumberofmodels havebeenproposedtodeterminetheflowrateofsolidsinthisregion,the acceptedanalysisisthatofDarnellandMol[4],whichispresentedherein simplifiedform. Aspolymerpelletsmovedownthechannel,theybecomecompactedintoa plugthatmovesatavelocityV p inthedownchannelorzdirection.Ingeneral, thereisslipbetweentheplugandboththebarrelandscrewsurfaces.Thebarrel movesatavelocityV b ,whichinmagnitudeequalspDN,whereNisthe revolutionsperunittimeofthescrew;thisvelocityvectormakesanangleyto thedown-channeldirection.Clearly,thevelocityofthebarrelrelativetotheplug is(V b ÀV p )anditmakesanangle(yþf)tothezdirection;thisistheangleat whichthebarrelappearstomoveforanobservermovingwiththeplug. WithreferencetoFigure15.4,wehave tanf¼V pl V b À V pl tany  À1 ð15:2:1Þ FIGURE15.3Polymerflowinthechannelbetweenthescrewandbarrelsurface. 634 Chapter 15 Copyright © 2003 Marcel Dekker, Inc. whereV pl isthecomponentoftheplugvelocityalongthescrewaxis.Thus, V pl ¼V b tanftany tanfþtany ð15:2:2Þ Thevolumetricflowrate,Q s ,ofthesolidplugistheproductoftheaxial componentoftheplugvelocityandtheareaforflowinthatdirection, Q s ¼V pl pDHð15:2:3Þ which,incombinationwithEq.(15.2.2),gives Q s ¼ðp 2 D 2 NHÞ tanftany tanfþtany ð15:2:4Þ whereallthequantitiesexceptfareknown;fisobtainedbyasimultaneous forceandmomentbalanceonasectionofthesolidplug,asshowninFigure15.5. Theforcethatcausestheplugtomoveistheforceoffriction,F 1 ,between thebarrelandtheplug.Asmentionedpreviously,thebarrelvelocityrelativeto theplugisinadirectionthatmakesanangleyþftothezdirection.This, therefore,isthedirectionofF 1 .ThemagnitudeofF 1 isgivenby F 1 ¼f b pWdzð15:2:5Þ wheref b isthecoefficientoffrictionbetweentheplugandthebarrelsurface,pis theisotropicpressurewithintheplug,anddzisthethicknessoftheplug.A pressuregradientdevelopsacrosstheplugandtheforceduetothisisasfollows: F 6 ÀF 2 ¼HWdpð15:2:6Þ FIGURE15.4Diagramshowingthedifferentvelocityvectors. Polymer Processing 635 Copyright © 2003 Marcel Dekker, Inc. As will be evident later, dp is a positive number, so pressure increases with increasing z. Normal forces act on the plug at the flights due to the presence of the isotropic pressure. Thus, we have F 8 ¼ pH dz ð15:2:7Þ The normal force that acts on the other flight is F 7 ¼ pH dz þ F* ð15:2:8Þ where F* is a reaction force. Finally, there are friction forces on the two flights and on the screw surface, as shown in Figure 15.5. Their magnitudes are as follows: F 3 ¼ f s F 7 ð15:2:9Þ F 4 ¼ f s F 8 ð15:2:10Þ F 5 ¼ f s pW dz ð15:2:11Þ where f s is the coefficient of friction between the plug and the screw surface. Each of the forces F 1 to F 8 can be resolved into a component parallel to the screw axis and a component perpendicular to the screw axis. These must sum to zero because the plug does not accelerate. The axial force balance takes the following form: F 1 sin f þðF 6 À F 2 Þsin y ÀðF 7 À F 8 Þcos y þðF 3 þ F 4 þ F 5 Þsin y ¼ 0 ð15:2:12Þ FIGURE 15.5 Forces acting on the solid plug. (From Ref. 3.) 636 Chapter 15 Copyright © 2003 Marcel Dekker, Inc. Introducing expressions for the various forces into Eq. (15.2.12) and rearranging, we find that F* ¼ A 1 pdzþ A 2 dp cos y À f s sin y ð15:2:13Þ where A 1 ¼ f b W sin f þ 2Hf s sin y þWf s sin y ð15:2:14Þ A 2 ¼ HW sin y ð15:2:15Þ Another expression for F* can be obtained by a moment balance about the axis of the screw, D 2 ½F 1 cos f ÀðF 6 À F 2 Þcos y ÀðF 7 À F 8 Þsin y ÀðF 3 þ F 4 þ F 5 Þcos y¼0 ð15:2:16Þ with the result that F* ¼ B 1 pdzÀ B 2 dp sin y þf s cos y ð15:2:17Þ where B 1 ¼ f b W cos f À 2Hf s cos y À Wf s cos y ð15:2:18Þ B 2 ¼ HW cos y ð15:2:19Þ Equating the two expressions for F* and rearranging yields dp dz ¼À ðA 1 K ÀB 1 Þ ðA 2 K þB 2 Þ p ð15:2:20Þ where K ¼ sin y þf s cos y cos y À f 2 sin y ð15:2:21Þ Integrating Eq. (15.2.20) from z ¼ 0, near the hopper base, where p ¼ p B to z gives p ¼ p B exp À ðA 1 K ÀB 1 Þ ðA 2 K þB 2 Þ z  ð15:2:22Þ and pressure rises exponentially with distance z.Ifz b is taken as the length of the solids-conveying zone in the down-channel direction, p at z b gives the pressure at the end of this zone. We began this analysis seeking the angle f so that we could calculate the solids-conveying rate from Eq. (15.2.4). Now, f is found to be given implicitly in Polymer Processing 637 Copyright © 2003 Marcel Dekker, Inc. terms of the pressure rise by Eq. (15.2.22), because it is hidden in the constants A 1 and B 1 . Thus, a knowledge of Dp is needed for the determination of Q s .IfDp is nonzero, we would have to calculate this quantity. In principle, we can work backward from the extruder exit and compute p at z b using the condition that the mass flow rate has to be the same throughout the extruder. Note from Eq. (15.2.4) that the volumetric flow rate increases linearly with the screw speed. If, on the other hand, the flow rate is known, we can determine f from Eq. (15.2.4) and the pressure rise from Eq. (15.2.22). To calculate the pressure at the end of the solids- conveying region, though, we need the value of p B , the pressure at the base of the hopper. For this, it is necessary to examine the flow of granular solids in a conical bin. This has been done, and results are available in the literature [5,6]. Example 15.1: What is the maximum possible rate of solids conveying through an extruder? Solution: The flow rate is maximum when there is no obstruction at the extruder exit (i.e., Dp ¼ 0) and when there is no friction between the polymer and the screw surface. Under these conditions, Eq. (15.2.12) becomes F 1 sin f À F* cos y ¼ 0 whereas Eq. (15.2.16) takes the form F 1 cos f À F*siny ¼ 0 with the result that tan f ¼ cot y which when inserted into Eq. (15.2.4) leads to the desired result: Q max ¼ p 2 D 2 NH sin y cos y In closing this section we mention that Chung has observed that the model of Darnell and Mol is strictly valid only up to the point that the polymer begins to melt [7]. Because the barrel temperature is kept above the melting point of the polymer, a layer of liquid forms fairly quickly and coats the solid plug. As a consequence, Eqs. (15.2.5) and (15.2.9)–(15.2.11) have to be modified and the forces F 1 , F 3 , F 4 , and F 5 calculated using the shear stress in the molten polymer film. A result of this modification is that f becomes a function of the screw revolutions per minute (rpm) [7,8]. We also mention that Campbell and Dontula have proposed a new model that does not require us to assume that the screw is stationary [9]. This model appears to give better agreement with experimental data. 638 Chapter 15 Copyright © 2003 Marcel Dekker, Inc. 15.2.3 Melting Zone Melting of the polymer occurs due to energy transfer from the heated barrel and also due to viscous dissipation within the polymer itself. This melting does not happen instantly but takes place over a significant part of the screw length. The purpose of any analysis of the melting process is to predict the fraction of polymer that is melted at any down-channel location and to relate this quantity to material, geometrical, and operating variables. Maddock [10] and Tadmor and Klein [3] studied the melting process by ‘‘carcass analysis’’: They extruded colored polymer and stopped the extruder periodically. By cooling the polymer and extracting the screw, they could track the progress of melting and also determine the sequence of events that ultimately resulted in a homogeneous melt. They found that a thin liquid film was formed between the solid bed of the polymer and the barrel surface. This is shown in Figure 15.6. Because of the relative motion between the barrel and the polymer bed, the molten polymer was continually swept from the thin film in the x direction into a region at the rear of the bed between the flight surface and the bed. Liquid lost in this manner was replaced by freshly melted polymer so that the film thickness d and the bed thickness both remained constant. As meltin g proceeded, the solid polymer was transported at a constant velocity V sy to the thin film–solid bed interface and, correspondingly, the bed width X decreased with increasing down-channel distance. A large number of models, of varying degrees of complexity, exist for calculating X as a function of distance z [3,8,11,12]. In the simplest case, it is assumed that the solid polymer is crystalline with a sharp melting point T m and a latent heat of fusion l and that the molten polymer is a Newtonian liquid. It is also assumed that the solid and melt physical proper ties such as the density, specific FIGURE 15.6 Melting of a polymer inside the extruder. (From Ref. 2.) Polymer Processing 639 Copyright © 2003 Marcel Dekker, Inc. [...]... solid polymer to the TSE at any desired rate, and as a consequence, the conveying channels of the TSE are only partially filled with polymer; the degree of fill is typically 25–50%, and it changes as the screw pitch changes There is, therefore, no pressurization of melt in the screw bushings A consequence of this is a decoupling of the different parts of the extruder, and what happens in one portion of. .. immense amount of plastic, particularly nylon, polycarbonate, and polyester, is compounded with short glass fibers The addition of up to 40 wt% glass to the polymer significantly increases the heat distortion temperature and allows the compounded product to be used for under-the-hood automotive applications The process of compounding polymers with glass fibers is typically carried out with the help of twin-screw... short glass fibers Make polymeric alloys using two miscible plastics Carry out reactive extrusion for the synthesis of copolymers For all of these purposes, it is generally desirable that the dispersed phase leaving the extruder be in the form of primary particles (not agglomerates), if solid, and that it have molecular dimensions, if liquid Also, it is necessary that the concentration of the dispersed phase... zT needed for this is zT ¼ 2W 1=2 c2 ð15:2:39Þ The foregoing analysis combines experimental observations with fundamentals of transport phenomena to analytically relate X to z It is obviously quite restrictive Given the known behavior of polymeric fluids, we can immediately think of a number of modifications, such as making the shear viscosity in Eq (15.2.26) depend on temperature and shear rate We can... has been the development of the extensional flow mixer [19] Here, material is made to flow through a series of converging and diverging regions of increasing intensity This again results in a fine and well-dispersed morphology, but at the expense of a higher pressure drop Note that a single-screw extruder in combination with a static mixer is well suited for the manufacture of polymer alloys or blends... Copyright © 2003 Marcel Dekker, Inc Polymer Processing F IGURE 15.13 15.3 651 Isometric view of a bilobal kneading block (From Ref 22.) INJECTION MOLDING One of the conceptually simplest methods of fabricating a plastic component, though complex in geometry, is to make a mold or cavity that is identical in shape and size to the article of interest and to fill it with a molten polymer, which then solidifies... added to reduce the density of the molded article By simultaneous or sequential injection of two polymers into the same cavity, it is possible to make a part with a foamed core and a dense skin During injection molding (as shown in Fig 15.14), we can also inject an inert gas such as nitrogen into the mold so that it channels through the less viscous sections of the molten polymer This results in weight... Other moldable polymers that are frequently encountered are polymethyl methacrylate for lenses and light covers, and polycarbonates and ABS for appliance housings and automobile parts As shown in schematic form in Figure 15.14, an injection-molding machine is essentially a screw extruder attached to a mold The action of the extruder results in a pool of molten polymer directly in front of the screw tip,... (oxygenfree N2) is injected through the nozzle into the center of the still-molten polymer (From Ref 24.) Copyright © 2003 Marcel Dekker, Inc Polymer Processing 653 the part is removed, and the cycle of operations is repeated Typical cycle times range from a few seconds to a minute Injection-molding machines are normally described in terms of the screw diameter, the maximum shot size in ounces, and the... and can cost several tens of thousands of dollars To consistently make moldings having the correct dimensions, it is necessary that the mold material be wear resistant and corrosion resistant and not distort during thermal cycling; chrome and nickel plating are common Details of mold design and of the mechanical aspects of opening and closing molds and ejecting solidified parts are available in the . manufacture of high-value-added sophisticated items such as compact disks or optical lenses, on the other hand, requires a great deal of engineering and an intimate knowledge of the fundamentals of polymer behavior. In. point of the polymer, the region very close to the base of the hopper is often water cooled to prevent polymer from melting in the hopper and forming a solid plug, which would block additional polymer. instantly but takes place over a significant part of the screw length. The purpose of any analysis of the melting process is to predict the fraction of polymer that is melted at any down-channel