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Polymer Processing Modeling and Simulation Tim Osswald Juan P. Hernández-Ortiz Polymer Processing Modeling and Simulation Sample Chapter 2: Processing Properties ISBNs 978-1-56990-398-8 HANSER 1-56990-398-0 Hanser Publishers, Munich • Hanser Publications, Cincinnati CHAPTER 2 PROCESSING PROPERTIES Didyou ever considerviscoelasticity? —Arthur Lodge 2.1THERMAL PROPERTIES The heat flowthrough amaterial can be definedbyFourier’s lawofheat conduction. Fourier’s lawcan be expressedas ∂T q = − k (2.1) x x ∂x where q x is theenergy transport perunitarea in the x direction, k x thethermal conductivity and ∂T/∂x thetemperaturegradient. At theonset of heating, thepolymerresponds solely as aheat sink, and theamount of energy per unit volume, Q ,stored in thematerialbefore reaching steady state conditions can be approximated by Q = ρCp ∆ T (2.2) where ρ is thedensity of thematerial, C p the specific heat, and ∆ T thechange in temperature. Thematerial properties found in eqns.(2.1) and(2.2) areoften written as onesingleproperty, 38 PROCESSING PROPERTIES Table2.1: ThermalPropertiesfor Selected Polymeric Materials Polymer Specific Specific Thermal Coeff. Thermal Max gravity heat conduc. therm. diffusivity temp. expan. kJ/kg/KW/m/K µ m/m/K(m 2 /s)10− 7 o C ABS1.04 1.47 0.3901.7 70 CA 1.28 1.50 0.15 100 1.04 60 EP 1.9 -0.23 70 -130 PA66 1.14 1.67 0.24 90 1.01 90 PA66-30% glass1.38 1.26 0.52 30 1.33 100 PC 1.15 1.26 0.2 65 1.47 125 PE-HD0.95 2.3 0.63 120 1.57 55 PE-LD 0.92 2.3 .33200 1.17 50 PET 1.37 1.05 0.24 90 -110 PF 1.4 1.3 0.35 22 1.92 185 PMMA 1.18 1.47 0.2 70 1.09 50 POM 1.42 1.47 0.2800.7 85 coPOMa 1.41 1.47 0.2 95 0.72 90 PP 0.905 1.95 0.24 100 0.65 100 PPOb 1.06 -0.22 60 -120 PS 1.05 1.34 0.15 80 0.6 50 PTFE 2.1 1.00.25 140 0.7 50 uPVCc 1.4 1.0 0.16 70 1.16 50 pPVCd 1.31 1.67 0.14 140 0.7 50 SAN 1.08 1.38 0.17 70 0.81 60 UPE1.20 1.2 0.2 100 -200 Steel 7.854 0.434 60 -14.1 800 a Polyacetal copolymer; b Polyphenylene oxide copolymer; c Unplasticized PVC; d Plasticized PVC namely thethermaldiffusivity, α ,which for an isotropic material is defined by k α = (2.3) ρCp Typical values of thermal propertiesfor selected polymers are showninTable 6.1[7, 17]. Forcomparison, thepropertiesfor stainlesssteel are also shownatthe endofthe list. It shouldbepointed out that thematerial propertiesofpolymers are not constant and may vary with temperature, pressure or phasechanges.This section will discusseach of these propertiesindividually and presentexamples of some of themostwidelyusedpolymers and measurement techniques.For amore in-depthstudy of thermal propertiesofpolymers the reader is encouraged to consult theliterature [24,46, 66]. 2.1.1 Thermal Conductivity When analyzing thermal processes, thethermal conductivity, k ,isthe mostcommonlyused propertythat helpsquantifythe transport of heat through amaterial.Bydefinition,energy is transported proportionally to thespeed of sound. Accordingly, thermal conductivity THERMAL PROPERTIES 39 0.22 PET 0.20 PMMA PBMA , W/m/k 0.18 0.16 NR PVC 0.14 Thermal conductivity PIB 0.12 100 150 200 250 300 Temperature (K) Figure 2.1: Thermal conductivity of various materials. follows therelation k ≈ C p ρul (2.4) where u is thespeed of sound and l themolecular separation. Amorphous polymers show an increaseinthermal conductivity with increasingtemperature, up to theglasstransition temperature, T g .Above T g ,the thermal conductivity decreases with increasingtemperature. Figure 2.1[24]presents the thermal conductivity,belowthe glasstransition temperature, for various amorphous thermoplasticsasafunctionoftemperature. Due to theincreaseindensity upon solidificationofsemi-crystalline thermoplastics, thethermal conductivity is higher in thesolid statethan in themelt. In themeltstate, however, thethermalconductivity of semi-crystallinepolymersreduces to that of amorphous polymers as can be seen in Fig.2.2 [40]. Furthermore, it is not surprising that thethermal conductivity of melts increases with hydrostatic pressure. Thiseffect is clearlyshown in Fig. 2.3[19]. As long as thermosetsare unfilled, theirthermal conductivity is very similartoamorphous thermoplastics. Anisotropy in thermoplastic polymers also plays asignificant roleinthe thermal conductivity.Highly drawn semi-crystalline polymer samples can have amuch higher thermal conductivity as a resultofthe orientationofthe polymerchains in thedirectionofthe draw. Foramorphous polymers,the increaseinthermal conductivity in thedirectionofthe drawisusually not higher than two. Figure 2.4[24]presentsthe thermal conductivity in the directions paralleland perpendicular to thedrawfor high density polyethylene, polypropy- lene, andpolymethyl methacrylate. Asimplerelationexistsbetween theanisotropicand the isotropic thermal conductivity [39]. This relation is written as 1 2 3 + = (2.5) k k ⊥ k where thesubscripts and ⊥ represent thedirections paralleland perpendicular to thedraw, respectively. 40 PROCESSING PROPERTIES 0.5 HDPE 0.4 PA 6 LDPE 0.3 PC 0.2 PS PP Thermal conductivity k, W/m/K Thermal conductivity 0.1 0 0 50 100 150 200 250 Temperature, T (°C) Figure 2.2: Thermal conductivity of various thermoplastics. 1.2 PP ) T = 230°C HDPE 1bar LDP E (k/k 1.1 PS PC riation in thermal Va conductivity 1.0 1250 500 750 1000 Pressure, P(bar) Figure 2.3: Influence of pressureonthermal conductivity of various thermoplastics. THERMAL PROPERTIES 41 8.0 6.0 PE-HD iso 4.0 PP /k || k 2.0 1.5 PMMA 1.0 PMMA iso 0.8 PP /k | 0.6 k PE-HD 0.4 1 2 3 4 5 6 Draw ratio Figure 2.4: Thermal conductivity as afunctionofdrawratio in thedirections perpendicular and paralleltothe stretchfor various oriented thermo-plastics. 0.7 0.6 PE-LD +Quartz powder 60% wt 0.5 PE-LD +GF (40% wt) ll -orientation , k (W/m/K) 0.4 PE-LD +GF (40% wt) -orientation 0.3 PE-LD 0.2 Thermal conductivity 0.1 0 050100 150 200 250 Temperature, T (°C) Figure 2.5: Influence of filleronthe thermal conductivity of PE-LD. The higher thermal conductivity of inorganicfillers increases thethermal conductivity of filledpolymers.Nevertheless, asharp decreaseinthermal conductivity around the melting temperature of crystallinepolymers can still be seen with filled materials. Theeffectof filler on thermal conductivity for PE-LD is showninFig.2.5 [22]. This figure showsthe effect of fiber orientationaswellasthe effect of quartzpowder on thethermal conductivity of lowdensity polyethylene. Figure 2.6demonstrates theinfluence of gascontent on expanded or foamed polymers, and theinfluence of mineral content on filled polymers. There arevarious models availabletocomputethe thermal conductivity of foamed or filled plastics[39,47, 51]. Aruleofmixtures,suggestedbyKnappe [39], commonlyused 42 PROCESSING PROPERTIES 103 102 Polymer in metal W/m/K ,k 101 vity 100 Closed Metal in cells polymer -1 Thermal conducti Thermal 10 Opencells Foams Polymer/meta 10-2 10.5 00.5 1 Volume fraction Volume fraction of gas of metal Figure 2.6: Thermal conductivity of plasticsfilledwith glass or metal. to compute thermal conductivity of composite materials is written as 2 k + k − 2 φ ( k − k ) k = m f f m f k (2.6) c 2 k + k + φ ( k ) m m f f m − k f where, φ f is thevolume fractionoffiller,and k m , k f and k c are thethermal conductivity of thematrix, filler and composite,respectively. Figure 2.7compares eqn. (2.6) with experimental data[2] for an epoxy filled with copper particlesofvarious diameters.The figure also compares thedatatothe classicmodel given by Maxwell [47] whichiswritten as ⎛ ⎞ k f − 1 ⎜ k ⎟ k = ⎜ 1+3φ m ⎟ k (2.7) c ⎝ f k ⎠ m f +2 k m In addition, amodel derivedbyMeredith andTobias [51]appliestoacubicarray of spheres inside amatrix. Consequently,itcannot be used for volumetric concentration above 52% since thespheres will touch at that point.However,theirmodel predictsthe thermal conductivity very well up to 40% by volume of particle concentration. When mixing severalmaterials thefollowingvariationofKnappe’s model applies n k m − k i 1 − i =1 2 φ i 2 k m + k i k c = (2.8) n k m − k i 1+ i =1 2 φ i 2 k m + k i where k i is thethermal conductivity of thefiller and φ i its volume fraction. Thisrelationis useful for glassfiberreinforced composites(FRC) with glassconcentrations up to 50% by volume.Thisisalsovalid for FRCwith unidirectional reinforcement.However,one must differentiate between thedirectionlongitudinal to thefibers andthattransversetothem. THERMAL PROPERTIES 43 1.6 1.4 Experimental data (Temperature 300K) 1.2 100 µm 46 µm 25 µm , k(W/m/K) 1.0 11 µm 0.8 0.6 Knappe 0.4 Maxwell Thermal conductivity 0.2 0 0 10 20 30 40 50 60 Concentration, φ (%) Figure 2.7: Thermal conductivity versus volume concentrationofmetallic particlesofanepoxy resin. Solid lines represent predictions using Maxwelland Knappe models. Forhighfiber content,one can approximatethe thermal conductivity of thecomposite by thethermal conductivity of thefiber.The thermal conductivity can be measured usingthe standard testsASTMC177 and DIN52612. Anew method currently being balloted (ASTM D20.30) is preferred by most peopletoday. 2.1.2 SpecificHeat The specific heat, C ,represents theenergy required to change thetemperatureofaunit massofmaterial by one degree. It can be measured at either constant pressure, C p ,or constant volume, C v .Since thespecific heat at constant pressure includes theeffect of volumetric change, it is larger than thespecific heat at constant volume.However,the volume changes of apolymerwith changing temperatures have anegligible effect on the specific heat.Hence,one can usually assume that thespecific heat at constant volume or constant pressure are thesame. It is usually true that specific heat onlychanges modestly in therange of practical processing and design temperatures of polymers.However,semi- crystallinethermoplastics display adiscontinuity in thespecific heat at themeltingpoint of thecrystallites.
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