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Shear stress T Figure 1.12 Comparison between measured values and power law Example Following values are given for a LDPE: nR = 0.3286 J3 =0.00863 ( 0 C" 1 ) KOR =135990 (N s" R <strong>•</strong> nT 2 ) The viscosity 7]a at T = 200 0 C and ya = 500 s" 1 is calculated from Equation 1.32 7JZ = 373.1Pa-s Shear rate y 1.3.7.3 Muenstedt's Polynomial The fourth degree polynomial of MUENSTEDT [2] provides a good fit for the measured values of viscosity. For a specific temperature this is expressed as (1.33) where A0, A1, A2, A3, A4 represent resin-dependent constants. These constants can be determined with the help of the program of RAO [ 10], which is based on multiple linear regressions. This program in its general form fits an equation of the type and prints out the coefficients a0, ax and so on for the best fit. s" 1 measured PE-LO 19O 0 C N/m 2
Shift Factor for Crystalline Polymers The influence of temperature on viscosity can be taken into account by the shift factor aT [2]. For crystalline polymers this can be expressed as where bv b2 = resin dependent constants T = melt temperature (K) - reference temperature (K) T0 Shift Factor for Amorphous Polymers (1.34) The shift factor aT for amorphous polymers is derived from the WLF equation and can be written as where C1, C2 = resin dependent constants T = melt temperature ( 0 C) = reference temperature ( 0 C) T0 (1.35) The expression for calculating both the effect of temperature and shear rate on viscosity follows from Equation 1.33. With Equation 1.31 we get (1.36) (1.37) The power law exponent is often required in the design work as a function of shear rate and temperature. Figure 1.13 illustrates this relationship for a specific LDPE. The curves shown are computed with Equation 1.34 and Equation 1.37. As can be inferred from Figure 1.13, the assumption of a constant value for the power law exponent holds good for a wide range of shear rates.
Page 1 and 2: Natti S. Rao Gunter Schumacher D e
Page 3 and 4: Preface Today, designing of machine
Page 5 and 6: Figure 1.1 Deformation of a Hookean
Page 7 and 8: Figure 1.4 Shear flow The shear or
Page 9 and 10: where c and m are empirical constan
Page 11 and 12: Apparent viscosity 7\Q True viscosi
Page 13: Pa Shear stress T Figure 1.11 Deter
Page 17 and 18: The power law exponent is obtained
Page 19 and 20: 1.3.7.5 Klein's Viscosity Formula [
Page 21 and 22: where Mw = molecular weight JC —
Page 23 and 24: Steady state shear compliance J°t
Page 25 and 26: Ig 6\ Ig G" lga; Figure 1.22 Storag
Page 27 and 28: Characterization of the Transient S
Page 29 and 30: Time/ Figure 1.28 Tensile creep at
Page 31 and 32: 1.4.2.2 Nonlinear Viscoelastic Beha
Page 33 and 34: Figure 1.37 Maxwell fluid [1 ] The
Page 35 and 36: Die swel B1 Figure 1.40 Dependence
Page 37 and 38: 2 Thermodynamic Properties of Polym
Page 39 and 40: where u = internal energy T = tempe
Page 41 and 42: Enthalpy /7-/720 kWh kg Kl kg polyn
Page 43 and 44: Table 2.1 Approximate Values for th
Page 45 and 46: Analogous to Ohm's law in electric
Page 47 and 48: Figure 3.3 One-dimensional heat tra
Page 49 and 50: Figure 3.5 Heat flow in a multilaye
Page 51 and 52: The combination of convection and c
Page 53 and 54: The equation for an infinite cylind
Page 55 and 56: The foregoing equations apply to ca
Page 57 and 58: Figure 3.11 Midplane temperature fo
Page 59 and 60: Figure 3.12 Temperature distributio
Page 61 and 62: For drag flow the velocity gradient
Page 63 and 64: Lewis number: ratio of thermal diff
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The Reynolds number ReL, based on t
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At thermal equilibrium according to
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The rate of heat generation in a pl
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The mass of the fluid permeating th
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4 Designing Plastics Parts The defo
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The parabolic failure criterion is
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Figure 4.3 Secant modulus [4] Stres
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where P = load (N) Lybyd = dimensio
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5 Formulas for Designing Extrusion
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for values of the ratio n (R0 + R1)
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melt density pm = 0.7 g/cm 3 Soluti
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and finally the pressure drop Ap fr
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Pressure drop Ap from Equation 5.2
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Proportionality factor K from Equat
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n = 3.043 RTh= 1.274 mm Shear rate
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The relation of a slit is H = heigh
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Shear stress (N/m 2 ) Figure 5.5 Ef
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Figure 5.7 Surface distortion on a
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Residence time t (s) LDPE m = 40 kg
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Table 5.1 Dimensions of Square Scre
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Pressure drop in screen Mesh size F
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In practice, this efficiency is als
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5.2.2 Melt Conveying Starting from
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md = 46.42 kg/h Equation 5.31 and E
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Solution Power Zc in the screw chan
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Indices: m: melt f: melt film b: ba
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Example with symbols and units a) S
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X- Q Figure 5.24 Velocity and tempe
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(5.54) As seen from the equations a
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Figure 5.26 Three-zone screw [8] Th
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The profiles of stock temperature a
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where HF = feed depth H = metering
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Screw speed (rpm) Screw diameter D
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5.2.6 Mechanical Design of Extrusio
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Next Page Screw Vibration When the
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5.3.1 Pressure Drop in Runner As th
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Examples of calculating pressure dr
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Example with symbols and units The
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Flow length L mm Spiral height H Fi
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Solution The conversion factors for
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Figure 5.43 shows a sample plot of
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With the values for the properties
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Cooling time mm Distance between mo
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Table 5.2 Results of Optimization o
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Per cent unmelted Axial distance al
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A* [%] LDPE Axial length (screw dia
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A* [%] A* [%] LDPE LDPE Axial lengt
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Flow length (mm) Flow length (mm) F
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Flow length L (mm) Flow length L (m
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[26] VDI Warmeatlas, VDI Verlag, Du
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A Final Word The aim of this book i
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166 Index terms Links Index terms L
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168 Index terms Links Index terms L
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viii Contents 1.4 Viscoelastic Beha
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x Contents 5.2 Extrusion Screws ...
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