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Shear stress r Figure 1.10 Comparison between measurements and values calculated with Equation 1.25 [8] 1.3.7.2 Power Law of Ostwald and de Waele The power law of OSTWALD [6] and DE WAELE [7] is easy to use, hence widely employed in design work [10]. This relation can be expressed as or (1.26) (1.27) where K denotes a factor of proportionality and n the power law exponent. Another form of power law often used is or Shear rate f s" 1 PE-LD 19O 0 C (1.28) (1.29) In this case, nR is the reciprocal of n and KR = K~ nR . From Equation 1.26 the exponent n can be expressed as N/m 2 (1.30)
Pa Shear stress T Figure 1.11 Determination of the power law exponent n in Equation 1.30 As shown in Figure Ll 1, in a double log plot the exponent n represents the local gradient of the curve y^ vs. T. Furthermore (1.31) The values of K and n determined from the flow curve of LDPE at 190 0 C shown in Figure 1.12 were found to be K = 1.06 <strong>•</strong> 10" 11 and n = 2.57 [8]. As can be seen from Figure 1.12, the power law fits the measured values much better than the hyperbolic function of EYRING [4] and PRANDTL [5]. The deviation between the results from the power law and from the experiment is a result of the assumption that the exponent n is constant throughout the range of shear rates considered, whereas in fact n varies with the shear rate. The power law can be extended to consider the effect of temperature on the viscosity as follows: where Shear rate f K0R = consistency index P = temperature coefficient T = temperature of melt. s" 1 PE-LD r=150°C tana= dl^ =/7 (1.32)
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: Apparent viscosity 7\Q True viscosi
Page 15 and 16: Shift Factor for Crystalline Polyme
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
Page 159 and 160:
Flow length (mm) Flow length (mm) F
Page 161 and 162:
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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