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1 Formulas of Rheology One of the most important steps in processing polymers is melting the resin, which is initially in the solid state, and forcing the melt through a die of a given shape. During this operation, the melt, whose structure plays a key role in determining the quality of the product to be manufactured, undergoes different flow and deformation processes. The plastics engineer has therefore to deal with the melt rheology, which describes the flow behavior and deformation of the melt. The theory of elasticity and hydromechanics can be considered the frontier field of rheology, because the former describes the behavior of ideal elastic solids, whereas the latter is concerned with the behavior of ideal viscous fluids. Ideal elastic solids deform according to Hooke's Law and ideal viscous fluids obey the laws of Newtonian flow. The latter are also denoted as Newtonian fluids. Plastic melts exhibit both viscous and elastic properties. Thus, the design of machines and dies for polymer processing requires quantitative description of the properties related to polymer melt flow. Starting from the relationships for Hookean solids, formulas describing viscous shear flow of the melt are treated first, as far as they are of practical use in designing polymer machinery. This is followed by a summary of expressions for steady and time-dependent viscoelastic behavior of melts. 1.1 Ideal Solids The behavior of a polymer subjected to shear or tension can be described by comparing its reaction to external force with that of an ideal elastic solid under load. To characterize ideal solids, first of all it is necessary to define certain quantities as follows [I]: The axial force Fn in Figure 1.1 causes an elongation A/ of the sample of diameter dQ and length I0 that is fixed at one end. Following equations apply for this case: Engineering strain: Hencky strain: Tensile stress: (Li) (1.2) (1.3)
Figure 1.1 Deformation of a Hookean solid by a tensile stress [1 ] Reference area: Poisson's ratio: (1.4) Figure 1.2 shows the influence of a shear force Fv acting on the area A of a rectangular sample causing the displacement AU. The valid expressions are defined by: Shear strain: Shear stress: Figure 1.2 Deformation of a Hookean solid by shearing stress [1 ] (1.5) (1.6) (1.7)
Page 1 and 2: Natti S. Rao Gunter Schumacher D e
Page 3: Preface Today, designing of machine
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 and 14: Pa Shear stress T Figure 1.11 Deter
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
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The foregoing equations apply to ca
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Figure 3.11 Midplane temperature fo
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Figure 3.12 Temperature distributio
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For drag flow the velocity gradient
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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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