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Nomenclature: a: thermal diffusivity (m 2 /s) g: acceleration due to gravity (m/s 2 ) /: characteristic length (m) p: ressure (N/m ) t. time (s) Indices D, P: memory and process of polymer respectively AT: Temperature difference (K) w: Velocity of flow (m/s) O1: Outside heat transfer coefficient [W/(m 2 • K)] /J: Coefficient of volumetric expansion (K" ) /J1: Temperature coefficient in the power law of viscosity (KT 1 ) (5S: Mass transfer coefficient (m/s) & Diffusion coefficient (m 2 /s) Tj: Viscosity (Ns/m 2 ) A: Thermal conductivity (Index i refers to the inside value) [W/(m K)] v: Kinematic viscosity (m 2 /s) tv: Residence time (s) p: Density (kg/m 3 ) 3.4.1 Physical Meaning of Dimensionless Groups Biot number: Ratio of thermal resistances in series: (/ / A1) / (1 / oQ Application: heating or cooling of solids by heat transfer through conduction and convection Brinkmann number: ratio of heat dissipated (T] w ) to heat conducted (AAT) Application: polymer melt flow Fourier number: ratio of a characteristic body dimension to an approximate temperature wave penetration depth for a given time [16] Application: unsteady state heat conduction Deborah number: ratio of the period of memory of the polymer to the duration of processing [13]. At Deb > 1 the process is determined by the elasticity of the material, whereas at Deb < 1 the viscous behavior of the polymer influences the process remarkably. Grashof number: ratio of the buoyant force g /5 • AT f to factional force (v) Application: heat transfer by free convection Graetz number: ratio of the time to reach thermal equilibrium perpendicular to the flow direction (I 2 Ia) to the residence time (tv) Application: heat transfer to fluids in motion
Lewis number: ratio of thermal diffusivity (a) to the diffusion coefficient (S) Application: phenomena with simultaneous heat and mass transfer. Nusselt number: ratio of the total heat transferred (a • /) to the heat by conduction (X) Application: convective heat transfer. Pedet number: ratio of heat transfer by convection (pc -w-l) to the heat by conduction a) Application: heat transfer by forced convection. Nahme or Griffith number: ratio of viscous dissipation (j3T W 2 J]) to the heat by conduction (X) perpendicular to the direction of flow Application: heat transfer in melt flow Prandtl number: ratio of the kinematic viscosity (v) to thermal diffusivity (a) Application: convective heat transfer Reynolds number: ratio of the inertial force (p w I) to viscous force (T]) Application: The Reynolds number serves as a criterium to judge the type of flow. In pipe flow, when Re is less than 2300 the flow is laminar. The flow is turbulent at Re greater than about 4000. Between 2100 and 4000 the flow may be laminar or turbulent depending on conditions at the entrance of the tube and on the distance from the entrance [2] Application: fluid flow and heat transfer. Sherwood number: ratio of the resistance to diffusion (/ / S) to the resistance to mass transfer (l/j8s) Application: mass transfer problems Schmidt number: ratio of kinematic viscosity (v) to the diffusion coefficient (S) Application: heat and mass transfer problems Stokes number: ratio of pressure forces (p • Z) to viscous forces (rj • w) Application: pressure flow of viscous media like polymer melts. The use of dimensionless numbers in calculating non Newtonian flow problems is illustrated in Section 4.3.3 with an example. 3.5 Heat Transfer by Convection Heat transfer by convection, particularly by forced convection, plays an important role in many polymer processing operations such as in cooling a blown film or a part in an injection mold, to mention only two examples. A number of expressions can be found in the literature on heat transfer [3] for calculating the heat transfer coefficient a (see Section 3.1.6). The general relationship for forced convection has the form (3.53)
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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
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Figure 1.4 Shear flow The shear or
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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
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: For drag flow the velocity gradient
Page 65 and 66: The Reynolds number ReL, based on t
Page 67 and 68: At thermal equilibrium according to
Page 69 and 70: The rate of heat generation in a pl
Page 71 and 72: The mass of the fluid permeating th
Page 73 and 74: 4 Designing Plastics Parts The defo
Page 75 and 76: The parabolic failure criterion is
Page 77 and 78: Figure 4.3 Secant modulus [4] Stres
Page 79 and 80: where P = load (N) Lybyd = dimensio
Page 81 and 82: 5 Formulas for Designing Extrusion
Page 83 and 84: for values of the ratio n (R0 + R1)
Page 85 and 86: melt density pm = 0.7 g/cm 3 Soluti
Page 87 and 88: and finally the pressure drop Ap fr
Page 89 and 90: Pressure drop Ap from Equation 5.2
Page 91 and 92: Proportionality factor K from Equat
Page 93 and 94: n = 3.043 RTh= 1.274 mm Shear rate
Page 95 and 96: The relation of a slit is H = heigh
Page 97 and 98: Shear stress (N/m 2 ) Figure 5.5 Ef
Page 99 and 100: Figure 5.7 Surface distortion on a
Page 101 and 102: Residence time t (s) LDPE m = 40 kg
Page 103 and 104: Table 5.1 Dimensions of Square Scre
Page 105 and 106: Pressure drop in screen Mesh size F
Page 107 and 108: In practice, this efficiency is als
Page 109 and 110: 5.2.2 Melt Conveying Starting from
Page 111 and 112: 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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