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3 Formulas of Heat Transfer Heat transfer and flow processes occur in the majority of polymer processing machinery and often determine the production rate. Designing and optimizing machine elements and processes therefore require knowledge of the fundamentals of these phenomena. The flow behavior of polymer melts has been dealt with in Chapter 2. In the present chapter, the principles of heat transfer relevant to polymer processing are treated and explained with examples. 3.1 Steady State Conduction Fourier's law for one-dimensional conduction is given by where Q = heat flow thermal conductivity A — area perpendicular to the direction of heat flow T = temperature x = distance (Figure 3.1) Figure 3.1 Plane wall [1] 3.1.1 Plane Wall Temperature profile (Figure 3.1) [I]: Heat flow: (3.1) (3.2) (3.3)
Analogous to Ohm's law in electric circuit theory, Equation 3.3 can be written as [2] in which where AT = temperature difference 8 — wall thickness jR = thermal resistance Example (3.4) (3.5) The temperatures of a plastic sheet (30 mm thick) with a thermal conductivity X = 0.335 W/(m K) are TWi = 100 0 C and TWi = 40 0 C accordingto Figure 3.1. Calculate the heat flow per unit area of the sheet. Solution Substituting the given values in Equation 3.3 we obtain 3.1.2 Cylinder Temperature distribution (Figure 3.2) [I]: Figure 3.2 Cylindrical wall [1] (3.6)
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 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: Table 2.1 Approximate Values for th
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
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
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Shear stress (N/m 2 ) Figure 5.5 Ef
Page 99 and 100:
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
Page 163 and 164:
[26] VDI Warmeatlas, VDI Verlag, Du
Page 165 and 166:
A Final Word The aim of this book i
Page 167 and 168:
166 Index terms Links Index terms L
Page 169 and 170:
168 Index terms Links Index terms L
Page 171 and 172:
viii Contents 1.4 Viscoelastic Beha
Page 173 and 174:
x Contents 5.2 Extrusion Screws ...
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