Unformatted text preview: UNIVERSITY OF
UNIVERSITY OF FLORIDA EMA 4666C - Polymer Processing
Rheology of Polymer Melts
Dr. Anthony Brennan
University of Florida
Department of Materials Science &
Department
Engineering
Engineering
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UNIVERSITY OF Agenda FLORIDA Introduction
Newtonian Behavior
Hagen-Poiseuille Flow
Couette Flow
Normal Stresses in Shear Flow
Deborah Number
Thermosets Rheometry
Melt Flow Index
Capillary Viscometer
07/17/11 EMA 4666C Viscometer
Cone and Plate- Polymer Processing Copyright Protected 2 UNIVERSITY OF
UNIVERSITY OF FLORIDA Rheology Viscosity is the internal friction of
Viscosity
a fluid or the resistance to flow
under mechanical stress
under
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UNIVERSITY OF FLORIDA Simple Shear Rheometer 07/17/11 EMA 4666C - Polymer Processing Copyright Protected 4 UNIVERSITY OF
UNIVERSITY OF FLORIDA Simple Shear Viscosity Shear Rate Dependence τ =κ γ
Doolittle’s Equation η = Ae
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B
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UNIVERSITY OF FLORIDA Assumptions: No slip at walls
Melt is incompressible
Flow is steady, laminar and time
Flow
independent
independent
Fluid viscosity is pressure
Fluid
independent
independent
End effects are neglible
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UNIVERSITY OF FLORIDA Pressure Flow Thru a Slit
o L y h z v ∆ p = p1 − p2 p1 p2 The pressure flow through the slit is common to dies for films.
The
Newtonian Behavior is given as:
Newtonian 2 y 2 h ∆p vz ( y) = 1 − 8 µ L h W h3 ∆ p Q = 12 µ L 2 07/17/11 EMA 4666C - Polymer Processing Copyright Protected 7 UNIVERSITY OF
UNIVERSITY OF FLORIDA Pressure Flow Thru a Slit Recall that the viscous flow of a fluid follows a power law
Recall
relationship for Non-Newtonian Behavior which is given as:
relationship h h ∆ p vz ( y) = 2m L 2( s + 1) s 2 y s +1 1 − h s W h 2 h ∆ p Q = 2 ( s + 2 ) 2 m L Where s = 1/n and n is the Power Law Index that describes
.
( n −1)
deviation from Newtonian: η = m (T ) γ Where m = “consistency” and if n =1, then Newtonian Fluid
Behavior:
η = µ (T )
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UNIVERSITY OF FLORIDA Pressure Flow Thru a tube
o L v y
z
p1 R ∆ p = p1 − p2
p2 The pressure flow through the slit is common to dies for films.
The
Newtonian Behavior is defined as Hagen-Poiseuille flow and given
by:
by:
2
2 R ∆p r vz (r ) = 1 − 4 µ L R π R 4 ∆ p Poiseuille Flow
Poiseuille
Q =
(Hagen-Poiseuille) (Hagen-Poiseuille)
8 µ L 07/17/11 EMA 4666C - Polymer Processing Copyright Protected 9 UNIVERSITY OF
UNIVERSITY OF FLORIDA Pressure Flow Thru a Tube In a very similar fashion, one can analyze the flow through a
In
tube and show that:
tube
s vz ( r ) = r s +1 R R ∆ p 1− ( s +1) 2 m L R π R R ∆ p Q = ( s + 3) 2m L 3 s Where s = 1/n and n is the Power Law Index that. describes deviation
from Newtonian:
( n −1) η = m (T ) γ Where m = “consistency” and if n =1, then Newtonian Fluid
Behavior:
η = µ (T )
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UNIVERSITY OF POWER LAW BEHAVIOR FLORIDA Estimation of m and n Generalized behavior
Generalized
or LLDPE at 170C
or Log
Log η (Pa sec) 5 η = m (T ) γ 4 .
( n −1) Estimated values of n
Estimated
from tangent of
response:
response: 3 n ~ - 0.58
2
-2
07/17/11 -1 1 3 2 log shear rate (sec )
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UNIVERSITY OF FLORIDA Stress Analysis in Flow Re < 2100 Re > 2100 Normal Stresses in Flow •
2
xy N1 =τ xx −τ yy = −ψ 1 γ , T γ •
• 2
N 2 =τ yy −τ zz = −ψ 2 γ , T γ xy • ψ1 And ψ2 are material parameters that define the primary and
secondary normal stress coefficients (Tensor notation that
correlate both strain rate and temperature).
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UNIVERSITY OF FLORIDA 07/17/11 Shear Rate Dependence EMA 4666C - Polymer Processing Copyright Protected 13 UNIVERSITY OF
UNIVERSITY OF FLORIDA Deborah Number Stress analyses are based upon: Time,
Stress
Temperature, Rate
Temperature, λ τ characteristic diffusion
De = = =
tt
processing time
Dimensionless number, related directly to the relaxation
time or diffusion time of the polymer chains and the
experimental time. Critical to consider with respect to the
processing design. 07/17/11 EMA 4666C - Polymer Processing Copyright Protected 14 UNIVERSITY OF
UNIVERSITY OF FLORIDA Reynolds Number A descriptor for the boundary between “laminar” or plug flow
descriptor
and “turbulent” flow. The Re value must exceed 2100 to 2300
to reach turbulent flow character.
to D Vρ
Re =
η
Where D is the diameter (cm) of the channel, V is the velocity of the
fluid, ρ is the density (kg/m3), and η is the viscosity ( Pa.sec). Velocity
is expressed as: Q
V=
A Where Q is output (cm3/sec) and A is cross sectional area of
channel.
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UNIVERSITY OF FLORIDA Estimation of Re Given the following information, estimate the Re.
Diameter of channel = 0.375 cm η = 175 Pa.sec
175
ρ = 980 kg/m3
980
Q = 275 cm3/sec
First solve for V: Q/A where A is: 2.75 x10−4
2.75 x10−4
Q /π r2 =
=
= 24.9 m s −1
π (3.75 x10−3 / 2)2 1.1 x10−5
Then Re: 0.00375 x 24.9 x 980
Re =
= 0.52
175
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UNIVERSITY OF FLORIDA Normal Stresses – Die Swell
Do Die Swell in Parison is
Die
given by:
given D1 B1 = D1 / Do L1 Die Swell in Parison
Die
Wall thickness is given
by:
by:
Do D2 B2 = H1 / H o L2 07/17/11 EMA 4666C - Polymer Processing Copyright Protected 17 UNIVERSITY OF
UNIVERSITY OF FLORIDA Thermosets γ ,T , c η =η C +C c
• cg η =ηo e RT c − c
g E 1 2 Cg ~ Gel Point (M → ∞ )
C~ degree of cure (consumed
C~
functional groups)
functional
C1, C2 ~ constants for the data
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UNIVERSITY OF FLORIDA 07/17/11 TTr Phase Diagram EMA 4666C - Polymer Processing Copyright Protected 19 UNIVERSITY OF
UNIVERSITY OF FLORIDA Ram Accumulator Blow Molding
Hopper
Die Control/Power Unit Heater Extrudate Ram
Drive Blower
Zone
4 07/17/11 Zone
3 Zone
2 Zone
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UNIVERSITY OF FLORIDA 07/17/11 Barrel Configuration EMA 4666C - Polymer Processing Copyright Protected 21 UNIVERSITY OF
UNIVERSITY OF FLORIDA Screw Geometry Terms:
– e - width of flight; δ - Gap between Flight and Barrel
width
Wall
Wall
– w - spacing of flight; D - Diameter of Barrel;
spacing
or Helical Angle; L - total channel length
or
e φ - Flight φ
δ h
w D
L 07/17/11 EMA 4666C - Polymer Processing Copyright Protected 22 UNIVERSITY OF
UNIVERSITY OF FLORIDA Drag Flow Analysis Pitch = π D tan φ
Vd = π D N cos φ
w = ( π D tan φ − e ) cos φ Ls
tan φ =
πD Qd = (1 / 2 w h ) Vd
Net Drag Flow for the Extruder 1 π 2 D 2 N h sin φ cos φ
Qd =
2
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UNIVERSITY OF FLORIDA Pressure Flow Analysis Pitch = π D tan φ w = ( π D tan φ − e ) cos φ
Ls
tan φ =
πD Vd = π D N cos φ
1 dP
Qp =
. w h3
12 µ dz Net Pressure Flow for the Extruder π D h sin φ dP
Qp = dL
12 µ 3 07/17/11 2 EMA 4666C - Polymer Processing Copyright Protected 24 UNIVERSITY OF
UNIVERSITY OF FLORIDA Extruder Output (Mass Throughput) π 2
Eh 2
m = ρ bulkπ D tan φ ( DB − Ds ) − sin φ 4
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UNIVERSITY OF FLORIDA Extruder/Die Characteristics
Extruder/Die 07/17/11 EMA 4666C - Polymer Processing Copyright Protected 26 UNIVERSITY OF
UNIVERSITY OF FLORIDA Flow and Pressure Profiles 07/17/11 EMA 4666C - Polymer Processing Copyright Protected 27 UNIVERSITY OF
UNIVERSITY OF FLORIDA 07/17/11 Melting Behavior in Screw EMA 4666C - Polymer Processing Copyright Protected 28 UNIVERSITY OF
UNIVERSITY OF FLORIDA 07/17/11 Examples of DIes EMA 4666C - Polymer Processing Copyright Protected 29 UNIVERSITY OF
UNIVERSITY OF FLORIDA Typical Pressures Vbz W ( h − δ ) π n D cos φ dP
Qp = dL
12 µ Vbz = π n D cos φ
Solve for P (Assume Poiseuille Flow) π D h3 sin 2 φ dP µ= 12 Q dL
p 07/17/11 EMA 4666C - Polymer Processing Copyright Protected 30 UNIVERSITY OF
UNIVERSITY OF FLORIDA References
Introduction to Physical Polymer Science, 2nd
Introduction
Edition, Lesley H. Sperling, Wiley
Interscience (1992) ISBN 0-471-53035-2
Interscience
Some figures were reproduced from: Polymer
Some
Processing Fundamentals, Tim A. Osswald
(1998), Hanser/Gardner Publications,
Cincinnati, OH.
Cincinnati,
The Physics of Polymers, Gert Strobl (1996)
The
Springer-Verlag, New York.
Springer-Verlag, 07/17/11 EMA 4666C - Polymer Processing Copyright Protected 31 ...
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