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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
HagenPoiseuille Flow
Couette Flow
Normal Stresses in Shear Flow
Deborah Number
Thermosets Rheometry
Melt Flow Index
Capillary Viscometer
07/22/11 Cone and Plate Viscometer Processing Copyright Protected
EMA 4666C  Polymer 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
F 07/22/11 dl EMA 4666C  Polymer Processing Copyright Protected 3 UNIVERSITY OF
UNIVERSITY OF FLORIDA Simple Shear Rheometer 07/22/11 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
07/22/11 n Vo
B
V f EMA 4666C  Polymer Processing Copyright Protected 5 UNIVERSITY OF
UNIVERSITY OF FLORIDA Reduced Viscosity Curve LDPE 07/22/11 EMA 4666C  Polymer Processing Copyright Protected 6 UNIVERSITY OF
UNIVERSITY OF FLORIDA Rheology  Parallel Plate Analysis Simple Parallel Plate with Velocity
Simple
Profile
Profile 07/22/11 EMA 4666C  Polymer Processing Copyright Protected 7 UNIVERSITY OF
UNIVERSITY OF FLORIDA A Single Element for Analysis High Pressure End F3 R F1 r F2 dz
07/22/11 dr Low Pressure End EMA 4666C  Polymer Processing Copyright Protected 8 UNIVERSITY OF
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 Parallel Plate Analysis:
Parallel
Analyze Forces
Analyze Force Balance: ∂P F1 = π dr P − dz ∂z 2
F2 = π dr P
2 F3 = 2π dr dz dτ
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UNIVERSITY OF FLORIDA Parallel Plate Analysis:
Parallel
Analyze Forces – Steady State
Analyze Steady State Flow: ∑F z =0 One can see that: ∂P π dr P = π dr P − dz − ∂z 2 2 2π dr dz dτ
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UNIVERSITY OF FLORIDA Parallel Plate Analysis:
Parallel
Analyze Forces – Steady State
Analyze where: dr ∂P dτ = 2 ∂z By integration, τ r at any radius r can be
By
defined by:
defined r dP τr = 2 dz The pressure gradient is by definition
The
uniform, so for a pressure drop over L,
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UNIVERSITY OF FLORIDA Parallel Plate Analysis:
Parallel
Analyze Forces – Steady State
Analyze max τ will be at the wall where r
max
= R and so:
and PR τw = 2L given:
given: 07/22/11 . τ = ηγ
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UNIVERSITY OF FLORIDA Simplified Flow in Polymer Processing
Rectangular Channel o v
y h
z Flow Velocity Flow Rate
07/22/11 y v z ( y ) = vo h vo h W Q = 2 EMA 4666C  Polymer Processing Copyright Protected 14 UNIVERSITY OF
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 07/22/11 EMA 4666C  Polymer Processing Copyright Protected 15 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 NonNewtonian 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
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 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 HagenPoiseuille 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 =
(HagenPoiseuille) (HagenPoiseuille)
8 µ L 07/22/11 EMA 4666C  Polymer Processing Copyright Protected 17 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
07/22/11 1 1 3 2 log shear rate (sec )
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UNIVERSITY OF FLORIDA 07/22/11 Shear Rate Dependence EMA 4666C  Polymer Processing Copyright Protected 20 UNIVERSITY OF
UNIVERSITY OF FLORIDA Non Isothermal Flow Viscous heating versus convection heating: ηV
∆T = 8k 2
o s Brinkman number is a reduced temperature analysis
which provides a relative diffusion parameter for
analyzing the differences in heating. η Vo2 Br = (T − T )k h g
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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 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. 07/22/11 EMA 4666C  Polymer Processing Copyright Protected 23 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 References
Introduction to Physical Polymer Science, 2nd
Introduction
Edition, Lesley H. Sperling, Wiley
Interscience (1992) ISBN 0471530352
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
SpringerVerlag, New York.
SpringerVerlag, 07/22/11 EMA 4666C  Polymer Processing Copyright Protected 26 UNIVERSITY OF
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 07/22/11 EMA 4666C  Polymer Processing Copyright Protected 27 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
1, 07/22/11 EMA 4666C  Polymer Processing Copyright Protected 28 UNIVERSITY OF
UNIVERSITY OF FLORIDA 07/22/11 TTr Phase Diagram EMA 4666C  Polymer Processing Copyright Protected 29 ...
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This note was uploaded on 07/22/2011 for the course EMA 6166 taught by Professor Staff during the Fall '08 term at University of Florida.
 Fall '08
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