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Unformatted text preview: Slide Set 9 Rheology of Polymer Melts
Rheology
Dr. Anthony Brennan
University of Florida
Department of Materials Science &
Department
Engineering
Engineering
EMA 4161C  Polymer Physical Properties 1 Learning Objectives
Learning
• General Rheological Principles:
– Newtonian Behavior
– HagenPoiseuille Flow
– Couette Flow
•
•
• Normal Stresses in Shear Flow
Deborah Number
Thermosets – Rheometry
• Melt Flow Index
• Capillary Viscometer
• Cone andEMA 4161C  Polymer Physical Properties
Plate Viscometer 2 Rheology
Rheology
• Viscosity is the internal friction of a
Viscosity
fluid or the resistance to flow under
mechanical stress
mechanical
F dl EMA 4161C  Polymer Physical Properties 3 Simple Shear Rheometer
Simple EMA 4161C  Polymer Physical Properties 4 Simple Shear Viscosity
Simple
Shear Rate Dependence τ =κ γ
•Doolittle’s Equation η = Ae n Vo
B
V f EMA 4161C  Polymer Physical Properties 5 Reduced Viscosity Curve LDPE
Reduced EMA 4161C  Polymer Physical Properties 6 Rheology  Parallel Plate Analysis
Rheology
• Simple Parallel Plate with Velocity
Simple
Profile
Profile EMA 4161C  Polymer Physical Properties 7 A Single Element for Analysis
Single
High Pressure End F3 R F1 r dr F2 dz EMA 4161C  Polymer Physical Properties Low Pressure End 8 Assumptions:
Assumptions:
•
•
• No slip at walls
Melt is incompressible
Flow is steady, laminar and time
Flow
independent
independent
• Fluid viscosity is pressure independent
• End effects are neglible EMA 4161C  Polymer Physical Properties 9 Parallel Plate Analysis:
Analyze Forces
Analyze •Force Balance: ∂P F1 = π dr P − dz ∂z 2
F2 = π dr P
2 F3 = 2π dr dz dτ
EMA 4161C  Polymer Physical Properties 10 Parallel Plate Analysis:
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
EMA 4161C  Polymer Physical Properties 11 Parallel Plate Analysis:
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,
EMA 4161C  Polymer Physical Properties 12 Parallel Plate Analysis:
Analyze Forces – Steady State
Analyze • max τ will be at the wall where r =
max
R and so:
and PR τw = 2L given:
given: . τ = ηγ
EMA 4161C  Polymer Physical Properties 13 Simplified Flow in Polymer Processing
Simplified
Rectangular Channel o v
y h
z Flow Velocity Flow Rate y v z ( y ) = vo h vo h W Q = 2 EMA 4161C  Polymer Physical Properties 14 Pressure Flow Thru a Slit
Pressure
o L y h z v ∆ p = p1 − p2 p1 p2 •The pressure flow through the slit is common to dies for
The
films. Newtonian Behavior is given as:
films. 2 y 2 h ∆p vz ( y) = 1 − 8 µ L h W h3 ∆ p Q = 12 µ L 2 EMA 4161C  Polymer Physical Properties 15 Pressure Flow Thru a Slit
Pressure
•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
Where
deviation from Newtonian:
deviation
= m (T ) ( n −1) η γ Where m = “consistency” and if n =1, then Newtonian Fluid
Where
Behavior:
η = µ (T )
EMA 4161C  Polymer Physical Properties 16 Pressure Flow Thru a tube
Pressure
o L v y
z R p1 ∆ 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 EMA 4161C  Polymer Physical Properties 17 Pressure Flow Thru a Tube
Pressure
•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
Where
deviation from Newtonian:
deviation
( n −1) η = m (T ) γ Where m = “consistency” and if n =1, then Newtonian Fluid
Where
Behavior:
η = µ (T )
EMA 4161C  Polymer Physical Properties 18 POWER LAW BEHAVIOR
POWER
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 1 1 3 2 log shear rate (sec )
1 4 EMA 4161C  Polymer Physical Properties 19 Shear Rate Dependence
Shear EMA 4161C  Polymer Physical Properties 20 Non Isothermal Flow
Non
•Viscous heating versus convection heating: ηV
∆T = 8k 2
o s Brinkman number is a reduced temperature
Brinkman
analysis which provides a relative diffusion
parameter for analyzing the differences in heating.
parameter η Vo2 Br = (T − T )k h g
EMA 4161C  Polymer Physical Properties 21 Stress Analysis in Flow
Stress 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 • ψ 1 And ψ 2 are material parameters that define the primary and
secondary normal stress coefficients (Tensor notation that
correlate both strain rate and temperature).
EMA 4161C  Polymer Physical Properties 22 Deborah Number
Deborah
•Stress analyses are based upon: Time,
Stress
Temperature, Rate
Temperature, λ τ characteristic diffusion
De = = =
tt
processing time
Dimensionless number, related directly to the
Dimensionless
relaxation time or diffusion time of the polymer
chains and the experimental time. Critical to
consider with respect to the processing design. EMA 4161C  Polymer Physical Properties 23 Reynolds Number
Reynolds
•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
Where
the fluid, ρ is the density (kg/m3), and η is the viscosity
),
( Pa.sec). Velocity is expressed as:
Pa sec). Q
V=
A Where Q is output (cm3/sec) and A is cross sectional area of
/sec)
channel.
channel.
EMA 4161C  Polymer Physical Properties 24 Estimation of Re
Estimation
•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
EMA 4161C  Polymer Physical Properties 25 References
References
• Introduction to Physical Polymer Science, 4th
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, EMA 4161C  Polymer Physical Properties 26 Normal Stresses – Die Swell
Normal
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 EMA 4161C  Polymer Physical Properties 27 Thermosets
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, EMA 4161C  Polymer Physical Properties 28 TTr Phase Diagram
TTr EMA 4161C  Polymer Physical Properties 29 ...
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This note was uploaded on 07/17/2011 for the course EMA 4161c taught by Professor Staf during the Fall '10 term at University of Florida.
 Fall '10
 Staf

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