Slide-Set-9-Rheology-Polymer-Melts

Slide-Set-9-Rheology-Polymer-Melts - Slide Set 9 Rheology...

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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 – Hagen-Poiseuille 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 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 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 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 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 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, 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.

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