Rheology - UNIVERSITY OF UNIVERSITY OF FLORIDA EMA 4666C -...

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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 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 1 UNIVERSITY OF 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 07/22/11 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 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 9 UNIVERSITY OF 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 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 10 UNIVERSITY OF 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 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 11 UNIVERSITY OF 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, 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 12 UNIVERSITY OF 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 EMA 4666C - Polymer Processing Copyright Protected 13 UNIVERSITY OF 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 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 deviation from Newtonian: ( n −1) η = m (T ) γ Where m = “consistency” and if n =1, then Newtonian Fluid Behavior: η = µ (T ) 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 16 UNIVERSITY OF 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 07/22/11 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 ) 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 18 UNIVERSITY OF 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 ) -1 4 EMA 4666C - Polymer Processing Copyright Protected 19 UNIVERSITY OF 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 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 21 UNIVERSITY OF 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). 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 22 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. 07/22/11 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. 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 24 UNIVERSITY OF 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 07/22/11 EMA 4666C - Polymer Processing Copyright Protected 25 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, 07/22/11 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.

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