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Rheometry CM4650 4/27/2018 Chapter 10: Rheometry Capillary Rheometer piston CM4650 F entrance region Polymer Rheology r A barrel z 2Rb polymer melt L 2R reservoir B exit region Q Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University 1 © Faith A. Morrison, Michigan Tech U. Rheometry (Chapter 10) measurement All the comparisons we have discussed require that we somehow measure the material functions on actual fluids. 2 © Faith A. Morrison, Michigan Tech U. 1 Rheometry CM4650 4/27/2018 Rheological Measurements (Rheometry) – Chapter 10 Tactic: Divide the Problem in half Modeling Calculations Experiments Dream up models RHEOMETRY Calculate model Build experimental predictions for Standard Flows apparatuses that stresses in standard allow measurements flows in standard flows Calculate material Calculate material functions from Compare functions from model stresses measured stresses Pass judgment on models U. A. Morrison,© Tech Faith Michigan Collect models and their report 3 cards for future use Chapter 10: Rheometry Capillary Rheometer piston CM4650 F entrance region Polymer Rheology r A barrel z 2Rb polymer melt L 2R reservoir B exit region Q Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University 4 © Faith A. Morrison, Michigan Tech U. 2 Rheometry CM4650 4/27/2018 Rheological Measurements (Rheometry) – Chapter 10 Tactic: Divide the Problem in half Modeling Calculations Experiments Dream up models RHEOMETRY Calculate model Build experimental predictions for Standard Flows apparatuses that stresses in standard allow measurements in standard flows As withflows Advanced Rheology,Calculate material there is way Calculate material functions from Compare functions from too muchmodel stresseshere to cover in measured stresses the time we have. Pass judgment on models U. A. Morrison,© Tech Faith Michigan (we will be taking a tour) Collect models and their report 5 cards for future use Rheological Measurements (Rheometry) – Chapter 10 Thumbnail of Rheometry Need to create the flow Measure the stresses accurately Shear (not too hard) Elongation (hard) • Capillary (Weissenberg- • Capillary entrance losses – Rabinowich and slip corrections, many assumptions good for high ; Ψ from die swell, • Filament stretching – hard to Ψ) produce; research only • Torsional flow • Counter-rotating rolls – pretty • Parallel plate (correction accessible needed; low ; Ψ, Ψ) • Cone and plate (low ; can do Ψ,Ψ • Couette (cup and bob; low ; Ψ, Ψ) © Faith A. Morrison, Michigan Tech U. Tech Michigan A. Morrison, © Faith 6 3 Rheometry CM4650 4/27/2018 Rheological Measurements (Rheometry) – Chapter 10 Standard Flows Summary Choose velocity field: Symmetry of flow alone implies: con 11 12 0 0 v (x ) x ≡ H 1 2 2 0 0 21 22 x 0 1 0 0 33123 x3 2 11 0 0 ≡ 0 22 0 x1, x2 2 0 0 33123 To measure the stresses we need for material functions, we must produce the defined flows 7 © Faith A. Morrison, Michigan Tech U. Rheological Measurements (Rheometry) – Chapter 10 Simple Shear flow (Drag) (z-plane ≡ 0 section) y 0 x Challenges: •Sample loading •Maintain parallelism •Producing linear motion •Stress measurement (Edge effects) •Signal strength From the McGill website (2006): Hee Eon Park, first-year postdoc in Chemical Engineering, works on a high-pressure sliding plate rheometer, the only instrument of its kind in the world. J. M. Dealy and S. S. Soong “A Parallel Plate Melt Rheometer Incorporating a Shear Stress Transducer,”J. Rheol. 28, 8 355 (1984) © Faith A. Morrison, Michigan Tech U. 4 Rheometry CM4650 4/27/2018 Rheological Measurements (Rheometry) – Chapter 10 Although we stipulated simple, homogeneous shear flow be produced throughout the flow domain, can we, perhaps, relax that requirement? ≡ 0 0 9 © Faith A. Morrison, Michigan Tech U. Rheological Measurements (Rheometry) – Chapter 10 Viscometric flow: motions that are locally equivalent to steady simple shearing motion at every particle •globally steady with respect to some frame of reference •streamlines that are straight, circular, or helical •each flow can be visualized as the relative motion of a sheaf of material surfaces (slip surfaces) •each slip surface moves without changing shape during the motion •every particle lies on a material surface that moves without stretching (inextensible slip surfaces) Viscometric Flows: 1. Steady tube flow 2. Steady tangential annular flow 3. Steady torsional flow (parallel plate flow) 4. Steady cone-and-plate flow (small cone angle) 5. Steady helical flow Wan-Lee Yin, Allen C. Pipkin, “Kinematics of viscometric flow,” Archive for Rational Mechanics and Analysis, 37(2) 111-135, 1970 R. B. Bird, R. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, 2nd edition, Wiley (1986), section 3.7. 10 © Faith A. Morrison, Michigan Tech U. 5 Rheometry CM4650 4/27/2018 Rheological Measurements (Rheometry) – Chapter 10 Viscometric Flows: Experimental Shear Geometries (viscometric flows) 1. Steady tube flow 2. Steady tangential annular flow 3. Steady torsional pp flow (z-plane 4. Steady cone-and-plate flow section) 5. Steady helical flow y x H r (z-plane r section) y piston F (-plane (plane x section) section) A barrel 2Rb polymer melt reservoir B (-plane (-plane section) (z-plane section) (z-plane section) section) r Q 11 © Faith A. Morrison, Michigan Tech U. Rheological Measurements (Rheometry) – Chapter 10 Types of Shear Rheometry Mechanical: •Mechanically produce linear drag flow; Measure (shear strain transducer): 1. planar Couette Shear stress on a surface •Mechanically produce torsional drag flow; Measure: (strain-gauge; force rebalance) 1. cone and plate; Torque to rotate surfaces 2. parallel plate; Back out material functions 3. circular Couette •Produce pressure-driven flow through conduit Measure: 1. capillary flow Pressure drop/flow rate 2. slit flow Back out material functions 12 © Faith A. Morrison, Michigan Tech U. 6 Rheometry CM4650 4/27/2018 Shear Rheometry– Capillary Flow Capillary Rheometer piston F entrance region r A barrel z 2Rb polymer melt L 2R reservoir B exit region Q 13 © Faith A. Morrison, Michigan Tech U. Shear Rheometry– Capillary Flow Capillary Rheometer piston The basics F entrance region A barrel r z 2Rb polymer melt 2R well-developed flow reservoir B exit region Q 14 © Faith A. Morrison, Michigan Tech U. 7 Rheometry CM4650 4/27/2018 Shear Rheometry– Capillary Flow Exercise: • What is the shear stress in capillary flow, for a fluid with r unknown constitutive equation? •What is the shear rate in capillary z flow? 15 © Faith A. Morrison, Michigan Tech U. Shear Rheometry– Capillary Flow To calculate shear rate, shear stress, look at EOM: P p gz v v v P t steady Assume: unidirectional state •Incompressible fluid •no -dependence •long tube P 1 r 0 rr •symmetric stress tensor r r r r •Isothermal 1 r 2 •Viscosity independent of pressure 0 0 r r 2 r P 1 r 0 rz rz z rz r r rz 16 © Faith A. Morrison, Michigan Tech U. 8 Rheometry CM4650 4/27/2018 Shear Rheometry– Capillary Flow Shear stress in capillary flow, for a fluid with unknown constitutive equation Shear stress in capillary flow: P P r r 0 L rz R P 2L R constant z (varies with position, i.e. inhomogeneous flow) What was the shear stress in drag flow? 17 © Faith A. Morrison, Michigan Tech U. Shear Rheometry– Capillary Flow Viscosity from capillary flow – inhomogeneous shear flow z x2 R r x Shear coordinate system 1 near wall: eˆ1 eˆz It is not the 21 rz rR R same shear eˆ2 eˆr v v rate z z 0 rR R everywhere, eˆ3 eˆ (r) r but if we focus on the 21 R wall shear stress wall we can still get ( ) R 0 R wall shear rate 18 © Faith A. Morrison, Michigan Tech U. 9 Rheometry CM4650 4/27/2018 Shear Rheometry– Capillary Flow Viscosity from Wall Stress/Shear rate Note: we are assuming no-slip at the wall Wall shear stress in capillary flow: P P r PR 0 L rz rR 2L 2L rR P constant z What is shear rate at the wall in capillary flow? If is known, this is easy to calculate. 19 © Faith A. Morrison, Michigan Tech U. Shear Rheometry– Capillary Flow If is known, is easy to calculate. Velocity Fields, Flow in a Capillary 2 2Q r Newtonian fluid: vz (r) 2 1 R R Power-law GNF fluid: 1 1 1 1 1 P P n 1 r n n 0 L vz (r) R 1 2mL 1 n 1 R 20 © Faith A. Morrison, Michigan Tech U. 10 Rheometry CM4650 4/27/2018 Shear Rheometry– Capillary Flow If is known, is easy to calculate. Wall shear-rate for a Newtonian fluid Hagen-Poiseuille: PR4 Q 8L 44Q 4Q 1 PR a R3 R3 2L slope = 1/ PR R 2L 21 © Faith A. Morrison, Michigan Tech U. Shear Rheometry– Capillary Flow If is known, is easy to calculate. Wall shear-rate for a Power-law GNF PL-GNF flow rate: 10 1 3 44Q PR n 1 nR Q 1 n a 2L m 1 3n R3 1/n 1 4m 1/ n 3 intercept = slope = 1/n 0.1 0.1 1 10 PR R 2L 22 © Faith A. Morrison, Michigan Tech U. 11 Rheometry CM4650 4/27/2018 Shear Rheometry– Capillary Flow If is known, is easy to calculate. For an unknown, non-Newtonian fluid, is not known, and we need to take special steps to determine the wall shear rate The wall shear rate is generally greater than for a Newtonian fluid. vz(r) ,R,nonNewtonian Newtonian velocity profile non-Newtonian velocity profile ,R,Newtonian r 23 © Faith A. Morrison, Michigan Tech U. Shear Rheometry– Capillary Flow If is not known, we must take extra steps to determine . For a General non-Newtonian fluid Q ? Something wall shear-rate-ish ? Something wall shear-stress-ish 24 © Faith A.
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