Two main standard flows in rheology:

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1 CM4655 Polymer Rheology Lb Elongtionl Flow Mesurement Prof. Fith A. Morrison Michign Technologicl University x 3 x 1, x 1 Two min stndrd flows in rheology: Sher Flow Elongtionl Flow cpillry flow; torsionl flow die entry flow this is next 1

2 Elongtionl flow occurs when there is stretching - die exit, flow through contrctions fluid 3 Unixil Elongtionl Flow x x 3 x 1 x 1, x pth lines ( t) 1 ( x t) v ( x t) x3 13 ( t) 4

3 Unixil Elongtionl Flow x 3 x x 1, x x 1 velocity field ( t) 1 ( x t) v ( x t) x3 13 ( t) 5 How does the stress tensor simplify for elongtionl flow? x 3 x x 1, x x 1 There is 18 o of symmetry round ll three coordinte xes. 6 3

4 Becuse of symmetry, there re only 3 nonzero components of the extr stress tensor in elongtionl flows. ELONGATION: This gretly simplifies the experimentlists tsks s only three stress components must be mesured insted of 6. 7 Stedy Elongtionl Flow Mteril Functions Imposed Kinemtics:, constnt Mteril Stress Response: Mteril Functions: Elongtionl Viscosity Alterntively, 8 4

5 fluid 9 Experimentl Elongtionl Geometries x 1 x 3 ir-bed to support smple fluid x 1 x 3 t o t o +t t o +t funnel-flow region thin, lubricting lyer on ech plte x 3 x 1 R(to) h(t o ) R(t) h(t) R(z) z y corner vortex Ro 5

6 Unixil Extension zz rr f ( t) A( t) tensile force time-dependent crosssectionl re lod cell mesures force f (t) For homogeneous flow: zz rr t A( t) A e f ( t) e A t fluid smple z r 11 Experimentl Difficulties in Elongtionl Flow idel elongtionl deformtion experimentl chllenges end effects initil inhomogeneities initil finl effect of grvity, drfts, surfce tension finl finl 1 6

7 Severl specilized elongtionl rheometers hve been developed nd commercilized over the lst yers 1. Filment Stretching Elongtionl Rheometer (FiSER). Metl Belt Elongtionl Rheometer (MBER) 3. Sentmnt Extension Rheometer (SER) 4. Cpillry Brekup Elongtionl Rheometer (CBER) 13 Filment Stretching Rheometer (FiSER) Tirttmdj nd Sridhr, J. Rheol., 37, (1993) Opticlly monitor the midpoint size Very susceptible to environment End Effects McKinley, et l., 15th Annul Meeting of the Interntionl Polymer Processing Society, June

8 Stedy nd strtup flow Recovery Good for melts RHEOMETRICS RME 15 Achieving commnded strin requires gret cre. Use of the video cmer (lthough tedious) is recommended in order to get correct strin rte. 16 8

9 Sentmnt Extension Rheometer Originlly developed for rubbers, good for melts Mesures elongtionl viscosity, strtup, other mteril functions Two counter-rotting drums Esy to lod; reproducible 17 Sentmnt et l., J. Rheol., 49(3) 585 (5) Comprison on different host instruments Comprison with other instruments (literture) 18 9

10 CBER Extensionl Rheometer Polymer solutions Works on the principle of cpillry filment brek up Cmbridge Polymer Group nd HAAKE For more on theory see: cmpoly.com/notes/7.pdf Brochure: Opertion Impose rpid step elongtion form fluid filment, which continues to deform flow driven by surfce tension lso ffected by viscosity, elsticity, nd mss trnsfer mesure midpoint dimeter s function of time Use force blnce on filment to bck out n pprent elongtionl viscosity 19 Cpillry brekup experiments Comments Must know surfce tension Trnsient greement is poor Filment stretching pprtus Stedy stte greement is cceptble Be wre of effect modeling ssumptions on reported results Ann nd McKinley, J. Rheol. 45, 115 (1). 1

11 We do not hve n elongtionl rheometer We cn estimte n elongtionl viscosity with cpillry results Die Entry Flow Cogswell Anlysis Binding Anlysis 1 Elongtionl Viscosity vi Contrction Flow: Cogswell/Binding Anlysis Fluid elements long the centerline undergo considerble elongtionl flow By mking strong ssumptions bout the flow we cn relte the pressure drop cross the contrction to n elongtionl viscosity R(z) R o z y funnel-flow region corner vortex 11

12 Entrnce nd exit effects - Bgley correction R PR L L P R Constnt t constnt Q R Run for different cpillries r z R entrnce region well-developed flow exit region P R L R This is the result when the end effects re negligible. 3 Bgley Plot P end effects f ( Q) f ( ) Pressure drop (psi) ( s ) P end effects 5s L/R e(5, s -1 ) Figure 1.8, p. 394 Bgley, PE 4 1

13 Assumptions for the Cogswell Anlysis incompressible fluid funnel-shped flow; no-slip on funnel surfce unidirectionl flow in the funnel region well developed flow upstrem nd downstrem -symmetry pressure drops due to sher nd elongtion my be clculted seprtely nd summed to give the totl entrnce pressure-loss neglect Weissenberg-Rbinowitsch correction sher stress is relted to sher-rte through power-lw elongtionl viscosity is constnt shpe of the funnel is determined by the minimum generted pressure drop no effect of elsticity (sher norml stresses neglected) neglect inerti R(z) R o 5 z n m R y constnt F. N. Cogswell, Polym. Eng. Sci. (197) 1, F. N. Cogswell, Trns. Soc. Rheol. (197) 16, Cogswell Anlysis elongtion rte R o 11 R 4 Q 3 R m n1 elongtion norml stress 3 8 ( n 1) 11 p ent elongtion viscosity 11 o 9 3 ( n 1) p R ent 6 13

14 Cogswell Anlysis using Excel From sher: m n1 4 Q R o 3 R p ent p ent R RAW DATA RAW DATA Cogswell Cogswell gmmdota deltpent(psi) deltpent(p) sh stress(p) N1(P) e_rte elongvisc 3*sherVisc E E+5-6.7E+5.5E+1.79E E E+5 7.9E E E E+4.7E E E+4-3.7E E+ 3.4E+4.65E E E E E+ 3.79E+4 3.3E E E E E+.14E+4 4.E ( n 1) 11 p ent 11 o 7 Assumptions for the Binding Anlysis incompressible fluid funnel-shped flow; no-slip on funnel surfce unidirectionl flow in the funnel region well developed flow upstrem nd downstrem -symmetry sher viscosity is relted to sher-rte through power-lw elongtionl viscosity is given by power lw shpe of the funnel is determined by the minimum work to drive flow no effect of elsticity (sher norml stresses neglected) the quntities dr dz nd d R dz, relted to the shpe of the funnel, re neglected; implies tht the rdil velocity is neglected when clculting the rte of deformtion neglect energy required to mintin the corner circultion neglect inerti R(z) R o R m l D. M. Binding, JNNFM (1988) 7, z n t1 o y 14

15 Binding Anlysis p ent 3t 1 (1 t) t m(1 t) lt(3n 1) n Int t ( n 1) (1 t ) 1 3 t ( n 1) (1 t ) R o (1 n) m l, elongtionl prefctor I nt 1 t1 11 n 3n 1 d n (3n 1) Q R o 3 nro m n1 R ( cpillry) R ( brrel) 1 elongtion viscosity l t 1 o 9 Binding Anlysis Note: there is non-itertive solution method described in the text; The method using Solver is slightly preferble, since it uses ll the dt in finding optiml vlues of l nd t. Evlution Procedure 1. Sher power-lw prmeter n must be known; must hve dt for p ent versus Q. Guess t, l 3. Evlute I nt by numericl integrtion over 4. Using Solver, find the best vlues of t nd l tht re consistent with the p ent versus Q dt 3 15

16 Binding Anlysis using Excel Solver I nt 1 t1 11 n 3n 1 d n Evlute integrl numericlly phi f(phi) res E re 1 ( b1 b ) h Summing: Int= Binding Anlysis using Excel Solver Optimize t, l using Solver By vrying these cells: t_guess= l_guess= ******* SOLVER SOLUTION ******** predicted exptl DeltPent DeltPent difference 1.6E E E- 6.88E E E E E+5 6.E E E+5.14E-.78E+5.54E+5 9.8E-3 trget cell 5.57E- predicted ctul ctul Sum of the differences: Minimize this cell 3 16

17 Exmple clcultion from Bgley s Dt sher or elongtinl viscosity (P s) 1.E+5 1.E+4 1.E+3 1.E+ 1.E+1 (Cogswell) Solver solution sher viscosity Cogswell elong visc Trouton prediction Binding elong visc Binding Solver Power (sher viscosity) y = 698.5x R =.9998 (Binding) This curve ws clculted using the procedure in the text 3 Bgley's dt from Figure 1.8 Understnding Rheology Morrison; ssumed contrction ws 1.5:1 1.E+ 1.E+ 1.E+1 1.E+ 1.E+3 rte of deformtion (1/s) 33 Assignment: Estimte the elongtionl viscosity of your polymer s function of temperture. Compre your results with Trouton s rule. Trouton s Rule