Lecture 12 - Non-Newtonian Pipe Flow

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Unformatted text preview: ric flow rate: L PR 2 vz 8 L PR 4 Q 8 L 2 L v P f D 2 v v z dA A Q A.vz Rearrange equation for average velocity This gives the pressure drop (energy loss) along the pipe section due to viscous forces (not accounted for in Bernoulli) . We use this in the MFEE: f laminar 64 Re hf P g Laminar Pipe Flow of Power Law fluid Time independent non-Newtonian Fluid ; Substitute fluid constitutive model (Power Law), No slip Boundary condition, i.e. r=R, vz = 0 Average pipe velocity Volumetric flow rate: Q A.vz r vz vz 3nn11 1 R n n1 n1 1 P Q 2LK n 3nn1 R 1 3 n1 n Rearrange equation for average velocity to get pressure drop in terms of velocity, pipe dimensions, etc.. Then obtain frictional losses to use in MFEE (hf ); 2 L v P f D 2 Re PL v 2 n D n 4n 8n 1 K 3n 1 n P v 2LK n 3nn1 R n vz dA A v dvz dr s K n s K hf f L v2 ; D 2g n f 64 / Re PL Laminar Pipe Flow of Power Law Fluid n 1 n Power Law Fluid: s K v z 3n 1 r n 1 vz n 1 R 2.5 n=2 2.0 v/vaverage vz vz n=1 n=1 This demonstrates how ‘shear thinning’ (n < 1) and ‘shear thickening’ (n > 1) significantly affect the velocity profile in pipe flow n=0.5 1.5 n=0.25 n=0.1 n = 0.1 1.0 r/R vs n=0.1 r/R vs n =0.25 r/R vs n=0.5 r/R vs n=1 r/R vs n=2 0.5 0.0 -1.0 -0.5 0.0 0.5 1.0 r/R r/R Laminar Pipe Flow : Shear Rate at Wall r 2 v z 2vz 1 R Newtonian Fluid Shear rate 2r dv z 2v z 2 R dr At Wall, r = R dv z 2 2v z dr R Generalised Shear rate at wall in pipe flow Shear stress at wall in pipe flow 4Q 8v 3 R D sw PR 2L Laminar Pipe Flow : Shear Rate at Wall r vz vz 3nn11 1 R n Power Law Fluid n1 Shear rate dv 3n 1 1 z dr n R At Wall, r = R n 1 n dv z 3n 1 1 vz dr n R Shear rate at wall in pipe flow for PL fluid r n vz 1 W 3n 1 8v z 4n D 3n 1 W 4n Shear stress at wall in pipe flow for PL fluid sw PR 2L Determining power law relationship in Laminar Other key equations for Power law fluid; Pipe Flow Shear stress and Shear rate at the wall, r = R P sw s K n s w K w n PR 2L n w 34 n1 . Q 4Q 8v 3 This is the shear rate for a Newtonian fluid R D and Q in terms hear stress P v 2LK n 3nn1 R 1 n 1 n Example, Steffe p 143 Determine the rheological properties of bread dough using the data from a ‘capillary viscometer’ (i.e Q and P from pipe flow using various L & Ds Measured Q & P Link to excel sheet Example, Steffe p 143 Determine the rheological properties of bread dough using the data from a ‘capillary viscometer’ (i.e Q and P from pipe flow using various L & Ds Measured Q & P Link to excel sheet Example, Steffe p 143 Determine the rheological properties of bread dough using the data from a ‘capillary viscometer’ (i.e Q and P from pipe flow using various L & Ds 4Q 3 R Link to excel sheet 220 5.4 Wall Shear Stress ( Ln( w) w) 200 5.2 180 n = 0.28 K’=eIntercept K’ = 27.6 sw=27.6 w0.28 y = 27.6 x0.28 Intercept = 3.32 slope = 0.28 r ² 0.9764522388 5.0 160 4n K K ' 3n 1 140 4.8 n 120 K= 24.07 4.6 100 s K n 4.4 80 60 4.2 0 3.5 s w 24w 4.0 200 4.5 400 5.0 5.5 600 6.0 800 6.5 7.0 0.28 s 24 0.28 1000 7.5 -1 Generalised shear rate ( s ) Ln Example, Steffe p 143 Determine the rheological properties of bread dough using the data from a ‘capillary viscometer’ (i.e Q and P from pipe flow using various L & Ds n w 34 n1 . My answer: s 24 0.2 8 Link to excel sheet Reynolds Number: Power Law Fluid For Power Law Fluid: Kw n 1 Re v D Re PL Still valid, but now viscosity depends on shear rate v 2n D n 8n1 K 34nn1 n Q n n V w 34 n 1 4R 34 n 1 8D...
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## This note was uploaded on 04/17/2013 for the course CHEE 2003 taught by Professor Jasonstokes during the One '12 term at Queensland.

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