Lecture 12 - Non-Newtonian Pipe Flow

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Unformatted text preview: s Pr 2L 2 sw PR 2L P P P2 1 Newtonian fluids = > Q D 4 P 128 L 2 L v P f D 2 or hf 2 P L v f= fn [ReD, roughness] f g D 2 g pipesection Frictional losses during flow in pipes P P gz1 P2 gz2 1 s Pr 2L sw PR 2L Newtonian fluids = > Q D 4 P 128 L or P P gz1 P2 gz2 1 Fluids flow from high to low pressure (from 1 to 2 if P > 0) 2 L v P f D 2 hf 2 P L v f= fn [ReD, roughness] f g D 2 g pipesection Laminar Flow in a Pipe of Bingham: General Approach 1 Q 3 3 R s w General equation for Q for time-independent fluids: In a Bingham Fluid, shear rate is defined as: f (s ) sw 0 s 2 f (s )ds s s y B The function is discontinuous because there is no shearing flow is regions when the stress is less than the yield stress f (s ) 0 f (s ) s s y B for 0 s s y (inner portion of tube) (outer portion of tube) for s y s s w Need to integrate for each region to get Q or average velocity, and since the shear rate = 0 when the stress is below the yield stress. s Q 1 w 2 s s y ds s R 3 s w sy B 4 R 4 P 4 s y 1 s y 1 3 Q sw 3 sw 8 B L Laminar Flow in a Pipe of NNF: Yield Stress Fluids Pipe flow of Yield Stress Fluids (e.g. Bingham or H-B) : • When fluid does flow, there will be a ‘solid’ plug-like core in the centre of the tube (where the shear stress < yield stress) s Stress as fn of radius: Radius (R0 ) when s > sy: R0 Velocity Profile, Bingham: vz , B Pr 2L s y 2L P 2 L Pr 2 s y s w s y P B 2L Velocity of plug determined by setting r=R0=sy2L/P Ls w s y 2 vz , Bingham plug P B Laminar Flow in a Pipe of NNF: Yield Stress Fluids s s y B Bingham model s s y K n Herschel-Buckley model sw R0 s y 2L P s PR 2L Flowing region s>sy Pr 2L Solid plug ssy Ls w s y 2 1 1 1 2L n vz , Bingham plug 2 Ls w s y n Pr 1 1 n s y vz ,HBplug P B 1 1n 1 s w s y 2L P(1 1 ) K n P(1 n ) K n ..reduces to Bingham if K = B and n=1 1 vz , HB Laminar Flow in a Pipe of NNF: Yield Stress Fluids s s y B Bingham model s s y K n Herschel-Buckley model sw R0 s y 2L P s PR 2L Flowing region s>sy Pr 2L Solid plug ssy 1 n Pr 1 1 s y s w s y n 1 2L P(1 1 ) K n n 1 vz , HB 2L vz , HBplug 2 Ls w s y 1 1 n 1 P(1 1 ) K n n ..reduces to Bingham if K = B and n=1 MFEE – Mechanical For of energy equation: NNF Bingham Reynolds number Hedstrom number Re B He v D B s y D 2 B 2 Friction factor for Bingham, Laminar flow: 64.(6 Re B He) f 2 6 Re B L v2 hf f D 2g s at onset of flow 2 v v D (v D) 1 ( 8D D 2 ) 1 (w ) D 2 1 (s y / B ) D 8 8 8 Re flowonset 1 He 8 B B B B B Supplementary slides Laminar Pipe Flow of NonNewtonian Fluids Step – by – step Laminar Pipe Flow of Power Law Fluid NEWTONIAN: Substitute fluid constitutive model, i.e. Newtonians Law of viscosity: dv s z dr vz 2 PR 2 r 1 4 L R Parabolic velocity profile L Time independent non-Newtonian Fluid: Power Law fluid: From previous page Integration gives: dv s K s K z dr n n 1 n Pr dv Pr s K z dr K n .dvz 2L dr 2L n 1 1 1 1 n 1r P n n P v z c r dr dvz 2LK n 1 1 2LK n 1 No slip Boundary condition, i.e. r=R, v = 0…=>c =…… \ Laminar Pipe Flow of Power Law Fluid Integration gives: P vz 2LK n nn 1 R n r 1 n1 n1 n constant...
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