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# Non newtonian models

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### Non newtonian models

1. 1. Friction Factors & Head Loss Mario Cerda Cortés Mario.cerda.c@gmail.com
2. 2. Mathematical Model Description - Power law τ w = K * γ n - Pseudoplastic (n<1) - Dilatant (n>1) - Yield power lawτ w = τ y + K * γ n - Yield pseudoplastic (n<1) - Yield dilatant (n>1) -Bingham plastic τ w = τ y + η * γ -Bingham (n=1)τw : Shear stress (Pa)τy : Yield stress (Pa)η : Bingham viscosity (Pa*s)K : Consistency index (Pa*sn)γ : Shear rate (1/s)n : Flow behaviour index
3. 3. ητy
4. 4. Basic Parameters EquationReynolds n ρ m *V 2 − n * D n  4n  Re PL =   8n −1 * K  1 + 3n   n+2 Critical Reynolds 6464n * ( n + 2 )  n +1    Re PLc = (1 + 3n ) 2Critical Velocity 1  8n −1 * K * Re PLc  1 + 3n  n  2−n VPLc =     ρm * D n  4n     ε fTRoughness Reynolds correction Reε = D 2 Re PLTorrance’s Reynolds ReTo = ρ m *V 2 − n * D n 8n −1 * K ρ m : Slurry density (kg/m3) D : Pipe internal diameter (m) V : Fluid velocity (m/s) ε : Pipe roughness (m)
5. 5. Regimen Friction model Conditions 16 Re PL ≤ Re PLcLaminar fL = Re PL Re PL > Re PLcTurbulent withsmoothed pipe 1 fTS = 2.69 n − 2.95 − 4.53 n ( 2 ) * log ReTo * f TS− n + 0.68 n Reε < 5 Re PL > Re PLcTurbulent with 1  D  2.65 = 4.07 * log +6− Reε > 70roughness pipe fTR  2 *ε  n Re PL > Re PLcTransition with f RoughnessH 2O fTR = fTS 5 < Reε < 70roughness pipe f SmoothedH 2O : Fanning friction factor f τy : Yield stress (Pa) η : Bingham viscosity (Pa*s) K : Consistency index (Pa*sn)
6. 6. Basic Parameters EquationReynolds n ρ m *V 2 − n * D n  4n  ReYPL =   8n −1 * K  1 + 3n   n+ 2 Critical Reynolds 6464 * n * ( n + 2)  n +1    ReYPLc = *α (1 + 3n ) n 2−n  (1 − xc ) 2 2 xc (1 − xc ) xc 2     1 + 3n + 1 + 2n + 1 + n  α=  (1 − xc ) n 2−n n 3232 n+ 2  xc  n  1  HeYPL = * ( 2 + n ) n +1 *   (1 − x )  1+ n  * 1− x  n  c   c  2 D 2 * ρ m *τ y  τ  n −2 HeYPL = * y  K K2   ρ m : Slurry density (kg/m3) D : Pipe internal diameter (m) V : Fluid velocity (m/s) ε : Pipe roughness (m)
7. 7. Basic Parameters EquationCritical Velocity 1  8n −1 * K * Re PLc  1 + 3n  n  2−n VPLc =     ρm * D n  4n    Roughness Reynolds correction ε fT Reε = Re PL D 2Torrance’s Reynolds Re PL = ρ m *V 2 − n * D n 8n −1 * K ρ m : Slurry density (kg/m3) D : Pipe internal diameter (m) V : Fluid velocity (m/s) ε : Pipe roughness (m)
8. 8. Regimen Friction model Conditions 16 ReYPL ≤ ReYPLcLaminar fL = ψ * Re PL  (1 − x ) 2 2 x (1 − x ) n x2  ψ = (1 + 3n ) * (1 − x ) n +1 n  + +   1 + 3n 1 + 2n 1+ n    1 n +1  (1 − x ) 2 2 x (1 − x ) x2  3 D τ n Q = π *   * n *  w  * (1 − x ) n *   1 + 3n + +  2 K  1 + 2n 1+ n   ReYPL > ReYPLcTurbulent with 1 = 2.69 − 2.95 − 4.53 ( * log ReTo * f TS− n + 2 + ) 0.68 4.53 * log(1 − x ) Reε < 5smoothed pipe fTS n n n n ReYPL > ReYPLcTurbulent with 1  D = 4.07 * log  +6− 2.65  2 *ε Reε > 70roughness pipe fTR  n ReYPL > ReYPLcTransition with f RoughnessH 2O fTR = f TS 5 < Reε < 70roughness pipe f SmoothedH 2O
9. 9. Basic Parameters Equation ρ m *V * DReynolds & Hedstrom Re B = η D 2 *τ y * ρ m He = η2Critical Reynolds Re Bc = He  4 x4  1 − xc + c  8 xc  3  3 xc He = (1 − xc ) 16800 3Critica Velocity VBc = η * Re Bc ρm * DRoughness Reynolds correction Reε = ε fT Re PL D 2Torrance’s Reynolds ReTo = ρ m *V 2 − n * D n 8n −1 * K
10. 10. Regimen Friction model Conditions Re BI ≤ Re BIcLaminar 16  He He 4  fL = * 1 +  6 Re + 3  Re B  B 3 f L * Re 3  B  Re BI > Re BIcTurbulent withsmoothed pipe 1 fTS = 2.69 n − 2.95 − 4.53 n ( * log ReTo * f TS− n + 2 )0.68 4.53 n + n * log(1 − x ) Reε < 5 Re BI > Re BIcTurbulent with 1  D  2.65 = 4.07 * log +6− Reε > 70roughness pipe fTR  2 *ε  n Re BI > Re BIcTransition with f RoughnessH 2O fTR = f TS 5 < Reε < 70roughness pipe f SmoothedH 2O
11. 11. ( P2 − P1 ) + (V22 − V12 ) + ( Z Vi 2  f Darcy * Leq  Vi 2 TDH = 2 − Z1 ) + ∑ k i + ∑  γm 2g i 2g i   D  2g i f Darcy = 4 f Fanning γ m = g * ρm m g = 9.81 2 ski: Minor losses o Singularities (homework!!) Start up Pressure for Bingham & Yield Power Law Fluids 4τ y * L Pstart −up = D
12. 12. 1. Peker & Helvaci, “Solid liquid two phase flow”2. Chhabra & Richarson, “Non newtonian flow and applied rheology”3. Abulnaga , “Slurry system handbook”4. Nayyar, “Piping Handbook”5. Wasp, “Solid liquid flow – Slurry pipeline transportation”6. Couper et al, “Chemical process equipment”7. Darby, “Chemical engineering fluid mechanics”
13. 13. slurry, tailing, Bingham, Hedstrom, Metzner, Darby, Tomita, Fanning, Torrance,Irvine, mining, mineria, Darby, codelco, rheological, Rheology, dilatant, rio tinto,barrick, power law, non newtonian, bhp billiton, tailing, head loss, pseudoplastic,xtrata, pulpa, relave, Buckingham, freeport macmoran, anglo america, antofagastaminerals, viscosity, yield, friction, Darcy