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# 5. Modeling of non-stratified mixture flows

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### 5. Modeling of non-stratified mixture flows

1. 1. oe4625 Dredge Pumps and Slurry Transport Vaclav Matousek October 13, 2004 1 Dredge Pumps and Slurry Transport Vermelding onderdeel organisatie
2. 2. 5. MODELING OF NON-STRATIFIED MIXTURE FLOWS (Pseudo-Homogeneous Flows) NEWTONIAN SLURRIES NON-NEWTONIAN SLURRIESOctober 13, 2004 2Dredge Pumps and Slurry Transport
3. 3. Repetition: Newtonian vs non-Newtonian  dv x Newton’s law of liquid viscosity (valid for laminar flow) τ = µf −   dr  October 13, 2004 3 Dredge Pumps and Slurry Transport
4. 4. Repetition: Newtonian vs non-NewtonianNewton’s law of liquid viscosity F  dVx (valid for laminar flow): τ = = µf −  A  dy  October 13, 2004 4 Dredge Pumps and Slurry Transport
5. 5. MODELING OF NEWTONIAN SLURRY FLOW EQUIVALENT-LIQUID MODEL October 13, 2004 5 Dredge Pumps and Slurry Transport
6. 6. Equivalent-Liquid ModelA. Physical background• Slurry flow behaves as a flow of a single-phase liquid having the density of the slurry.• The "equivalent liquid" has the density of the mixture but other properties (as viscosity) remain the same as in the liquid (water) alone. October 13, 2004 6 Dredge Pumps and Slurry Transport
7. 7. Equivalent-Liquid ModelB. Construction of the model for Im:• The model suggests that all particles contribute to the increase of the suspension density.• The increase of the mixture density is responsible for the increase of the liquid-like shear stress resisting the flow at the pipe wall.• The model equation is obtained in the same way as the Darcy-Weisbach equation with one exception: the density of mixture is considered instead of the liquid density. October 13, 2004 7 Dredge Pumps and Slurry Transport
8. 8. Repetition: Conservation of Momentumin 1D-flowFor additional conditions :- incompressible liquid,- steady and uniform flow in a horizontal straight pipe dP dP 4τ o − A = τ oO , i.e. − = dx dx D for a pipe of a circular cross section and internal diameter D. October 13, 2004 8 Dredge Pumps and Slurry Transport
9. 9. Repetition: Water Friction in 1D Flow in Pipe 8τ o A comparison of the Darcy-Weisbach friction coefficient equation λf = ρ f Vf2 + with the linear momentum eq. dP 4τ o (driving force = resistance force) − = for pipe flow dx D = dP λ f ρ f V 2 gives the general pressure-drop f equation for the pipe flow − = (Darcy-Weisbach equation, 1850) dx D 2 October 13, 2004 9 Dredge Pumps and Slurry Transport
10. 10. Equivalent-Liquid ModelB. Construction of the model for Im (cont’):• The shear stress at the pipe wall: water flow equivalent-liquid flow λf λf τ o, f = ρ fV 2 τ o ,m = ρ mV 2 8 8• The hydraulic gradient for slurry (equiv.-liquid) flow, Im, is dP λ f Vm2 ρ m ρm Im = − = = If dx ρ f g D 2 g ρ f ρf October 13, 2004 10 Dredge Pumps and Slurry Transport
11. 11. Very fine sand (MTI) in 158-mm horiz pipe (x): water (*): Cvd = 0.18 ( ): Cvd = 0.24 (+): Cvd = 0.30 (*): Cvd = 0.36 (o): Cvd = 0.42 October 13, 2004 11 Dredge Pumps and Slurry Transport
12. 12. Fine sand (DUT) – vertical flow (+): water (*): Cvd = 0.10 (∆): Cvd = 0.14 (◊): Cvd = 0.21 (∇): Cvd = 0.27 ( ): Cvd = 0.34October 13, 2004 12
13. 13. Fine sand (DUT) – vertical flow (+): water (*): Cvd = 0.10 (∆): Cvd = 0.14 (◊): Cvd = 0.21 (∇): Cvd = 0.27 ( ): Cvd = 0.34October 13, 2004 13
14. 14. Generalized Equivalent-Liquid ModelModel for both fine and coarse pseudo-homogeneous flows (e.g. vertical flows):The hydraulic gradient for slurry flow, Im, is Im − I f = A If , Cvd ( S s − 1)If A’=1, then Im = SmIf , i.e. the equivalent-liquid model.If A’=0, then Im = If , i.e. the liquid (water) model.If 0 < A’ < 1, then Im according to the generalized model with calibrated A’ value. October 13, 2004 14 Dredge Pumps and Slurry Transport
15. 15. MODELING OF NON-NEWTONIAN SLURRY FLOW MANY DIFFERENT MODELSOctober 13, 2004 15Dredge Pumps and Slurry Transport
16. 16. Non-Newtonian Slurry ModelA. Physical background• Very fine particles increase the viscosity of the pseudo-homogeneous mixture.• The suspension does not obey Newton’s law of viscosity and its constitutive rheological equation has to be determined experimentally. Each particular mixture obeys its own law of viscosity.• Modeling of non-Newtonian mixtures is even more complex than the modeling of Newtonian mixtures in pipes. October 13, 2004 16 Dredge Pumps and Slurry Transport
17. 17. Non-Newtonian Slurry Model A. Physical background • Internal structure of flow compared with Newtonian: - different velocity distrib. - different viscosity distrib. - identical shear-stress distribution October 13, 2004 17 Dredge Pumps and Slurry Transport
18. 18. Non-Newtonian Slurry ModelB. Construction of the model for ImA general procedure for a determination of the frictional head loss, Im:1. the rheological parameters of a slurry2. the slurry flow regime (laminar or turbulent)3. the pressure drop using a scale-up method or an appropriate friction model. October 13, 2004 18 Dredge Pumps and Slurry Transport
19. 19. Repetition: Newtonian vs non-Newtonian  dv x Newton’s law of liquid viscosity (valid for laminar flow) τ = µf −   dr  October 13, 2004 19 Dredge Pumps and Slurry Transport
20. 20. Step 1. Rheological ParametersTable. Typical values of dynamic viscosity µ at room temperature Substance Dyn. Viscosity µ [mPa.s] Air 10-2 Water 1 Olive oil 102 Honey 104 Molten glass 1015 (source: Chhabra & Richardson, 1999) October 13, 2004 20 Dredge Pumps and Slurry Transport
21. 21. Step 1. Rheological Parameters Rotational viscometerOctober 13, 2004 21Dredge Pumps and Slurry Transport
22. 22. Step 1. Rheological Parameters Viscometers for determination of rheological parameters: Rotational CapillaryOctober 13, 2004 22Dredge Pumps and Slurry Transport
23. 23. Step 1. Rheological Parameters Rheogramsfrom rotational viscometers:  dvx   dvx  τ = µf − dVx τ = fn  −   dr   dy  dr from tube (capillary) viscometers:  dv x  τ = fn  −   dy  October 13, 2004 23 Dredge Pumps and Slurry Transport
24. 24. Step 1. Rheological Parameters: Tube Viscometer Conditions : - incompressible and homogeneous slurry, - steady and uniform flow in horizontal straight tube, - laminar flow (will be required in next slide) dP 4τ o ∆P D − = ⇒ τo = dx D L 4 The wall shear stress τ0 obtained from -dP/dx = ∆P/L measured in a tube of the internal diameter D over the length L. October 13, 2004 24 Dredge Pumps and Slurry Transport
25. 25. Step 1. Rheological Parameters: Tube Viscometer Conditions : - incompressible and homogeneous slurry, - steady and uniform flow in horizontal straight tube, - laminar flow λm = 64/Re !!! λm 64 64 µ m τo = ρ mV and λm = = 2 Re Vm D ρ m m 8 Combination of these two equations gives the relationship representing the pseudo-rheogram τ0=fn(Vm/D) obtained from the tube viscometer. October 13, 2004 25 Dredge Pumps and Slurry Transport
26. 26. Step 1. Rheological Parameters: Tube Viscometer Conditions : - incompressible and homogeneous slurry, - steady and uniform flow in horizontal straight tube, - laminar flow 8Vm  dvx  τo = µm ( pseudo)equivalent to τ = µm  −  D  dy  The method must be found that transforms a pseudo- rheogram to a real rheogram. October 13, 2004 26 Dredge Pumps and Slurry Transport
27. 27. Intermezzo. Rabinowitsch-Mooney TransformationPrincipleThe Rabinowitsch-Mooney transformation transforms:- the pseudo-rheogram [To, 8Vm/D] to- the normal rheogram [T, dvx/dy] and vice versafor a laminar flow of non-Newtonians. October 13, 2004 27 Dredge Pumps and Slurry Transport
28. 28. Intermezzo. Rabinowitsch-Mooney TransformationProcedure:1. The pseudo-rheogram: determined experimentally in a tube viscometer: points [To, 8Vm/D] plotted in the ln-ln d ( ln τ o ) n co-ordinates give  8V m  , where τo = K   n= K  D   8V  d  ln m   D  dv x  3 n + 1  8V m2. Transformation rules: T=To and =  dy  4n  D (for Newtonians n’=1)3. The normal rheogram: the [T, dvx/dy] points are fitted by a suitable rheological model to determine the rheol. parameters October 13, 2004 28 Dredge Pumps and Slurry Transport
29. 29. Step 1. Rheological ParametersPseudo-rheograms by capillary or tube viscometers: wall shear stress, To dVx mean velocity, Vm dy October 13, 2004 29 Dredge Pumps and Slurry Transport
30. 30. Step 1. Rheological Parameters Transformed to the normal rheograms using the Rabinowitsch-Mooney dVx transformation: dy shear stress, T shear rate, dv/dy Result: the slurry behaves as aBingham plastic liquid, its internal friction is described by two rheological parameters, Ty, nB. October 13, 2004 30 Dredge Pumps and Slurry Transport
31. 31. Step 1. Rheological Parameters Rheograms of various non-Newtonian slurries: dVx dy Dilatant (2 parameters) Pseudo-plastic (2 par.) Bingham (2 par.)Yield dilatant (3 parameters) Yield pseudo-plastic (3 par.) October 13, 2004 31 Dredge Pumps and Slurry Transport
32. 32. Step 1. Rheological Parameters Example of rheogram obtained from viscometer testOctober 13, 2004 32Dredge Pumps and Slurry Transport
33. 33. Step 1. Rheological Parameters Rheograms of various non-Newtonian slurries: dVx dy Dilatant (2 parameters) Pseudo-plastic (2 par.) Bingham (2 par.)Yield dilatant (3 parameters) Yield pseudo-plastic (3 par.) October 13, 2004 33 Dredge Pumps and Slurry Transport
34. 34. Step 1. Rheological ParametersRheological model with two-parametersThe Bingham-plastic model contains rheological parametersTy, nB. The rheogram is a straight line with the slope nB (calledplastic viscosity or tangent viscosity). Ty is called the yield stress. dvx τ = τ y + ηB dyThis model satisfies the flow behaviour of the majority of finehomogeneous high-concentrated slurries that are dredged(aqueous mixtures of non-cohesive clay etc.). October 13, 2004 34 Dredge Pumps and Slurry Transport
35. 35. Step 1. Rheological ParametersRheological model with two-parametersThe Power-law model contains rheological parameters K, n.The rheogram is a curve. n is called the flow index (it is not theviscosity). K is called the flow coefficient. n  dvx  τ = K   dy Different types of fluid behavior occur:0 < n < 1: pseudo plastic slurry (becomes thinner under the increasing shear rate)n > 1: dilatant slurry (becomes thicker under the increasing dv/dy). October 13, 2004 35 Dredge Pumps and Slurry Transport
36. 36. Step 1. Rheological ParametersRheological model with three-parametersThe Yield Power-law model (Buckley-Herschel model)contains rheological parameters Ty, K, n. The rheogram is a curvewhich does not pass through origin. n  dvx  τ =τy + K    dy Different types of fluid behavior occur:0 < n < 1: pseudo plastic slurry (becomes thinner under the increasing shear rate)n > 1: dilatant slurry (becomes thicker under the increasing dv/dy). October 13, 2004 36 Dredge Pumps and Slurry Transport
37. 37. Step 2. Flow Regime (Lam/Turb)Threshold for Bingham plastic slurriesis given by the value of the modified Bingham Reynolds number ρ mVm D Re B = = 2100  τ yD  ηB 1 +   6η BVm in which Ty and nB are the rheological parameters of the Binghamplastic slurry. October 13, 2004 37 Dredge Pumps and Slurry Transport
38. 38. Step 3. Pressure drop due to frictionTwo Approaches: Scale Up or Friction model.There are different scale-up methods for laminar and turbulent flows.There are different friction models for laminar and turbulent flows. October 13, 2004 38 Dredge Pumps and Slurry Transport
39. 39. Step 3. Pressure drop: Scale UpPrincipleThe Im-Vm measurement results from tube viscometers can bescaled up to prototype pipes without an intermediary of arheological model (for the flow of identical slurry).The principle of the scaling-up technique is that in non-Newtonianflows the wall shear stress is unchanged in pipes of different pipesizes D. October 13, 2004 39 Dredge Pumps and Slurry Transport
40. 40. Step 3. Pressure drop: Scale UpLaminar flowusing the Rabinowitsch-Mooney transformation: D.∆P 8Vm τo = = idem = idem 4L DThis says that if a To versus 8Vm/D relationship is determinedexperimentally for a laminar flow of certain slurry in one pipe it isvalid also for pipes of all different sizes.Scaling up between two different pipeline sizes [e.g. between atube viscometer (index 1) and a prototype pipeline (index 2)]: D1 D2 I m2 = I m1 Vm2 = Vm1 D2 D1 October 13, 2004 40 Dredge Pumps and Slurry Transport
41. 41. Step 3. Pressure drop: Scale UpTurbulent flowThe Rabinowitsch-Mooney transformation not applicable: D.∆P 8Vm τo = = idem ≠ idem 4L Dbecause the near-wall velocity gradient is not described by 8Vm/D,instead the wall-friction law uses an empirical friction coefficientto relate the wall shear stress with the mean velocity.Scaling up between two different pipeline sizes [e.g. between atube viscometer (index 1) and a prototype pipeline (index 2)]: D1  λ f  D2   I m2 = I m1 Vm 2 = Vm1 1 + 2.5 ln   D2   8  D1    October 13, 2004 41 Dredge Pumps and Slurry Transport
42. 42. Step 3. Pressure drop: Scale UpTurbulent flowThe example of the scaleup of the turbulent flow ofslurry:data measured inthe 77-mm pipeand predicted forthe 154-mm pipe. October 13, 2004 42 Dredge Pumps and Slurry Transport
43. 43. Step 3. Pressure drop: Friction ModelLaminar flowIntegrating of a rheological model over a pipe cross section givesthe mean velocity and the relation between Vm and the pressuredrop (the same procedure as for the water flow).The obtained equation can be rearranged to the standard eq. ∆P λnN Vm 2 Im = = ρ f gL D 2 g 64 in which the wall-friction law for the laminar flow: λnN = and thus all effect on the frictional pressure drop RenN due to slurry rheology are expressed in the modified Reynolds number. October 13, 2004 43 Dredge Pumps and Slurry Transport
44. 44. Step 3. Pressure drop: Friction ModelLaminar flow – EXAMPLE: Bingham plastic slurry flowThe rheological model after integrating dv x 8Vm τ o  4τ y τ y  4 τ = τ y + ηB = 1 − + 4 dy D ηB  3τ o 3τ o    compared with the basic force-balance and Darcy-W. equations ∆P 4τ o ∆P λnN Vm 2 = and Im = = L D ρ f gL D 2 g leads to the wall-friction law for the Bingham-slurry flow => October 13, 2004 44 Dredge Pumps and Slurry Transport
45. 45. Step 3. Pressure drop: Friction ModelLaminar flow – EXAMPLE: Bingham plastic slurry flowleads to the wall-friction law for the Bingham-slurry flow −1 64ηB  4τ y τ  4 64 λnN = 1 − +  = y DVm ρm  3τ o 3τ    ReB 4 oThe modified Bingham Reynolds ρ mVm Dnumber is Re B =  τ yD  η B 1 +(the fourth-power term in the above  6η BVm  eq. is neglected) October 13, 2004 45 Dredge Pumps and Slurry Transport
46. 46. Step 3. Pressure drop: Friction ModelTurbulent flowIn turbulent flow, a velocity profile differs from that determinedby a rheological model (analogy with a Newtonian flow).The standard Darcy-Weisbach equation is valid ∆P λnN Vm 2 Im = = ρ f gL D 2 g but the wall-friction law is an empirical function λ nN = fn (R e nN , k ,...) October 13, 2004 46 Dredge Pumps and Slurry Transport
47. 47. Step 3. Pressure drop: Friction ModelTurbulent flow - EXAMPLEThe most successful models for turbulent flows of non-Newtonianslurries are: Wilson – Thomas model Slatter model. October 13, 2004 47 Dredge Pumps and Slurry Transport