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1b. Basic principles of flow in pipe

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1b. Basic principles of flow in pipe

1. 1. oe4625 Dredge Pumps and Slurry Transport Vaclav Matousek October 13, 2004 1 Dredge Pumps and Slurry Transport Vermelding onderdeel organisatie
2. 2. 1. BASIC PRINCIPLES OF FLOW IN PIPESOLID PARTICLES IN QUIESCENT LIQUIDSOLID PARTICLES IN FLOWING LIQUID October 13, 2004 2 Dredge Pumps and Slurry Transport
3. 3. PARTICLES IN LIQUID BUOYANCY DRAG LIFT TURBULENT DISPERSION INTERPARTICLE CONTACTSOctober 13, 2004 3Dredge Pumps and Slurry Transport
4. 4. SOLID PARTICLE IN QUIESCENT LIQUID Terminal settling velocity of sphereTerminal settling velocity of non-spherical particle (particle shape effect) Hindered settling velocity of particle in cloud (solids concentration effect) October 13, 2004 4 Dredge Pumps and Slurry Transport
5. 5. Terminal Settling Velocity of SphereForces acting on a solid spherical particle submerged in aquiescent water column: π d3Gravitational force: Fg = ρs g [N] 6 π d3Buoyancy force: Fb = ρf g [N] 6Drag force: FD = fn( ρ f , µ f , d , vts ) [N]The balance of the three forces acting on the submerged solid body determines the settling velocity, vts, of the body.October 13, 2004 5Dredge Pumps and Slurry Transport
6. 6. Terminal Settling Velocity: Buoyancy ForceExample: The hydrostatic force acts on the top and the bottom of a solid cylinder submerged in the liquid. Top of cylinder: Force downwards Ftop=(p0+h1ρfg)dA Bottom of cylinder: Force upwards Fbot=-(p0+h2ρfg)dA Buoyancy force: Ftop+Fbot=ρfg (h1-h2)dA=-ρfgVolumecylind October 13, 2004 6 Dredge Pumps and Slurry Transport
7. 7. Drag Force The drag force is a product of the pressure differential developed over a sphere due to the flow round the sphere. Total drag is composed of skin-friction drag and pressure drag. Figure: Pressure distribution around a smooth sphere for laminar and turbulent-layer flow, compared with theoretical inviscid flow.October 13, 2004 7Dredge Pumps and Slurry Transport
8. 8. Terminal Settling Velocity: Drag ForceThe pattern of the flow round a particle (sphere) is characterized by developments in the boundary layer (BL) at the particle surface. The BL can be laminar or turbulent. October 13, 2004 8 Dredge Pumps and Slurry Transport
9. 9. Terminal Settling Velocity: Drag ForceDrag acting on a solid particle (sphere) depends on a development of flow in the boundary layer. Flow separation and with the separation associated development of a turbulent wake affect the drag force. October 13, 2004 9 Dredge Pumps and Slurry Transport
10. 10. Terminal Settling Velocity: Drag Force Separation of flow from the sphere surface can occur as a result of the adverse pressure gradient (dp/dx > 0). The separation increases pressure drag on sphere. The effect of separation is to decrease the net amount of flow work that can be done by a fluid element on the surrounding fluid at the expense of its kinetic energy, with the net result that pressure recovery is incomplete and flow losses (drag) increase.October 13, 2004 10Dredge Pumps and Slurry Transport
11. 11. Terminal Settling Velocity: Drag ForceThe dimensional analysis of FD = fn(ρf, vts, µf, d) provides two dimensionless groups: 8FD drag. force CD = 2 2 = πd vts ρf hydrodynamic. force vtsdρf inertia. force Rep = = µf viscous. forceThe relationship CD = fn(Rep) is determined experimentally. October 13, 2004 11 Dredge Pumps and Slurry Transport
12. 12. Terminal Settling Velocity: Drag ForceThe relationship CD = fn(Rep) is determined experimentally: vts for a spherical particle is measured. Regimes Laminar: Rep < 1 CD = 24/Rep Transitional: CD = fn(Rep) Turbulent: 3 x 105 > Rep > 500 CD = 0.445. October 13, 2004 12 Dredge Pumps and Slurry Transport
13. 13. Terminal Settling Velocity: Drag Force Laminar regime: Rep<1 (Stokes flow): • laminar flow round a sphere, no flow separation from a sphere; wake is laminar • drag is predominantly due to friction • pressure differential due to viscosity between the forward (A) and rearward (E) stagnation points: p(A) > p(E)October 13, 2004 13Dredge Pumps and Slurry Transport
14. 14. Terminal Settling Velocity: Drag Force Transitional regime: 1000 > Rep>1: • the flow separates and forms vortices behind the sphere; • drag is a combination of friction and pressure dragOctober 13, 2004 14Dredge Pumps and Slurry Transport
15. 15. Terminal Settling Velocity: Drag Force Inertial regime: 3x105 > Rep>103: • the boundary layer on the forward portion of the sphere is laminar; separation occurs just upstream of the sphere midsection; wide turbulent wake downstream • the pressure p(E) in the separated region is almost constant and lower than p(A) over the forward portion of the sphere • drag is primarily due to this pressure differential, no viscous effectOctober 13, 2004 15Dredge Pumps and Slurry Transport
16. 16. Terminal Settling Velocity: Drag Force Critical Rep ˜ 3 x 105 : • the boundary layer becomes turbulent and the separation point moves downstream, wake size is decreased • the pressure differential is reduced and CD decreases abruptly; • rough particles – turbulence occurs at lower Rep, thus Rep,cr is reduced.Turbulent boundary layer has more momentum than laminar BL and can better resist an adverse pressure gradient. It delays separation and thus reduces the pressure drag. October 13, 2004 16 Dredge Pumps and Slurry Transport
17. 17. Terminal Settling Velocity: Drag ForceStreamlined bodies are so designed that the separation point occurs as far down- stream as possible. If separation can be avoided the only drag is skin friction. October 13, 2004 17 Dredge Pumps and Slurry Transport
18. 18. October 13, 2004 18
19. 19. Terminal Settling Velocity of SphereThe balance of the gravitational, buyoancy and drag forces π d3 CD 6 ( ρs − ρ f ) g = 8 π d 2vts ρ f 2 [N]produces an eq. for the terminal settling velocity of a spherical particle, vts 4 ( ρ s − ρ f ) gd vts = [m/s]. 3 ρf CDThe vts formula is an implicit equation and must be solved iteratively for settling in the transitional regime.October 13, 2004 19Dredge Pumps and Slurry Transport
20. 20. Terminal Settling Velocity of SphereIn the laminar regime (obeying the Stokes law, Rep < 0.1, i.e.sand-density particles of d < 0.05 mm approximately) CD = 24/Rep, so that vts = (ρ s − ρf ) gd 2 18 µfIn the turbulent regime (obeying the Newtons law, Rep >500, i.e. sand-density particles of d > 2 mm approximately) CD = 0.445, and v ts = 1.73 (ρ s − ρf ) gd ρfOctober 13, 2004 20Dredge Pumps and Slurry Transport
21. 21. Terminal Settling Velocity of SphereThe two regimes are connected via the transition regime, CD = fn(Rep).The determination of vts requires an iteration.Grace (1986) proposed a method for adetermination of vts without necessityto iterate. The Grace method uses twodimensionless parameters ρ f ( ρs − ρ f ) g d = d. * 3 µ2 f ρ2 f v ts = v ts 3 * µ f (ρs − ρ f ) g October 13, 2004 21 Dredge Pumps and Slurry Transport
22. 22. Terminal Settling Velocity of non-S ParticleThe non-spherical shape of a particle reduces its settling vtvelocity. This can be quantified by the velocity ratio called theshape factor. ξ = vtsThe shape factor is a function of :•the volumetric form factor k (k=0.26 for sand, gravel) ρf ( ρs − ρf ) g•the dimensionless particle diameter, d* d* = d.3 µ2 f The terminal velocity for sand particles is typically 50-60 % of the value for the sphere of the equivalent diameter. October 13, 2004 22 Dredge Pumps and Slurry Transport
23. 23. Terminal Settling Velocity of Sand ParticleIn the laminar regime (sand particles smaller than 0.1 mm) the Stokes equation : v = 424 (S s − Sf ) d2 t SfIn the transition regime (0.1 mm<d< 1 mm) the Budryck equation : 8.925 vt = 1 + 95 (S s − Sf ) d 3 −1 d SfIn the turbulent regime (sand particles larger than 1 mm) the Rittinger equation : v t = 87 (S s −Sf )d SfRemark: input d in [mm], output vt in [mm/s]. October 13, 2004 23 Dredge Pumps and Slurry Transport
24. 24. Terminal Settling Velocity of Sand Particle 0,1 valsnelheid in water v (mm/s) 0,2 0,3 0,4 0,5 0,6 Terminal settling velocity 0,7 0,8 0,9 1,0 of sand & gravel 2,0 klasse 1 particles using Stokes, 4,0 3,0 Budryck and Rittinger 5,0 6,0 7,0 v= . 8925 d ⋅ 1 − 95 ⋅ ( 2.65 − 1) ⋅ d3 − 1 [Budryck] equations. 8,0 9,0 10,0 v= 87 ⋅ ( 2,6 klasse 2 20,0 5− 1) ⋅ d [Ri 30,0 ttin ger 40,0 ] 50,0 60,0 70,0 80,0 90,0 v= 100,0 klasse 3 424 ⋅ (2 200,0 ,65 − 300,0 1) ⋅ 400,0 d 6 500,0 [ St 600,0 oke 700,0 800,0 s] 900,0 1000,0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 2,00 3,00 4,00 5,00 6,00 7,00 8,00 9,00 10,00 korrelgrootte d (mm) October 13, 2004 24 Dredge Pumps and Slurry Transport
25. 25. Hindered Settling Velocity of ParticleWhen a cloud of solid particles settles in a quiescent liquid additional hinderingeffects influence the settling velocity, vth, of particles in the cloud:•the increased buoyancy due to the presence of other particles at the same verticallevel•the upflow of liquid as it is displaced by the descending particles, and•the increased drag caused by the proximity of particles within the cloud.The hindering effects are strongly dependent on the volumetric concentration ofparticles in the cloud, Cv, and described bythe Richardson & Zaki equation for which the Wallis eq. determines the index m 4.7 (1 + 0.15 Re0.687 ) vth = vt (1 − Cv ) m p m= 1 + 0.253Re0.687 p October 13, 2004 25 Dredge Pumps and Slurry Transport
26. 26. SOLID PARTICLE IN FLOWING LIQUID Particle – liquid interaction: Hydrodynamic lift Turbulent dispersion Particle – particle interaction: Permanent contact Sporadic contact October 13, 2004 26 Dredge Pumps and Slurry Transport
27. 27. Particle-Liquid Interaction: LiftThe lift force, FL, on a solid particle is a product of simultaneous slip (given byrelative velocity vr = vf - vs) and particle rotation. The velocity differentialbetween liquid velocities above and below the particle produces a pressuredifferential in the vertical direction over the particle and thus the vertical force. A. Magnus lift due to external rotation B. B. Saffman lift due to velocity gradient October 13, 2004 27 Dredge Pumps and Slurry Transport
28. 28. Particle-Liquid Interaction: LiftThe Saffman lift force: F Saff = 1,61⋅ µ f ⋅ D ⋅ ur ⋅ ReG ρ f D2 du with the shear Reynolds number: ReG = ⋅ µ f dy The Magnus lift force:   1  ur ×ωr  FMag = ⋅ ρ f ⋅ ur ⋅ CLR ⋅ A⋅   2 1  ωp − ∇× u f   2  with the lift coefficient: Saffman lift Magnus lift D ⋅ ωp CLR = ur October 13, 2004 28 Dredge Pumps and Slurry Transport
29. 29. Lift Application: Initiation of Sediment MotionDriving forces: Resisting forces: •Drag •Particle weight (gravity) •Buoyancy •Grain packing •Lift Lift force •Downslope weight Drag force Submerged weightOctober 13, 2004 29Dredge Pumps and Slurry Transport
30. 30. Particle-Liquid Interaction: Turb DispersionAn intensive exchange of momentum and random velocity fluctuations in alldirections are characteristic of the turbulent flow of the carrying liquid in a pipeline.A turbulent eddy is responsible for the transfer of momentum and mass in a liquidflow. The length of the turbulent eddy is called the mixing length.The turbulent fluctuating component v of the liquid velocity v is associated with aturbulent eddy.Turbulent eddies are responsible for solid particle suspension. The ability of acarrying liquid to suspend the particles is determined by- the intensity of liquid turbulence (depends on liquid velocity)- the size of the turbulent eddy (depends on pipe diameter)- the size of the solid particle. October 13, 2004 30 Dredge Pumps and Slurry Transport
31. 31. Particle-Liquid Interaction: Turb Dispersion Turbulent diffusion model of Schmidt and RouseThe model is a balance of upwards and downwards solids fluxes composed of the volumetric settling rates and the diffusion fluxes: v’y + vt cv -(ML/2).dc v/dy ML Reference level cv+(ML/2).dcv/dy v’y - vt October 13, 2004 31 Dredge Pumps and Slurry Transport
32. 32. Particle-Liquid Interaction: Turb Dispersion Turbulent diffusion model of Schmidt and RouseThe model is a balance of upwards and downwards solids fluxes composed of the volumetric settling rates and the diffusion fluxes: The upwards flux per unit area = The downwards flux per unit area 1   ML  dcv  1   ML  dcv  cv +  2  dy  ( v y − vt ) 2    cv − 2  dy  ( v y + vt ) 2    d cv ML gives −ε s = v t .c v where εs = v y dy 2  vt  and the integration provides cv ( y) = Cvb .exp − ( y − yb )   εs  October 13, 2004 32 Dredge Pumps and Slurry Transport
33. 33. Real Turbulent-Suspension ProfilesMedium sand in a 150-mm pipe (horizontal): October 13, 2004 33 Dredge Pumps and Slurry Transport
34. 34. Real Turbulent-Suspension ProfilesMedium sand in a 150-mm pipe (horizontal): October 13, 2004 34 Dredge Pumps and Slurry Transport
35. 35. Particle-Liquid Interaction: Turb Dispersion Turbulent diffusion model modified for hindered settlingThe model is a balance of upwards and downwards solids fluxes composed of the volumetric settling rates and the diffusion fluxes: The upwards flux per unit area = The downwards flux per unit area gives d cv = v th .c v = v t (1 − c v ) .c v ML m −ε s where εs = v y dy 2 and the integration must be carried out numerically (there is no analytical solution). October 13, 2004 35 Dredge Pumps and Slurry Transport
36. 36. Example: Measured concentr’n profile October 13, 2004 36
37. 37. Example: Local solids dispersion coef. October 13, 2004 37
38. 38. Example: Measured concentr’n profile October 13, 2004 38
39. 39. Example: Local solids dispersion coef. October 13, 2004 39
40. 40. Example: Solids dispersion coefficient October 13, 2004 40
41. 41. Particle-Particle Interaction: ContactsSand/gravel particles are transported in dredging pipelines often in aform of a granular bed sliding along a pipeline wall at the bottom of apipeline. A mutual contact between particles within a bed gives arise tointergranular forces (i.e. stresses=force/area) transmitted throughouta bed and via a bed contact with a pipeline wall also to the wall. Flow Bed Pipeline October 13, 2004 41 Dredge Pumps and Slurry Transport
42. 42. Particle-Particle Interaction: ContactsDEM simulation of coarse slurry flow with particles inpermanent contact, the granular bed slides en bloc. October 13, 2004 42 Dredge Pumps and Slurry Transport
43. 43. Particle-Particle Interaction: ContactsThe stress distribution in a granular bodyoccupied by non-cohesive particles incontinuous contact is a product of theweight of grains occupying the body. Theintergranular stress has twocomponents:- an intergranular normal stress and- an intergranular shear stress. October 13, 2004 43 Dredge Pumps and Slurry Transport
44. 44. Particle-Particle Interaction: Contacts The intergranular stress has two components: - intergranular normal stress and - intergranular shear stress. According to Coulombs law these two stresses are related by the coefficient of friction. Du Boys τs τs (1879) applied Coulombs law totanφ = = sheared river beds. He related the σs ρf g( Ss −1) CvbHs normal stress and shear stress at the bottom of a flowing bed by the internal-friction coefficient (see eq.) October 13, 2004 44 Dredge Pumps and Slurry Transport
45. 45. Particle-Particle Interaction: CollisionsColliding particles in shear flow exercise also intergranularnormal and shear stresses. October 13, 2004 45 Dredge Pumps and Slurry Transport
46. 46. Particle-Particle Interaction: CollisionsDEM simulation of coarse slurry flow with colliding particles. October 13, 2004 46 Dredge Pumps and Slurry Transport
47. 47. Particle-Particle Interaction: Collisions The normal and shear stresses in a granular body experiencing the rapid shearing are τ sb related by using the coefficient of dynamic tan φ = friction tanΦ instead of its static equivalent σs tanΦ. Bagnold (1954,1956) measured and described the normal and tangential (shear) stresses in mixture flows at high shear rates (velocity gradients).Bagnolds dispersive force is a product of intergranular collisions ina sheared layer rich in particles. The direction of the force is normal tothe layer boundary on which it is acting. October 13, 2004 47 Dredge Pumps and Slurry Transport
48. 48. Bagnold’s experiment on collisional stress The classical rotational viscometer (see Fig.) was modified: Rotating inner cylinder (RIC), Stationary outer cylinder (SOC). Measured: - Revolutions of RIC (Velocity gradient) - Torque of RIC (Shear stress) - Pressure at RIC wall (Normal stress) October 13, 2004 48 Dredge Pumps and Slurry Transport
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