oe4625 Dredge Pumps and Slurry Transport  Vaclav Matousek  October 13, 2004                                      1  Dredge...
1. BASIC PRINCIPLES OF FLOW IN PIPESOLID PARTICLES IN QUIESCENT LIQUIDSOLID PARTICLES IN FLOWING LIQUID October 13, 2004  ...
PARTICLES IN LIQUID                         BUOYANCY                           DRAG                            LIFT       ...
SOLID PARTICLE IN QUIESCENT LIQUID             Terminal settling velocity of sphereTerminal settling velocity of non-spher...
Terminal Settling Velocity of SphereForces acting on a solid spherical particle submerged in aquiescent water column:     ...
Terminal Settling Velocity: Buoyancy ForceExample: The hydrostatic force acts on the top and the  bottom of a solid cylind...
Drag Force                                    The drag force is a product of                                      the pres...
Terminal Settling Velocity: Drag ForceThe pattern of the flow round a particle (sphere) is  characterized by developments ...
Terminal Settling Velocity: Drag ForceDrag acting on a solid particle (sphere) depends on a  development of flow in the bo...
Terminal Settling Velocity: Drag Force                                    Separation of flow from                         ...
Terminal Settling Velocity: Drag ForceThe dimensional analysis of FD = fn(ρf, vts, µf, d) provides  two dimensionless grou...
Terminal Settling Velocity: Drag ForceThe relationship CD = fn(Rep) is determined experimentally:  vts for a spherical par...
Terminal Settling Velocity: Drag Force                                    Laminar regime: Rep<1                           ...
Terminal Settling Velocity: Drag Force                                    Transitional regime: 1000 > Rep>1:              ...
Terminal Settling Velocity: Drag Force                                    Inertial regime: 3x105 > Rep>103:               ...
Terminal Settling Velocity: Drag Force                                         Critical Rep ˜ 3 x 105 :                   ...
Terminal Settling Velocity: Drag ForceStreamlined bodies are so designed that the separation point occurs as far down-    ...
October 13, 2004   18
Terminal Settling Velocity of SphereThe balance of the gravitational, buyoancy and drag forces          π d3              ...
Terminal Settling Velocity of SphereIn the laminar regime (obeying the Stokes law, Rep < 0.1, i.e.sand-density particles o...
Terminal Settling Velocity of SphereThe two regimes are connected via the transition regime, CD = fn(Rep).The determinatio...
Terminal Settling Velocity of non-S ParticleThe non-spherical shape of a particle reduces its settling                    ...
Terminal Settling Velocity of Sand ParticleIn the laminar regime (sand particles smaller   than 0.1 mm) the Stokes equatio...
Terminal Settling Velocity of Sand Particle                                        0,1        valsnelheid in water v (mm/s...
Hindered Settling Velocity of ParticleWhen a cloud of solid particles settles in a quiescent liquid additional hinderingef...
SOLID PARTICLE IN FLOWING LIQUID                    Particle – liquid interaction:                          Hydrodynamic l...
Particle-Liquid Interaction: LiftThe lift force, FL, on a solid particle is a product of simultaneous slip (given byrelati...
Particle-Liquid Interaction: LiftThe Saffman lift force: F                                        Saff   = 1,61⋅ µ f ⋅ D ⋅...
Lift Application: Initiation of Sediment                 MotionDriving forces:                     Resisting forces:    •D...
Particle-Liquid Interaction: Turb DispersionAn intensive exchange of momentum and random velocity fluctuations in alldirec...
Particle-Liquid Interaction: Turb Dispersion             Turbulent diffusion model of Schmidt and RouseThe model is a bala...
Particle-Liquid Interaction: Turb Dispersion             Turbulent diffusion model of Schmidt and RouseThe model is a bala...
Real Turbulent-Suspension ProfilesMedium sand in a 150-mm pipe (horizontal):   October 13, 2004                          3...
Real Turbulent-Suspension ProfilesMedium sand in a 150-mm pipe (horizontal):   October 13, 2004                          3...
Particle-Liquid Interaction: Turb Dispersion       Turbulent diffusion model modified for hindered settlingThe model is a ...
Example: Measured concentr’n profile  October 13, 2004               36
Example: Local solids dispersion coef.  October 13, 2004                 37
Example: Measured concentr’n profile  October 13, 2004               38
Example: Local solids dispersion coef.  October 13, 2004                 39
Example: Solids dispersion coefficient  October 13, 2004                 40
Particle-Particle Interaction: ContactsSand/gravel particles are transported in dredging pipelines often in aform of a gra...
Particle-Particle Interaction: ContactsDEM simulation of coarse slurry flow with particles inpermanent contact, the granul...
Particle-Particle Interaction: ContactsThe stress distribution in a granular bodyoccupied by non-cohesive particles incont...
Particle-Particle Interaction: Contacts                                      The intergranular stress has two             ...
Particle-Particle Interaction: CollisionsColliding particles in shear flow exercise also intergranularnormal and shear str...
Particle-Particle Interaction: CollisionsDEM simulation of coarse slurry flow with colliding particles.  October 13, 2004 ...
Particle-Particle Interaction: Collisions                                      The normal and shear stresses in a granular...
Bagnold’s experiment on collisional stress                                          The classical rotational              ...
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Oe4625 _lecture_01_b

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Oe4625 _lecture_01_b

  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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