Your SlideShare is downloading. ×
4b. Modeling of stratified mixture flows
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

4b. Modeling of stratified mixture flows

1,161
views

Published on

Published in: Technology, Business

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
1,161
On Slideshare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
20
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. oe4625 Dredge Pumps and Slurry Transport Vaclav Matousek October 13, 2004 1 Dredge Pumps and Slurry Transport Vermelding onderdeel organisatie
  • 2. 4. MODELING OF STRATIFIED MIXTURE FLOWS (Heterogeneous Flows) EMPIRICAL MODELING THEORETICAL MODELING October 13, 2004 2 Dredge Pumps and Slurry Transport
  • 3. THEORETICAL MODELING MACROSCOPIC (Large Control Volume) MICROSCOPIC (Infinitesimal Control Volume)October 13, 2004 3Dredge Pumps and Slurry Transport
  • 4. MACROSCOPIC MODEL A TWO-LAYER MODEL FLOW COMPOSED OF TWO LAYERS EACH LAYER = CONTROL VOLUME CONSERVATION LAWS FOR EACH LAYER: Conservation of mass Conservation of momentumOctober 13, 2004 4Dredge Pumps and Slurry Transport
  • 5. Stratified flows Example of stratified flowOctober 13, 2004 5Dredge Pumps and Slurry Transport
  • 6. 2LM: Flow PatternA. Simplified Flow Pattern October 13, 2004 6 Dredge Pumps and Slurry Transport
  • 7. 2LM: Flow PatternA. Real Flow Pattern October 13, 2004 7 Dredge Pumps and Slurry Transport
  • 8. 2LM: Flow PatternA. Real Flow Pattern October 13, 2004 8 Dredge Pumps and Slurry Transport
  • 9. 2LM: Flow PatternA. Real Flow Pattern October 13, 2004 9 Dredge Pumps and Slurry Transport
  • 10. 2LM: Flow Pattern Fully - stratified flow Partially - stratified flow C1=0 A1 C1 C2 A2 C2c Particle-free layer Suspended load Contact load Figure: Schematic cross-section for two-layer model.October 13, 2004 10Dredge Pumps and Slurry Transport
  • 11. 2LM: PrinciplesA. Simplified Flow Pattern- STRATIFICATION.B. Particle Support Mechanism- the CONTACT- the SUSPENSION = NO CONTACT. October 13, 2004 11 Dredge Pumps and Slurry Transport
  • 12. 2LM: PrinciplesA. Simplified Flow Pattern- the real/virtual interface at the top of a contact layer- the homogeneous distribution of velocity (V) and concentration (C) within each layer (C1, C2, V1, V2)- no slip between phases within a layer October 13, 2004 12 Dredge Pumps and Slurry Transport
  • 13. 2LM: PrinciplesB. Particle Support MechanismINTERGRANULAR CONTACT & PARTICLE SUSPENSION (contact load) (suspended load)Contacts: continuous (Coulombic contacts within a stationary or sliding granular bed)or sporadic (collisions within a transition shear layer) October 13, 2004 13 Dredge Pumps and Slurry Transport
  • 14. 2LM: PrinciplesB. Particle Support MechanismMECHANISMS FOR SOLID PARTICLE SUSPENSIONDiffusive effect of carrier turbulence (no interparticle contacts within suspended layer)Dispersive effect of repulsion forces due to interparticle collisions (within shear layer) October 13, 2004 14 Dredge Pumps and Slurry Transport
  • 15. 2LM: Flow PatternA. Simplified Flow Pattern Fully - stratified flow Partially - stratified flow V1 C1=0 A1 C1 V1 V2 C2 A2 C2 V2 Particle-free layer Suspended layer Contact layer October 13, 2004 15 Dredge Pumps and Slurry Transport
  • 16. 2LM: Flow Pattern Fully - stratified flow Partially - stratified flow O1 Particle-free layer Suspended layer A1 Contact layer O12 β A2 β Position of an interface: β Perimeters: O1, O2, O12 Areas: A1, A2 O2 Figure: Geometry of schematic cross-section.October 13, 2004 16Dredge Pumps and Slurry Transport
  • 17. 2LM: Concentration distributionA. Simplified Flow PatternSolids fractions in both layers: Cvi.A = C1.A1 + C2.A2Solids fraction in contact layer: Cc.A = C2.A2Solids fraction in suspension layer: Cs.A = C1.A1 October 13, 2004 17 Dredge Pumps and Slurry Transport
  • 18. 2LM: Conservation of massA. Continuity equationsSlurry flow rate: A.Vm = A1.V1 + A2.V2Solids flow rate: As.Vs = Cvi.A.Vs Cvi.A.Vs = Cvd.A.Vm Cvd.A.Vm = C1.A1.V1 + C2.A2.V2 October 13, 2004 18 Dredge Pumps and Slurry Transport
  • 19. Flow-Stratification ParameterThe overall relationship(for both turbulent suspension and shearing action) Cc  Vm  = exp  −Coeff Cvi  vt  Coeff = const = 0.018 (Gillies et al, 1991) 0.024 (D=150 mm; Matousek, 1997) 0.0212 (Gillies and Shook, 2000) October 13, 2004 19 Dredge Pumps and Slurry Transport
  • 20. Repetition: Conservation of Momentum in 1D-flowFor- 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 20 Dredge Pumps and Slurry Transport
  • 21. Repetition: C. of Momentum in 1D-flow P − P2 4τ oFor a straight horizontal circular pipe 1 = L D P1 P2 t0 V D t0 L October 13, 2004 21 Dredge Pumps and Slurry Transport
  • 22. 2LM: Flow PatternA. Simplified Flow Pattern Fully-stratified flow Heterogeneous flow O1 A1 O 12 A2 β β O2 October 13, 2004 22 Dredge Pumps and Slurry Transport
  • 23. 2LM: Conservation of momentumA. Force-balance equations (for the unit length L of a pipe)Upper layer: dP.A1 = τ1.O1 + τ12.012Lower layer: dP.A2 = - τ12.012 + τ2.02. τ2.O2 = τ2f.O2 + τ2s.O2 and τ2s.O2 = µs.FN.τ2f is the shear stress due to flow at a pipe wall of perimeter O2 (velocity- dependent viscous friction)τ2s is the shear stress due to sliding at a pipe wall of the solids occupying a contact layer (velocity-independent mechanical friction). October 13, 2004 23 Dredge Pumps and Slurry Transport
  • 24. 2LM: ApplicationsFully Stratified FlowPrediction of frictional pressure drop (hydraulic gradient) October 13, 2004 24 Dredge Pumps and Slurry Transport
  • 25. 2LM: ApplicationsFully Stratified Flow Relative excess pressure gradient:Im − I f Im − I f = I pg 2µs ( Ss − S f ) Cvb Relative velocity: Vm Vr = VsmRelative concentration: Cvd Cr = Cvb October 13, 2004 25 Dredge Pumps and Slurry Transport
  • 26. 2LM: ApplicationsFully Stratified Flow The result ofprediction of frictional pressure drop (hydraulic gradient) October 13, 2004 26 Dredge Pumps and Slurry Transport
  • 27. 2LM: Applications Fully & Partially Stratified Flows Prediction ofthe maximum velocity at the limit of stationary deposition (the demi-McDonald’s diagram) October 13, 2004 27 Dredge Pumps and Slurry Transport
  • 28. 2LM:Applications Max velocity at limit of stationary deposit: Vsm = fn(d,D)October 13, 2004 28Dredge Pumps and Slurry Transport
  • 29. 2LM: Applications Vsm = fn(d,D,Ss)October 13, 2004 29Dredge Pumps and Slurry Transport
  • 30. 2LM: ApplicationsPartially Stratified Flows Prediction ofthe deposition-limit velocity the hydraulic gradient the thickness of the bed the velocity of the bed the slip ratioOctober 13, 2004 30Dredge Pumps and Slurry Transport
  • 31. 2LM: ApplicationsPartially Stratified Flows EXAMPLEOctober 13, 2004 31Dredge Pumps and Slurry Transport
  • 32. Example: ExperimentsMeasurements in the 150-mm pipe: Pressure drop Mean velocity of slurry Mean concentration of solids Concentration distribution. October 13, 2004 32 Dredge Pumps and Slurry Transport
  • 33. Example: Modeling of suspensionThe concentration gradient the Schmidt-Rouse model with the implemented hindered settling effect dcv  λ  = vt (1 − cv ) cv ε s = fn  D, u* = m εs Vm  dy 8  The hydraulic gradient the two-layer model with the stratification-ratio equation October 13, 2004 33 Dredge Pumps and Slurry Transport
  • 34. Example: Measured concentr’n profile October 13, 2004 34
  • 35. Example: Local solids dispersion coeff. October 13, 2004 35
  • 36. Example: Measured concentr’n profile October 13, 2004 36
  • 37. Example: Local solids dispersion coeff. October 13, 2004 37
  • 38. Example: Solids dispersion coefficient October 13, 2004 38
  • 39. Example: Solids dispersion coefficient October 13, 2004 39
  • 40. Example: Construction of simplifiedprofile Inputs: • Measured concentration profile • Measured mean concentration Cvi. Outputs: • The value of the solids dispersion coefficient • The position of the interface between two layers • The mean concentration of solids in the bed • The stratification-ratio value. October 13, 2004 40 Dredge Pumps and Slurry Transport
  • 41. Example: Stratification evaluated Position of the interface: • The position at which the concentration profile of turbulent suspension is linked to the granular bed. Mean concentration in the bed: • The mean concentration tend to vary slightly with Cvi and Vm. Stratification ratio: • The portions of solids that contribute to contact or suspended loads. October 13, 2004 41 Dredge Pumps and Slurry Transport
  • 42. Medium sand: concentr. profile October 13, 2004 42
  • 43. Medium sand: concentr. profile October 13, 2004 43
  • 44. Medium sand: concentr. profile October 13, 2004 44
  • 45. Medium sand: concentr. profile October 13, 2004 45
  • 46. Hydraulic gradient Inputs to the two-layer model: • Parameters of simplified concentration profile • Measured mean velocity (Vm) • Friction coefficients (µs, λ1f , λ2f , λ12). Outputs: • The value of the hydraulic gradient • The value of the slip ratio. October 13, 2004 46 Dredge Pumps and Slurry Transport
  • 47. Medium sand: hydraulic gradient October 13, 2004 47
  • 48. Medium sand: hydraulic gradient October 13, 2004 48
  • 49. Example: Conclusions• The concentration gradients in slurry flows of fine to medium sands in a 150-mm pipe are due dispersive action of carrier turbulence.• The concentration gradients can be predicted using the Schmidt-Rouse turbulent diffusion model with the implemented hindered settling effect. The dispersion coefficient can be considered constant across the suspension flow. October 13, 2004 49 Dredge Pumps and Slurry Transport
  • 50. Example: Conclusions• The concentration gradient can be used for the determination of the simplified concentration profile in the two-layer flow pattern.• The hydraulic gradient determined using the two-layer model fits reasonably the measured value. October 13, 2004 50 Dredge Pumps and Slurry Transport

×