Stoke’s Law and Settling Particles

1. Stoke’s Law and Settling Particles Lecture 9 – MINE 292 - 2013

2. Terminal Velocity of Settling Particle 1. Coarse particles settle faster than fines of the same S.G. 2. High-density particles settle faster than low-density ones of the same size. 3. When pulp density is below 2% solids, free-settling behaviour occurs and settling is much faster than when pulp density is high (Hindered settling). These terms also apply to centrifugal classification in hydrocyclones and centrifuges. 4. Particle shape has a major effect on settling - a sphere settles much faster than a cube of the same relative size and S.G. Some particles are shaped like flakes - these settle much slower than a particle of same relative size and S.G.

3. Terminal Velocity of Settling Particle Discrete particle settling rate in a fluid at constant temperature is given by Newton’s Equation (for turbulent conditions): vt = [(4g(s - )dp) / (3Cd )] 0.5 where vt= terminal settling velocity (m/s) g= gravitational constant (m/s2) s=density of the particle (kg/m3) =density of the fluid (kg/m3) dp= particle diameter (m) Cd= Drag Coefficient (dimensionless) Terminal settling velocity derives from balancing drag, buoyant, and gravitational forces that act on the particle

4. Reynolds Number Fluid Mechanics term Reynolds Number, Re (or NR), is a dimensionless number Ratio of inertial forces to Viscous forces Re describes relative importance of these two forces for a given set of flow conditions where: v = mean velocity of an object relative to a fluid (m/s) L = characteristic dimension, (fluid length; particle diameter) (m) μ = dynamic viscosity of fluid (kg/(m·s)) ν = kinematic viscosity (ν = μ/ρ) (m²/s) ρ = fluid density (kg/m³)

5. Drag Coefficient and Reynolds Number Cd is determined from Stokes Law which relates drag to Reynolds Number

6. Drag Coefficient and Reynolds Number Cd is determined from Stokes Law which relates drag to Reynolds Number

7. Drag Coefficient and Reynolds Number Type I Free-Settling Velocity Cd is determined from Stokes Law which relates drag to Reynolds Number - Particle Settling in a Laminar (or Quiescent Liquid) Momentum Balance fb fd fd where m = particle mass v = particle velocity fg= gravity force fb = buoyancy force fd = drag force fg

8. Drag Coefficient and Reynolds Number Cd is determined from Stokes Law which relates drag to Reynolds Number • Particle concentration (pulp density) is very small (< 0.1%weight) • Discrete spherical particle will act unhindered by other particles • Velocity will increase until drag force = effective weight of particle • When settling rate remains constant >>> terminal velocity • By combining these equations and solving for v, we get:

9. Drag Coefficient and Reynolds Number Cd is determined from Stokes Law which relates drag to Reynolds Number • Particle is assumed solid and spherical with diameter d, so: • Also applies to stationary particle with water flowing up or buoyant particle with water flowing down • Problem is to define CDwhich depends on theflow regime • This requires knowledge about Reynold's Number • For Region 'a' (10-4 < Re < 0.2) in the previous diagram, CD = 24/Re, and we obtain the famous Stokes' Law: Laminar Flow Regime

10. Drag Coefficient and Reynolds Number Cd is determined from Stokes Law which relates drag to Reynolds Number • For Region 'b' (0.2 < Re < 1,000), the generally accepted relationship for Re is: • For Region 'c' (1,000 < Re < 2 x 105), flow regime is now turbulent and the value of CD is essentially constant at 0.44, so vtis described by Newton's Equation : Transition Flow Regime Turbulent Flow Regime

11. Drag Coefficient and Reynolds Number Cd is determined from Stokes Law which relates drag to Reynolds Number • For Region 'd' (Re > 2 x 105), drag decreases considerably • Turbulent conditions enter the boundary layer around the particle • The drag coefficient, CD, declines to 0.1 • This regime can speed up reactions between liquids and solids • The regime is unlikely to be found in thickeners or clarifiers • However, what happens in the boundary layer affects how particles interact with the fluid and settle in the fluid

12. Drag Coefficient and Reynolds Number Cd is determined from Stokes Law which relates drag to Reynolds Number • Particle shape has a major influence on terminal velocity • A non-spherical shape increase the value of CD at any Re value • Settling velocity is lower than a sphere of equal volume and density • A general shape factor is often determined and used as follows: • Typical values of Θ are: sand 2.0; coal 2.25; gypsum 4.0; graphite flakes 22.0

13. Terminal Velocity of Settling Particle Terminal velocity of a particle is affected by: • Temperature T ü • Fluid Density ρwü • Particle Density ρpü • Particle Size d ü • Particle Shape Θü • Degree of Turbulence Reü • Volume fraction of solids Cvü • Solid surface charge and pulp chemistry IEPü • Magnetic and electric field strength • Vertical velocity of fluid

14. Temperature Effect on Settling • Owen* in 1972 concluded temperature affects velocity through viscosity changes in accord with Stokes Law (free-settling) • So it is common to correct for temperature assuming that settling velocity is inversely proportional to kinematic viscosity. • However, sediment deposition also depends on Aggregation • Turbulence and shear stresses play major roles in helping particles come together to settle more quickly • Weak flocs may break-up near the bed due to higher shear stresses at higher temperature and so particles are re-entrained • van der Waals Attraction Forces do not vary with temperaturebut Repulsive Forces are temperature dependent * M.W. Owen, 1972. "Effect of Temperature on the Settling Velocities of an Estuary Mud", Report No. INT 106, Hydraulics Research Station, Wallingford.

15. Temperature Effect on Settling • van der Waals Attraction Forces do not vary with temperaturebut Repulsive Forces are temperature dependent. Potential energy V(R) between two identical particles versus separation distance (R).

16. Temperature Effect on Settling • Lau, 1994* shows results exactly opposite to Owen • For kaolinite in distilled water, velocity increases by ~30% as temperature drops from 26 to 5°C • Stoke's Law suggests viscosity should decrease velocity by ~32% Kaolinite (5 µm) – Distilled Water * Y. L. Lau , 1994. "Temperature effect on settling velocity and deposition of cohesive sediments", Journal of Hydraulic Research, 32:1, 41-51

17. Terminal Velocity of Settling Particle Type I Settling of Spheres For a density of 2.65: Diameter of 120 µm has a velocity of 1.0 cm/sec Diameter of 10 µm has a velocity of 0.01 cm/sec For a diameter of 10 µm S.G. = 1.50; vs = 3x10-3cm/sec S.G. = 1.10; vs = 8x10-4cm/sec S.G. = 1.01; vs = 7x10-5 cm/sec

18. Terminal Velocity of Settling Particle potash, soda ash

19. Terminal Velocity under Hindered Settling Conditions McGhee’s (1991) equation estimates velocity for spherical particles under hindered settling conditions for Re ≈0.3: Vh/V = (1 - Cv)4.65 where Vh = hindered settling velocity V= free settling velocity Cv = volume fraction of solid particles For Re≈1,000, the exponent decreases to 2.33 So, exponent = 4.65 - 0.00232Re McGhee, T.J., 1991. Water Resources and Environmental Engineering. 6th Edition. McGraw-Hill, New York.

20. Terminal Velocity under Hindered Settling Conditions McGhee, T.J., 1991. Water Resources and Environmental Engineering. Sixth Edition. McGraw-Hill, New York.

21. Relationship between Cv and Weight% Wt% = 100ρCv / (ρCv + 1 - Cv) Cv = ρWt% / (ρWt% + 100 – Wt%)

22. Effect of Alum on IEP - Coagulation • Surface Charge Issues • Like-charged particles repel one another • Particles with no surface charge remain dispersed in suspension • Chemicals in solution affect surface charge of mineral particles • IEP = iso-electric point • pH is potential-determining

23. Design of Settling Vessels

24. Ideal Rectangular Settling Vessel Side view Settling pond or waste water treatment vessel

25. Ideal Rectangular Settling Vessel Model Assumptions • Homogeneous feed at low Cv • Distributed uniformly over tank cross-sectional area • Liquid in settling zone moves in plug flow at constant velocity • Particles settle according to Type I settling (free settling) • Particles small enough to settle according to Stoke's Law • When particles enter sludge region, they exit suspension

26. Ideal Rectangular Settling Vessel Side view Vs Vo u = average (constant) velocity of fluid flowing across vessel vs = settling velocity of a particular particle vo = critical settling velocity of finest particle recovered at 100%

27. Retention Time Average time spent in the vessel by an element of the suspension and W, H, L are the vessel dimensions; u is the constant velocity

28. Critical Settling Velocity If to is the residence time of liquid in the tank, then all particles with a settling velocity equal to or greater than the critical settling velocity, vo, will be settled out at or prior to to, i.e., So all particles with a settling velocity equal to or greater than v0(i.e., coarser in size than do) will be separated from the fluid in the tank

29. Critical Settling Velocity Since Note: this expression for vo has no H term. This defines the Overflow Rate or Surface-Loading Rate - Key parameter to design ideal Type I settling clarifiers - Cross-sectional area A is calculated to get desired v0

30. The Significance of “H” Side view Vo Vx The value of H can be used to estimate the fractional recovery of particles with a settling velocity below vo

31. The Significance of “H” Only a fraction of particles with a settling velocity vx (less than vo) will settle out. The fraction Fx of particles of size dx (with velocity vx) that settle out is:

32. The Significance of “H” Only a fraction of particles with a settling velocity vx (less than vo) will settle out. The fraction Fx of particles of size dx (with velocity vx) that settle out is:

33. Cumulative Distribution Curve for Particle Velocities Fraction of particles with a velocity below vs settling velocity vs (mm/sec) ò Total Fraction Removed:

34. Ideal Circular Settling Vessel Side view Can apply to a thickener, but at low feed %solids

35. Ideal Circular Settling Vessel At any particular time and distance ò

36. Ideal Circular Settling Vessel In an interval dt, a particle having a diameter below do will have moved vertically and horizontally as follows: For particles with a diameter dx (below do), the fractional removal is given by: ò

37. Sedimentation Thickener/Clarifier Top view Side view

38. Plate or Lamella Thickener/Clarifier

39. Continuous Thickener (Type III)

40. Thickener (Type III) Control System

41. Continuous Thickener (Type III) Solid Flux Analysis

42. Continuous Thickener (Type III) Solid Movement in Thickener

43. Continuous Thickener (Type III) Experimental Determination of Solids Settling Velocity

44. Continuous Thickener (Type III) Solids Settling Velocity in Hindered Settling

45. Continuous Thickener (Type III) Solids Gravity Flux

46. Continuous Thickener (Type III) Bulk Velocity where ub = bulk velocity of slurry Qu = volumetric flow rate of thickener underflow A = Surface area of thickener

47. Continuous Thickener (Type III) Total Solids Flux

48. Continuous Thickener (Type III) Limiting Solids Flux, GL – Dick’s Method

49. Continuous Thickener (Type III) Limiting Solids Flux, GL – Dick’s Method - In hindered settling zone, solids concentration ranges from feed concentration to underflow concentration Xu - Within this range, a concentration exists that gives smallest (or limiting) value, GL, of the solid flux G - If thickener is designed for a G value such that G > GL, solids will build-up in the clarifying zone and overflow

50. Continuous Thickener (Type III) Limiting Solids Flux, GL – Dick’s Method - The point where the total gravity flux curve is minimum gives GL and XL - GL is highest flux allowed in the thickener - At bottom of thickener, there is no gravity flux as all solid material is removed via bulk flux, i.e.,