The document discusses terminal velocity and settling of particles in fluids. It defines the equation for terminal settling velocity as a function of gravitational constant, particle and fluid densities, particle diameter, and drag coefficient. The drag coefficient is related to the particle Reynolds number, which quantifies the ratio of inertial to viscous forces on the particle. Ideal rectangular and circular settling vessels are examined in terms of retention time, critical settling velocity, and fractional particle removal. Continuous thickeners are also discussed, including solid flux analysis, mass balances, and methods for determining cross-sectional area based on underflow concentration and settling tests.

1. Lecture 12 – MINE 292 - 2012

2. Terminal Velocity of Settling Particle
Rate at which discrete particles settle in a fluid at constant temperature
is given by Newton’s equation:
vs = [(4g(s - )dp) / (3Cd )] 0.5
where
vs = 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)
The terminal settling velocity is derived by balancing drag, buoyant,
and gravitational forces that act on the particle.

3. Reynolds Number
In fluid mechanics, the Reynolds Number, Re (or NR), is a dimensionless
number that is the ratio of inertial forces to viscous forces.
It quantifies the relative importance of these two types of forces for a
given set of flow conditions.
where:
v = mean velocity of an object relative to a fluid (m/s)
L = characteristic dimension, (length of fluid; particle diameter) (m)
μ = dynamic viscosity of fluid (kg/(m·s))
ν = kinematic viscosity (ν = μ/ρ) (m²/s)
ρ = fluid density (kg/m³)

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

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
Cd is determined from Stokes Law which relates drag to Reynolds Number

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

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

10. Terminal Velocity of Settling Particle
Terminal velocity is affected by:
 Temperature
 Fluid Density 
 Particle Density 
 Particle Size 
 Particle Shape
 Degree of Turbulence 
 Volume fraction of solids
 Solid surface charge and pulp chemistry
 Magnetic and electric field strength
 Vertical velocity of fluid

11. Drag Coefficient of Settling Particle

12. Terminal Velocity of Settling Particle

13. Type I Free-Settling Velocity
Particle Settling in a Laminar (or Quiescent Liquid)
Momentum Balance

14. Type I Free-Settling Velocity
Particle Settling in a Laminar (or Quiescent Liquid)

15. Type I Free-Settling Velocity
Integrating gives the steady state solution:
For a sphere:

16. Terminal Velocity of Settling Particle
Type I Settling of Spheres

17. Terminal Velocity of Settling Particle

18. 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 is only 2.33
McGhee, T.J., 1991. Water Resources and Environmental Engineering. Sixth Edition. McGraw-Hill, New York.

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

20. Relationship between Cv and Weight%

21. Effect of Alum on IEP

22. Ideal Rectangular Settling Vessel
Side view

23. Ideal Rectangular Settling Vessel
Model Assumptions
1. Homogeneous feed is distributed uniformly over tank cross-
sectional area
2. Liquid in settling zone moves in plug flow at constant velocity
3. Particles settle according to Type I settling (free settling)
4. Particles are small enough to settle according to Stoke's Law
5. When particles enter sludge region, they exit the suspension

24. Ideal Rectangular Settling Vessel
Side view
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%

25. 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

26. 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 settle out at
or prior to to, i.e.,
So all particles with a settling velocity equal to or greater
than v0 will be separated in the tank from the fluid

27. Critical Settling Velocity
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
Since

28. Ideal Circular Settling Vessel
Side view

29. Ideal Circular Settling Vessel
At any particular time and distance


30. 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:

31. Sedimentation Thickener/Clarifier
Top view
Side view

32. Plate or Lamella Thickener/Clarifier

33. Continuous Thickener (Type III)

34. Thickener (Type III) Control System

35. Continuous Thickener (Type III)
Solid Flux Analysis

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

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

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

39. Continuous Thickener (Type III)
Solids Gravity Flux

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

41. Mass Balance in a Thickener

42. Thickener Cross-Sectional Area

43. Thickener Cross-Sectional Area
Talmadge – Fitch Method

44. Thickener Cross-Sectional Area
Talmadge – Fitch Method
- Obtain settling rate data from experiment (determine
interface height of settling solids (H) vs. time (t)
- Construct curve of H vs. t
- Determine point where hindered settling changes to
compression settling
- intersection of tangents
- construct a bisecting line through this point
- draw tangent to curve where bisecting line intersects the curve

45. Thickener Cross-Sectional Area
Talmadge – Fitch Method
- Draw horizontal line for H = Hu that corresponds to the
underflow concentration Xu, where
- Determine tu by drawing vertical line at point where
horizontal line at Hu intersects the bisecting tangent line

46. Thickener Cross-Sectional Area
Talmadge – Fitch Method
- Obtain cross-sectional area required from:
- Compute the minimum area of the clarifying section
using a particle settling velocity of the settling velocity
at t = 0 in the column test.
- Choose the largest of these two values

47. Thickener Cross-Sectional Area
Coe – Clevenger Method
- Developed in 1916 and still in use today:
where
A = cross-sectional area (m2)
F = feed pulp liquid/solids ratio
L = underflow pulp liquid/solid ratio
ρs = solids density (g/cm3)
Vm = settling velocity (m/hr)
dw/dt = dry solids production rate (g/hr)

48. Thickener Depth and Residece Time
- Equations describing solids settling do not include tank
depth. So it is determined arbitrarily by the designer
- Specifying depth is equivalent to specifying residence
time for a given flow rate and cross-sectional area
- In practice, residence time is of the order of 1-2 hours
to reduce impact of non-ideal behaviour

49. Typical Settling Test

50. Type II Settling (flocculant)
- Coalescence of particles occurs during settling (large
particles with high velocities overtake small particles
with low velocities)
- Collision frequency proportional to solids concentration
- Collision frequency proportional to level of turbulence
(but too high an intensity will promote break-up)
- Cumulative number of collisions increases with time

51. Type II Settling (flocculant)
- Particle agglomerates have higher settling velocities
- Rate of particle settling increases with time
- Longer residence times and deeper tanks promote
coalescence
- Fractional removal is function of overflow rate and
residence time.
- With Type I settling, only overflow rate is significant

52. Primary Thickener Design
- Typical design is for Type II characteristics
- Underflow densities are typically 50-65% solids
- Safety factors are applied to reduce effect of
uncertainties regarding flocculant settling velocities
• 1.5 to 2.0 x calculated retention time
• 0.6 to 0.8 x surface loading (overflow rate)

53. Primary Thickener Design
Non-ideal conditions
• Turbulence
• Hydraulic short-circuiting
• Bottom scouring velocity (re-suspension)
All cause shorter residence time of fluid and/or particles

54. Primary Thickener Design Parameters
Depth (m) 3 - 5 m
Diameter (m) 3 - 170 m
Bottom Slope 0.06 to 0.16 (3.5° to 10°)
Rotation Speed
of rake arm 0.02 - 0.05 rpm

55. Hindered (or Zone) Settling (Type III)
- solids concentration is high such that particle interactions
are significant
- cohesive forces are so strong that movement of particles
is restricted
- particles settle together establishing a distinct interface
between clarified fluid and settling particles

56. Compression Settling (Type IV)
- When solids density is very high, particles provide partial
mechanical support for those above
- particles undergo mechanical compression as they settle
- Type IV settling is extremely slow (measured in days)