Lecture 9 – MINE 292 - 2013
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.
Terminal Velocity of Settling Particle
Discrete particle settling rate in a fluid at constant temperature
is given by Newton’s Equation (for turbulent conditions):
where
vt = [(4g(s - )dp) / (3Cd )] 0.5
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
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
vL vL
Re 



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)
Drag Coefficient and Reynolds Number
Cd is determined from Stokes Law which relates
drag to Reynolds Number
Drag Coefficient and Reynolds Number
Cd is determined from Stokes Law which relates
drag to Reynolds Number
Drag Coefficient and Reynolds Number
Cd is determined from Stokes Law which relates
drag to Reynolds Number
Type I Free-Settling Velocity
- Particle Settling in a Laminar (or Quiescent Liquid)
Momentum Balance
fb
fd
fd
dv
m
 f g  fb  fd
dt
where
fg
m = particle mass
v = particle velocity
fg = gravity force
fb = buoyancy force
fd = drag force
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
fd  f g  f b
C Dv 2 w A
fd 
2
 When settling rate remains constant >>> terminal velocity
f g  f b Vg( p  w )
 By combining these equations and solving for v, we get:
 2 g(  p  w )V 
vt  

C

A
D w


0 .5
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:
0 .5




 4 g(  p  w ) d 
vt  

3
C

D w


Also applies to stationary particle with water flowing up or
buoyant particle with water flowing down
Problem is to define CD which depends on the flow 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:
g(  p  w ) d 2
vt 
18 
Laminar Flow
Regime
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:
24
3
CD 

 0.34
Re
Re
Transition Flow
Regime
 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 vt is described
by Newton's Equation :
 g(  p  w ) d 
v t 1.74 


w


0.5
Turbulent Flow
Regime
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
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:
CD 
24 
Re
 Typical values of Θ are:
sand 2.0; coal 2.25; gypsum 4.0; graphite flakes 22.0
Terminal Velocity of Settling Particle
Terminal velocity of a particle is affected by:










Temperature
T
Fluid Density
ρw
Particle Density
ρp
Particle Size
d
Particle Shape
Θ
Degree of Turbulence
Re
Volume fraction of solids
Cv
Solid surface charge and pulp chemistry IEP
Magnetic and electric field strength
Vertical velocity of fluid








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 temperature
but 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.
Temperature Effect on Settling
• van der Waals Attraction Forces do not vary with temperature
but Repulsive Forces are temperature dependent.
Potential energy V(R) between two identical particles
versus separation distance (R).
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
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 = 3x10-3 cm/sec
S.G. = 1.10; vs = 8x10-4 cm/sec
S.G. = 1.01; vs = 7x10-5 cm/sec
Terminal Velocity of Settling Particle
potash,
soda ash
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.
Terminal Velocity under
Hindered Settling Conditions
McGhee, T.J., 1991. Water Resources and Environmental Engineering. Sixth Edition. McGraw-Hill, New York.
Relationship between Cv and Weight%
Wt% = 100ρCv / (ρCv + 1 - Cv)
Cv = ρWt% / (ρWt% + 100 – Wt%)
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
Design of Settling Vessels
Ideal Rectangular Settling Vessel
Side view
Settling pond or waste water treatment vessel
Ideal Rectangular Settling Vessel
Model Assumptions
1. Homogeneous feed at low Cv
2. Distributed uniformly over tank cross-sectional area
3. Liquid in settling zone moves in plug flow at constant velocity
4. Particles settle according to Type I settling (free settling)
5. Particles small enough to settle according to Stoke's Law
6. When particles enter sludge region, they exit suspension
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%
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
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
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
The Significance of “H”
Side view
Vx
Vo
The value of H can be used to estimate the fractional
recovery of particles with a settling velocity below vo
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:
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:
Fraction of particles
with a velocity below vs
Cumulative Distribution Curve
for Particle Velocities
settling velocity vs (mm/sec)
Total Fraction Removed:

Ideal Circular Settling Vessel
Side view
Can apply to a thickener, but at low feed %solids
Ideal Circular Settling Vessel
At any particular time and distance

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:

Sedimentation Thickener/Clarifier
Top view
Side view
Plate or Lamella Thickener/Clarifier
Continuous Thickener (Type III)
Thickener (Type III) Control System
CCD Thickeners
Continuous Thickener (Type III)
Solid Flux Analysis
Continuous Thickener (Type III)
Solid Movement in Thickener
Continuous Thickener (Type III)
Experimental Determination of Solids Settling Velocity
Continuous Thickener (Type III)
Solids Settling Velocity in Hindered Settling
Continuous Thickener (Type III)
Solids Gravity Flux
Continuous Thickener (Type III)
Bulk Velocity
where
ub = bulk velocity of slurry
Qu = volumetric flow rate of thickener underflow
A = Surface area of thickener
Continuous Thickener (Type III)
Total Solids Flux
Continuous Thickener (Type III)
Limiting Solids Flux, GL – Dick’s Method
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
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.,
Mass Balance in a Thickener
Mass Balance for solids in a thickener-clarifier:
Solids in = Solids out (sludge) + Solids out (effluent)
QoXo = QuXu + QeXe
where
QoXo = solids flow into the thickener-clarifier
Very little of the solids exits with the effluent, so:
QoXo ≈ QuXu
Thickener Cross-Sectional Area
At steady-state:
AGL = QuXu ≈ QuXu
and so:
A
Qo X o
GL
where
GL = limiting solids flux at concentration XL
Note that QoXo is generally fixed for a given wastewater stream. So, the greater the value of GL, the
smaller the area A. However, a greater GL implies a
lower Xu which may not be desired.
Typical Settling Test
Thickener Cross-Sectional Area
Talmadge – Fitch Method
- Obtain settling rate data from experiment (determine
interface height of settling solids (H) vs. time (t)
- Construct a 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
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
Thickener Cross-Sectional Area
Talmadge – Fitch Method
Thickener Cross-Sectional Area
Talmadge – Fitch Method
- Obtain one value for cross-sectional area from:
- Compute a second value for the minimum area of the
clarifying section using the free-settling velocity in the
column test.
- Choose the larger of these two values
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 (t/m3)
Vm = settling velocity (m/hr)
dw/dt = dry solids production rate (t/hr)
Thickener Depth and Residence 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 the impact of non-ideal behaviour
Type II Settling (flocculant)
- Coalescence occurs during settling (large particles with
high velocities overtake smaller ones with low velocities)
- Collision frequency proportional to solids concentration
and turbulence in feed well (too high promotes break-up)
- Cumulative number of collisions increases with time
- Agglomerates have higher settling velocities
- Rate of particle settling increases with time
- Longer residence times (deep tanks) promote coalescence
- Fractional removal  O/F rate and residence time
- With Type I settling, only O/F rate is determinant
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)
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
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
Hindered Zone Settling (Type III)
- High solids value leads to significant particle interactions
- Cohesive forces are so strong that particle movement is
restricted
- Particles settle together establishing a distinct interface
between clarified fluid and settling particles
Compression Settling (Type IV)
- Very high solids density so particles provide mechanical
support for those above
- particles undergo mechanical compression as they settle
- Type IV settling is extremely slow (measured in days)
- Formula describing compression settling:
Ht  H  ( Ho  H  )e
m( t t o )
where: Ht = height of solids at time t
H = height of solids at infinite time
Ho = height of solids at time to
m = slope of ln[(Ht-H)/(Ho-H)] vs. t