Lecture 12
–
MINE 292

2012
Terminal Velocity of Settling Particle
R
ate at which discrete particles settle in a fluid at constant temperature
is given by Newton’s equation:
v
s
= [(4g(
s

)d
p
) / (3C
d
)]
0.5
where
v
s
= terminal settling velocity (m/s)
g
= gravitational constant (m/s
2
)
s
=
density of the particle (kg/m
3
)
=
density of the fluid (kg/m
3
)
d
p
= particle diameter (m)
C
d
= Drag Coefficient (dimensionless)
The terminal settling velocity is derived by balancing drag, buoyant,
and gravitational forces that act on the particle.
Reynolds Number
In fluid mechanics, the Reynolds Number,
Re
(or
N
R
), 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)
Drag Coefficient and Reynolds Number
C
d
is determined from Stokes Law which relates
drag to Reynolds Number
Drag Coefficient and Reynolds Number
C
d
is determined from Stokes Law which relates
drag to Reynolds Number
Drag Coefficient and Reynolds Number
C
d
is determined from Stokes Law which relates
drag to Reynolds Number
Drag Coefficient and Reynolds Number
C
d
is determined from Stokes Law which relates
drag to Reynolds Number
Drag Coefficient and Reynolds Number
C
d
is determined from Stokes Law which relates
drag to Reynolds Number
Drag Coefficient and Reynolds Number
C
d
is determined from Stokes Law which relates
drag to Reynolds Number
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
Drag Coefficient of Settling Particle
Terminal Velocity of Settling Particle
Type I Free

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

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

Settling Velocity
Integrating gives the steady state solution:
For a sphere:
Terminal Velocity of Settling Particle
Type I Settling of Spheres
Terminal Velocity of Settling Particle
Terminal Velocity under
Hindered Settling Conditions
McGhee’s (1991) equation estimates velocity for spherical
particles under hindered settling conditions for Re < 0.3:
V
h
/V = (1

C
v
)
4.65
where
V
h
= hindered settling velocity
V
= free settling velocity
C
v
= 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.
Terminal Velocity under
Hindered Settling Conditions
McGhee, T.J
.,
1991.
Water Resources and Environmental Engineering
. Sixth Edition. McGraw

Hill, New York.
Relationship between C
v
and Weight%
Effect of Alum on IEP
Ideal Rectangular Settling Vessel
Side view
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
Ideal Rectangular Settling Vessel
Side view
u = average (constant) velocity of fluid flowing across vessel
v
s
= settling velocity of a particular particle
v
o
= 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 t
o
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, v
o
, will settle out at
or prior to t
o
, i.e.,
So all particles with a settling velocity equal to or greater
than v
0
will be separated in the tank from the fluid
Critical Settling Velocity
Note: this expression for
v
o
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
v
0
Since
Ideal Circular Settling Vessel
Side view
Ideal Circular Settling Vessel
At any particular time and distance
Ideal Circular Settling Vessel
In an interval
dt
, a particle having a diameter below
d
o
will have moved vertically and horizontally as follows:
For particles with a diameter
d
x
(below
d
o
),
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
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
u
b
= bulk velocity of slurry
Q
u
= volumetric flow rate of thickener underflow
A
= Surface area of thickener
Mass Balance in a Thickener
Thickener Cross

Sectional Area
Thickener Cross

Sectional Area
Talmadge
–
Fitch Method
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
Thickener Cross

Sectional Area
Talmadge
–
Fitch Method

Draw horizontal line for
H
=
H
u
that corresponds to the
underflow concentration
X
u
, where

Determine
t
u
by drawing vertical line at point where
horizontal line at
H
u
intersects the bisecting tangent line
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
Thickener Cross

Sectional Area
Coe
–
Clevenger Method

Developed in 1916 and still in use today:
w
here
A
= cross

sectional area (m
2
)
F
= feed pulp liquid/solids ratio
L
= underflow pulp liquid/solid ratio
ρ
s
= solids density (g/cm
3
)
V
m
= settling velocity (m/hr)
dw
/
dt
= dry solids production rate (g/hr)
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
Typical Settling Test
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
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
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 (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
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)
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