# Stoke's Law and Settling Particles

Mechanics

Oct 30, 2013 (4 years and 6 months ago)

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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
-

-

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

Fitch Method

Thickener Cross
-
Sectional Area

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

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

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

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)