CHEE2940 Lecture 8 - Multiple Particle Settling

CHEE2940: Particle Processing

Lecture 8: Multiple Particle Settling This Lecture Covers Hindered settling of particle suspensions Batch settling Continuous settling

Chee 2940: Multiple Particle Settling

GENERAL CONSIDERATIONS

Settling of individual particles is affected by the presence of other particles in suspensions. Two important effects: hydrodynamic and non-hydrodynamic (colloidal interaction forces) Hydrodynamic effects: dependence of suspension viscosity and drag force on particle concentration. Chee 2940: Multiple Particle Settling 1

Colloidal interparticle forces are significant at very high particle concentration. Attractive interaction produces aggregation, causing two or more particles to settle as an effectively larger entity and, thereby, increase the velocity. Repulsive interaction produces dispersion and hinders particle settling.

Hydrodynamic effects are dealt with in this lecture. Colloidal effects will be described later. Chee 2940: Multiple Particle Settling 2

8.1 HINDERED SETTLING OF PARTICLES Solid volume fraction, ε,

Volume of particlesTotal volume of particles & liquid

ε =

Liquid volume fraction = 1 - ε Particle concentration is considered in effective suspension density, ρe, and viscosity, µe. Chee 2940: Multiple Particle Settling 3

Suspension density

( )1e s fρ ρ ε ρ ε= + − Suspension viscosity

Einstein equation: ( )1 2.5eµ µ ε= + for ε < 0.01

µ … liquid viscosity

Extended Einstein equation (Batchelor, 1977): ( )21 2.5 6.2eµ µ ε ε= + +

Chee 2940: Multiple Particle Settling 4

For ε > 0.3, non-Newtonian shear thinning or thickening occurs and the effective viscosity may depend on the shear stress. Empirical correlations have to be used (Quemada, 1984):

2

1em

εµ µε

−

= −

εm … maximum packing volume fraction (= 0.63)

General correlation: ( )e fµ µ ε=


Stokes law for relative velocity in suspension - Fluid density is replaced by ρe - Fluid viscosity is replaced by µe

- Particle velocity, Vrel, relative to liquid gives

( )2

18s e

rele

D gV

ρ ρµ

−=

Inserting equations for ρe and µe yields

( )( )

2 1 18

s frel

D gV

fρ ρ ε

µ ε− −

= ×

Stokes velocity for single particles Effect of concentration

6 Chee 2940: Multiple Particle Settling

( )rel TV V F ε= ⋅ ; ( ) ( )3.652

1 11 2.5 6.2

F εε εε ε−

= ≅ −+ +

F(ε) is less than 1 Actual velocity is hindered.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

Particle volume fraction, ε

Cor

rect

ion

fact

or, F

Dependence of F on ε

1 ε−- Circles for 21 2.5 6.2ε ε+ +

- Red line for ( )3.651 ε− .


Generalisation for hindered settling velocity

( ),other propertiesrel TV V F ε= ⋅

Particle relative velocity

Particle velocity liquid velocityrel p fV V V= − = −

Particle hindered settling velocity

( ),...p f rel f TV V V V V F ε= + = + Vf depends on settling conditions (batch-wise or continuous). Chee 2940: Multiple Particle Settling 8

8.2 BATCH HINDERED SETTLING Hindered settling in a measuring container is batch-wise. There is no net flow through the vessel:

Q Q 0p f+ =

Qp … volume flow rate of particle settling Qf … volume flow rate of liquid moving upwards

( )p pQ V Aε= and 1f fQ V A ε= −

A … cross-sectional area of the vessel Chee 2940: Multiple Particle Settling 9

Velocity of upward flow liquid

1f pV V εε

= −−

Hindered settling velocity of particles

( ) ( ),... ,...1p f T p TV V V F V V Fεε ε

ε= + = − +

−

( ) ( )1 ,...p TV V Fε ε= − Stokes Law for Hindered Settling

( ) ( )3.65,... 1F ε ε= − ; ( )4.651p TV V ε= ⋅ − Chee 2940: Multiple Particle Settling 10

Empirical Equation for Batch-wise Hindered Settling

( )1 np TV V ε= ⋅ −

n is the Richardson-Zaki index (1954). n depends on the particle diameter and other parameters (Khan & Richardson, 1989):

0.57

0.57

4.8 0.1031 0.043

ArnAr

+=

+

( )3

2f s fD

Arρ ρ ρ

µ−

=g…Archimedes number.


Analysis of Solid Settling Flux

Volume flow rate of particle settling: p pQ V Aε=

Solid settling flux, Js, is defined as Qp

s pJ VA

ε≡ =

(Js = superficial particle velocity) In terms of terminal velocity, Js, is described as

( )1 ns TJ V ε ε= −

Typical plot for Js versus ε show a maximum. Chee 2940: Multiple Particle Settling 12

0

0.02

0.04

0.06

0.08

0 0.2 0.4 0.6 0.8 1Solid fraction concentration

Solid

flux

/Ter

min

al v

eloc

ity

Variation of solid flux, /s TJ V , versus solid

concentration, ε (D = 100 µm, ρs = 2500 kg/m3) Chee 2940: Multiple Particle Settling 13


Solid volume fraction, ε

Relationship between hindered settling and solid flux. - Low solid flux at low concentration (few particles exist) and at high concentration (settling is reduced).

14

Sharp Interface in Suspension Settling

VInt

ε1

ε2

Vp1

Vp2

Interface or discontinuity in concentration occurs in the settling of particle suspension. Mass balance over the interface gives

( ) ( )1 1 2 2p Int p IntV V V Vε ε− = −


Re-arranging yields

1 1 2 2 1 2

1 2 1 2

p p s s sInt

V V J J JVε εε ε ε ε ε

− − ∆= = =

− − ∆

dds

IntJVε

=

Significance on a flux plot (Js versus ε) 1) The gradient of the curve at ε is the velocity of a suspension layer of this ε. 2) The slop of a chord joining 2 points at ε1 and Chee 2940: Multiple Particle Settling 16

ε2 is the velocity of an interface between suspensions of the concentrations.

0

0.02

0.04

0.06

0.08

0 0.2 0.4 0.8 1Solid fraction c

olid

flux

/Ter

min

al v

eloc

itySlope = velocity of suspension layer at ε

Slope f interface between suspen and ε2.

ε2 ε ε1


0.6

= velocity osions of ε1

S

oncentration 17

The Batch Settling Test - A suspension of particles of known

concentration is prepared in a measuring cylinder.

- The cylinder shaken to mix the suspension and then placed upright to allow the suspension to settle.

- The positions of the interface are monitored in time.

Two types of settling occur depending on the initial concentration, εB, of the suspension.


Type 1 settling (hindered settling) - Occurs at low initial concentration - Three zones of constant concentrations:

zone A = clear liquid, ε = 0; zone B = suspension of the initial concentration, εB; and zone S = bed of settled particles of εS.

ε

ε ε ε εΒ

εΒ

εΑ=0 εΑ=0 εS S


Interface between clear liquid and initial suspension (Slope gives velocity)

Interface between clear liquid and settled bed

Interface between initial suspension and settled bed

Change in position of interface AB, BS, and AS.


Type 2 settling (zone settling) - Occurs at high initial concentration - Four zones of constant concentrations: in

addition to zone A, B, and S, a zone E of variable concentration is formed.

ε ε ε ε εΑ=0 εΑ=0 εΑ=0 εΒ εS εS εEmax εS εΕ


Zone E: Concentration changes with height but the minimum and maximum concentrations εEmax and εEmin are constant.

Change in position of interfaces in type 2 settling. Chee 2940: Multiple Particle Settling 22

ε

Intercept point JS

Tangent point

ε εΒ1 εΒ2 S

Flux plot showing determining if settling will be type 1 or type 2. εs is known.


Determination of the solid flux from the interface height versus time

The experimental data for the interface height vs. time can be used to determine the velocity of interface: V dh dt . /Int =

Equation for interface velocity and solid flux: dds

IntJVε

=


d dd dsJ h

tε=

Mass conservation gives Mass of solid = constant at any time

0

mass at = 0mass at time

B

tt

dhh A h t Adt

ε ε ε = + −

εB

ε

t = 0

h 0 t h

-dh/dt

0Bh h dh tdt

ε ε ε= −

h1…intercept of tangent to h(t)

1 1

0dh h hdt

h ht t

− −= −

− 0 1/Bh hε ε=


Solid flux vs concentration

J εBh0

ε Interface height vs time

h1

Diagram showing the construction of flux curve from a bath settling Chee 2940: Multiple Particle Settling 26

test

Construction of flux curves (K&S, p. 334) - The curve h(t) is given by the settling test - We want to determine ε and J at time t.

The procedure is based on Eqs: d dds

dJ h

tε= &

1 0

B

h hε ε

= .

1) Calculate: scale for the J –axis = time scale * height scale*concentration scale.

2) Draw a vertical line at ε = εB and a horizontal line at h = h0.

3) Draw a tangent to the curve h(t) at time = t & Chee 2940: Multiple Particle Settling 27

a parallel line from the origin of the J diagram 4) From the intercept point D draw a horizontal

line to cut the εB line at E. 5) Produce a line FE to cut AB at G to give ε. 6) Draw a vertical line at G to cut the parallel

line at H, which is a point of the J(ε) curve 7) Repeat step 2 to 6 to obtain enough points

for the flux curve.


8.3 CONTINUOUS HINDERED SETTLING Occurs in the industrial dewatering in thickeners Modelling is based on the steady continuous mass balance. Three cases are considered: settling with down flow only, up flow only and combined down and up flows.

εh Js Jf

Q, εF Down flow settling Feed: Volume flow rate, Q, and solid concentration εF.


The mass balance gives

( )s fQ A J J= +

Js … solid flux; Jf … liquid flux A … cross-sectional area of the vessel The general theory of hindered settling gives

( )1

fsrel p f T

JJV V V V F εε ε

= − = − =−

Combining yields ( ) ( )1s TQJ V FAε ε ε ε= + −


Total solid flux = flux due to settling+flux due to bulk flow

s setQJ JAε

= + where ( ) ( )1set TJ V Fε ε ε= −

Jset can be determined by a bath settling test

m

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1Solid volu e fraction

Solid

flux

/Ter

min

al v

eloc

ity

Flux due to bulk flow

Flux due to settling

Total continuous downward flux

εF

εB


Continuous settling with upward flow only

Feed: Volume flow rate, Q,

εh

Js Jf

Q, εF

and solid concentration εF.

The mass balance gives

( )f sQ A J J= −

The theory of hindered settling gives


( )1

fsrel T

JJV V F εε ε

= + =−

Combining yields ( ) ( )1s TQJ V FAεε ε ε= − −

Total solid flux = flux due to settling-flux due to bulk flow

s setQJ JAε

= −

where ( ) ( ) ( )1 1 nset T TJ V F Vε ε ε ε ε= − = −

Jset can be determined by bath settling test


-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1

Solid volume fraction

Solid

flux

/Ter

min

al v

eloc

ity

Flux due to bulk flow

Flux due to settling

εB εF

Total continuous upward flux


Settling in a real thickener

(with upflow and downflow sections)

Q, εF

εΤ O, εο εΒ

U, εu

Feed flow rate Q, concentration εF Underflow rate U, concentration εu Chee 2940: Multiple Particle Settling 35

Overflow rate O, concentration εo Volumetric flow rate and concentration balances

Q O U= + F o uQ O Uε ε ε= +

Solving gives

F o

u o

U Q ε εε ε

−=

− & u F

u o

O Q ε εε ε

−=

−


Knowing the feed flow rate and the solid concentrations, the underflow and overflow rates can be calculated.

36

The feed flow is split at the feed inlet into the downward flow (below the feed inlet) and upward flow (above the feed inlet) considered previously. Flux below the feed inlet Total downward solid flux: u

down setUJ JAε

= + The flux plot is obtained using the technique used in the construction of the continuous settling with down flow only. An example diagram Chee 2940: Multiple Particle Settling 37

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1Solid volume fraction

Dow

nwar

d so

lid fl

ux

Underflow withdrawn flux

Settling flux

is shown below.

Net flux below feedwell

ε crit ε max

J crit

ε (+)crit


There is a minimum solid flux at εcrit. Since all solid must pass this point, position with ε < εcrit will receive more particles, and position with

ε > εcrit will receive fewer particles, until εcrit is reached. At equilibrium, Jcrit must equal to the feed and underflow fluxes so that a thickener can be designed from the conditions

uFcrit

UQJA A

εε= = and 0oε =

The condition of minimum is given by / 0down udJ dε =

dJ U=> set

d Aε= −


The thickening at εcrit is called the critically loaded thickening (feed flux = underflow flux). Flux above the feed inlet Total downward solid flux: o

up setOJ JAε

= − The flux plot is obtained with the same technique.

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1

pwar

d)

S

olid

flux

(D

ownw

ard)

Overflow widrawal flux

settling flux

Net flux over the feedwell

40 Chee 2940: Multiple Particle Settling

Underloaded thickeners When the feed concentration, εF, is less than the critical concentration, εcrit, the thickener is said to be underloaded. The normal operation of thickeners is under the regime of slightly underloaded thickening.


Overloaded thickeners When the feed concentration, εF, is greater than the critical concentration, εcrit, the thickener is said to be overloaded. The overloaded operation will return to the critically loaded regime at long time to reach equilibrium.


Practical Applications




Settling facilities in drinking water treatment


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CHEE2940 Lecture 8 - Multiple Particle Settling

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