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 particles ε= Total volume of particles & liquid 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

ρe = ρ sε + ρ f (1 − ε ) Suspension viscosity Einstein equation: µe = µ (1 + 2.5ε ) for ε < 0.01 µ … liquid viscosity Extended Einstein equation (Batchelor, 1977): 2 µe = µ (1 + 2.5ε + 6.2ε ) 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):

 ε  µe = µ  1 −   εm 

−2

εm … maximum packing volume fraction (= 0.63) General correlation: µe = µ f ( ε ) Chee 2940: Multiple Particle Settling

5

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

D g ( ρ s − ρe ) Vrel = 18µe Inserting equations for ρe and µe yields 2

D g ( ρs − ρ f ) 2

Vrel =

18µ

Stokes velocity for single particles Chee 2940: Multiple Particle Settling

1− ε × f (ε ) Effect of concentration 6

1− ε 3.65 Vrel = VT ⋅ F ( ε ); F ( ε ) = ≅ (1 − ε ) 2 1 + 2.5ε + 6.2ε F(ε) is less than 1 Actual velocity is hindered. 1

Correction factor, F

0.8 0.6 0.4 0.2

Dependence of F on ε - Circles for 1 + 2.51ε−+ε 6.2ε - Red line for (1 − ε ) . 2

3.65

0 0

0.1

0.2

0.3

0.4

Particle volume fraction, ε Chee 2940: Multiple Particle Settling

7

Generalisation for hindered settling velocity

Vrel = VT ⋅ F ( ε ,other properties ) Particle relative velocity

Vrel = Particle velocity − liquid velocity = V p − V f Particle hindered settling velocity

V p = V f + Vrel = V f + VT 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: Qp + Q f = 0 Qp … volume flow rate of particle settling Qf … volume flow rate of liquid moving upwards Q p = V p Aε and Q f = V f A (1 − ε ) A … cross-sectional area of the vessel Chee 2940: Multiple Particle Settling

9

Velocity of upward flow liquid

V f = −V p

ε

1− ε Hindered settling velocity of particles V p = V f + VT F ( ε ,...) = −V p

ε

+ VT F ( ε ,...)

1− ε V p = VT (1 − ε ) F ( ε ,...)

Stokes Law for Hindered Settling

F ( ε ,...) = (1 − ε ) Chee 2940: Multiple Particle Settling

3.65

; V p = VT ⋅ (1 − ε )

4.65

10

Empirical Equation for Batch-wise Hindered Settling V p = VT ⋅ (1 − ε )

n

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

4.8 + 0.103 Ar n= 0.57 1 + 0.043 Ar

0.57

Ar =

D3 ρ f ( ρ s − ρ f ) g

Chee 2940: Multiple Particle Settling

µ

2

…Archimedes number. 11

Analysis of Solid Settling Flux Volume flow rate of particle settling: Q p = V p Aε Solid settling flux, Js, is defined as Qp Js ≡ = V pε A (Js = superficial particle velocity) In terms of terminal velocity, Js, is described as

J s = VT ε (1 − ε )

n

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

12

Solid flux/Terminal velocity

0.08

0.06

0.04

0.02

0 0

0.2

0.4

0.6

0.8

1

Solid fraction concentration

Variation of solid flux, J s / VT , versus solid 3 concentration, ε (D = 100 µm, ρs = 2500 kg/m ) 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).

Chee 2940: Multiple Particle Settling

14

Sharp Interface in Suspension Settling Vp1

ε1 ε2

VInt Vp2

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

(V

Chee 2940: Multiple Particle Settling

p1

− VInt ) ε1 = (V p 2 − VInt ) ε 2

15

Re-arranging yields

VInt =

V p1ε1 − V p 2ε 2

ε1 − ε 2

J s1 − J s 2 ∆J s = = ε1 − ε 2 ∆ε

dJ s VInt = dε 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.

Solid flux/Terminal velocity

0.08

Slope = velocity of suspension layer at ε

0.06

Slope = velocity of interface between suspensions of ε1 and ε2.

0.04

0.02

ε2

ε

ε1

0 0

0.2

0.4

0.6

0.8

1

Solid fraction concentration Chee 2940: Multiple Particle Settling

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. Chee 2940: Multiple Particle Settling

18

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.

εΒ

Chee 2940: Multiple Particle Settling

ε

εΑ=0

ε εΒ εS

εΑ=0 εS

ε

19

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.

Chee 2940: Multiple Particle Settling

20

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.

εΒ

ε

ε

εΑ=0 εΕ

Chee 2940: Multiple Particle Settling

εS

εΑ=0

ε εEmax εS

εΑ=0

ε εS

21

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

JS

Intercept point

Tangent point

εΒ1

εΒ2

εS

ε

Flux plot showing determining if settling will be type 1 or type 2. εs is known. Chee 2940: Multiple Particle Settling

23

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: VInt = dh / dt . Equation for interface velocity and solid flux: dJ s VInt = dε dJ s dh = dε dt Chee 2940: Multiple Particle Settling

24

Mass conservation gives Mass of solid = constant at any time t=0 -dh/dt

εB h0

t

ε

h1…intercept of tangent to h(t)

Chee 2940: Multiple Particle Settling

h

  dh   εN B h0 A = ε h + ε  − t A dt    

mass at t = 0   mass at time t

dh ε B h0 = ε h − ε t dt dh h1 − h h1 − h =−  0−t dt t ε = ε B h0 / h1 25

Solid flux vs concentration

J h0

εB 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: ddJε = ddht & εh = hε . s

B 1

0

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.

Chee 2940: Multiple Particle Settling

28

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. Q, εF h

Js

Chee 2940: Multiple Particle Settling

Jf

ε

Down flow settling Feed: Volume flow rate, Q, and solid concentration εF. 29

The mass balance gives

Q = A( J s + J f )

Js … solid flux; Jf … liquid flux A … cross-sectional area of the vessel The general theory of hindered settling gives Jf Js = VT F ( ε ) Vrel = V p − V f = − ε 1− ε Qε Combining yields J s = + VT (1 − ε ) ε F ( ε ) A Chee 2940: Multiple Particle Settling

30

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

Qε where J set = VT (1 − ε ) ε F ( ε ) J s = J set + A Jset can be determined by a bath settling test 0.1

Solid flux/Terminal velocity

Total continuous downward flux 0.08

0.06

0.04

Flux due to bulk flow

0.02

Flux due to settling

εF

0 0

Chee 2940: Multiple Particle Settling

εB

0.2

0.4

0.6

0.8

1

Solid volume fraction

31

Continuous settling with upward flow only Q, εF Jf

h Js

ε

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

The mass balance gives

Q = A( J f − J s )

The theory of hindered settling gives Chee 2940: Multiple Particle Settling

32

Vrel =

Js

ε

+

Jf

1− ε

= VT F ( ε )

Qε Combining yields J s = VT (1 − ε ) ε F ( ε ) − A Total solid flux = flux due to settling-flux due to bulk flow

J s = J set

Qε − A

where J set = VT (1 − ε ) ε F ( ε ) = VT ε (1 − ε )

n

Jset can be determined by bath settling test Chee 2940: Multiple Particle Settling

33

0.1 0.08

Solid flux/Terminal velocity

0.06 0.04 Flux due to settling

0.02 0 -0.02

εB

εF 0

0.2

0.4

0.6

0.8

1

Total continuous upward flux

-0.04 -0.06

Flux due to bulk flow

-0.08 -0.1 Solid volume fraction Chee 2940: Multiple Particle Settling

34

Settling in a real thickener (with upflow and downflow sections) Q, εF

εΤ εΒ

O, εο 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 Qε F = Oε o + U ε u

Solving gives

ε F − εo U =Q & εu − εo

εu − ε F O=Q εu − εo

Knowing the feed flow rate and the solid concentrations, the underflow and overflow rates can be calculated. Chee 2940: Multiple Particle Settling

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: J down

Uεu = + J set A

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

0.1 Net flux below feedwell

Downward solid flux

0.08

0.06

J crit

0.04

Settling flux

0.02

ε (+)crit Underflow withdrawn flux

ε crit

ε m ax

0 0

0.2

0.4

0.6

Solid volume fraction

0.8

1

is shown below.

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 Chee 2940: Multiple Particle Settling

38

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 J crit

Qε F U ε u = = A A

and ε o = 0

The condition of minimum is given by dJ down / d ε u = 0 => Chee 2940: Multiple Particle Settling

dJ set U =− dε A

39

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

Oε o − A

The flux plot is obtained with the same technique. (Downward)

0.1 0.08 0.06 settling flux

0.04

Solid flux

0.02 0 -0.02

pward)

-0.04 -0.06

Chee 2940: Multiple Particle Settling

0

0.2

0.4

Overflow widrawal flux

0.6

0.8

1

Net flux over the feedwell

40

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.

Chee 2940: Multiple Particle Settling

41

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.

Chee 2940: Multiple Particle Settling

42

Practical Applications

Chee 2940: Multiple Particle Settling

43

Chee 2940: Multiple Particle Settling

44

Chee 2940: Multiple Particle Settling

45

Settling facilities in drinking water treatment Chee 2940: Multiple Particle Settling

46