FLOTATION KINETICSA flotation model is similar to chemical kinetics

dN/dt =-k1 N1a- k2 N2

b

N - species (1 and 2) concentration t- time k - rate constant(s)a, b – process order-negative sign indicates that the concentration is diminishing due to the loss of particles being floated. -exponents a and b signify the order of the process

Since flotation seems to depend only on particles concentration

dN/dt =-k1 N1a

Model Relation

Classic first order = [1 – exp (–k1t)]

Modified first order = {1 – 1/(k2t)[1 – exp (–k2t)]}

For reactor with ideal mixing = [1 – 1/(1 + t/k3)]*

Modified for gas–solid adsorption

= k4t/(1 + k4t)*

Kinetics of second order = ()2

k5t/(1 +  k5t)

Modified second order = {1 – [ln (1 + k6t)]/(k6t)}

Two rate constants

= [1– { exp (–k7t) + (1 – ) exp(–k8t)}

Distributed rate constants = [1 – exp(–kt) f (k, 0) dk]

0

* Equivalent models because k3 = 1/k4. – flotation recovery after time t, – maximum recovery, – fraction of particles having lower flotation rate constant, k7, k – flotation rate constant.

Flotation kinetics models

Selected kinetic equations (ε – recovery of a component in separation product, εmax – maximum recovery of the same component in separation product, k – rate constant of separation, t – separation time

Model Formula

Zeroth-order model tkε (1)

First-order model tkeεε 1max (2)

First-order with rectangular distribution of floatabilities

tke

tkεε 1

11max

(3)

Fully mixed reactor model

k

tεε

1

11max

(4)

Improved gas/solid adsorption model

tk

tkεε

1max (5)

2

3 -order model

2

max

max

2

11

11

εtk

εε (6)

Second-order model tkε

tkεε

max

2max

1 (7)

Second-order model with rectangular of floatabilites

tk

tkεε 1ln

11max

(8)

A. Bakalarz, J. Drzymala, 2013, Interrelation of the Fuerstenau upgrading curve parameters with kinetics of separation, Physicochemical Problem of Mineral Processing, 49(2), 443-451

more

0

20

40

60

80

100

0 10 20 30

reco

very

of

a co

mpo

nent

in

conc

entr

ate,

ε, %

separation time, min

remaining components

component 1

Flotation kinetics of the whole mass and components

components (recovery vs time)

0

10

20

30

40

0 10 20 30

yiel

d of

con

cent

rate

, γ,

%

separation time, min

sum of kinetics of component 1 and

remaining components

Flotation results plotted as a relationship between recovery of each component in concentrate and separation time (a), yield of components forming concentrate vs. separation time (b)

product (yield vs time)

A. Bakalarz, J. Drzymala, 2013, Interrelation of the Fuerstenau upgrading curve parameters with kinetics of separation, Physicochemical Problem of Mineral Processing, 49(2), 443-451

0

20

40

60

80

100

0 10 20 30

reco

very

of

com

pone

nt 1

in

conc

entr

ate,

ε1,

c, %

separation time, min

component 1

0

20

40

60

80

100

0 20 40 60 80 100

reco

very

of

com

pone

nt 1

in

conc

entr

ate,

ε1,

c, %

recovery of component 2 in concentrate, ε2,c, %

ideal upgrading

idea

lup

grad

ing

Fuerstenau curve

0

20

40

60

80

100

0 10 20 30

reco

very

of

com

pone

nt 2

in

conc

entr

ate,

ε2,

c, %

separation time, min

component 2

a b

relation between flotation kinetics and upgrading curves

The kinetics of separation of feed components (a) provide separation results in the form of the Fuerstenau upgrading curve (b).A. Bakalarz, J. Drzymala, 2013, Interrelation of the Fuerstenau upgrading curve parameters with kinetics of separation, Physicochemical Problem of Mineral Processing, 49(2), 443-451

c,1

c,2

0 1 2

3 2

0 ,cεk,cε 21

100

2ln1

,cεk',cε

2)251(

111001

,cεk,cε

)2100(100

21

,cε

,cεk'

,cε

1

100

2ln1

,cεk,cε

k,cε

,cε100

210011001

2

100

2100ln51

111001

,cεk

,cε

1100

2100ln100

100

2100ln2100

1

,cεk

,cεk

,cε

2

3

2)251(

111001

,cεk',cε

2

100

2100ln51

111001

,cεk'

,cε

2

2100

)210010(1

111001

,cε

,cεk,cε

2

)2100(20

21

111001

,cε

,cεk',cε

2 )2100(100

21

,cε

,cεk

,cε

1100

2100ln100

100

2100ln2100

1

,cεk'

,cεk'

,cε

2

)2100(20

21

111001

,cε

,cεk,cε

100)1(2

2100

1

k,cε

,cεk

,cε

ugrading curves (here Fuerstenau’s) equations based on kinetics of flotation

c,1 recovery of component 1 in concentrate c,2 recovery of component 2 in concentrate

4

9

7

13

0

20

40

60

80

100

0 20 40 60 80 100

reco

very

of

com

pone

nt 1

in c

once

ntra

te, ε

1,c,

%

recovery of component 2 in concentrate, ε2,c, %

k=1.5

k=3

k=0.5

k=1

0

20

40

60

80

100

0 20 40 60 80 100

reco

very

of

com

pone

nt 1

in c

once

ntra

te, ε

1,c,

%

recovery of component 2 in concentrate, ε2,c, %

k=5

k=2

k=0.4

k=1

Theoretical shape of the separation data in the Fuerstenau plot

0

20

40

60

80

100

0 20 40 60 80 100

reco

very

of

com

pone

nt 1

in c

once

ntra

te, ε

1,c,

%

recovery of component 2 in concentrate, ε2,c, %

k=0.005

k=0.5

k=0.02

k=1

4 97

0

20

40

60

80

100

0 20 40 60 80 100

reco

very

of

com

pone

nt 1

in c

once

ntra

te, ε

1,c,

%

recovery of component 2 in concentrate, ε2,c, %

k=3

k=0.5k=0.2

k=1

13

*for a suitable equation see previous slide (more plots in A. Bakalarz, J. Drzymala, 2013, Interrelation of the Fuerstenau upgrading curve parameters with kinetics of separation, Physicochemical Problem of Mineral Processing, 49(2), 443-451

*

Remeber: for characterizing separation results we need either two parameter or a law governing separation and then you can use one parameter which can be called selectivity as in these plots selectivity k

An example of separation results approximation using the Fuerstenau plot

plant 3, trial 1 a=102.28

0 20 40 60 80 100

r

0

20

40

60

80

100

= a(100-r)/(a-r)

Polish copper ore – lab tests with xanthate

0 20 40 60 80 100

component 2 in product 2, %

0

20

40

60

80

100

(c

om

po

ne

nt

1 in

pro

du

ct

1)%

ideal upgrading

F = (89/89)

no upgrading a=~1000

a=100

a=~110

Homework

Calculate the rate constant and order of a set of yield flotation data

Microlaboratory cellsLaboratory cells Laboratory machines Industrial machines

Mechanical Pneumo-mechanical PneumaticPressurized (DAF)Other (sparged hydrocyclone, ASH)

 

 

gas

magnetic stirrer

porous glass

water level

froth product

x

gas

deflector

stirrer

flotaton product

water level

porous glass

Other laboratory flotation devicesa) cylindrical cell equipped with magnetic stirrer (Fuerstenau, 1964)b) laboratory flotation device of Partridge and Smith, 1971

Laboratory cells

air

drive

Laboratory Mechanobr flotation machine

Laboratory Denver flotation machine

EIMCO Product Leaflets, 2000

Industrial flotation

Flotation machines are used individually and as a group (bank)

Svedala Product Handbook, 1996

Flotation machines are rectangular and circular

Constructions and impellers of flotation machines are different

DenverMechanobrFagergreen (WEMCO-EIMCO)

DENVER

Wemco-Fagergreen (V=0.085 ÷ 85m3)

Kelly E.G., Spottiswood D.J., Introduction to mineral processing. J.Wiley& Sons, N.Jork 1985

Wemco-Fagergreen (WEMCO-EIMCO) mechanical flotation machines

EIMCO Product Leaflets, 2000

Denver Agitair Metso RCS (Metso Minerals) Outotec (Outokumpu) X-Cell (FLSmidth Minerals) Humbolt-Wedag IMN Gliwice

Wills B.A., Mineral processing technology. Pergamon Press

1983

Fragment of mechano-pneumatic flotation machine (continueous, multi-impeller tankless Denver D-R

Pneumo-mechanic multi-tank (15m3 each)

(Aker FM – Humbold Wedag)

Humbold-Wedag Product Leaflets, 1998

feed

taili

ng

Maszyna jednowirnikowa

Maszyna przepływowa

wielowirnikowa

Pneumo-mechanical flotation machines IMN

New machines: large volume and output, saving energy

New machines: large volume and output, saving energy

Flotation technologies. Outotec Leaflets 2007

Historyczny rozwój pojemności maszyn flotacyjnych

Outokumpu Oy Leaflets 2000

(Outokumpu OK-100, V= 100m3

TankCell 300 300m3

Flotation technologies, Outotec Oyj. Leaflets 2007

© 2012 Outotec Oyj. www.outotec.com

Outotec TankCell 500 (500m3)

RCS™ (Reactor Cell System) from 5 to 200 m3 (Metso Minerals/Svedala)

1-radial flow of slurry to tank wall

2-primary slurry stream to benith impeller

3-secondary recirculation towards upper part of tank

Basics in mineral processing. Metso Minerals 2003

RCS™ (Reactor Cell System) from 5 to 200 m3 (Metso Minerals)

Basics in mineral processing. Metso Minerals 2003

RCS™ (Reactor Cell System) from 260 m3 (Metso Minerals)

pneumo-machanic XCELL (FLSmidth Minerals)

XCELL™ Flotation Machines. FLSmidth Mineralss brochure 2008.

FLOTATION COLUMNS

Metso Outotec (Outokumpu)

Jameson Cell Imhoflot Pneuflot (Humbolt-Wedag)

Injection Jameson Cell

Pneumatic PNEUFLOT

Pneumatic flotation with PNEUFLOT® cells HUMBOLDT WEDAG leaflet 2009

Pneumatic cell Imhoflot. Maelgwyn Mineral Service leaflet 4/06 Chile 2006

Multi-injection Imhoflot 3 (centrifugal flotation)

concentrate

tailing

feed pump tailing pump

feed reagents

compressed air

feed

air plus suspension

Siemens SIMINE Hybrid Flot

Metals and Mining, Siemens VAI, No. 1, 2011

Injection columnInjection column

Dissolved air flotation (DAF)

Dissolved air flotation (DAF)

Flotation, ZWR Polkowice