Size Reduction

• Size reduction or comminution is an important step in the

processing of many solid materials

• It may be used to create particles of a certain size and

shape, to increase surface area available for chemical

reaction

• Size reduction of solids is an energy intensive and highly

inefficient process

• Design and scale-up of comminution processes is usually

based on experience and testing

1

Nature of the material to be crushed

Hardness.

Compressive load of pounds per square inch

Very soft ,10,000; soft 15000; medium 20,000;

Hard material 25,000; very hard 30,000

The Mohr Scale of Hardness

Soft Intermediate Hard

Hardness

1. Talc 5. Apatite 8. Topaz

2. Rock salt or gypsum 6. Felspar 9. Carborundum

3. Calcite 7. Quartz 10. Diamond.

4. Fluorspar

Comminution mechanisms

• Compression: between two solid surfaces,

• Attrition & impact: against a solid surface and

other particles,

• Cutting: of the particles,

• Shear: against surrounding fluid, particles and

surfaces,

• Non-mechanical: e.g. laser and plasma ablation.

Types of Comminution equipment

Stressing mechanisms

Stress applied between two surfaces

Stress applied at a single solid surface

Surface particle or particle -particle at high velocity

impact fracture plus attrition

Stress applied by carried medium

Jaw Crusher, gyratory Crusher, crushing roll

Hammer mill, Pin mill, Fluid energy mill

Sand mill, ball mill, colloid mill

Wet grinding

5

Particle Size

Size Range of products Term Used

1-.1m Coarse Crushing

.1m Crushing

1cm Fine Crushing/coarse Grinding

1mm Intermediate Grinding/ Milling

100 m Fine grinding

10 m Ultrafine grinding

6

Down to 3mm 3mm to 5 m <5 m

Crushers Ball mill, Rod mill Sand mills

Table Mills Pin Mill, Vibration

Mills

Colloid Mills

Fluid energy mills

Crushing rolls

Particle fed to crushing rolls

2 =angle of nip

20-300

Ball Mill

Optimum speed2r=g

Ball size

where B is the ball diameter (mm); F, 80% passing size of

the feed ( m); K, an empirical constant = 350 for wet

grinding; =335 for dry grinding; SG, specific gravity of

the material being ground; Wi, Bond Work Index of the

ore; %Cs, fraction of the critical speed; and D is the diameter

of the mill inside liners (m).

Ref: Minerals Engineering 20 (2007) 320–326

Sand mill

Colloid Mills

Feed Particles: 50 m

Product Particles: 1 m

Consists of a flat rotor and stator made of

chemically inert synthetic abrasive material

Power consumption: very high; the feed material should

be grounded as finely as possible

Fluid energy mill

Feed Particles: 500 m

Product Particles: 22 mPower consumption: very high

Energy Requirement and Product Size Distribution

• There are three well-known postulates predicting energy requirements for particle size reductions

• Rittinger (1867) proposed that the energy required for particle size reduction is directly proportional to the area of new surface created

• If initial and final particle sizes are x1 and x2 respectively, then assuming a volume shape factor kv independent of size,

• If the surface shape factor ks is also independent of size, then for each original particle, the new surface created upon reduction is given by:

• Which simplifies to:

14

• Therefore, new surface created per unit mass of original

particles

P is the particle density

• Hence assuming shape factors and density are

constant, Rittinger’s postulate may be expressed as:

• Where CR is a constant

• If this is the integral form, then in differential form, Rittinger’s

postulate becomes

15

• Based on stress analysis theory for plastic deformation

• Energy required in any comminution process was directly proportional to the ratio of the volume of the feed particle to the product particle

• Therefore, size ratio, x1/x2 fixes the volume ratio, x13/x2

3

which determines the energy requirement

• And so, if x1 is the change in particle size,

• Which fixes volume ratio, x13/x2

3 and determines the energy requirement

16

Kick (1885) Law

• So, x1/x1 determines the energy requirement for particle size reduction from x1 to x1 – x1

• As x1 → 0,

CK is the Kick’s law constant

• Integrating,

• This proposal is unrealistic in most cases since it predicts that the same energy is required to reduce 10 m particles to 1 m particles as is required to reduce 1 m boulders to 10 cm block

• Can be related to fc, the crushing strength of the material

17

• Bond (1952) suggested a more useful formula:

• However, Bond’s law is usually presented in the form shown below:

• Where EB is the energy required to reduce the top particle size of the material from x1 to x2 and WI is the Bond work index, Unit kWh/short ton~4000J/kg

• The law is based on data which Bond obtained from industrial and laboratory scale processes involving many materials

• Since top size is difficult to define, in practice X1 to X2 are taken to be the sieve size in micrometers through which 80% of the material in the feed and product respectively, will pass

18

• Bond’s formula gives a fairly reliable first approximation to

the energy requirement provided the product top size is not

less than 100 m

• In differential form Bond’s formula becomes:

• It can be seen from the above analysis that the three proposals

can be considered as being the integrals of the same

differential equation:

19

• It is common practice to assume

that Kick’s proposal is applicable

for large particle size (coarse

crushing and crushing),

• Rittinger’s for very small particle

size (ultra-fine grinding)

• Bond’s formula being suitable for

intermediate particle size, the

most common range for many

industrial grinding processes

20

Milling operation

M

C

p q

p-q

kf

Material Balance in a unit

Mill

Feed, fRemoval, p

Breakage function, B(y,x): The fraction by mass of breakage products

from size x that fall below size y, x>y

1)1exp(

1)/exp(),(

xyxyB Broadbent and Callcott

Distribution of product

N3(x)

x

Size range 1

Size range 2

Size range 3

4

x4, x3, x2, x1

Mill

Feed, fRemoval, p

Milling Circuit Matrix

b11=1-B(x2,x1)

b21=B(x2,x1)-B(x3,x1)

b31=B(x3,x1)-B(x4,x1)

b41=B(x4,x1)

Amounts entering differing

grades after milling

1111 fbp

2221212 fbfbp

3332321313 fbfbfbp

4443432421414 fbfbfbfbpfMp

From millFine cut

Classifier Matrix

Classifier

Coarse cut p-q

p q

44

33

22

11

000

000

000

000

c

c

c

c

C pCq

Close Loop

M

C

p q

p-q

kf

kMp

pCq

qpfk

pCpqp

fMCIIk

fkMCII

kMCIf

pCIfk

1

fMCIIMC

kMC

pCq

1

pCI

pCpI

Mixing

Mixing and Segregation• Achieving good mixing of particulate solids of different size

and density is important in many process industries

• A perfect mixture of two types of particles is one in which a group of particles taken from any position in the mixture will contain the same proportions of each particle as the proportions present in the whole mixture

• A random mixture is a mixture in which the probability of finding a particle of any component is the same at all locations and equal to the proportion of that component in the mixture as a whole

28

Segregation

• In many systems, particles to be mixed have different properties and tend to exhibit segregation

• Particles with the same physical property collect together in one part of the mixture and random mixture is not a natural state

• Even if particles are originally mixed by some means, they will tend to unmix on handling (moving, pouring, conveying, processing)

• Differences in size, density and shape of constituent particles of a mixture may give rise to segregation

• Difference in particle size is most important, density difference is comparatively unimportant except in gas fluidization

• Demixing or segregation can give rise to variations in bulk density of powder going to packaging

• Chemical composition of the product may be off specification (e.g. in blending of constituents for detergents or drugs)

29

Four mechanisms of segregation according to size may be identified:

(1) Trajectory segregation:

if a small particle of diameter x and density p, whose drag is governed by

Stokes’ law is projected horizontally with a velocity U into a fluid of

viscosity , the limiting distance that it can travel horizontally is

U px2/18

• A particle of diameter 2x would travel four times as far before coming to

rest

• This mechanism can cause segregation where particles are caused to move

through air or when powders fall from the end of a conveyor belt

30

(2) Percolation of fine particles:

• if mass of particles is disturbed in such a way that individual particles move, a rearrangement in the packing of the particles occurs

• The gaps created allow particles from above to fall and particles in some other place to move upwards

• If the powder is composed of particles of different size, it will be easier for small particles to fall down and so there will be a tendency for small particles to move downwards leading to segregation

• Even a very small difference in particle size can give rise to significant segregation

• Segregation by percolation of fine particles can occur whenever the mixture is disturbed causing rearrangement of particles

• This can happen during stirring, shaking, vibration or when pouring particles into a heap. Segregation by percolation occurs in charging and discharging storage hoppers

• As particles are fed into a hopper they generally pour into a heap resulting in segregation if there is a size distribution and the powder is free-flowing

31

• (3) Rise of coarse particles on vibration (‘Brazil-nut effect’):

• if a mixture of particles of different size is vibrated the larger particles move upwards

• The rise of the larger or denser ‘intruder’ within the bed of smaller particles has been explained in terms of creation and filling of voids beneath the intruder

32

• (4) Elutriation segregation: when a powder containing an

appreciable proportion of particles under 50 m is charged

into a storage vessel or hopper, air is displaced upwards

• The upward velocity of air may exceed the terminal freefall

velocity of some of the finer particles, which may then remain

in suspension after the larger particles have settled

• Thus a pocket of fine particles is generated in the hopper each

time solids are charged

33

Reduction of Segregation

• making the size of the components as similar as possible

• by reducing the absolute size of both components

• Make all particles than 30 m (for particle densities in the range 2000 – 3000 kg/m3).

– In such fine powders, interparticle forces generated by electrostatic charging, van der Waals forces and forces due to moisture are large compared with gravitational and inertial forces

• The mobility of particles in free-flowing powders can be reduced by addition of small quantities of liquid

• The reduction in mobility reduces segregation and permits better mixing

34

Three main mechanisms for mixing (J.C. Williams)

Convection

•Driven by bulk flow

•Fast macromixing

•Easy to scale up

•Limited by segregated flow structures (incomplete mixing)

Shear

•Caused by velocity gradients

•Required for micromixing of cohesive systems

•Scale-up criteria unknown

Dispersion/Diffusive

•Driven by individual particle motion

•Always slow

•Leads to complete macroscopic homogeneity

•Scale-up criteria unknown

Sampling

• To determine the quality of a mixture, it is generally necessary to take samples

• Sampling of mixtures and analysis of mixture quality require application of statistical methods

• Mean composition: the true composition of a mixture is often not known but an estimate may be found by sampling

Statistics relevant to random binary mixtures are as follows:

• For N samples of composition y1 to yN in one component, the estimate of the mixture composition is given by:

36

• Standard deviation and variance: the true standard

deviation, , and the true variance, 2, of the composition of

the mixture are quantitative measures of the quality of the

mixture

• The true variance is usually not known but an estimate S2 is

defined as:

• The important change is "N-1" instead of "N" (which is

called "Bessel's correction").

• The standard deviation is equal to the square root of variance37

unknownncompositiotruetheifN

yy

S

knownncompositiotruetheifN

y

N

i

i

N

i

i

;1

;

1

2

2

1

2

2

• Theoretical limits of variance: for a two-component system the theoretical upper and lower limits of mixture variance are:

• Where p and (1-p) are the proportions of the two components determined from samples and n is the number of particles in each sample

• Mixing indices: a measure of the degree of mixing is the Lacey mixing index

• In practical terms the Lacy mixing index is the ratio of ‘mixing achieved’ to ‘mixing possible’

• A Lacey mixing index of zero would represent complete segregation and a value of unit would represent a completely random mixture

• Practical values of this mixing index are found to lie in the range 0.75 to 1.0

• A further mixing index is defined as:

• This index gives better discrimination for practical mixtures and approaches unity for completely random mixtures

38

22

0

22

0

R

iL

Characteristic Curve of Mixing

39

Rate of Mixing

40

0; 0 tatcdt

d

tc

i eL 1

Effect of speed on rate of mixing in simple drum

mixerX is the percentage of the samples that is unmixed

Ref: COULSON and MAITRA, Ind. Chemist 26 (1950) 55.

Mixing and subsequent separation of solid

particles

Mixing depends of powder type

43

Cohesive powder : forms lumps

Free flowing: Restrain the movement of particles

both diffusive mixing and shear mixing give rise to size segregation For such powders, convective mixing is the major mechanism promoting mixing

Types of mixers: tumbling mixers, convective mixers, fluidized bed mixers, high shear mixers

• Tumbling mixers

• Agitated mixers• Paddle

Energy consumption: up to 150 kW/m3

44

(a) double cone, (b) V, and (c) bin

blenders.

Types of mixer

• Agitated mixers

Ribbon mixers (vertical & horizontal)

Power required as high as 6 kW/m3

Screw mixers (vertical and horizontal and orbiting types)

power consumption: up to 80 kW/m3

45

Types of mixer

• Agitated mixers

• Sigma-blade and Z-blade mixer

• Forberg mixer

• Gravity silo blenders

46

Mixing silo blender: Zeppelin Centro blender

Waeschle’s gravity blender and combine flow Blender

Types of mixer

• Pneumatic blenders

47

Types of mixer

• High intensity mixers

• Henschel mixer

• Paddle mixer

48

A random mixture consists of two components A and B in proportions

60 and 40% by mass, respectively. The particles are spherical and A

and B have particle densities 500 and 700 kg/m 3 , respectively. The

cumulative undersize mass distributions of the two components are

shown in Table 11W.1.

If samples of 1 g are withdrawn from the mixture, what is the

expected value for the standard deviation of the composition of the

samples? m N3A N3B

2057 1

1676 0.8

1405 0.5 1

1204 0.32 0.88

1003 0.19 0.68

853 0.12 0.44

699 0.07 0.21

599 0.04 0.08

500 0.02 0

422 0

Problem-1

Problem2The performance of a solids mixer was assessed by calculating the

variance occurring in the mass fraction of a component amongst a selection of samples withdrawn from the mixture. The quality was tested at intervals of 30 s and the data obtained are:

sample variance (−) 0.025 0.006 0.015 0.018 0.019

mixing time (s) 30 60 90 120 150

If the component analysed represents 20 per cent of the mixture by mass and each of the samples removed contains approximately 100 particles, comment on the quality of the mixture produced and present the data in graphical form showing the variation of the mixing index with time.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

0 50 100 150 200

sigmaR^2 0.0016

Sigma0^2 0.16

sigma^2

30 0.025 0.852273

60 0.006 0.972222

90 0.015 0.915404

120 0.018 0.896465

150 0.019 0.890152

Cumulative mass undersize

(%)

100 80 10

Particle dia ( m) 1000 600 200

1)1exp(

1)/exp(),(

xyxyB

MC

Mineral has the following breakage function, the fraction by

mass of breakage product from size x that fall below size y,

, and is classified and crushed in the circuit shown below.

Where M is the mill and C is the classifier. The feed to the

circuit is

Feed rate into the circuit is 5 tonnes per hour.

1.Construct a mill matrix based on the above diameter using

the given breakage function.

If the classifier can be represented by a leading diagonal matrix

of elements: 0.3, 0.5 and 0.7, what will be the size distribution

and the total flow rate of the course cut from the classifier?

Outline the solution procedure in terms of the various matrices.

Breakage and selection functions

Mill

Feed, fRemoval, p

Breakage function, B(y,x):

Selection Function , S(x)

The fraction by mass of breakage products

from size x that fall below size y, x>y

The fraction by mass of particles that are

selected and broken in time t

1)1exp(

1)/exp(),(

xyxyB Broadbent and Callcott