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
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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 Hardness 1. Talc 5. Apatite 2. Rock salt or gypsum 6. Felspar 3. Calcite 7. Quartz 4. Fluorspar
Hard 8. Topaz 9. Carborundum 10. Diamond.
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 Jaw Crusher, gyratory Crusher, crushing roll Stress applied at a single solid surface Surface particle or particle -particle at high velocity impact fracture plus attrition Hammer mill, Pin mill, Fluid energy mill Sand mill, ball mill, colloid mill Stress applied by carried medium
Wet grinding
Size Range of products
Term Used
1-.1m
Coarse Crushing
.1m
Crushing
1cm
Fine Crushing/coarse Grinding
1mm Particle
Size
Intermediate Grinding/ Milling
100 m
Fine grinding
10 m
Ultrafine grinding
5
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
6
Crushing rolls
Particle fed to crushing rolls 2 =angle of nip 20-300
Ball Mill Optimum speed 2r=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 m Power 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
is the particle density • Hence assuming shape factors and density are constant, Rittinger’s postulate may be expressed as: P
• Where CR is a constant • If this is the integral form, then in differential form, Rittinger’s postulate becomes 15
Kick (1885) Law • 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/x23 which determines the energy requirement • And so, if x1 is the change in particle size,
• Which fixes volume ratio, x13/x23 and determines the energy requirement
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• 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
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• 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
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• 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:
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• 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
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Milling operation
f
k
M C
p p-q
q
Material Balance in a unit Removal, p
Feed, f
Mill
Breakage function, B(y,x): The fraction by mass of breakage products from size x that fall below size y, x>y
B( y, x)
exp( y / x) 1 exp(1) 1
Broadbent and Callcott
Distribution of product Size range 1 Size range 2 N3(x) Size range 3 4 x4, x3, x2, x
x1
Milling Circuit Matrix b11=1-B(x2,x1)
b21=B(x2,x1)-B(x3,x1)
Amounts entering differing grades after milling
b31=B(x3,x1)-B(x4,x1) b41=B(x4,x1) Removal, p
Feed, f
p1 b11 f1
Mill
p2 p3
p
M
f
p4
b21 f1 b22 f 2 b31 f1 b32 f 2 b33 f3
b41 f1 b42 f 2 b43 f3 b44 f 4
Classifier Matrix
From mill
Fine cut Classifier
p
q
Coarse cut p-q
C
c11 0 0 c22 0 0
0 0
0 0
0 0
c33 0 0 c44
q
C p
Close Loop k
f
M C
p
q k
f
I
f I
k
I
p-q k
f
q
C
p
p
M
k
p
q I p I
p q
q
p C C p
C p
p
C
I
C p
I C M C M k I
C M
k f 1
f
p
C M k C M
I
I
C M
1
f
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
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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
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(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
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• (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
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• (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
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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
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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:
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• 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: N
2
yi 2
i 1
; if the true composition known
N N
yi S2
y
i 1
N 1
2
; if the true composition unknown
• 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 variance 37
•
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
•
Li
•
• • •
•
2 0 2 0
2 2 R
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
Characteristic Curve of Mixing
39
Rate of Mixing d dt
Li
c ;
1 e
0
at t
0
ct
40
Effect of speed on rate of mixing in simple drum mixer
X 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 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
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Types of mixers: tumbling mixers, convective mixers, fluidized bed mixers, high shear mixers • Tumbling mixers
(a) double cone, (b) V, and blenders. •
(c) bin
Agitated mixers • Paddle
Energy consumption: up to 150 kW/m3
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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
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Types of mixer • Agitated mixers • Sigma-blade and Z-blade mixer • Forberg mixer • Gravity silo blenders
Mixing silo blender: Zeppelin Centro blender
Waeschle’s gravity blender and combine flow Blender
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Types of mixer • Pneumatic blenders
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Types of mixer • High intensity mixers • Henschel mixer • Paddle mixer
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Problem-1 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 m N3A N3B samples? 2057 1676 1405 1204 1003 853 699 599 500 422
1 0.8 0.5 0.32 0.19 0.12 0.07 0.04 0.02 0
1 0.88 0.68 0.44 0.21 0.08 0
Problem2 The 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.
0.98
0.03 0.025
sigmaR^2
0.0016
0.96
Sigma0^2
0.16
0.94
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
0.02
0.92 0.015 0.9 0.01
0.88 0.86
0.005
0.84
0 0
50
100
150
200
Mineral has the following breakage function, the fraction by mass of breakage product from size x that fall below size y, B( y, x)
exp( y / x) 1 exp(1) 1
, and is classified and crushed in the circuit shown below. Where M is the mill and C is the classifier. The feed to the M C 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. Cumulative mass undersize 100 (%) Particle dia ( m) 1000
80
10
600
200
Breakage and selection functions Removal, p
Feed, f
Mill
Breakage function, B(y,x): The fraction by mass of breakage products from size x that fall below size y, x>y Selection Function , S(x)
B( y, x)
The fraction by mass of particles that are selected and broken in time t
exp( y / x) 1 exp(1) 1
Broadbent and Callcott
