CHEE2940: Particle Processing
Lecture 2: Particle Size and Shape This lecture covers Particle size and shape Particle size analysis Measurement techniques
Chee 3920: Particle Size and Shape
WHY IS PARTICLE SIZE ANALYSIS IMPORTANT? • .Determines the quality of final products • Establishes performance of processing • Determines the optimum size for separation • Determines the size range of loses.
Chee 3920: Particle Size and Shape 1
2.1 PARTICLE SIZE AND SHAPE • Particle size: refers to one particle. • Precise particle size is difficult to obtain due to the irregular shape of particles.
From M. Rhodes, Intro Part. Tech., Wiley, 1998
Chee 3920: Particle Size and Shape 2
• For spherical particles, defining particle size is easy; it is simply the diameter of the particle.
• For non-spherical particles, the term "diameter" is strictly inapplicable. For example, what is the diameter of a flake or a fiber?
• Also, particles of identical shape can have quite different chemical composition and, therefore, have different densities.
• The differences in shape and density could introduce considerable confusion in defining particle size.
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• Equivalent diameter is often used.
- Equivalent volume diameter – diameter of a sphere with the same volume (mass) as the particle:
3 6 /vd V π=
V … real particle volume. - Equivalent surface diameter - diameter of a sphere with the same surface area as the particle (BET isotherm):
Chee 3920: Particle Size and Shape 4
/sd A π= A … real particle surface area.
- Equivalent volume-surface (Sauter) diameter - diameter of a sphere with the same volume to surface area ratio as the particle.
6 /Sauterd V A=
V and A … real particle volume and surface area.
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Example of equivalent diameters for a particle with a shape of rectangular box Dimension (mm) 20 x 30 x 40 Surface area (mm2) 5200 Volume (mm3) 24000 dv (mm) 35.8 ds (mm) 40.7 dSauter (mm) 27.7
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- Stokes (hydraulic) diameter – from settling velocity (drag force and weight) – diameter of a sphere with the same density and terminal settling velocity (discussed later).
( )18
StokesUd
gµ
ρ δ=
−
U … real particle terminal settling velocity
Chee 3920: Particle Size and Shape 7
µ … liquid viscosity µ = 0.001 Pa/s for water µ = 0.00001 Pa/s for air
g … acceleration due to gravity (9.81 m/s2) ρ and δ … particle & liquid densities. ρ = 2500 kg/m3 for quartz (SiO2)
δ = 1000 kg/m3 for water.
- Sieve diameter – The smallest dimension of sieve aperture through which particles pass.
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• Microscopically Observed Shapes Martin’s diameter – bisects the area of the particle image –always taken in the same direction. Feret’s diameter – distance between parellel tangents –always taken in the same direction. Equivalent area – diameter of a circle with the same area of the particle image. Equivalent perimenter – diameter of a circle with the same perimeter of the particle image.
From M Rhode, IPT, 1998.
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• Deviation of irregular shape from spheres.
Is described by sphericity.
- Volume sphericity, Vψ (the same volume) ( )2 /V Vd Aψ π=
where is volume-equivalent diameter Vd A is the real surface area.
- Surface sphericity, Aψ (the same surface) ( ) )3 (/ 6A Ad Vψ π=
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where is surface-equivalent diameter Ad V is the real volume.
- Sauter-diameter sphericity, VAψ and AVψ
( )232 /VA d Aψ π=
( ) ( )3
32 / 6AV d Vψ π= where is Sauter diameter. 32d
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• Equivalent diameter of many particles
- Mean diameter, d
1
m
i ii
d dγ=
= ∑
where is diameter of i-th size range idiγ is mass fraction of i-th size range.
- Volume equivalent diameter, Vd
( )3 3
1
m
V i ii
d dγ=
= ∑
Chee 3920: Particle Size and Shape 12
- Surface equivalent diameter, Ad
( )2 2
1
m
A i ii
d dγ=
= ∑
- Sauter diameter, 32d
3
132
2
1
m
i iim
i ii
dd
d
γ
γ
=
=
=∑
∑
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2.2 METHODS OF PARTICLE SIZE ANALYSIS
Table 2.1 Some methods of particle size analysis
Method Equivalent sizeTest sieving 100 mm – 10 microns Elutriation 40 microns – 5 microns Gravity sedimentation 40 microns – 1 microns Centrifu. sedimentation 40 microns – 50 nano Microscopy 50 microns – 10 nano Ligth scattering 10 microns – 10 nano Sieve Analysis
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- Good for particle >25 µm, cheap & easy. - Carried out by passing sample via a series of
sieves (Fig 2.1) - Weighing the amount collected on each sieve - With wet or dry samples.
• Test sieves • Designed by the norminal aperture size (Fig 2.2)
• Popular designs: BSS (British), Tyler series (American), DIN (German).
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Largest apertur
Smallest aperture
Fig 2.1 Example of sieve arranChee 3920: Particle Size and Shape
gement (Wills)
16
Mesh = number of apertures per inch
Table 2.2 BSS 410 wire-mesh sieves (Wills)
Chee 3920: Particle Size and Shape 17
Fig 2.2 Examples of aperture designs (Wills)
Chee 3920: Particle Size and Shape 18
Presentation of results for sieve analysis
Table 2.3 Example of size distribution
Size range Mid-point Mass retained
Mass fraction Cumulative undersized
Cumulative Oversized
(micron) (micron) (g)+200 200 0 0 1.000 0.000
200 - 150 175 10 0.111 1.000 0.000150 - 100 125 40 0.444 0.889 0.111100 - 50 75 30 0.333 0.444 0.55650 - 0 25 10 0.111 0.111 0.889
0 0 0 0.000 1.000sum 90
Chee 3920: Particle Size and Shape 19
Gravity sedimentation technique • Uses the dependence of the settling velocity on the particle size (the Stokes law)
du'mg m g F mdt
− − =
where the 1st term is the particle weight, the 2nd is the buoyancy, 3rd is the drag force and the last term is the inertial force. u is particle velocity.
• Stokes law for drag: 3F duπµ=
Chee 3920: Particle Size and Shape 20
• (Terminal) settling velocity: ( ) 2
18gd
uρ δ
µ−
=
where ρ and δ are particle and liquid density, g is gravity acceleration and µ is liquid viscosity. • Experimental steps: - Sample is uniformly dispersed in water in a beaker.
- A siphon tube is immersed into 90% of the water depth.
Chee 3920: Particle Size and Shape 21
- Particle with size d is sucked from the beaker at time interval t calculated from the immersed depth and Stokes’ velocity: /t h u= .
Fig 2.3 Beaker decantation for gravity sedimentation size analysis (Wills)
Chee 3920: Particle Size and Shape 22
Pipette filler to collect the sample
Two-way stopcock
Fig 2.4 Andrean pipette for sedimentation size analysis (Wills)
Chee 3920: Particle Size and Shape
Elutriation technique • Uses an upward current of water or air for sizing the sample.
• Is the reverse of gravity sedimentation and Stokes’s law applies.
• Particles with lower settling velocity overflow • Particles with greater velocity sink to under flow.
• Sizing is achieved with a series of simple elutriators (Fig 2.5).
Chee 3920: Particle Size and Shape
Fig 2.5 Simple elutriator (Wills)
Chee 3920: Particle Size and Shape
• For fine particles (<10 microns), cyclosizer is usually used (Fig 2.6).
Fig 2.6 Warman cyclosizer (Wills) Chee 3920: Particle Size and Shape
Microscopy techniques • Used for small (dry) samples. • Particle size is directly measured. • Optical microscopes: 1 micron (wavelength of light is ~ 100 microns)
• Electron (TEM and SEM): ~ 10 nm.
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Light scattering techniques • Based on the capability of colloidal particles to scatter light.
• Useful for colloidal particles. • Static light scattering: Intensity ~ particle volume and particle concentration.
• Dynamic light scattering measurements give the r.m.s. of displacements, 2x .
• Brownian diffusivity, D, of particles is determined from the Einstein-Smoluchowski
Chee 3920: Particle Size and Shape 21
equation 2 2x Dt=
• Particle size is determined from Einstein’s equation
3 /Bd k T Dπµ = where µ is liquid viscosity kB is Boltzman’s constant T is absolute temperature.
Chee 3920: Particle Size and Shape 22
2.3 ANALYSIS OF SIZE DISTRIBUTION (Of many particles)
• Based on tabular results of size analysis (Table 2.2)
• Characteristic parameters: mean diameter, standard deviation, distribution functions, and cumulative curves.
• Mean diameter (shown previously)
1
m
i ii
d dγ=
= ∑
Chee 3920: Particle Size and Shape 23
• Standard deviation, σ,
( ) ( ) ( ) ( )2 2 2222
1 1
m m
i i i i ii i
d d d d d dσ γ γ= =
= − = − = −∑ ∑
• Frequency distribution - Histogram: mass of size range versus size range.
- Normalised histogram: mass fraction vs size range.
- Continuous distribution function: mid-points of mass fraction vs mid-points of size range
Chee 3920: Particle Size and Shape 24
0
10
20
30
40
0 - 50 50 - 100 100 - 150 150 - 200Size range (micron)
Mas
s re
tain
ed (g
)
Mass versus size range
Histogram for data in Table 2.3
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0
0.1
0.2
0.3
0.4
0.5
0 - 50 50 - 100 100 - 150 150 - 200Size range (micron)
Mas
s fra
ctio
n
Mass fraction versus size range
Normalised histogram (Table 2.3)
Chee 3920: Particle Size and Shape 26
0
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200d (microns)
f(d)
Midpoint of mass fraction versus midpoint of size range
Continuous distribution function (γ => f) (Data in Table 2.3)
Chee 3920: Particle Size and Shape 27
- Theoretical distribution functions (taken from theory on probability and statistics)
Nornal (Gaussian) distribution
( )2
1 1exp22d df d
σσ π
− = −
σ … standard deviation of the distribution d … mean (median) diameter
Property: 1( )df d d∞
−∞
=∫ or . ( ) 1if d d∆ =∑
Chee 3920: Particle Size and Shape 28
0
0.5
1
1.5
2
2.5
0 50 100 150 200d (microns)
f(d)
( )2
1 1exp22d df d
σσ π
− = −
Experiments
Fig 2.7a Example of Gaussian (normal) frequency distributions. 100 md µ= & 20σ =
Chee 3920: Particle Size and Shape 29
Log-normal distribution
( )21 1exp
22x xf x
σσ π
− = −
where ( )logx d= .
σ … standard deviation of the distribution d … mean (median) diameter
Property: 1( )df x x∞
−∞
=∫ or . ( ) 1if x x∆ =∑
Chee 3920: Particle Size and Shape 30
0
0.5
1
1.5
2
2.5
0.5 1.5 2.5log(d/microns)
f(d)
( )( ) ( ) 2
log log1 1exp22
d df d
σσ π
− = −
0
0.5
1
1.5
2
2.5
0 50 100 150 200d (microns)
f(d)
Fig 2.7b Example of log-normal frequency
distributions in the normal (left) and log-normal (right) diagrams. ( )log / m 1.6d µ = & 0.17σ = .
Chee 3920: Particle Size and Shape 31
Comments: Many size distributions do not follow the theoretical Gaussian and log-normal statistics. The theoretical concepts remain valid for describing the particle size distributions. We need the mean (median) diameter and the standard deviation. A number of approximate equations have used for the particle size distributions (shown later).
Chee 3920: Particle Size and Shape 32
• Viewing distributions
The log-normal plot gives more details of fines
No details of fines can be seen in the normal-normal plot
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• “Average” size of many particles Mode – most frequent size occurring Median – d50 (50% cumulative distribution) Means – different types for different uses - Arithmetic mean - Quadratic mean - Geometric mean - Harmonic mean
Chee 3920: Particle Size and Shape 34
Graphical correlations (M Rhode, 1998)
Chee 3920: Particle Size and Shape 35
Arithmetic mean 1 2 1...
n
in i
dd d dd
n n=+ + +
= =∑
Quadratic mean ( ) ( ) ( ) ( )2 2 22 1 2 ... nd d d
dn
+ + += ∴ ( )2
1
1 n
ii
d dn =
= ∑
Geometric mean ( )1/
1 2 1 2... ... nnn nd d d d d d d= ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅
(It presents the arithmetic mean of the lognormal distribution!)
Harmonic mean 1 2
1 1 1...1 nd d dd n
+ + += ∴ ( )
1i=1/
n
i
ndd
=
∑
Chee 3920: Particle Size and Shape 36
Modes of distributions f(d)
d
- Mono disperse particles - Mono modal distribution
0
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200d (microns)
f(d)
Chee 3920: Particle Size and Shape 37
- Bimodal distributions (Fig 2.8 – solid line)
0
1
2
3
0 50 100 150 200d (microns)
f(d)
Bimodal distribution occurs for mixtures of two minerals.
Chee 3920: Particle Size and Shape 38
For analysis, bimodal distribution is separated (using an appropriate mathematical technique called deconvolution) into the Gaussian/log-normal distributions.
• Cumulative distributions
- Undersized cumulative distribution
( )1
m
i ii
Q d γ=
= ∑
(Summing from the smallest size fraction) Chee 3920: Particle Size and Shape 39
- Oversized cumulative distribution
( ) ( )1
1i i ii m
P d Q dγ=
= = −∑ .
Example of determing cumulative distributions Size range Mid-point Mass
retainedMass fraction Cumulative
undersizedCumulative Oversized
(micron) (micron) (g)+200 200 0 0 1.000 0.000
200 - 150 175 10 0.111 1.000 0.000150 - 100 125 40 0.444 0.889 0.111100 - 50 75 30 0.333 0.444 0.55650 - 0 25 10 0.111 0.111 0.889
0 0 0 0.000 1.000sum 90
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0.000
0.200
0.400
0.600
0.800
1.000
0 50 100 150 200d (microns)
Cum
ulat
ive
mas
s fra
ctio
n
Oversized
Undersized
Cumulative distribution curves (Table 2.3)
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- Many curves of cumulative oversized and
undersized distributions versus particle size are S-shaped.
- Two approximations for cumulative distributions are known, i.e., Rosin-Rammler and Gates-Gaudin-Schuhmann distributions.
- Rosin-Rammler (RS) distribution
( ) exp'
ndP dd
= −
Chee 3920: Particle Size and Shape 42
where and n are parameters. 'd'd and n can be determined from the graph
of ( ){ }log ln P− versus ( )log d in the log-log diagram which gives a straight line
( ){ } ( ) ( )log ln log log 'P d n d n − = − d
n … the slope of the straight line. -nlog(d’) … intercept of the straight line.
Chee 3920: Particle Size and Shape 43
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200d (microns)
P(d
)
-4
-3
-2
-1
0
1
0 1 2 3log(d/micron)
log{
-ln[P
(d)]}
Example of data which can be described by the Rosin-Rammler distribution.
The slope of the log-log diagram gives n = 2.
The intercept is equal to –4 and gives d’ = 100 microns.
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- Gates-Gaudin-Schuhmann (GGS) distribution ( ) ( )/ ' nQ d const d d= ×
n >1 represents samples with increasing coarse fractions, and n < 1 represents samples with decreasing coarse fractions.
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d
Q(d)
n =1
n<1
n>1
• Relationship between frequency and cumulative distributions
( ) ( )max
d
d
P d f x x= ∫ d ; . ( ) ( )mind
d
Q d f x dx= ∫Differential relationships: P ( )d
df d
d= => f(d) is also called differential frequency distribution!
( )dQd
f dd
= −
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• Comparison of number, volume & surface distributions
Many instruments measure number distribution but we want surface area or volume distribution
(M Rhode, 1998)
Chee 3920: Particle Size and Shape 47
Conversions Surface distribution: ( ) ( )2
s S Nf d k d f d=
Volume (mass) distribution: ( ) ( )3v v Nf d k d f d=
And the condition of normalisation:
( )0
d 1f d d∞
=∫
We also have to assume constant shape and density with size.
Chee 3920: Particle Size and Shape 48