chee2940 lecture 2 - particle size

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


• 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.


• 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):


/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.


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


- 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


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.


• 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.


• 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=


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


• 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=

= ∑


- 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

=

=

=∑

∑


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


- 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).


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)


Fig 2.2 Examples of aperture designs (Wills)


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


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=


• (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.


- 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)


Pipette filler to collect the sample

Two-way stopcock

Fig 2.4 Andrean pipette for sedimentation size analysis (Wills)


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).


Fig 2.5 Simple elutriator (Wills)


• 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.


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


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.


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=

= ∑


• 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


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


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)


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)


- 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∆ =∑


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 =


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∆ =∑


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 = .


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).


• Viewing distributions

The log-normal plot gives more details of fines

No details of fines can be seen in the normal-normal plot


• “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


Graphical correlations (M Rhode, 1998)


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

=

∑


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)


- 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.


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


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)


- 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

= −


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.


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.


- 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.


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

= −


• Comparison of number, volume & surface distributions

Many instruments measure number distribution but we want surface area or volume distribution

(M Rhode, 1998)


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.


chee2940 lecture 2 - particle size

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