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 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. Chee 3920: Particle Size and Shape

3

• Equivalent diameter is often used. - Equivalent volume diameter – diameter of a sphere with the same volume (mass) as the particle:

d v = 6V / π 3

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

ds = 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.

d Sauter = 6V / A V and A … real particle volume and surface area. Chee 3920: Particle Size and Shape

5

Example of equivalent diameters for a particle with a shape of rectangular box Dimension (mm) 2 Surface area (mm ) 3 Volume (mm ) dv (mm) ds (mm) dSauter (mm)

Chee 3920: Particle Size and Shape

20 x 30 x 40 5200 24000 35.8 40.7 27.7

6

- Stokes (hydraulic) diameter – from settling velocity (drag force and weight) – diameter of a sphere with the same density and terminal settling velocity (discussed later).

d Stokes =

18µU g (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 2

g … acceleration due to gravity (9.81 m/s ) ρ and δ … particle & liquid densities. 3 ρ = 2500 kg/m for quartz (SiO2) 3 δ = 1000 kg/m for water. - Sieve diameter – The smallest dimension of sieve aperture through which particles pass. Chee 3920: Particle Size and Shape

8

• 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.

Chee 3920: Particle Size and Shape

9

• Deviation of irregular shape from spheres. Is described by sphericity. - Volume sphericity, ψ V (the same volume) 2 ψ V = π ( dV ) / A where dV is volume-equivalent diameter A is the real surface area.

- Surface sphericity, ψ A (the same surface) 3 ψ A = π ( d A ) / ( 6V ) Chee 3920: Particle Size and Shape

10

where d A is surface-equivalent diameter V is the real volume.

- Sauter-diameter sphericity, ψ VA and ψ AV

ψ VA = π ( d32 ) / A 2

ψ AV = π ( d32 ) / ( 6V ) 3

where d32 is Sauter diameter. Chee 3920: Particle Size and Shape

11

• Equivalent diameter of many particles - Mean diameter, d m

d = ∑ γ i di i =1

where di is diameter of i-th size range γ i is mass fraction of i-th size range.

- Volume equivalent diameter, dV

( dV ) Chee 3920: Particle Size and Shape

3

m

= ∑ γ i di

3

i =1

12

- Surface equivalent diameter, d A

(dA )

2

m

= ∑ γ i di

2

i =1

- Sauter diameter, d32 m

d32 =

∑γ d i =1 m

∑γ d i =1

Chee 3920: Particle Size and Shape

i

i

3 i 2 i

13

2.2 METHODS OF PARTICLE SIZE ANALYSIS Table 2.1 Some methods of particle size analysis Method Equivalent size Test 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 Chee 3920: Particle Size and Shape

14

- 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). Chee 3920: Particle Size and Shape

15

Largest apertur

Smallest aperture

Fig 2.1 Example of sieve arrangement (Wills) Chee 3920: Particle Size and Shape

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 Mass fraction Cumulative Cumulative retained undersized Oversized (micron) (micron) (g) +200 200 0 0 1.000 0.000 200 - 150 175 10 0.111 1.000 0.000 150 - 100 125 40 0.444 0.889 0.111 100 - 50 75 30 0.333 0.444 0.556 50 - 0 25 10 0.111 0.111 0.889 0 0 0 0.000 1.000 sum 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 = m dt st nd where the 1 term is the particle weight, the 2 rd is the buoyancy, 3 is the drag force and the last term is the inertial force. u is particle velocity.

• Stokes law for drag: F = 3πµ du Chee 3920: Particle Size and Shape

20

• (Terminal) settling velocity: 2 ρ − δ ) gd ( u= 18µ 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.

Chee 3920: Particle Size and Shape

20

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 2 the r.m.s. of displacements, x . • Brownian diffusivity, D, of particles is determined from the Einstein-Smoluchowski Chee 3920: Particle Size and Shape

21

equation

x

2

= 2 Dt

• Particle size is determined from Einstein’s equation 3πµ d = k BT / D 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) m

d = ∑ γ i di i =1

Chee 3920: Particle Size and Shape

23

• Standard deviation, σ, m

m

σ = ∑ γ i ( di − d ) = ∑ γ i di − ( d ) = ( di ) − ( d ) 2

i =1

2

2

2

2

2

i =1

• 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

40 Mass retained (g)

Mass versus size range

30 20 10 0 0 - 50

50 - 100 100 - 150 150 - 200 Size range (micron)

Histogram for data in Table 2.3 Chee 3920: Particle Size and Shape

25

0.5 Mass fraction versus size range

Mass fraction

0.4 0.3 0.2 0.1 0 0 - 50

50 - 100 100 - 150 Size range (micron)

150 - 200

Normalised histogram (Table 2.3) Chee 3920: Particle Size and Shape

26

0.5 0.4

Midpoint of mass fraction versus midpoint of size range

f(d)

0.3 0.2 0.1 0 0

50

100 d (microns)

150

200

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     d d − 1 1   f (d ) = exp −    σ 2π  2  σ   σ … standard deviation of the distribution

d … mean (median) diameter

Property:

∞

∫ f ( d ) dd = 1

−∞

Chee 3920: Particle Size and Shape

or ∑ f ( d ) ∆d = 1 . i

28

2.5

2  1  1 d −d    f (d ) = exp −    σ 2 σ 2π      

2 Experiments f(d)

1.5 1 0.5 0 0

50

100 d (microns)

150

200

Fig 2.7a Example of Gaussian (normal) frequency distributions. d = 100 µ m & σ = 20 Chee 3920: Particle Size and Shape

29

Log-normal distribution 2  1  1  x − x   exp −  f ( x) =   σ 2π  2  σ   where x = log ( d ).

σ … standard deviation of the distribution d … mean (median) diameter

Property:

∞

∫ f ( x ) dx = 1

−∞

Chee 3920: Particle Size and Shape

or ∑ f ( x ) ∆x = 1 . i

30

2.5

2.5

f (d ) =

2

1.5

1.5

2

   

f(d)

f(d)

2

  log ( d ) − log d  ( ) 1  1 exp −    2 σ σ 2π   

1

1

0.5

0.5

0

0

0

50

100 d (microns)

150

200

0.5

1.5 log(d/microns)

2.5

Fig 2.7b Example of log-normal frequency distributions in the normal (left) and log-normal (right) diagrams. log ( d / µm) = 1.6 & σ = 0.17 .

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

The log-normal plot gives more details of fines

Chee 3920: Particle Size and Shape

33

• “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

n

d + d 2 + ... + d n = d= 1 n

Arithmetic mean Quadratic mean ( d )

2

d1 ) + ( d 2 ) ( = 2

2

∑d i =1

i

n

+ ... + ( d n )

2

n

∴

1 n 2 d= d ( ) ∑ i n i =1

Geometric mean (It presents the arithmetic mean of the lognormal distribution!) d = n d1 ⋅ d 2 ⋅ ... ⋅ d n = ( d1 ⋅ d 2 ⋅ ... ⋅ d n )

1/ n

Harmonic mean

1 1 1 + + ... + dn 1 d1 d 2 = d n

∴

d=

n n

∑ (1/ d ) i

i=1

Chee 3920: Particle Size and Shape

36

Modes of distributions - Mono disperse particles

f(d)

- Mono modal distribution

d

0.5 0.4

f(d)

0.3 0.2 0.1 0 0

Chee 3920: Particle Size and Shape

50

100 d (microns)

150

200

37

- Bimodal distributions (Fig 2.8 – solid line) 3

f(d)

2

1

0 0

50

100 d (microns)

150

200

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/lognormal distributions.

• Cumulative distributions - Undersized cumulative distribution m

Q ( di ) = ∑ γ i i =1

(Summing from the smallest size fraction) Chee 3920: Particle Size and Shape

39

- Oversized cumulative distribution 1

P ( di ) = ∑ γ i = 1 − Q ( di ). i =m

Example of determing cumulative distributions Size range Mid-point Mass Mass fraction Cumulative Cumulative retained undersized Oversized (micron) (micron) (g) +200 200 0 0 1.000 0.000 200 - 150 175 10 0.111 1.000 0.000 150 - 100 125 40 0.444 0.889 0.111 100 - 50 75 30 0.333 0.444 0.556 50 - 0 25 10 0.111 0.111 0.889 0 0 0 0.000 1.000 sum 90

Chee 3920: Particle Size and Shape

40

Cumulative mass fraction

1.000

Oversized Undersized

0.800 0.600 0.400 0.200 0.000 0

50

100 150 d (microns)

200

Cumulative distribution curves (Table 2.3) Chee 3920: Particle Size and Shape

41

- 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   d  P ( d ) = exp −     d '  Chee 3920: Particle Size and Shape

n

   42

where d ' and n are parameters. 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  P ( d )  = n log ( d ) − n log ( d ') n … the slope of the straight line. -nlog(d’) … intercept of the straight line. Chee 3920: Particle Size and Shape

43

1

0.8

0

log{-ln[P(d)]}

P(d)

1

0.6 0.4

-1 -2

0.2

-3

0

-4

0

50

100 d (microns)

150

200

0

1 2 log(d/micron)

3

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.

Chee 3920: Particle Size and Shape

44

- Gates-Gaudin-Schuhmann (GGS) distribution Q ( d ) = const × ( d / d ')

n

n >1 represents samples with increasing coarse fractions, and n < 1 represents samples with decreasing coarse fractions. Q(d)

n<1 n =1 n>1 d

Chee 3920: Particle Size and Shape

45

• Relationship between frequency and cumulative distributions P(d ) =

d

∫

f ( x ) dx ; Q ( d ) =

d max

d min

∫

f ( x ) dx .

d

Differential relationships: dP = f ( d ) => f(d) is also called differential frequency distribution! dd dQ = − f (d ) dd Chee 3920: Particle Size and Shape

46

• 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: f s ( d ) = k S d f N ( d ) 2

Volume (mass) distribution: f v ( d ) = kv d f N ( d ) 3

And the condition of normalisation: ∞

∫ f ( d ) dd = 1 0

We also have to assume constant shape and density with size. Chee 3920: Particle Size and Shape

48