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CHAPTER 2 PARTICLE SIZE CHARACTERIZATION

2.1 Introduction to Particle Size and Shape

• In many powder handling and processing operations, particle size and size distribution play a key role in determining the bulk properties of the powder.

• Example of regular-shaped particles.

Shape Dimensions Sphere Radius Cube Side length

Cylinder Radius and height Cuboid Three side lengths Cone Radius and height

• In practice, it is important to use the method of size measurement, which directly gives the particle size, which is relevant to the situation, or process of interest.

2.2 Methods of Measurement: 2.2.1 Laser Diffraction

Malvern Mastersizer S (Malvern Instrument Inc.)

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2.2.2 Sieve

Sieve Shaker (Retsch)

2.2.3 Electrozone Sensing

Electrozone sensing

(Coulter Counter)

10

• Coulter counter can measure particle size ranging from 0.4 – 1200 micron. Its response is unaffected by particle color, shape, composition or refractive index.

• Particles suspended in a weak electrolyte solution are drawn through a small aperture, separating two electrodes between which an electric current flows.

• The voltage applied across the aperture creates a "sensing zone". As particles pass through the aperture (or "sensing zone"), they displace their own volume of electrolyte, momentarily increasing the impedance of the aperture.

• This change in impedance produces a pulse that is digitally processed in real time. The Coulter Principle states that the pulse is directly proportional to the tri-dimensional volume of the particle that produced it.

• Analyzing these pulses enables a size distribution to be acquired and displayed in volume (µm3 or fL) and diameter (µm).

• In addition, a metering device is used to draw a known volume of the particle suspension through the aperture; a count of the number of pulses can then yield the concentration of particles in the sample.

2.2.4 Sedimentation

• Suitable to be applied in paint and ceramic industry.

• Involves instrument that detects the settling rate of particles in a fluid by monitoring: o a change in concentration of particles at a known

depth in the fluid or;

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o the rate at which particles settle out onto the bottom of the container in which it confined.

• Parameter measured are Stoke’s diameter where Stoke’s law is obeyed.

• Sedimentation by gravity involves particles of size more than 2 micron

• Sedimentation under centrifugal force involve particles of size less than 2 micron

2.2.5 Scanning Electron Microscope (SEM)

• A microscope generating an electron beam scanning back and forth over a sample.

• Due to the interaction between the beam and the sample, several different signals are produced providing the user with detailed information about the surface structure, differences of atomic number within the sample or information about elemental content.

SEM image of grinded palm kernel shell particles at 21x and

2000x magnifications respectively.

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3.0 Sampling method

• Most laboratory tests use only a small sample and this has to be taken from a production stream or from an existing, stored material. This sample has to be the representative of the whole material.

• Powders are unlike fluid where the properties change easily under applied load.

• They may consolidate with time, and attrition and segregation may occur in transfer.

• Sampling is an important element of powder handling such that it demands careful scientific design and operation of the sampling system.

• The purpose is to collect a manageable mass of material (sample), which is representative of the total mass of powder from which it was taken.

• This is achieved by taking many small samples from all parts of total which, when combined, will represent the total with an acceptable degree of accuracy.

• This means that all particles in the total must have the same probability of being included in the final sample.

• To satisfy the above requirements, the following basic ‘golden’ rules of sampling should be followed wherever practicable.

o Sampling should be made preferably from a moving

stream (this applies to both powders and suspensions) but powder on a stopped belt also can be sampled.

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o A sample of the whole of the stream should be taken for many (equal-spaced) periods of time rather than part of the stream for the whole of the time.

• Re-combined primary sample taken from the whole is too large for most powder test and so that, it need to be sub-divided into secondary or tertiary sub-sample.

• Allen (1981) reviewed and tested most methods available for sampling splitting and found the best is equipment called spinning riffler (Figure 1).

• In spinning riffler, the sample is slowly conveyed by a vibratory feeder from the feed hopper to the rotating carousel where it is divided into many container ports via a machined rotary head.

• The sub-samples are collected in this depending on how many samples are required.

• Feed rate is controlled by varying the gap under the hopper and varying the vibration of the feeder.

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Spinning Riffler

Sample Splitter

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4.0 Types Of Particle Diameters, Mean Size And Size Distribution.

4.1 Types of particle diameters

• Some definitions of terms used:

dp = Sieve size, the width of the minimum square aperture through which the particle will pass

dv = Volume diameter, the diameter of a sphere having the same volume as the particle.

dsv = Surface/volume diameter, the diameter of a sphere having the same external surface area/volume ratio as the particle.

ds = Surface diameter, the diameter of a sphere having the same surface as the particle.

• Some of the diameters are related through Waddell’s

sphericity factor, ψ, defined as;

ψ = surface area of equivalent volume sphere surface area of the particle Some common diameters used in microscopy analysis are: (Figure 1.1, page 2)

• Equivalent circle diameter

• Martin’s diameter

• Ferret diameter

• Shear diameter

2

s

v

d

d

=ψ

(2.1)

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or

=ψ

v

sv

d

d (2.2)

• ψ, dv and dsv can be calculated exactly for geometrical shapes such as cuboids, rings etc.

• According to Geldart (1989), in practice, value of ψ for a specific material can be obtained as follows;

o Sieve out a narrow size fraction of the powder, usually

the middle of the size range or the coarser material, so as to obtain at least 0.5 – 1 kg.

o Split this down using a riffler to obtain a few hundreds

particles.

o Count the exact number (n) and weight them (M).

Knowing the particle density, ρp, calculate the equivalent sphere volume diameter, dv from

3/1

6

=

n

Md

p

vπρ (2.3)

o Put the 0.5 – 1 kg of powder in a circular tube 50 – 75

mm diameter and measure the pressure drop, ∆P, across the bed at a variety of low flowrates.

o The Carman-Kozeny equation relates pressure drop to

particle size, dsv, voidage and bed depth, L at low Reynolds number.

( )23

2

1150

svd

U

L

P µ

ε

ε−=

∆ (2.4)

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o Thus, dsv can be calculated from pressure drop data.

o From equation above, calculate ψ.

• For irregular particles:

o Particles more than 75 µm: sieving

o Particles less than 75 µm: Laser diffraction technique: e.g. Malvern Mastersizer, Coulter Counter etc.

• Different measurement techniques give different sizes if the particles are non-spherical, which is usually the case.

• dsv and dv are generally considered to be the most useful sizes where fluid/particle interactions are involved. (eg. Flow thru’ fixed and fluidized bed, pneumatic conveying etc.) and

they are related to each other thru’ the particle sphericity, ψ.

• Malvern Master Sizer S: it measures dv if sphericity is known, than dsv can be estimated.

• Other way to find dsv is by using Ergun equation.

• According to Abrahamsen & Geldart (1980):

o For quartz: ψ = 0.8 o dv = 1.13dp (2.5)

• The average sphericity for regular figures:

773.0

d

d

v

sv = (2.6)

• Thus, for non-spherical particles,

dsv ≈ 0.87dp (2.7)

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• For spherical or near-spherical particles; dv = dsv = dp (2.8)

ASSIGNMENT 1. Calculate the equivalent volume sphere diameter dv and the

surface volume equivalent sphere, dsv of a cuboid particle of side length 1, 2 and 4 mm.

2. Particles with density of 2000 kg/m3 with sphericity of are

poured into a container. A sample of 2000 particles are taken from the powder weighed 1000 mg. determine the volume diameter, dv and surface volume diameter, dsv of the particles.

4.2 Mean size and size distribution

• No industrial powder is monosized and it is usually necessary to characterize the powder by both the size distribution and a mean size.

• If a powder of mass M has a size range consisting of Np1 of size d1, Np2 of size d2 and so on, the mean surface/volume size is expressed by:

...dNdNdN

...dNdNdNd

233p

222p

211p

3

33p

3

22p

3

11p

sv+++

+++= (2.9)

∑∑

=dx

xdsv (2.10)

where x is the weight fraction of particles in each size range.

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• When sieving is used, d1, d2,… are replaced by the arithmetic averages of adjacent sieve apertures, dpi and the equation becomes:

∑=

pii

pdx

d1

(2.11)

and dp is mean particle size and not directly related to the dsv.

• Mean particle size, dp: emphasis to the important influence which small proportions of fines have.

• Equation (2.11) should not be used if the powder has an unusual distribution, for example bi- or tri-modal or has an extremely wide size range.

• This type of powder will not behave in a homogeneous way and cannot be characterized by a single number.

• Refer to Table 1 for example.

• The British Standard Sieve series is arranged in multiples of 2¼, and this is used as a basis in Table 2.

• This will give an idea of relative spread as judged from the number of sieves. (Refer Table 2).

• It is always advisable to first plot the size distribution of powder as a weight fraction, or percentage in a size range, against the average size, i.e. x vs. dpi because a plot of cumulative percentage undersize can conceal peculiarities of distribution.

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Table 1: Size Distribution of sieved sand

Sieve aperture,

µm

Size dpi,

µm

Wt % in range, xi

Cum. % undersize

-600 + 500 550 0.50 100 -500 + 420 460 11.60 99.5 -420 + 350 385 11.25 87.9 -350 + 300 325 14.45 76.65 -300 + 250 275 20.80 62.2 -250 + 210 230 13.85 41.4 -210 + 180 195 12.50 27.55 -180 + 150 165 11.90 15.05 -150 + 125 137 3.15 3.15

Table 2: Width of Size Distributions Based on Relative Spread

Number of sieves on which the middle 70%

(approx.) of the powder is found

pmd

σ

Type of distribution

1 0 Very narrow 2 0.03 Narrow 3 0.17 Fairly narrow 4 0.25 Fairly wide 5 0.33

}

6 0.41 7 0.48

9 0.60

} 11 0.70

> 13 > 0.80 Extremely wide

Wide

Very wide

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• Other method of characterizing powder, the median, dpm: corresponds to the 50% value on the graph of cumulative percentage undersize versus size.

• There is no universally agreed way of comparing the width of the size distribution of two powders having different mean sizes, nor of defining how wide a distribution is.

• In order to compare the width of the size distribution of two powders having different mean size or defining how wide a

distribution is; relative spread, pmd

σ

from cumulative

percentage undersize plot is used as suggested by Geldart (2003).

2

dd %16%84 −=σ

(2.12)

• See Figure 1 for example.

• In the example given, σ = 105 µm and σ/dpm = 0.39.

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Figure 1: Size distribution of a sand characterized in various

ways

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5.0 Particle Size Analysis 5.1 Equivalent of Means

• A population of particles is described by a particle size distribution.

• Particle size distributions may be expressed as frequency distribution curves, f(x) or dF/dx or cumulative curves, F(x).

• The distributions can be by number, surface, mass or volume (if particle density does not vary, then the mass distribution is the same as the volume distribution).

• In most practical applications, the population of powders must be described by a single number.

• Some of the central tendency expression which depending on the particular application are as below:

o Mode: The most frequently occurring size in the

sample. The mode has no practical significance as a measure of central tendency and is rarely used in practice.

o Median: Easily read from the cumulative distribution as

the 50% size, which splits the distribution into two equal parts. Also has no special significance as a measure of particle size.

• Many different means can be defined for a given size distribution. All the means below can be described by the equation;

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( )

( )

∫

∫=

1

0

1

0

dF

dFxg

xg (2.13)

but 1dF

1

0=∫ (2.14)

and thus, ( ) ( )dFxgxg

1

0∫= (2.15)

where x is the mean and g is the weighting function, which is different for each mean definition (refer Table 3).

Table 3: Definition of means

g(x) Mean and notation dp Arithmetic mean, dp,a

dp2 Quadratic mean, dp,q

dp3 Cubic mean, dp,c

Log dp Geometric mean, dp,g

1/ dp Harmonic mean, dp,h

• Equation (2.15) tells us that the mean is the area between the curve and the cumulative distribution, F(x) axis in a plot of F(x) versus the weighting function g(x) (refer Figure 2).

• Graphical determination of mean is always recommended because the distribution is more accurately presented as a continuous curve.

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Figure 2: Plot of cumulative frequency vs. weighting function,

g(x).