Data presentation and interpretation 2.1 Introduction I he behavior - - PDF document

Data presentation and interpretation

2.1 Introduction I he behavior and properties of particulate material are, to a large extent, dependent on particle morphology (shape, texture etc.) size and size

• distribution. Therefore proper measurement, informative data presentation

and correct data interpretation are fundamental to an understanding of powder handling and end-use properties. In this chapter the following questions will be addressed: What is meant by particle size? What is meant by particle diameter? For a single particle? For an assembly of particles? How is the average size of an assembly of particles defined? What is meant by particle shape? What is meant by particle size distribution? As well as answering these questions, methods of presenting data will be covered together with data analysis and interpretation. Physical characterization differs from chemical assay in that frequently a unique value does not exist. The determined amount of copper in an ore sample should not depend upon the analytical procedure employed whereas the measured size distribution is method dependent. Only homogeneous, spherical particles have an unambiguous size. The following story illustrates the problem. Some extra terrestrial beings (ETB) were sent to earth to study humans. Their homes were spherical and the more important the ETB the bigger the sphere. The ETB who landed in the Arctic had no problem in defining the shape of the igloos as hemispherical with a single (base) diameter. The ETB who landed in North America classified the wigwams as conical but required two dimensions, height and base diameter, to describe their size. The ETB

Data presentation and interpretation 5 7

who landed in New York classified the skyscrapers as cuboid with three dimensions mutually perpendicular. The one who landed in London gazed about him despairingly before committing suicide. One of the purposes of this chapter is to reduce the possibility of similar tragedies. 2.2 Particle size Ihe size of a spherical homogeneous particle is uniquely defined by its diameter. For regular, compact particles such as cubes or regular tetrahedra, a single dimension can be used to define size. With some regular particles it may be necessary to specify more than one dimension: For a cone the base diameter and height are required whilst for a cuboid three dimensions are needed. Derived diameters are determined by measuring size-dependent properties of particles and relating them to single linear dimensions. The most widely used of these are the equivalent spherical diameters. Thus, a unit cube has the same volume as a sphere of diameter 1.24 units; hence this is the derived volume diameter. The diameter therefore depends upon the measured property. Consider a cube of side 1 cm; its volume V = 1 cm^ and its superficial surface area S ^ 6 cm^, d^ is the diameter of a sphere having the same volume as the cube and d^ is the diameter of a sphere having the same surface area.

V^^-dl

so that d, 6 '

^6^'" \TIJ

1.241 5' = 7C(i; so that < ^ 5 = - I =1.382 The surface to volume ratio is of fundamental importance since it controls the rate at which a particle interacts with its surroundings. This is given by: S nd] „. S 6J2 . 6 — = —^ Thus — = —~ I.e. 5 ^ „ =

V {^/6)dl V dl d,.

Hence, for a unit cube d^^= 1. Thus a sphere of diameter 1.241 cm has the same volume as the cube, a sphere of diameter 1.382 cm has the same superficial surface area and a sphere of diameter 1cm has the same surface to volume ratio. Definitions of the symbols used are given in Table 2.1.

58 Powder sampling and particle size determination

If one were dealing with crystals of known shape it would be more sensible to relate the dimension to that shape, but this is not common practice; for the unit cube this procedure would make all the above-derived diameters equal to unity. A spherical homogeneous particle settling in a fluid rapidly reaches a constant 'terminal' velocity that is uniquely related to the diameter of the

• sphere. If an irregularly shaped particle is allowed to settle in a fluid, its

terminal velocity may be compared with that of a sphere of the same density settling under similar conditions. The size of the particle, defined as its free-falling diameter, is then equated to the diameter of that sphere. In the laminar flow region (i?^<0.25) irregularly shaped particles settle in random orientation and a single particle generates a range of equivalent diameters depending on its orientation. The Stokes diameter is some average of these. Outside the laminar flow region, such particles orientate themselves to give maximum resistance to motion and the free falling diameter that is generated will be the smallest of these (Figure 2.1). Thus the free-falling diameter for a non-spherical particle is smaller in the intermediate region than in the laminar flow region.

''s/^n^ 236 M m ^ 5 . ^ = 277 M m

Oo o

^a,max = 252Mm d, = 204Mm d^^^l'25m

• Fig. 2.1 Stokes diameter for an irregular particle of volume diameter

204 |Lim. With maximum resistance to drag the particle will fall at the same speed as a sphere of diameter 236 jim. With minimum resistance to drag the particle will fall at the same speed as a sphere of diameter 277 \im.

Data presentation and interpretation 59 Table 2.1 Definitions of particle diameters Symbol Diameter Definition Formula d^ Volume Diameter of a sphere having the same volume as the particle d^ Surface Diameter of a sphere having the same external surface area as the particle d^^, Surface- Diameter of a sphere having the same ratio of volume external surface area to volume as the particle (Sauter) d^ Drag Diameter of a sphere having the same resistance to motion as the particle in a fluid

• f the same viscosity and at the same velocity

{d^ approaches d^ when Re is small) df Free- Diameter of a sphere having the same free- falling falling speed as a particle of the same density in a fluid of the same density and viscosity d^f Stokes Free-falling diameter in the laminar flow region d^ Projected Diameter of a circle having the same area projected area as the particle in stable

• rientation

d Projected Diameter of a circle having the same area projected area as the particle in random

• rientation [for convex particles, mean value

for all orientations d == d^], d^ Perimeter Diameter of a circle having the same perimeter as the projected oufline of the particle d^ Sieve Width of the minimum square aperture through which the particle will pass

• df^

Feret The distance between pairs of parallel tangents to the projected outline of the particle in some fixed direction *^M Martin Chord length, parallel to some fixed direcfion, which divides the particle projected

• utline into two equal areas

y^ Unrolled Chord length through the centroid of the particle oufline

V^^d^

s = 71^3

• [dll4)

Fij =3;rd^rju *st = Wdd P = ltd,. statistical diameters, often defined in terms of the mean value for a particular particle.

60 Powder sampling and particle size determination

• Fig. 2.2 The projected area of a particle is orientation dependent. Martin's

diameter {dj^^) is 246 ^im, the Feret diameter {dp) is 312 | L i m and the projected area diameter in stable orientation (climax) * ^ ^52 | L i m (the particle is the same as in Figure 2.1) For irregular particles, the assigned size depends upon the method of measurement, hence the particle sizing technique should, whenever possible, duplicate the process one wishes to control. Thus, for paint pigments the projected area is important since this controls hiding power, whereas for chemical reactants the total surface area should be determined. The projected area diameter may be determined by microscopy for each individual particle, but surface area is usually determined for the powder as a whole. The magnitude of this surface will depend upon the method of measurement, permeametry for example, giving a much lower area than gas adsorption. Further, both of these methods depend upon the size of the gas molecules used in the determinations, since less surface may be accessible for larger molecules. The sieve diameter, for square mesh sieves, is the length of the minimum square aperture through which the particles can pass, though this definition needs modification for sieves which do not have square apertures. Microscopy is the only widely used particle sizing technique in which individual particles are observed and measured. A single particle can have an infinite number of linear dimensions and it is only when these are averaged that a meaningful value results. For an assembly of particles, each linear measurement quantifies the particle size in only one direction. If the particles are in random

• rientation, and if sufficient particles are counted, the size distribution of

these measurements reflects the size distribution of the particles perpendicular to the viewing direction. Because of the need to count a large number of particles in order to generate meaningful data these diameters are called statistical diameters.

Data presentation and interpretation 61

• Fig. 2.3 (a) Definition of unrolled diameter df^[ d^^ 2R] (b) unrolled

curve Projected area diameter (stable orientation) 2S2^m Martin diameter 246 ^m

• Fig. 2.4 Particle size of a quartz particle by microscopy using Feret, Martin

and projected area diameter If the projected areas of the particles are compared with the areas of series

• f circles the projected area diameters generated describe the particles in

two dimensions for the orientation in which they are measured. In microscopy this is usually the projected area in stable orientation but in certain cases the particles may be in a less stable orientation which generates a lower value (Figure 2.2). For a single particle the expectation of a statistical diameter and its coefficient of variation may be calculated from the following equations [1] (Figure 2.3).

62 Powder sampling and particle size determination

• Fig. 2.5 Electron microscope photomicrographs of two paint pigments

showing how particles can be aggregates of finer particles [2]. E{dj^)^- \d^i\e

(2.1) (2.2) E{d^)=^{dpddp (2.3) _2 _p,j2. E {dp) E{dp)

E(d^) = ^fd^dd,

^M

(2.4) (2.5)

E\d^)

(2.6)

Data presentation and interpretation 63 R6 is known as the shape descriptor (polar signature) [3]. Provided the ^increments are small enough a precise description of the boundary is

• btained. If the image has a perimeter that is encountered more than once

by R the descriptor can no longer regenerate the shape. For such a particle it is necessary to use equal length steps around the perimeter to define particle shape An illustration of two statistical diameters, Feret and Martin, and the projected area diameter is given in Figure 2.4. Anomalies can occur due to the state of aggregation of the particles. Figure 2.5a shows a single particle of Prussian blue about 1 | L i m in diameter. The nitrogen adsorption surface area is 61.3 m^g"^ from which the surface-volume mean diameter is calculated as 0.051 |im. This is the diameter of the primary particles of which the aggregate is made up. Similarly the micronized Prussian blue Figure 2.5b has approximately the same surface-volume mean diameter. With the red oxide Figure 2.5c the diameter is 0.21 fim that is approximately the same as the solid particle seen in the micrograph. 2.3 Average diameters The purpose of an average is to represent a group of individual values in a simple and concise manner in order to obtain an understanding of the

• group. It is important therefore that the average should be representative
• f the group. All averages are a measure of central tendency, which are
• nly modestly affected by the relatively few extreme values in the tails of

the distribution. The mode, the most commonly occurring (most popular) value in a distribution, passes through the peak of the relative distribution curve, i.e. it is the value at which the frequency is a maximum (Figure 2.6). More than one high density region may be present in which case the distribution is said to be multi-modal, i.e. bimodal, trimodal and so on. The median divides the distribution into two equal parts, i.e. it is the 50% size on the cumulative distribution curve. The mean {x) is the center of gravity of the distribution (Figure 2.7) i.e. For the mean, the moment of the sums of the elementary areas of the relative distribution, of width djc, about the ordinate, equals the sum of the moments of the elements about the ordinate:

^(x-x)—hx-Y^i^-x)—bx dx Y dx

64 Powder sampling and particle size determination %/^in 80 60

40 20

Mode

• •

Median Mean

dx

Particle size (^m)

• Fig. 2.6 Definitions of some average diameters

Particle size (^m)

• Fig. 2.7 Finding the center of gravity of a distribution by taking moments

Data presentation and interpretation 65

\ (2.7)

For a number distribution d^ = dN and:

IjcdTV

IdiV For a mass (volume) distribution d^ = dK = x^dTVgiving:

^ - V - = - V - — = i — - = % M (2.8)

ZdF Z^dTV Zx^dTV [The symbol x denotes size, as opposed to diameter, and includes a shape coefficient. This artifact is found to be useful for general treatment of data]. The mode and the median may be determined graphically but the above summation has to be carried out for the determination of the mean. For a slightly skewed distribution the approximate relationship, mean- mode ^ 3(mean-median) holds. For a symmetrical distribution, all three averages coincide. These means represent the distribution in only two of its

• properties. The characteristics of a particle size distribution are its total

number, length, surface, volume (mass) and moment. Note that: A system of unequally sized particles may be represented by a system of uniform particles having two, and only two, characteristics of the original distribution. The size of the particles in the uniform system is then the mean size of the non-uniform system with respect to these two characteristics.

66 Powder sampling and particle size determination

As a simple illustration, consider a system of (spherical) particles containing one particle of each diameter from one to ten (Figure 2.8). This distribution can be represented in number and length by a mean diameter

• f 5.5 (i.e. the number-length mean diameter dj^^ = 5.50) but ten particles

each of diameter 5.5 will not have the same surface or volume as the

• riginal distribution, i.e. Spheres each of diameter 8.37 will have the same

volume and the same moment as the original distribution (i.e. the mean of the mass distribution), but not the same number or length. Mean sizes are defined in Table 2.2 and values for the example above are given in Table 2.3. The size increases systematically with the order of the distribution; i.e. the mean of the volume distribution is greater than the mean of the surface distribution; the mean of the surface distribution is greater than the mean of the length distribution and the mean of the length distribution is greater than the mean of the number distribution. Table 2.2 Definitions of mean sizes Number, length Number, surface Number, volume

IdZ ZdTV

• ^A^^* -

\YAS_ 'zdTV "ZdK AN

• '

IxdTV ZdA^

• IdA^

ZdTV

Length, surface Length, volume Surface, volume Volume, moment

• Zd^

ZdZ

• ^SV

• '

I d F I d i ZdF ZdM ZdF Hs

IjcdA^

\YyAN_ ' XxdTV __ Y.x^dN l/dTV

YJX AN

Data presentation and interpretation 67

• Fig. 2.8 The homogeneous distribution that represents in number and

length a heterogeneous distribution of 10 particles of size 1 to 10 with unit separation in size. Table 2.3 Calculated values of mean sizes for a selection of particles of diameter 1 to 10 with one particle in each size class Number (AO A^=10 %L =5.50 1x^^=7.00 x^y = 7.86 \^VM "8.37 Length {L)

L=Y^xAN

i =55 ^Ns"^ 6.20 Xi^y=lA2

% M " 8 . 1 1

Surface

iS)

S=Y. x^dN x=\ 5 = 385 Xf^y =6.71

HM = 7-72

Volume {V)

F = Z x^dN

F= 3,025 %M=7.09 Moment {M)

M = S jc^dTV

M = 25,333 The arithmetic mean size of a number distribution (x^) is the sum of the sizes of the separate particles divided by the number of particle; it is most significant when the distribution is normal. TAN

• X NL

(2.9) The geometric mean size of a number distribution {x^ is the wth root of the product of the sizes of the n particles examined; it is of particular value with log-normal distributions:

68 Powder sampling and particle size determination

(ric"^)'

^^

A^lnjc^=ZlnjcdA^ lnx^= ^ (2.10) The harmonic mean size of a number distribution is the number of particles divided by the sum of the reciprocals of the sizes of the individual particles; this is related to specific surface and is of importance where surface area of the sample is concerned [4].

ZdiV

X^

Y^ANIx

^_^ -^^ dA^ ^"^

(2.11) The method of sizing may also be incorporated into the symbol. Hence, for particle sizing by microscopy, the arithmetic mean diameter becomes ^a ~ ^NL a' ^^^ volume-moment mean diameter calculated from the results of a sedimentation analysis isd^^ = <^vM,St • ^^^ mean diameter of a cumulative mass percentage distribution obtained by sieving is d^ or J^^ ^ . International Standards Organization is preparing a Standard, ISO/FDIS 9276-2 Calculation of average particle sizes/diameters and moments from particle size distributions. For a discussion of this draft Standard see Alderliesten [5]. 2.4 Particle dispersion The spread of a distribution may be expressed in terms of a range, i.e. the difference between its minimum and maximum sizes; the inter-quartile range is the difference between the 25% and 75% size (25^75); the inter- percentile range between two percentages, usually 10 and 90 doXgo). The least significant of these is the first, since a stray undersize or oversize

Data presentation and interpretation 69 particle can greatly affect its value. The most significant are the standard deviations. The standard deviation and geometric standard deviation are statistical measures of spread. The former is more commonly used with powders having a narrow size range and is the difference between the 50% and the (50±16)% sizes. The latter is more commonly used with powders having a wide size range and is the ratio of the 50% and the non-fractional (50±]6)%sizes. The standard deviation is defined as:

^^ \{l:{x-xf^<|>~^

T^(f> (2.12) Hence: ^2^Ix-A<^_-2 (2.13) where o^ is called the variance. The geometric standard deviation (<T ) is defined by:

l^kzlM

,2.4)

where: (X= Incr^ and z=\nx 2.5 Particle shape F^article shape is a fundamental powder property affecting powder packing and thus bulk density, porosity, permeability, cohesion, flowability, caking behavior [6] attrition, interaction with fluids and the covering power of pigments, although little quantitative work has been carried out on these

• relationships. Davies [7] gives other examples where information on shape

is needed to describe powder behavior. Many papers have been written on shape determination but there are few articles that relate the measurements to powder behavior and end-use

• properties. Hawkins [8] critically reviews nearly 300 articles on particle

shape measurement and Singh and Ramakrishnan [9]review seventy-six articles on powder characterization by particle shape assessment.

70 Powder sampling and particle size determination

Table 2.4 Qualitative terms for particle shape Acicular needle-shaped Angular sharp-edged or having a rough polyhedral shape Crystalline freely developed in a fluid medium of geometric shape Dentritic having a branched crystalline shape Fibrous regularly or irregularly thread-like Flaky plate-like Granular having approximately an equi-dimensional irregular shape Irregular lacking any symmetry Modular having rounded, irregular shape Spherical global shape

Rod Cube Hake Needle Prism Rake

• Fig. 2.9 Form and proportions

Qualitative terms [10] may be used to give some indication of particle shape but these are of limited use as a measure of particle properties (Table 2.4). Such general terms are inadequate for the determination of shape factors that can be incorporated as parameters into equations concerning particle properties where shape is involved as a factor. In order to do this, it is necessary to be able to measure and define shape quantitatively. Particle shape analysis can be carried out using pattern recognition techniques [11-14] in which input data are categorized into classes. The potential use of these techniques [15] and the use of the decision function in morphological analysis have been introduced. There are two points

Data presentation and interpretation 71

regarding the assessment of particle shape. One is that the actual shape is unimportant and all that is required is a number for comparison purposes. The other is that it should be possible to regenerate the original particle shape from the measurement data. The numerical relationships between the various 'sizes' of a particle depend on particle shape, and dimensionless ratios of these are called shape factors; the relations between measured sizes and particle volume or surface area are called shape coefficients. Heywood [16] recognized that the word 'shape' in common usage refers to two distinct characteristics of a particle, form and proportions. The former refers to the degree to which a particle approaches a definite form, such as a cube, tetrahedron or sphere, and the latter by the relative proportions of the particle which distinguish one cuboid, tetrahedron or spheroid from another of the same class (Figure 2.9). When three mutually perpendicular dimensions of a particle may be determined, Heywood's ratios [17] may be used: Elongation ratio n = = L/B (2.15) Flakiness ratio m = B/T (2.16) (a) thickness T is the minimum distance between two parallel planes which are tangential to opposite surfaces of the particle, one plane being the plane of maximum stability. (b) breadth B is the minimum distance between two parallel planes which are perpendicular to the planes defining the thickness and are tangential to opposite sides of the particle. (c) length L is the distance between two parallel plane which are perpendicular to the planes defining thickness and breadth and are tangential to opposite sides of the particle. Consider a particle circumscribed by a rectangular parallelepiped of dimensions Z by 5 by 7, then the projected area of the particle is given by: A = ~dl=a,BL (2.17) where a^ is the area ratio. The particle volume equals the projected area multiplied by the mean thickness:

a^jl^a^BLp.T

(2.18)

72 Powder sampling and particle size determination

• Fig. 2.10 Heywood's dimensions

where/7,. is the prismoidal ratio (see Figure 2.10). Combining equations (2.17) and (2.18), and eliminating B, L and T from the resulting equation using equations (2.15) and (2,16), gives:

T m

V ^

If the particle is equi-dimensional i.e. B = L = T and n = m volume coefficient takes on a special value a^ where:

8 (2.19) 1, then the

(2.20)

Thus, a^ may be used to defme particle form. When the particle is not equi-dimensional, the appropriate value of a^^ is a/m^n which substantiates the earlier statement that shape is a combination of form and proportions. Heywood classified particles into tetrahedral, prismoidal, sub-angular and rounded. Values of a^ and/?^ are given in Table 2.5 [18].

Data presentation and interpretation 73

Table 2.5 Values of a^ and/?^ for particles of various shapes Shape group . rtetrahedral ^"^"'^"•Iprismoidal sub-angular Rounded Table 2.6 Values of a Shape group Geometrical forms tetrahedral cubical spherical Approximate forms , ^tetrahedral ^"^"'^'Vrismoidal sub-angular Rounded «« 0.50-0.80 0.50-0.90 0.65-0.85 0.72-0.82 Pr 0.40-0.53 0.53-0.90 0.55-0.80 0.62-0.75 g^ and C for particles of various shapes

0.328 0.696 0.524 0.38 0.47 0.51 0.54 C 4.36 2.55 1.86 3.3 3.0 2.6 2.1 a^^ can be calculated using equation (2.19) combined with direct

• bservation to determine the shape group into which the particle fits and

the values of w and n. This is practical down to sizes as small as 5 | L i m by measurements of the number, mean size, weight and density of closely graded fractions. Indeed, a^^ may be determined directly by weighing a known number of particles of known mean size. a^^ is more difficult to determine, but Heywood developed the following relationship on the basis of a large number of experimental measurements: «,,^=1.57 + C

^ (2.21) in which C is a constant depending on geometrical form. Table 2.6 shows the values of a^^ and C for various geometrical forms and also for irregular particles.

74 Powder sampling and particle size determination

Macroscopically, shape may be derived using shape coefficients or shape factors. Microscopically particle texture may be defined using fractals or Fourier transforms. The introduction of quantitative image microscopy has made such approaches to particle texture analysis practical. Laser diffraction not only provides a particle size distribution it can also provide a value of volume concentration. For spherical and granular particles, this volume is in accord with measured volume concentration. For plate-like particles, the measured volume is an over-estimation that is related to the thinness and aspect ratio. The thickness calculated, using this method, has been used to rank several graphite powders. Results agree closely with those obtained by the BET method [19]. By using a laser diffraction particle size analyzer as a light scattering spectrometer, the fractal dimensions of clusters can be determined. This follows on the work

• f Prod and Kratky [20] who used neutron and x-ray diffraction scattering.

They demonstrated that the plot of log scattering against wavenumber exhibited fractal behavior. 2.5.1 Shape coefficients There are two especially important properties of particles, surface and volume, and these are proportional to the square and cube respectively of some characteristic dimension. The constants of proportionality depend upon the dimensions chosen to characterize the particles; the projected area diameter is used in the following discussion. The surface area and volume of a particle are given by the following equations: Surface of a particle, S = %d^^ - cCs,a^a ^ ^a Volume of a particle, V = —dl =a^ ^d^ = x\ (2.22) 6 where a^ and a^ are the surface and volume shape coefficients, the additional suffix denoting that the measured diameter is the projected area diameter, (for a sieve analysis for example, S - oc^^d^ and so on). It must be noted that, in general, particles do not have a unique surface area, the measured surface depending on the method of measurement that is the degree of discrimination. At a low level of discrimination the area of the convex envelope of the particle is measured; at higher levels the areas

• f concavities in the surface are included. Similarly, unless the particles

Data presentation and interpretation 75

are homogeneous, their measured volume also depends on the measurement technique. The surface area per unit volume (volume-specific surface) is the ratio

• f iS to V\ For example, for a single particle:

(2.23)

• ^v

S 6d^ <^s,ada ^v,a4 6

da

(2.24) where d^y, the surface-volume diameter is the diameter of a sphere having the same surface to volume ratio as the particle, asv_a is the surface-volume shape coefficient by microscopy. Similarly the volume-specific surface of a single particle by microscopy is: S.,a=6/d, (2.25) and the volume-specific surface of a single particle by sieving is defined as:

^vj^^ldA

(2.26) Thus, the manner in which the surface area changes from sample to sample can be investigated on the assumption that shape is size independent. Surface-volume shape coefficients have been determined for quartz and silica from surface area measurements using nitrogen adsorption giving \A<asya^\i with no significant variation with particle size. Fair and Hatch [21] found that, by measuring smoothed surfaces, values of a^^^ as low as 7 were found. Crowl [2] found that with Prussian blue, the specific surface by nitrogen adsorption applied to the primary particles of which each single particle was made up (Figure 2.5). With red iron oxide the specific surface applied to the single primary particle. Volume shape coefficients may be determined from knowledge of the number, volume mean size, weight and density of the particles comprising a fraction graded between close limits e.g. by sieving. Further, if surface areas are also determined by permeametry, surface shape coefficients may

76 Powder sampling and particle size determination

also be determined though this will differ from coefficients based on gas adsorption measurements. Hence, when any shape coefficient is quoted, the method of obtaining it should also be stated. 2.5.2 Shape factors The size of a particle may be expressed by a single dimension using one of the diameters defined in Table 2.1. The differences between these dimensions increase as the particle diverges in shape from a sphere. For a population of particles whose shape is not size dependent, distributions

• btained using different methods of analysis may be homologous.

Multiplying the sizes of one distribution by a constant (shape) factor will therefore generate the other distribution. One of the earliest defined shape factors is the sphericity xj/^ that was defined by Wadell as [22-24]. surface area of a sphere having the same volume as the particle ^^ ~ surface area of the particle Ww (2.27) This is less than unity for non-spherical particles. The reciprocal of the sphericity has been termed the coefficient of rugosity [25] or the coefficient of angularity [26]. At low Reynolds number and with convex particles, the drag diameter equals the surface diameter and Stokes diameter may be defined as:

dst^

dst-¥wd,

(2.28) Thus, for non-spherical particles d^<d^. Further, the surface-volume diameter is given by:

\^s J

Data presentation and interpretation 77 ds.=y^wd. (2-29) Hence, for non-spherical particles d^^< d^^<d^. Krumbein's [27,28] definition of sphericity is:

¥\ =

fB^^

\^J

(2.30) where L is the longest dimension of the particle, the breadth B is measured perpendicular to this and C is the particle thickness. (Note: these definitions differ from Heywood's). For microscope analysis Laird [29] prefers the definition: V,-^ (2.31) Where S^ is the surface area of a sphere with a diameter equal to the equivalent diameter {S^ = 4A) and S is the surface area of the particle computed from the measured surface area. For rounded images, whose principle dimensions in two directions at right angle to each other are a and b. Heywood [16] quotes the semi- empirical formula for the equivalent diameter: d = ^' \n

^-xOJ7ab\ (2.32) For rectangular images the following equation yields a result that is only 1% different to the one calculated above:

d,={abf^

(2.33)

Particles usually rest on microscope slides in the position of greatest

• stability. Cylindrical particles, of length kd where d is the cross-sectional

diameter, would be expected to rest with the axes horizontal for k>\ and vertical for k<]. Laird [29] found that this was so and that a region existed, 0.85<A:<1.5 where both orientations were adopted.

78 Powder sampling and particle size determination

For discs or cylinders with A:<0.85:

VL--^^^

(2.34) ^^ 1 + 2)5: For cylinders with k>\.S\

¥ L - - ^

(2.35)

Laird also determined sphericity from sieving and sedimentation studies. For two-dimensional images the proximity of the image to the outline

• f a circle is defined by circularity where:

, . An X cross - sectional or projected area of particle outline circularity = (perimeter of particle) _ AnA

d

\dcj 2

(2.36) This has a maximum value of unity for a circle and decreases with increasing departure from sphericity. (It can also be used inverted). It is also known as compactness or roundness. Fuchs [30] introduced a dynamic shape factor, y/p', to relate volume and Stokes diameters:

y/f^ (d.

\^st J

(2.37)

From equation (2.28), for non-spherical particles d^>d^f, so that i//>\. 2.5.3 Applications of shape factors and shape coefficients As an illustration of the application of shape coefficients and shape factors consider a cuboid of side x, x, kx, where A : is a variable. In this case, from equation 2.27: [A.Snk^] ¥w=- — (2-38) ^ (l + 2yt)

Data presentation and interpretation 79

and, assuming d^^ = ^dl/d^ (see Table 2.1 and equation 2.28).

(4.571^2 y

A25

(2.39) ^sv,st is the surface-volume shape coefficient by Stokes diameter (d^^). Both l/i//^ and a^^ are plotted against k in Figure 2.11; the shape of the curve will vary according to the ordinate units employed, e.g. if Stokes diameter were chosen for the ordinate instead of k. From this graph it can be seen that both factors are at a minimum when the shape is most compact (i.e. a cube) and increase as the particles become either rod-shaped or flaky. 2.5

2 h 1.5 h

0.1 10

• Fig. 2.11 Relationship between shape factors and particle dimensions.

If an analysis is carried out by two different techniques, provided tht the particle shape is not size dependant, the two results can be brought into coincidence by multiplying by a shape factor provided that particle shape does not change with particle size. For example, if the medium Stokes diameter is 29.2 |im and the median Coulter diameter is 32.0 |im for particles of cuboid form, multiplying the Stokes diameter by [32.0/29.2]

80 Powder sampling and particle size determination

will yield the Coulter distribution. Thus the shape factor for this powder, relating Stokes and Coulter diameter is 0.913. From equation 2.28, d^ = 38.4 | L i m and, from equation 2.27 i//^= 0.694. Thus the particles are discs (k = 0.329) or cylinders (k = 3.55) (Figure 2.10). It can be deduced from this that unless the particles are grossly irregular, the difference between Stokes diameter and Coulter diameter is very small. Ellison [31] obtained a value of 0.9 for the ratio of the sizes of silica particles determined by settling experiments and mounted in agar in random orientation. Hodkinson [32] found, from measurements on quartz particles by light scattering, a diameter ratio of 0.8 between particles in a liquid suspension and settled particles. Cartwright [33] attempted to find the magnitude of the difference in mean projected diameters between quartz particles in random and stable orientation by microscopy. He used four different mounting techniques and found no significant differences. He attributed this to the difficulty of mounting particles in random

• rientation.

These factors for the mean ratio of projected diameters for random and stable orientation are indicative of the properties of the powder and therefore of use to the analyst. Respirable coal mine dust samples from three different US mines were classified into four fractions using a Bahco classifier [34]. Shape factors were determined as ratios of the following mean diameters. From microscopy:

da

Y.ndl

From photosedimentation (2.40)

dp =

(2.41) From krypton gas adsorption (2.42)

^BET K-^wj

'es.^'"

yTlN^y

Data presentation and interpretation 81 dsy=-—- (2 A3) 2pAp p is from density measurements.

N\^, is the number of particles per gram of powder,

Ap is the projected area by photosedimentation. It was postulated that a relationship might exist between shape and the incidence of pneumoconiosis. For narrowly classified mica flakes and carbon fibers in the 1 to 100 ]xm size range it was found that particle shape could be estimated from the ratio of median sizes by laser diffraction and sedimentation [35]. Austin found that conversion between Sedigraph and sieve analyses depends not

• nly on a mean shape factor, but also on size distribution. He generated an

equation that applies for overall conversion when the sieve distribution followed a Schuman form (Equation 2.97) [36] Hausner [37] proposed a method of assessing particle shape by comparing the particle with an enveloping rectangle of minimum area. If the rectangle length is a and its width b, three characteristics are defined: Elongation ratio x = a/h Bulkiness factor j^ = A/ab Surface factor z = C^I\2.6A (2.43) where A is the projected area of the particle and C is the perimeter. Medalia [38] represents the particle in three dimensions as an ellipsoid with radii of gyration equal to those of the particle and defines an anisometry in terms of the ratio of these radii. Church [39] proposed the use of the ratio of the expected values of Martin's and Feret's diameters as a shape factor for a population of elliptical particles. Cole [40] introduced an image-analyzing computer (the Quantimet 720) to compare longest chord, perimeter and area for a large number of particles. Other parameters have been proposed by Pahl

• et. al [7,41], Beddow [42] and Laird [29].

Barreiros et. al. [43] tested three different particle shape materials; glass beads, crushed glass and mica using four particle sizing instruments; Coulter Multisizer, Sedigraph, Malvern 2600C and microscope. Not surprisingly they found wide variations between the derived distributions.

82 Powder sampling and particle size determination

In essence the Coulter and the Sedigraph median diameters were almost identical in all cases. For glass beads all the instruments generated similar median values. For crushed glass the median diameter with the Malvern and microscope were, respectively 28%, 45% greater, and for the mica the ratios were 1.96 and 2.43. They also found that the Malvern broadened the

• distribution. These differences are to be expected since, unless the particles

are greatly irregular, the volume diameter is similar to the Stokes. The projected area diameter by microscopy neglects the smallest particle dimension and one would expect a much greater median size for flaky particles.

The Malvern does not measure particle size as such but

measures the forward light flux distribution that it converts to the distribution of (opaque) spheres that would generate the same light flux

• distribution. They also calculated shape factors based on surface area

determination by krypton adsorption. Unfortunately the equations they applied are incorrect. 2.5.4 Shape indices Tsubaki et. al. [44,45] argue that many of the proposed shape factors have little practical relevance to the analysis of real powders until the advent of electronic equipment and the computer. They define six shape indices based on the following diameters (see Table 2.1): d^, d^, dp^, dj^. The shape indices are: i//^^, y/^f,, y/^j^, y/^^, y/^^^, y/^^ where, for example, y/^^ = djd^. They define statistical diameters and coefficients of variation according to equations (2.1) to (2.4). Further, elongation Z was also studied where: Z = —^^^ (2.44) dp .

dp is the minimum value of the Feret diameter and dp is the diameter perpendicular to this. Three more indices were added later, y/p^^, y/^^^ and

The arithmetic average of breadth and length is defined as follows:

^F=iK.„+^F.,,)

(2.45)

Data presentation and interpretation 83

The dynamic shape factor K\S defined as the ratio of the resistance to motion of a given particle divided by the resistance of a spherical particle

• f the same volume.

Under laminar flow conditions: 'd^

K^st J

(2.46) Since this is a squared term that, for comparison purposes, they wished to reduce to unit power they introduced the shape factor y/^^y={d^/d^). [Note

K ~y/w where if/^ is the Wadel sphericity factor]. For non-re-entrant particles, according to Cauchy's theorem, d^ = E{dp) and i//p^^ has a maximum value of 1.0 for circles, rectangles and other convex shapes; it is therefore very useful for indicating the extent of concavities, y/j^^, y/^p, ¥RF^ ¥R^ ¥F ^^^ ^ where found to show mainly the slimnness of the particles, the best indicators being y/p and Z in that order.

y/^f^ and y/^p^ were found to correspond poorly with particle morphology.

2.5.5 Shape regeneration Briefly, this method consists of finding the center of gravity (G) of a particle and its perimeter (particle outline), from which a polar coordinate system is set up. A fixed angular coordinate interval, say InlN radians, is chosen and the distances of G to the boundary at these A ^ points are determined so that the shape of the particle is represented by these distances {A^ and their angular coordinates {0^. Thus for 7V(= 8, 16, 32) measurements, determined on the image of the particle, the Fourier coefficients {A^ and {6^ can be estimated. For each particle a Fourier representation is obtained, giving pattern vectors of 8, 16 or 32 elements, which are subject to feature extraction, classification and recognition [46]. In 1969 and 1971 Meloy presented papers in which fast Fourier transforms were used to process particle silhouettes as signals [47,48] and this work was extended in 1977 [49,50]. One of their main conclusions was that particles have 'signatures' which depend on A^ and not on 6^ and they proposed the equation:

84 Powder sampling and particle size determination

"^n -A ' • (2.47)

A plot of A^ against log n yields a straight line of slope s that depends on particle shape, rounder particles having lower s values [51]. Beddow [42] showed how a number of particle silhouette shapes could be analyzed and reproduced by Fourier transforms. Gotoh and Finney [52] proposed a mathematical method for expressing a single, three-dimensional body by sectioning as an equivalent ellipsoid having the same volume, surface area and projected area as the original body. The sliape of individual particles can be characterized using Fourier grain analysis or morphological analysis [53-55]. The method has been used to analyze beach sand silhouettes, to relate attrition rate in a milling

• peration to particle shape [56,57] and it has been extended to measuring

the shape mix in powders [58]. Particle shape grouping by the co-ordinate detection function with Fourier analysis has been discussed by Shibata et. al [59]. Orford and Whalley [60] discuss various quantitative grain form analysis techniques. However most of these shape descriptors either do not have the pre-requisites of ideal shape parameters, namely, invariance to translation, rotation, scale and starting point, or are reported as a single- value factor that results in low discriminating power. To overcome these problems an image analysis technique of using Fourier descriptors was developed [61]. The technique, referred to as the Zahn-Roskies (ZR) Fourier descriptor representation, is based on the Fourier expansion of the angular bend of the periphery of a particle as a function of its arc length. The authors state that the technique of applying image analysis, ZR Fourier descriptors and neural networks, in conjunction with fiber optic imaging techniques seems promising and well suited to on-line monitoring and control of particulate processes and in particular crystallization since both size and shape parameters are made available. 2.5.6 Fractal dimensions characterization

• f textured surfaces

Rough (textured) particles do not have a unique surface. The measured surface depends upon the method of measurement and will increase as the degree of scrutiny increases. For example, corrugating a one-acre field by plowing it will increase its superficial area to V2 acres. Texture is difficult to define and quantify. Davies [7] for example, defined texture as the number of asperities possessed by a particle outline.

Data presentation and interpretation 85

He generated a shape distribution histogram by plotting asperity frequency against particle perimeter and defined a mean texture for a powder in terms

• f asperities per mm. Using these definitions it is possible to monitor

particle abrasion and quantify the potential of conveyed powders to generate dust [62]. Texture has also been defined in terms of roughness or rugosity where the rugosity coefficient is defined as the perimeter of the particle outline divided by the perimeter of the convex hull [63]. Mandelbrot introduced a new geometry in a book, which was first published in French in 1975, with a revised English edition [64] in 1977. In 1983 he published an extended and revised edition that he considered to be the definitive text [65]. Essentially he stated that there are regions between a straight line that has a dimension of 1, a surface that has a dimension of 2 and a volume that has a dimension of 3, and these regions have fractional dimensions between these integer limits, Kaye has presented an excellent review of the importance of fractal geometry in particle characterization [66]. If an irregular outline is enclosed by a polygon of constant side length X, the perimeter P^^ will increase as the side length decreases. For a polygon with n sides: Pp^ = nl (2.48) Mandelbrot showed that: P;^ = k2}-^ (2.49) Hence plots of log P^ against log /I will have a slope of \-D, The parameter D is characteristic of the texture of the particle and was called by Mandelbrot the fractal dimension. The fractal dimension for the outline

• f a particle lies between 1 and 2, the more irregular the outline, the higher

the value. As an illustration of the technique consider a map of the British coastline (Figure 2.12). The fractal dimension is found to depend upon the degree of scrutiny [67,68] (Figure 2.13), having a dual value of 1.41 for large step lengths decreasing to 1.28 for short step lengths. The method has found applications in determining the ruggedness of particle profiles and can be modified for use with image analyzers.

86 Powder sampling and particle size determination

• Fig. 2.12 Fractal analysis of the British coastline.

• Fig. 2.13 Relationship between step length {X) and perimeter {P^^for the

British coastline The texture of surfaces can also be described using fractals, the more irregular the surface, the higher its fractal dimension; a fractal surface is defined as being a surface in which increasing and similar detail is

Data presentation and interpretation 8 7

revealed with increasing magnification, i.e. the dimension is scale invariant [69]. This parameter has been used to examine rice hull (husks) [70]. Because of its high energy content, rice hull can be a source of energy for the rice milling process. The major obstacle has been its non-uniform flow in the reactor, attributable in part to its irregular geometry. Kaupp [71] fitted a polynomial to describe the surface irregularities as observed with an electron microscope by tracing the circumference with a computer guided image analyzer. Chan and Page developed an algorithm to calculate the boundary fractal dimensions of boundary segments [72]. The algorithm was validated by accurate calculation of the fractal dimension of the Koch island fractal. The algorithm was applied to measurements on three atomized copper powders having major differences in the roughness of their profiles. The fractal dimension was found to vary significantly reaching a peak at a magnification of 800 to 2000 and falling appreciably as the magnification was increased to 14,000. This sensitivity to magnification indicates that these powders do not have the self-similar profile characteristics of true fractals. This does not exclude fractals as a useful shape descriptor at some magnifications since they have been shown to be more sensitive than conventional shape factors in correlating particle packing, flow and inter-particle friction [73]. However the results indicate that care needs to be taken in their evaluation. Fan et. al [74] determined surface area by nitrogen gas adsorption and stated that the fractal dimension could be determined by adsorbing different sized molecules on similar surfaces or, keeping the adsorbate the same, but changing the size of the adsorbent. They adopted the latter procedure by using between sieve fractions. In the first case the specific surface S is related to the fractal dimension D^f hy equation (2.50) and in the second the relationship given in equation (2.51) applies. ^^^_0.5(D-'/-2) (2.50) 5 = r-<^-^'-3> (2.51) where cris the cross-sectional area of the adsorbate molecule and r is the particle radius. Avnir et. al [75] used molecules of different sizes and foundl<Z)<2. Fractal analysis is also being applied to computer modeling of aerosol agglomeration [76] and the reaction kinetics of pigment formation [77,78].

88 Powder sampling and particle size determination

The fractal surface of particles has been linked to their shedding propensity [79] and to erosion during pneumatic conveying [80]. For a recent review

• f the application of fractals to particle morphology readers are referred to

[81]. Pore wall roughness has also been determined by fractal geometry using mercury intrusion/retraction [82]. Pore wall roughness has also been determined by fractal geometry using mercury intrusion/retraction. [83]. Fractal analysis has been found to be a useful tool for not only characterizing the irregularities in surfaces but also for correlating these with the physical description and flow properties of pharmaceutical solids [84]. Size and shape factors in combination with fractal dimension and rheological tests provide, prior to preformulation assays, a reliable technique for the appropriate selection of materials and detection of undesirable properties related to shape-surface characteristics. 2.5.7 Other methods of shape analysis Gotoh and Finney [85] proposed a mathematical method for expressing a single, three-dimensional body by sectioning as an equivalent ellipsoid with the same volume, surface area and average projected area as the

• riginal body.

Micro-cuboids in three-dimensional turbulent flow have been examined using Fraunhofer diffraction [86]. Both dynamic and static three dimensional particle shape features could be obtained and served as a basis for particle shape analysis by pattern recognition. The use of wedge-shaped photodetectors to measure forward light scattering intensity has also been explored for determination of crystal shape [87]. 2.5.8 Sorting by shape Particle may be sorted by shape by taking advantage of their behavior on a sloping, vibrating surface. Rounded particle tend to roll, granular particles to hop and flaky particles to shuffle. Ridgeway and Rupp [88,89] used a table designed for sorting industrial diamonds (Jeffrey-Galion) for their work whereas others used a rotating disc [90,91]. Slotted sieves have also been used for shape sorting. These have the advantage that they sort quantitatively on the basis of two of the particles' principal dimensions and they sort more selectively than a sorting table [92]. Identical square mesh sieves have also been used to sort different shaped particles by residence

Data presentation and interpretation 89 time [93]. The Jeffrey-Galion table has also been used to sort milled pasta and milled gelatin by shape [94]. 2.6 Determination of specific surface from size distribution data Surface to volume (or mass) ratio is a fundamental property of a powder since it governs the rate at which the powder interacts with its

• surroundings. As examples, small crystals in a mother-liquor, with high

surface to volume ratios, dissolve or grow more rapidly than large ones. Medication, in powder form, can pass through the body relatively unadsorbed if the active ingredients are composed of large particles, whereas fine particles are rapidly adsorbed; in the former case there is little reaction to the medication whereas in the latter it could be toxic. Specific surface can be determined directly by permeametry, gas diffusion, gas adsorption and adsorption from solution and can also be calculated from size distribution data. 2.6.1 Determination of specific surface from a number count Consider a number count carried out by microscopy, where the measured diameter is the projected area diameter {d^. For an assembly of particles:

S=-^

'-^^

(2.52)

^

y max where there are An^ particles of projected area diameter da,-. Assuming that the surface-volume shape coefficient by projected area,

• csv.a, is size independent over the size range under consideration:

^^-y-^s.,"^

(2-53)

H^rdlr

90 Powder sampling and particle size determination

Thus if . S * ^ , is determined by some independent procedure, the manner in which the surface-volume shape coefficient by microscopy, asv,a, changes with projected area diameter can be investigated. Alternatively, assuming ^sva ^ ^' ^^ volume-specific surface by microscopy, S^^ can be determined. 2.6.2 Determination of specific surface from a surface count Consider a size distribution obtained by a surface analysis method, e.g. photosedimentation, where the measured diameter is the Stokes diameter {d^^ and the surface fraction between two diameters centered on d^^^is Sf.. Let where then and therefore Pr S = Sr Vr Sr

= s,/s

I s,

^PrS = a,p,An,dl = (^y,p,Mrdlt,r

s,p,r

Pr Vr

^^v,p,r^St,r ^

Assuming that a^^^, the surface-volume shape coefficient by photosedimentation, is size independent over the size range under consideration:

^^^ (2.54)

Z Pr^St^r

2.6.3 Determination of specific surface from a volume (mass) count Consider a particle size distribution determination by mass where the fractional masses (volume =V^ of particles of mean diameters {d^ are

• determined. For a sieve analysis, for example, the measured diameters are

Data presentation and interpretation 91

20 dW/dx 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Particle size (x) in microns

• Fig. 2.14 Frequency distribution presented as a histogram and as a

continuous curve. The ordinate is presented as percentage per micron so that the area under the curve is 100 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Particle size (x) in |im

• Fig. 2.15 Cumulative percentage undersize curve.

92 Powder sampling and particle size determination

10 15 20 25 30

Particle size(x) in microns 35 dF(jc)/(k

40

• Fig. 2.16 Frequency distribution presented as continuous curve with the

abscissa on a linear scale. The right ordinate is presented as percentage/micron so that the area under the curve is 100.

• f -

• i

f

• |

• Fig. 2.17 Frequency distribution presented as continuous curve with the

abscissa on a logarithmic scale. The right ordinate is presented as percentage log-micron so that the area under the curve is 100.

Data presentation and interpretation 93 sieve diameters {d^ r^ ^Ar) ^^^ the fractional volume residing between two sieves of mean aperture d^ ^ is V^.

q^ =-^ where V = Y. K

then

V^ =qf.V = a^^^An^d:

and 5 * ^ =a^^^An^d^^ q,V

therefore 5 * ^ = ^sv,x,r

and: 5,= a,,, E - ^ (2.55) Example: Determination of specific surface from sieve analysis data Table 2.7 Calculation of mass specific surface from sieve analysis data Sieve size Mean sieve Mass fraction Surface area factor (|Lim) size residing on sieve (cm^ cm-3) ^^ (^V) [qrldA,)

33.7 40.0

rL2 From the data in Table 2.7, the volume specific surface by sieving is 509 cm^ cm"^. For a powder of density 2500 kg m-^ this becomes 204cm2^i. 2.7 Tabular presentation of particle size distribution Having defined the relevant particle size the frequency of occurrence of each size can be found. Number distributions can be determined by microscopy, electrical and light sensing zone methods; surface 105 150 210 89 125 178 0.30 0.50 0.20

94 Powder sampling and particle size determination

distributions by photosedimentation and mass distributions by sieving and x-ray sedimentation. Table 2.8 shows an example of size analysis data (^ can be number, surface or weight). Column 1 gives the interval of the size classes; column 2 gives the mean size of each of these classes; column 3 gives the percentage frequency of occurrence within these classes; these sets of data are accumulated in column 4; percentage/|Lim is presented in column 5, percentage/log(|Lim) (base 10) in column 6 and percentage/ln(|Lim) (base e) in column 7. Size limits (urn) 1 - ^ 3 - 5 5 - 7 7 - 9 9-11 11-13 13-15 15-17 17-19 19-21 21-23 23-25 25-27 27-29 29-31 31-33 33-35 Table 2.8 Tabular presentation of Mean size (nm) 2~ 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Relative frequency (%) d^ 0^00 0.04 0.22 0.88 2.70 6.48 12.10 17.60 19.95 17.60 12.10 6.48 2.70 0.88 0.22 0.04 0.01 Cumulative frequency (%) < ^ 0^00 0.04 0.27 1.14 3.84 10.32 22.42 40.02 59.97 77.57 89.67 96.15 98.85 99.72 99.95 99.99 100.00 size distribution data % per Hm

d(t)

Ax 0.00 0.02 0.12 0.44 1.35 3.24 6.05 8.80 9.98 8.80 6.05 3.24 1.35 0.44 0.12 0.02 0.01 % per log(|Lim)

d(l)

dlog;c 1 3 13 39 85 141 179 176 133 78 35 12 3 1 % per ln(|im) d^ dlnx 2 8 31 89 195 324 413 405 306 179 81 28 8 1 For narrow size distributions it is usually preferable to use an arithmetic size progression (as in Table 2.8) and present the frequency data in the manner given in column 5. For wide size ranges a geometric size progression (e.g. 2, 2V2, 4) is usually preferred with frequency data as presented as in column 6. For data manipulation the presentation in column 7 is preferred. The relationships between these data are as follows:

Data presentation and interpretation 95 d^ d^ d^ = X— = log(^) dln(x) dx dlog(jc)

0.434-

d^ dlog(x) (2.56) 2.8 Graphical presentation of size distribution data 2,8.1 Presentation on linear graph paper The simplest presentation of this data is in the form of a histogram on a relative and/or cumulative basis (Figures 2.14 and 2.15). Since the distribution is continuous, straight lines can be drawn through the histogram to produce one of two curves: frequency/micron and cumulative percentage distribution (either undersize or oversize). It is common practice to normalize particle size distributions (i.e. summate to 100%). In

• rder to compare distributions it is equally essential to normalize the

frequency distribution so that the area under this curve is 100%. The

• rdinate should always be plotted as percentage per micron if the size is

plotted on a linear scale (Figure 2.16). For the linear scale, the relationship between size and frequency may be written:

f{x)

6F{x) dx

JdF(x)= j/(x>k

(2.57)

[(/) = ¥{x)] may be replaced by W for a mass distribution, 5 * for a surface distribution and A ^ for a number distribution. Alternatively, the frequency can be plotted with a logarithmic scale for the abscissa. In this case the ordinate is calculated as percentage per log(micron) in order to normalize the area under the frequency curve to 100 (Figure 2.17). This type of presentation is particularly useful for size analysis data since many powders are logarithmically distributed and many instruments present data with sizes in a geometric progression. A linear plot of geometrically tabulated data compresses the data points at the fme end of the distribution so that detail is lost« If further data manipulation is envisaged Naperian logarithms are preferred (i.e. to base e rather than base 10) since data in this form lend themselves to computer manipulation, particularly if the notation popular in Germany is used [95]. If a log to the base 10 abscissa is used the area under the frequency curve decreases from 100 to 43.4 units in this case [see equations (2.24)]. It is common practice to plot size distribution data in such a way that a straight line results, with all the advantages that follow from such a

96 Powder sampling and particle size determination

• presentation. Some of the mathematical expressions for achieving this are

given below. 2.9 Standard forms of distribution functions Particle size, like other variables in nature, tends to follow well-defmed mathematical laws in its distribution. This is not only of theoretical interest since data manipulation is made much easier if the distribution can be described by a mathematical law. Experimental data tends to follow the Normal law or Gaussian frequency distribution in many areas of statistics and statistical physics. However, the log-normal law is more frequently found with particulate systems. These laws suffer the disadvantage that they do not permit a maximum or minimum size and so, whilst fitting real distributions in the middle of the distribution, fail at each of the tails. 2.10 Arithmetic normal distribution The normal law or Gaussian frequency distribution occurs when the measured value of some property of a system is determined by a large number of small effects, each of which may or may not operate. If a large number of the measurements of the value are made and the results plotted as a frequency distribution the well known Gaussian bell-shaped curve results. The equation representing the normal distribution is: d^ 1

— = —T=^y^\i

dx 0-V271

' ~ 2 ^

(2.58) and ^=j/(jc)djc (2.59) i.e. the distribution is normalized (sums to unity or 100%) (J is the standard deviation, I' is the mean size, (j) is the general term for the frequency; number, length, surface or volume (mass).

Data presentation and interpretation 9 7

The mean size, x, is defined as:

The standard deviation a; is defined as: (2.60)

cr = . \Y.{x-xf^(|)

ZA^

(2.61) Let ^ = • x-x -, then adt = dx and equation (2.57) becomes: d^ At exp

^ /^^

(2.62) Hence: exp

W

(2.63) A plot of d^/d/ against ^ results in the well-known 'dumb-bell' shape of the normal probability curve [Figure 2.18]. Table 2.9 Tabular solution to the normal probability equation t 0.5 1.0 1.5 2.0 3.0 Integral {(/)) 0.5000 0.6915 0.8413 0.9332 0.9987 0.9997

98 Powder sampling and particle size determination

• Fig. 2.18 The normal probability curve; relative frequency against standard
• deviation. 68.26% of the distribution lies within 1 standard deviation of

the mean [-!</<+!].

99.9 99 95 90 80 70 50 30 ?0 10 5 1 .1

• ~
• \

^ . ^

• ^50

^ ^

\^

2.

5 10 15

Particle size in microns

• Fig. 2.19 Graphical presentation of size analysis data on normal

probability paper

20

Data presentation and interpretation 99

2.10.1 Manipulation of the normal equation The fraction undersize the mean size 3 c is obtained by inserting the limits jc = -00, jc = 3c, i.e. / = - 00, / = 0, in equation (2.62) to give ^= 0.5, i.e. the mean and median are coincident. Similarly it can be shown that the mode is the same as the median. Writing -X'^ =-t^ / 2 , exp(-X ) is defined as: Qxp{-X^)= lim 1--

• X ^

Expanding by the binomial theorem: Qxp[- X^)=l + n\ (-X1 \ n ) n(n -1) 2!

• XI

+

n(n - \){n - 2) 3!

• XI

\ n ) . + - n\

r-x^y

(n-r)\r\ The fraction lying within one standard deviation of the mean is obtained by inserting the integral limits (jc = 3c, / = 0, X = 0 ) and (x = x + a, t =\, X = \H2). Therefore:

fd^ = - L {exp{-X^yX

^ X^ \ X^ \X^ (-ir^X^'^-i 3 2! 5 3! 7 r-\ Ir-\

1/V2

100 Powder sampling and particle size determination

Thus: 1 1 + 2!5 4i 3!7

V^, + ....

^(x + cr)-^(x) = 0.5642[0.707-0.118 + 0.018-0.002 + ] ^(x + cr)-^(3c) = 0.3413 The fractional area under the curve between the mean size and one standard deviation from the mean, 0.8413-0.5000 - 0.3413. 68% of the distribution falls within one standard deviation from the mean (see Table 2.9) and all but 0.26% lies within two standard deviations from the mean. Therefore the standard deviation equals the difference between the 84.13% size and the mean (median) size. Equation 2.63 is the basis for arithmetic probability graph paper and the solution is widely available in tabular form. A fundamental property of the normal distribution is that differences from the mean are equally likely, i.e. the probability of finding particles 10 units larger than the mean is the same as finding particles 10 units smaller. An example, plotted on probability axes, is shown in Figure 2.19 with a mean of 12 jiim and a standard deviation of 3 M ^ m so that the (84%-50%) size interval equals the (50%-16%) size interval = 3 |Lim. The distribution is completely described by these two parameters. Although it might be expected that this type of distribution would be common, it seems to occur

• nly for narrow size ranges of classified material. Most distributions are

skewed, usually to the right. 2.11 The log-normal distribution According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. With the log-normal law, it is ratios of equal amounts that are equally likely. In order to maintain a symmetrical bell-shaped curve, it is therefore necessary to plot the relative frequency against size in a geometric progression. The equation of the log-normal distribution is obtained by replacing x with z = In jc, in equation (2.58). Then:

y^

A(t)

dinjc 1 exp

Data presentation and interpretation 101

• r:

y =

1 dim: ^J2nln(J, -exp \lrvc - live J IXxi^c (2.64) X g is the geometric mean of the distribution (i.e. the arithmetic mean of the logarithms) and a^ is the geometric standard deviation; a^ is the standard deviation of z. is the product of the group data in which the frequency of particles of size jc is d^

• ^1
• ^O

[(/) = T V (number), S (surface) or ^(weight)] Zzd^

Zd^

_ Zzd^ z = lnx« Xlnxd^ Thus:

x,=[U:

(2.65) Since the particle size is plotted on a logarithmic scale, the presentation of data on a log-probability graph is particularly useful when the range of sizes is large. The geometric standard deviation can be read from the graph, as with the arithmetic distribution, and is given by: log cr =logX5o-logX]

log cr =logXg4-logx

(2.66) logag =Jlog(x84/X5o)

/ 02 Powder sampling and particle size determination The geometric mode, the geometric mean and the median coincide for a log-nomial distribution. In Figure 2.20 the median size x is 18 |Lim, the 84% size (7^^ is 25.2 |im and the 16% size xJa is 12.86 |Lim. The ratio of the 84% size to the 50% size is the same as the ratio of the 50%) size to the 16% size and equals the geometric standard deviation cr^ = 1.40. The distribution is completely described by the two numbers (18 and 1.40).

99.99 99.9 99

c

95 90 80 70 50 30

10

s

.1 h .01

[

i ^16

• ^50

^84

"

10 20

Particle size in microns

40

Fig. 2.20 Graphical presentation

• f

size analysis data

• n

log- normalprobability paper. 2.11.1 Relationship between number mean sizes for a log-normal distribution Consider a log-normal distribution by number such that: d^ = —T= |exp

^27tlnc7^J In jc - In X

dlrLx: = l (2.67) i.e. the distribution is normalized. dN^ is the number of particles in a narrow size range centered on x^. XQ and x^ are the smallest and largest

Data presentation and interpretation 103

particles present in the distribution and Cg is the geometric standard deviation [which is the same for number, length, surface and volume (mass) distribution]. The number-length mean diameter is defined as:

Normalizing i.e. equating the denominator to unity gives:

— J '

Inlna exp

In jc - In jc

Inlna g J

jcdlruc (2.68) Also

X^'(W.

^13 = '-^^ =I^.'dA^.

• ^A'i-

\ I^'^P

Inx-lnx.

lliXxiG ^

x^dlnx (2.69) 3 _ r=0

Y.^]^^r

= Ix'cW.

1 2n\n(j exp

In J C - In X

x^dlnx

(2.70)

104 Powder sampling and particle size determination Making the substitution: X In X - In X

^l2\n(

so that V2lncr^dZ = d(lnjc) Then: x.rr =

• JQxp(^[2X\nag-X^)iX

• ""g^

4^

Jexp(2V2 A' In a^-X^^X (2.71) (2.72)

,3 _ V

4^

Jexp(3V2Xlno-g-X2)lY (2.73) Making the transformations: V2 \\ = X hi a^ in equation (2.71) ^2 = X - V2 In (j^ in equation (2.72) }^=X X\x\<j in equation (2.73) Then ^gN f 1

• ^exp^lln2^^jjexp(-r,2)dl^

(2.74) tls =^exp(2ln2.Tjfexp(-r22)ir2 (2.75)

In^cT I}exp(-r32>ir3 (2.76)

Data presentation and interpretation 105 The integration yields a value / = VTT giving: \nx^^^^\nXgj^+05\n^cTg (2.77) ln.x-;v^-lnx^;v'+10'n^^g (2-78) \ux^y = \nxg^ + 1

. 5 In^ a^

(2.79) Similarly ln.v,;^v/ - In V + 2.0In^ cj^. (2.80) 2.11.2 Derived mean sizes If the number size distribution of a particulate system is found to be log- normal, equations (2.77) to (2.80) can be used to determine other mean

• sizes. For example, the volume-moment mean size is the mean size of a

volume (mass) distribution:

I 4AN,

Therefore: In xyj^ - 4In x^j^ - 3.0In x^vj/ (2.81) Substituting from equations (2.79) and (2.80): \\\xyj^^ = \nxg^j + 3.5In^ a^ (2.82) Similarly, the mean size of a surface distribution is given by: \nxsy = Inx^^ +2.51n^ a^ (2.83)

106 Powder sampling and particle size determination

Using this equation, the volume specific surface of the particulate system may be determined since: 2.11.3 Transformation between log-normaldistrihutions If the number distribution is log-normal, the mass distribution is also log- normal with the same geometric standard deviation. Using the same treatment as was used to derive equation (2.77) gives, for a mass analysis:

\nxyf,^ - \nx^y + 3.5In^ a^

(2.84) Comparing with equation (2.82) gives: \x\x^y - Injc^yv +3.01n^ a^ (2.85) Since the relations between the number average sizes and the number geometric mean are known [equations (2.77) to (2.80)], these can now be expressed as relationships between number average sizes and the weight geometric mean jc^^ to give a similar set of equations. \nx^f ^\nx^y-2.5\n^cj^ (2.86) Injcy^.s =lnjc^j -2.01n^cr^ (2.87) \nx^y=\x\Xgy-\.5\n^c7g (2.88)

\nx^,f^\nx^y-\mn'-cj^

(2.89) Other average sizes can be derived from the above using a similar procedure to that used to derive equations (2.84) and (2.85) to give: Inx^y - In x^y - 0.5In^ a^ (2.90) Similarly, for a surface distribution, the equivalent equation to equation (2.80) is: I n A

• ^

^ ^ -IHA^V +2.01n2cr (2.91)

Data presentation and interpretation 107

Substituting this relationship into equations (2.86) to (2.89) yields the equivalent relationships relating surface average sizes with the surface geometric mean diameter. 2.11.4 Relationship between median and mode of a log-normal equation The log-normal equation may be written: 1 d^_ dx X v27r In cr,

• exp(-X^)

(2.92)

where Incr =lnx-lnjc so that: V21n( AX _ 1 dx X At the mode {x = x^) and: —y = 0

djc

I.e. dx

d^ dx

— = 0 where Y = -j=^ expl-X^) Ax ^2%x\n(jg so that dx iexp(-X^) = 0 2(lnx:^

• \nxg)

V2I

x^42\ r\Gr

g J^--mv-«"-gy In x^ = In Xg - In cj^ (2.93)

These relationships are summarized in Table 2.10.

108 Powder sampling and particle size determination Table 2.10 Relationship between average sizes for log- normal In %^ = In x^+ 0.5 In^cr^ In Xj^^ = In x^+ 1.0 In^cr^ \nxj^y=\nx^^+ 1.5 In^cr^ \n Xj^j^= \n x^r+ 2.0 \n^ag In Xi^^^ In X g y y

• H 1.5 In^cr^

In x^,/= In X g y ^ / + 2.0 In^cr^ In x^^-= In x^-^ 2.51n2o^ equations In x^i^ In x^-^ 2.5 In^cr^ In Xsj^= In X g y v + 3.0 In^cr^ In Xyj^= In Xg^+ 3.5 In^cr^ In Xgs= In X g y v + 2.0 In^cr^ Inx^^- lnXg^+3.01n2(T^ lnx^= '"^g~^ 10 ^^^^g 2.1 J. 5 An improved equation and graph paper for log-normal evaluations Using equation (2.59) and the relationship [94] : x-^ -exp(lnjc) (2.94) the log-normal equation can be written:

d(f> _

1

dx yf2nx^\na. -exp

2'" ^^ exp

In jx/x^)

V " 21nH y

(2.95) This form of the log-normal equation is more convenient for use since the variable (x) only appears once. Equation (2.95) may be written in the simplified form: dl dx

• ^exp -Z>ln^(x/x^)

(2.96) The relationship between geometric mean and mode [equation (2.93)] takes the form:

C

:exp J_ lb {1S)1)

Data presentation and interpretation 109 This modified form of the log-normal equation simplifies parameter determination from log-probability plots of experimental data. The graph paper may be furnished with additional scales of b and C both being determined by drawing a line parallel to the distribution through the pole (0.25 ^m, 50%). 2.11.6 Application Consider a log-normal equation with a geometric mean x^ = 6.75 jiim and a standard deviation a^ = 1.64. According to equation (2.63) the mode x^ will be 5.27 | L i m making:

• ^=0.1344exp

djc

• 2.021n^(jc/5.27)

This form is particularly useful when further mathematical computation is envisaged, such as for grade efficiency GJx) since: ' ^ dFix) G,(x) = Er^ (2.98) dx where (d^^dx:) is the frequency of the coarse stream, (d^dx) is the frequency of the feed and E^ is the total efficiency (see Chapter 5). 2.12 Johnson's Sg distribution Johnson's S^ distribution is a bounded log-normal distribution, i.e. a truncated log-normal equation with a minimum and maximum size so that the whole size distribution is directly considered. This differs from conventional correlation analysis as summarized by Herdan [97]. It has been presented, in a series of papers by Yu and Standish [98-101] as a function that can represent all unimodal size distributions of particles. It may be written: dF(z)

110 Powder sampling and particle size determination

y

• ^min /

27c(^-^min)(^max-^) exp

^ 1 ^

vV2y //^+o-Jn (^-^min)

(2.99)

(^min<^<^max)

The log-normal distribution is a special case of the SQ distribution which can be obtained by extending the size limits to ±<x). If we let so that t^fj^ +(T^ In 2-z„

d/ =

) :('

• zV max

Az then: AF{z) AF{t) 1 y - —^^— = = .— expl dr dt ( t^\ V 2y If the theoretical fraction undersize the actual smallest size is negligible it is usually safe to assume that z^^^ =0 so that:

'(z)= I exp

( , i \

dz

• 00

where ^-(J^ In z-z^

In This reduces to the log-normal equation if <^is simplified t o: ^-G^{^xvz-\xvz^ Distributions that fit the Sg distribution will also fit the log-normal distribution but differences will occur in the end regions. These differences arise since the log-normal distribution does not admit a

Data presentation and interpretation 111

minimum or maximum size that must exist in any real system [102]. These small discrepancies can lead to large differences in their transformed size distributions. The modified log-normal equation has been applied to particle size determination in acoustic spectroscopy using the Pen Kem 8000 Acoustophor [103]. The authors state that the unmodified equation over- estimates the fraction of large particles and under-estimates the median

• size. They also state that the median size generated using the truncated

log-normal distribution for fitting the acoustic spectrum is not the 50% size. 2.13 The Rosin-Rammler-Bennett-Sperling formula This formula was derived originally for broken coal and has since been found to apply to many other materials [104]. The equation may be written: i?=100exp(-Z)x«) (2.100) where R is the weight percentage retained on a sieve of aperture x, \og{\miR)^bx^\oge Taking logs gives: log[log(l 00/i?)] ^ log b +wlog x + log(loge) i.e. log[log(100/ /?)] ^n\ogx + constant A plot of the log of the log of reciprocal percentage weight retained against the log of particle size generates a straight line. The slope /? is a measure

• f the particle size dispersion (size range) of the powder.

The peak of the frequency curve for « = 1 is at (100/^) = 36.8% and, denoting the mode by x^ equation 2.69 gives b = \lx^. The sieve aperture for R = 36.8% is used to characterize the degree of comminution of the material and, since the slope of the line on the Rosin- Rammler graph depends on the particle size range, the ratio of tan~' n and x^ is a form of variance. This equation is useful for monitoring grinding operations for highly skewed distributions, but should be used with caution since the device of taking logs always reduces scatter hence, taking logs twice is not to be

• recommended. An alternative form of the Rosin-Rammler equation is:

7?=100exp[-(jc/jCo)«] (2.101)

/12 Powder sampling and particle size determination

2.14 Other distribution laws Yu and Standish [100] list the laws presented in Table 2.11 that have been used for particulate systems. The cumulative form of the above distributions may also be presented as cumulative two-parameter equations: Gates-Gaudin-Schumann [105-107] F (JC) - {bxY Gaudin-Meloy [108] F (x) = [ 1 - (1 - to) «] b^ Roller [109-111] Svensson [112] F{x) - a^x exp

Fix) = xEpiy)

(2.102) (2.103) (2.104) (2.105) where: Ep{y) = — j(l//7)exp(-j;)dj; y = {x I x^Y where x^ is the mode of the distribution. Three and four parameter equations have also been proposed, e.g.: Harris [113]. F{x) = \-{\-bx'Y (2.106) For s = 1 this degenerates to the Gaudin-Meloy equation. 2.14.1 Simplification of two parameter equations. Tarjan [114] converted a two-parameter size distribution function from the form (f> ^f(x) to the form (/> ^fx/x^ 5), where J

C Q 5 is the median. The result

is an easy to handle function with a high degree of correspondence to the more complicated logarithmic function (t)j^ below:

Data presentation and interpretation 113

^^ =(/>(bxr

where (2.107) 1 u (p = f—— \ exp

^ «2 271 -a

(2.108) Table 2.11 The properties of some commonly used density distribution functions Name Equation Range of x Normal Log-nomial Rosin-Rammler [104] Gates-Gaudin-Schumann [105-107] Gaudin-Meloy[108] Roller [109-111] Harris [113] Martin [115] Gamma function [116,117] Weinig[118] Heywood[119] Griffith [120] Klimpel-Austin [121] Beta function [122] 1 Ax

llncj

1

• exp
• exp

V2

• V2ln(T„

dlnx V27ilncr^,

(^x ^

= nb{bxr'

Ax (t> = \-{\-bx)"

= C ^0.5

b ' +

' = Cx

(/>^Cx'e-'" (/> = Cx''-^e-' </> = Cx'e-''" (t>=^Cx-''e-'"''^ (t> = Cx^ <t) = Cx

\~n{\~Cxf \-n{\-Cxy-'

• 00,+ O

0,+ 00] ^' ^maxl ^' -^maxJ ^' ^maxJ

• ^maxJ

114 Powder sampling and particle size determination

Let the parameter b in equations (2.100), (2.102) and (2.103) be expressed in terms of J

C Q 5 when ^= 0.50.

Rosin-Rammler (RR) 0.50=exp(-AjCo 5), Gates-Gaudin-Schumann (GGS) 0.50=(^jCo 5) Gaudin-Meloy (GM): 0.50=l-(l - bxQ^)" Substituting back for b gives: RR ^^^=l-exp

In 2 (2.109) GGS

<I>CGS=^-^{XIXQS)"

(2.110) GM

• ( l - ( x / X o . 5 ) ( l - ^ )

(2.111) 2.14.2 Comments All the above equations are attempts to fit a straight line relationship to a frequency distribution. This procedure is worthwhile only if some benefit is derived. Narrow range size distributions are usually best presented as cumulative percentage undersize by mass or number on linear graph paper: The curve should then be differentiated (rather than using raw data) in

• rder to smooth out experimental error. A simple visual presentation of

this kind is preferred, for reasons of clarity, for presenting information to non-mathematicians. These distributions can also be plotted on arithmetic normal probability paper to examine deviations from normality. Wide size distributions are best plotted using a logarithmic scale for the size axis. Log-probability paper is extremely useful for examining data. Multi- modality is easy to discern as is the removal of fines (by classification) from a grinding process. Grinding processes often generate parallel distributions on log-probability paper and the time to reach a desired endpoint can be predicted. Simple mathematical relationships exist for

Data presentation and interpretation 115

Table 2.12 Illustration of the law of compensating errors Mean size (cl) 0.84 1.09 1.30 1.54 1.83 2.18 2.59 3.08 3.67 4.36 5.19 6.17 7.34 8.72 10.37 12.34 14.67 17.45 20.75 24.68 29.34 34.90 41.50 49.35 58.69 " %in interval (c2) 0.00 0.02 0.05 0.14 0.32 0.68 1.34 2.41 3.94 5.90 8.06 10.07 11.50 11.99 11.42 9.94 7.91 5.74 3.81 2.31 1.28 0.65 0.30 0.13 0.05 0.02 0.5c2 (c3) 0.01 0.03 0.07 0.16 0.34 0.67 1.20 1.97 2.95 4.03 5.04 5.75 5.99 5.71 4.97 3.95 2.87 1.91 1.16 0.64 0.32 0.15 0.06 0.02 0.01 0.25 X coarse (c4) 0.00 0.01 0.03 0.08 0.17 0.34 0.60 0.99 1.47 2.02 2.52 2.87 3.00 2.86 2.49 1.98 1.44 0.95 0.58 0.32 0.16 0.08 0.03 0.01 0.00 0.00 0.25 X fine (c5) 0.02 0.00 0.01 0.01 0.03 0.08 0.17 0.34 0.60 0.99 1.47 2.02 2.52 2.87 3.00 2.86 2.49 1.98 1.44 0.95 0.58 0.32 0.16 0.08 0.03 0.01 Unbiased error % (c6) = c2+c3+c4 0.02 0.07 0.16 0.36 0.76 1.44 2.52 4.05 5.95 8.02 9.93 11.26 11.72 11.19 9.80 7.87 5.80 3.92 2.43 1.38 0.72 0.34 0.15 0.06 0.02 Biased error % (c7) = 0.75c2+c4 0.00 0.03 0.07 0.18 0.41 0.85 1.61 2.79 4.43 6.44 8.57 10.43 11.62 11.85 11.05 9.43 7.37 5.26 3.44 2.06 1.12 0.56 0.26 0.11 0.04 0.01

116 Powder sampling and particle size determination True distribution 25% wrongly placed in the size category below

• Fig. 2.21 Illustrations of the law of compensating errors

Mean size {x) in microns

• Fig. 2.22 Effect of displacing half the distribution, without bias, in each

size interval, together with misplacing 25% in each size interval into the next finest interval.

Data presentation and interpretation 11V

log-noiTnal distributions so that size distributions may be rapidly converted from number to surface to volume. As demonstrated by Yu and Standish [100] distribution transformations can lead to unacceptable errors, particularly with distribution laws with no limiting maximum or minimum sizes. Even direct transformation of a number count to a volume count for example is unacceptable, unless the distribution is narrow, the count is excessively high or special procedures are employed, as with carrying out volume counts in microscopy. 2.15 The law of compensating errors In any method of particle size analysis it is always possible to assign the wrong size to some of the particles. If this error is without bias, possibility

• f assigning too great a size is equally as probable as assigning too small a
• size. This will modify the distribution but will have little effect on the

central region. An illustration is shown, in Table 2.12 and Figure 2.21, with 25% in each size category wrongly placed in the size categoiy below and 25% in the size category above, together with a biased distribution in which 25% in each size interval is displaced to the adjacent finer interval

• above. These are plotted as percentage per In (^im) against particle size in

Figure (2.22). The original log-normal distribution has a median size of 7.29 |im with a geometric standard deviation of 1.78; the unbiased distribution has a median size of 7.29 |am with a geometric standard deviation of 1.80 and the biased distribution has a median size of 6.99 |im (an error of 4%) with a standard deviation of 1.79. For measurements in arithmetic progression

• f sizes the effect is small, provided sizing is carried out at 10 or more size

intervals, and for a log-normal distribution the position of the mode is only slightly affected. 2.16 Evaluation of nonlinear distributions on log-normal paper A bimodal distribution is detectable when plotted on log-probability axes by a change in the slope of the line. It is also possible to deduce other features. 2.16.1 Bimodal intersecting distributions. Figure 2.23 shows relative plots of two intersecting (overlapping), log- normal distributions with medians of 16 and 10 jim and standard deviations of 1.4 and 2.0 respectively. Relative plots of blends of these parent distributions are shown in Figure 2.24 and these look to be mono-

118 Powder sampling and particle size determination

Particle size(jc) in microns

• Fig. 2.23 Parent distributions for bimodal log-normal intersecting

mixtures. AW 120 dlog(x) c

• e

a.

100 80 60 40 20 Mix

^

• • -x-

ratios

• 10:90
• 30:70
• 70:30
• 90:10

Particle size(x) in microns(log scale)

• Fig. 2.24 Frequency plots of bimodal intersecting log-normal distributions

in various mix ratios.

Data presentation and interpretation 119

99.99 99.9

• ^ 99

C

C

95 90 80 70 50 30 20 10 5 1 . 1 . 1

• o— Parent
• I— Parent 2

X- - 30:70 Mix

• ^— 70:30 Mix

10 Particle size(x) in microns

100

• Fig. 2.25 Cumulative plots of bimodal intersecting log-normal distributions

200

dln(jc) 150

100 h

50

10 Particle size(jc) in microns

• '

1 / \

* 1 aicnt 1

• I aicni z

H H

IOC

• Fig. 2.26 Frequency plots of bimodal non-intersecting log-normal

distributions in the ratios 30:70, 50:50 and 70:30.

120 Powder sampling and particle size determination 99.999 99.99 99.9 99 95 (u 90 .N 80 ^ 70

• 8 50

§ 30

^ 10 .01 .001

• I

• '^— Parent

"•— Parent

• »—30:70
• ^ ^ 50:50
• ^—70:30
• 1

• »

10 100

Particle size(x) in microns

• Fig. 2.27 Log-probability plots of mixtures of two non-intersecting log-

normal distributions.

100

Particle size in microns (log scale)

• Fig. 2.28 Cumulative plots of trimodal non-intersecting log-normal

distribution together with parent distributions (1:1:1 mixture).

Data presentation and interpretation 121 Particle size {%) in microns

• Fig. 2.29 Relative plot of trimodal non-intersecting log-normal distribution

together with parent distributions (1:1:1 mixture). Particle size in microns

• Fig. 2.30 Andreasen analyses monitoring a grinding operation.

122 Powder sampling and particle size determination

Particle size in microns

• Fig. 2.31 Effect of classification on a log normal distribution. (Full line =

parent distribution, dotted line = sub 6 |im particles removed). modal distributions. A log-probability plot (Figure 2.25) shows the parent distributions together with mixtures: It can be seen that the mixtures generate curves that are asymptotic to the parent with the wider distribution (spread). Thus a log-probability plot picks out bimodalities that would not otherwise be detected. 2.16.2 Bimodal non- intersecting distributions. Figure 2.26 presents relative plots of blends of two non-intersecting log- normal distributions with medians of 8 and 13 | L i m and standard deviations

• f 1.3 and 1.5 respectively. The areas under the two quite distinct curves

give the proportions of the two components. On a log-probability plot (Figure 2. 27) the mixtures are asymptotic to both parents and have a point

• f inflection where the two distributions overlap.

2.16.3 Other distributions Figure 2.28 shows a trimodal distribution together with the parent distributions (1:1:1 mixture). This may be easily resolved into its component parts if the parent distributions do not intersect Figure 2.29).

Data presentation and interpretation 123

Table 2.13 Andreasen analyses monitoring a grinding operation Median size 5.30 4.10 3.75 3.42 Grinding time (T hours) 9 13 15 16 Tx^ 47.7 53.3 56.3 54.7 2.16.4 Applications of log-normal plots Figure 2.30 shows Andreasen analyses monitoring a grinding operation. Since, in this case, the product of the median and the grinding time approaches a constant value it is possible to predict the grinding time required to attain a desired end point median size (Table 2.13). Figure 2.31 shows a log-normal distribution of geometric mean size 10 |im. The distribution is deficient in sub 6 |im particles, probably due to classification during a comminution operation, and is asymptotic to this size. If (x-6) is taken as the particle size the parent distribution is

• btained. A similar sort of plot occurs when there is a deficiency of coarse

particles. 2.16.5 Curve fitting An application of curve fitting using Kalaidograph [123] is shown in Figure 2.32. The experimental data are plotted as a differential lo-g- probability graph and the program asked to fit a trimodal log-normal equation to the graph: Estimates are made of the nine variables (percentages under the 3 modes, 3 means and 3 geometric standard deviations) and the curve fitting procedure determines the true values. The time to carry out this iteration depends upon how near the estimates are to the true values. The 34% widths of the modes are obtained by estimating the ratios of the two sizes having 8% of the distribution above and 8% below so that these two sizes contain 84% of the distribution. The geometric standard deviation is the square root of the ratio of these sizes. Thus, the percentages under each mode can be determined. The program can be used for other distributions having nine or less variables.

124 Powder sampling and particle size determination dW^ /dln(x)F

Particle size(jc) in microns

• Fig. 2.32 Curve fitting a trimodal log-normal equation to Microtrac SPA

data

W/ /^ln(jc)i

(b) Particle size(x) in microns

• Fig. 2.33 (a) Size distribution of a slurry pumped at 25 gpm (b) size

distribution of a slurry pumped at 100 gpm

Data presentation and interpretation 125 2.16.6 Data interpretation The curve fitting procedure is a useful aid in data interpretation. It has been used, for example, to demonstrate that a unimodal log-normal distribution with a substantial sub-micron fraction appears as a bimodal log-normal distribution when measured using a gravitational sedimentation technique; this bimodality being due to the effects of thermal diffusion (Brownian motion) [124]. Figures 2.33 illustrate the differences in the size distribution of a product pumped at different flow rates. Increasing the flow rate caused the slurry to plug on-line filters more rapidly. The differences are subtle: The main effect of increasing the flow rate is to decrease the size of the coarse mode (agglomerates) from 1.65 ^lm to 1.22 | L i m while the fine mode remains constant at 0.47 ± 0.01 fim and the spread (o-p decreases from 2.17 to 2.12. The percentage under the fine mode remains constant at 58% ± 1%. 2.17 Alternative notations for frequency distribution The notation given here is widely used but alternative notations have also been developed {German Standard DIN 66141, (1974)} [125-129] Although elegant, the German notation requires memorization and is most suitable for frequent usage and computer applications. 2.17.1 Notation Let the fractional number smaller than size x be:

aW= l^oWd^ (2-112)

• ^min

dx Hence ^oC^) is the fractional number in the size range x to x + dx. Further:

eoWmax= j 9 o W d ^ = l (2-113)

126 Powder sampling and particle size determination The subscript may be varied to accommodate other distributions, namely qr and Qr where: r = 0 for a number distribution; r = 1 for a length distribution; r = 2 for a surface distribution; r = 3 for a volume distribution. 2.17.2 Moment of a distribution The moment of a distribution is written as:

• ^max

A/,,= I x''q,{x)6x (2.114)

• ^min

Note: MQ^ = j ^^(jc)dx: = l

• ^min

For an incomplete distribution:

Mk.r{^e^^u)- JxVWdx

(2.115)

2.17.3 Transformation from q/^x) to qj(x) If ^X^) is known qf{x) may be determined using the following:

• ^max
• ^max
• ^max

J q^{^x)&K- J x^qQ[x)dx= J x^~^qf{^x)Ax

^ i n

• "-min

Hence: q^ ix)

^^-^

j ^'^~^9^(^)djc

x' 'q^ {x)

qAA- J

(2-'16)

The denominator is necessary in order to normalize the distribution function.

Data presentation and interpretation 127 Examples. To convert from a number to a volume (mass) distribution put / = 0, r = 3. To convert from a surface to a mass distribution put / = 2, r = 3. (a) Effect of particle shape This transformation is derived with the assumption that particle shape does not change with size. More correctly, a shape coefficient a{x) needs to be introduced:

,.(.)= ^^y"^'^')

(2.117) j a{x)x^~^q^{x)Ax 2.17.4 Relation between moments Putting ^ = 0 in equation (2.116) Substituting x^^^'q^^ix) for x^q^ix) in equation (2.114) gives:

J x^q^[x^Ax

More

generally:

(2.118) (2.119) Examples. To determine the surface-volume mean diameter from a number distribution, put / = 0, r = 2, A : = 1.

128 Powder sampling and particle size determination

To determine the surface-volume mean diameter from a surface distribution, put t='2,r = 2,k= 1.

Ml 2

2.17.5 Means of distributions (a) Distribution means:

• ^nax

X|^= — = ^ = A/, ^ (2.120)

(b) Arithmetic means

• ^max

M

• ^min

(^*,o) -

* _ ^kfi

, 0 = ^ ; ^ (2.121)

Data presentation and interpretation 129 (c) More generally: (^^,o) =

I X q^[^x)dx

2.17.6 Standard deviations For a number distribution, the variance is defined by: 0 - 0 = 1 (x-x,o) ?oWd^ (2.122)

0-0= I x'qQ{x)<ix-x^Q I qQ{x)dx

More generally: (T,= I {x-x,) q,{x)<ix (2.123) Alternatively: < T ^ =Mi^\ M

M,

• M,

(2.124)

130 Powder sampling and particle size determination

Replacing r by r + 1, and putting / = r and A : = 1 in equation (2.119), and substituting in the above equation gives: al^M^^,[My,^,-M^^,] (2.125) From equations (2.121) and (2.125): ^^r=\r\_\r^\ "^u]

(2.126)

2.17.7 Coefficient of variation

C]^-^^ {2A21) C , ^ = % ^ - 1 (2.128) 2.17.8 Applications (a) Calculation of volume-specific surface surface area S = ^ volume

I a,{x)x^qo{x)dx

J a^{x)x qo{x)dx where a^ and a^ are the surface and volume shape coefficients. Assuming that these are independent of particle size and defining the ratio of a^ to a^ as a^y, the surface-volume shape coefficient is:

Data presentation and interpretation 131 M20 a

(2.129)

Also, since M] 2 =^12 ^1,2 (a) Calculation of the surface area of a size increment ^v v^e^^u)''

(2.130)

^v\^e^^u)~'^s

Now:

^2,o('^e'^«) % 0

^2,0 ^3,o(^e'^J Sr(^,)= 1 ?.(^)d^ J jv'^^oC^)'^

a^)^

M.

Qrixi) _^r,o(^min»^/) M-,0

132 Powder sampling and particle size determination

so:

QA\)-QA^e)-

7^3,0 (^e'^«)

M

Hence:

• Ji, — OC^

%0 ft(^J-ft(^e)

(2.131) The application of this equation enables a surface area to be calculated from a summation of increments, i.e. M

M

M

j A: 9Q(x)dx:+ ^x^q^{x)&K + •

^ . = a.

M

(2.132) e^ = mm, e^ = max 2.17.9 Transformation of abscissae Suppose, in an analysis, < ^ , which is a function of x, is measured e.g. (^= ln(x). Since the amount of material between sizes x and x ^ Ax is constant, there must be a simple relationship between q{^ and q{x). Let jc - / ^ ) so that ^ - ^x); then: q,{^)A^^q^{x)(\x

Data presentation and interpretation 133 i.e. the quantity in the interval d^ is the same as the quantity in the interval

• dx. The new ordinate ^^(<^) is calculated from qf.(x) and the differential

ratio djc/di^. q;(^ = q,(x)^ (2.133) For example, suppose the following relationship holds: ^-x* dx and q;(^ = -L^q^(x) (2.134) Similarly for ^ = ln(x)

• 1
• 41 / ^

— = x I.e. dln(x) = — dx X and 9*[ln(jc)] = x9^(A:) In general, suppose we have the q^^ix) distribution and wish to find the q^ix^) distribution. From equation (2.118):

Substituting in equation (2.136) gives: M he'' g ; ( ^ = . / " ^ \ ; _ , (2.135)

• Example. In a Coulter Counter analysis, the pulse height V is proportional

to particle volume, i.e. ^= V = p^x^

134 Powder sampling and particle size determination (a) Calculation of M^ Q = (

• ^

r O )'^ By definition: ^r,o= \x'qQ(x)dx Using the transformation from the above equation:

• ^max

M.

^ / • o = — {p''x'-qQ{x)dx

Pr J.

A/,.

Pr

(2.136) x'' a* (x^ Also, since qr{x) = — , substituting in equations (2.135) and M.

rfi

(2.136):

q,{x) = x'-ql{4)^^ P'

<Jr(x)

(2.137)

Data presentation and interpretation 135 (b) Calculation ofMj^ ^ From equation (2.114):

• ^max

1

f^^^^^

^a+r)/3.0(^) ^ ^-'--(...)/3,0V9; (2.138) (b) Calculation of volume specific surface c _ ^2,0 which, using equation (2.138), can be written: ^ v - ^ - / ^ % ^ (2.139) Thus, specific surface can be determined from moments calculated directly from Coulter Counter data. (b) Calculation of mean size xj^ y Equation (2.121) may be written: Substituting for M^^ from equation (2.140):

'"^'--tT'i^^

(2.140)

136 Powder sampling and particle size determination

2.18 Phi-notation In geological literature dealing with particle size distribution [130,131] a very advantageous transformation of particle size is commonly used. This transformation replaces scale numbers based on linear millimeter values by the logarithms of these values. Because it is a logarithmic transformation it simplifies computation. The transformation is: ^--log2X, or X , - 2 - ^ (2.141) where X^ is a dimensionless ratio of a given particle size, in millimeters, to the standard particle size of 1mm. Phi - values can be found if the common decadic logarithms of X/ are multiplied by (loglO^)"^ = 3.322 For easy manipulation a conversion chart [132] or a conversion tale [133,134] can be used. The standard deviation cr^used in this notation is defined as: ^^ =0.5(^4-«>16) (2-142) The skewness is defined as:

^,^84+^16-2^50 (2.145)

^84-^16

Other statistical measurements used in geology for particle size distribution characterization (moment, quartile and others) have been defined [135,136]. References

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• Yu. A.B. and Standish, N. (1987), Powder Technol., 52, 233, 109

99

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