estimation of mineral grain size using automated mineralogy

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Minerals Engineering 20 (2007) 452–460

Estimation of mineral grain size using automated mineralogy q

David Sutherland

Mineralurgy P/L, P.O. Box 818, Toowong, Brisbane 4066, Australia

Received 24 August 2006; accepted 13 December 2006Available online 15 February 2007

Abstract

The sizes of mineral grains control liberation and the subsequent separation. Automated mineralogy provides a basis for making auseful estimation of this very important parameter. But the estimation is not straightforward, partly due to the stereological aspects ofthe problem, and partly due to the ill-defined nature of mineral grains.

The present paper gives suggestions for making these estimates. The suggestions are made on the basis of past experimental studies,supplemented by computer simulation of different shapes and sizes of grains. It is concluded that phase specific surface area provides thebest ‘‘size basis’’ for ranking different ores. The magnitude of shape effects and size distribution effects are discussed. But the next step ofpredicting the extent of liberation from grain size measurements is still difficult and needs experimental support.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Ore mineralogy; Liberation

1. Introduction

Mineral grain size is a critical parameter in any processinvolving the liberation and separation of minerals. Tradi-tionally it has been possible to get semi-quantitative esti-mates of the grain size by observation using an opticalmicroscope. The increasing application of SEM basedautomated imaging systems, e.g. QEMSCAN (see Intellec-tion website, 2006) and MLA (see JKMLA website, 2006),permits more reliable estimates to be made with the possi-bility of assisting process optimization.

Ore texture can be imaged relatively easily using modernsystems. The problem is to interpret these measurements ina way that can be related to the grain size as it controls theliberation of the particles.

There are two major difficulties:

• The estimate of the ‘‘size’’ is always biased since, due tosectioning, the image of the grain is always less than orequal to the true ‘‘size’’ of the grain;

0892-6875/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mineng.2006.12.011

q Paper presented at Automated Mineralogy 06.E-mail address: [email protected]

• Also, the characteristics of the sections observed dependon both the shape of the grains as well as the distribu-tion of sizes of these grains.

If all the grains were perfect spheres it would be possibleto estimate the size distribution of the grains in the ore.With particles, of a relatively regular shape, it is also pos-sible to estimate particle size distributions, e.g. see King(1984). But, unfortunately, mineral textures are notoriouslyhaphazard and have ill-defined shapes making it extremelydifficult to estimate the size distribution of grains. A prac-tical approach is to estimate a mean size then possiblymake some qualitative assessment of the grain size distribu-tion based on images of the ‘‘grains’’ within the texture.

2. Sampling

Grain sizes should ideally be estimated by measurementson rock samples or particles of ore significantly coarserthan the mineral grains. There are severe sampling prob-lems introduced if the measurements are based on selectedrocks so that the best practicable method is to take a suit-able sample that is sufficiently large, crush and sub-sample,then look at particles of about 1 mm in size. It is important

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D. Sutherland / Minerals Engineering 20 (2007) 452–460 453

during this size reduction that the well-known minimumsample mass criterion should always be observed.

This is a compromise size, small enough to permit a suf-ficient number of particles to be measured, yet largeenough to allow viewing of essentially unbroken ore. Thislatter criterion may not be true for very coarse ores withgrain sizes greater than 100 lm. But this is a very nice prob-lem to have!

3. The use of line scan measurements

The use of line scan measurements is preferred for grainsize estimation. The interpretation of area images of unbro-ken rock is more difficult. Grains are seldom neatly sepa-rated like raisins in a plumb pudding. Veins or layers ofmineralisation may control the liberation. The line scan isa much more straightforward measure and is readilydefined by the transition along the line scan from the min-eral in question to any other mineral. The result of such ameasurement is an intercept length distribution from whicha mean intercept length, or a mineral surface area or meanvolume can be calculated. The following discussion givessome indication of the effects of shape and size distributionon these measurements.

3.1. Phase specific surface area

Both the mineral surface area and the mineral volumecan be estimated in an unbiased way from a line scan mea-surement. This means that the surface area per unit volumeof a mineral – or its phase specific surface area (PSSA) –can be reliably estimated. The following useful relationshipexists between the mean intercept length L and the PSSA

PSSA ¼ 4=L ð1Þ

This is used for measurement but can also be used forinvestigations with regular shapes. In particular, the PSSAof a unit cube (or sphere of unit diameter) is 6 – there are

Fig. 1. Intercept length dist

6 unit square faces on the cube of unit volume. So a ‘‘size’’of mineral grains can be reported by assuming that thegrains are cubes or spheres. This size DPSSA is given by:

DPSSA ¼ 6=PSSA ð2Þ

This is an unbiased measure of a mean grain size assum-ing a simple shape.

4. Experimentally measured intercept length distributions

An example is given below (Fig. 1) of the run length orintercept length distributions for a series of copper ores.

These results are typical – such plots generally show thatthe most probable intercept lengths are small with a drop-off in number of intercepts for longer intercept lengths.This leads to a measurement problem since the data is verysparse for long run lengths, yet these contribute signifi-cantly to the total amount of mineral present. It is neces-sary to measure a reasonable number of run lengths toget reliable results. An indication of these statistical effectswill be given later.

Experimental difficulties can arise with the measurementof intercept length distributions. When there is poor min-eral identification long intercepts may be ‘‘broken’’ by falsemineral identifications. If this is seen to be a problemit may be worth estimating the PSSA using area mea-surements.

5. Effects of grain shape on intercept length distributions

5.1. Sphere

The sphere is the simplest shape. The result of makingone thousand random intercepts of equally sized spheresis shown in Fig. 2.

The theoretical distribution in this case is linear in form.A best-fit line is shown, and by increasing the number ofintercepts it would be possible to get closer and closer to

ribution of copper ores.

Intercept Length Distribution for Sphere

0

40

80

120

160

200

240

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1

Normalised intercept length

Nu

mb

er o

f in

terc

epts

per

100

0

Fig. 2. Intercept length distribution of a sphere.

Intercept Length Distribution of Square Rod (5:1:1)

0

50

100

150

200

250

300

350

400

450

500

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1


Nu

mb

er o

f in

terc

epts

Fig. 4. Intercept length distribution of a square rod.

454 D. Sutherland / Minerals Engineering 20 (2007) 452–460

the true linear form. The mean intercept length is 0.67 forthe unit diameter sphere.

The maximum intercept length is the sphere diameterand this is also the most probable intercept length. DPSSA

for a sphere is, obviously, the sphere diameter and thus acorrect estimate.

Clearly the distribution for a sphere looks nothing likethe measured distributions.

5.2. Cube

A similar plot for equal cubes is shown in Fig. 3. Thenumber of intercepts is again approximately 1000.

It is immediately obvious that the shape of this distribu-tion is very different from that of the sphere. In particular,with a sphere the most likely intercept length is the maxi-mum (sphere diameter), while for the cube this is the least

likely. The maximum intercept length is equal to the dis-tance between opposite corners of the cube or

p3 for a unit

cube. The probability for all sizes up to about 2/3 maxi-mum is approximately equal, but then falls away rapidlyfor longer intercepts.

Intercept Length Distribution for Cube

0

40

80

120

160

200

240

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1

Normalised intercept length distribution

Nu

mb

er o

f in

terc

epts

Fig. 3. Intercept length distribution of a cube.

The most probable intercept appears to be between 0.6and 0.7 or a little longer than the cube side.

The mean intercept length is about 0.385 on the normal-ized scale or 0.67 of the cube edge. Once again, DPSSA isequal to the cube size and is an accurate estimate.

It appears that the edges and corners of a cube contrib-ute strongly to small intercepts, making it more like themeasured distributions of real systems.

5.3. Rod

By compressing or stretching the cube along its axes,some idea can be obtained of the shape effects of flat orelongated mineral grains. Two examples are given: firstan elongated square rod (Fig. 4) and then a square flatplate (Fig. 5) are considered.

Consider a square rod of unit thickness and length 5.The number of intercepts shown is about 900.

The distribution now shows a reasonably flat shape forthe lower quarter of possible lengths then drops off sharplyfor longer lengths.

The maximum intercept length, used for the normaliza-tion, is 5.20 times the rod thickness. On the normalized

Intercept Length Distribution of Square Plate (5:5:1)

0

50

100

150

200

250

300

350

400

450

500

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1


Nu

mb

er o

f in

terc

epts

Fig. 5. Intercept length distribution of a square plate.

Table 1Summary of shape effects

Shape Size (mm) Maximum intercept Mean intercept PSSA (mm2/mm3) PSSA size (mm)

Sphere Dia = 1 1.00 0.67 6.00 1.00Cube Side = 1 1.73 0.67 6.00 1.00Square rod – 5:1 Thickness = 1 5.20 0.91 4.40 1.36Infitite rod Thickness = 1 Infinite 1.00 4.00 1.50Square plate – 5:1 Thickness = 1 7.14 1.42 2.80 2.14Infinite plate Thickness = 1 Infinite 2.00 2.00 3.00


scale the rod length is 0.96 and the rod thickness 0.19. Themost probable intercept length falls in the range 0.2–0.3, sois a little longer than the rod thickness.

The mean intercept length is 0.21 on the normalizedscale or 1.11 times the rod thickness.

The size as estimated by DPSSA is 1.36. This is againbetween the thickness and length dimensions of the rod,but quite close to the rod thickness.

Using Eqs. (1) and (2) it is possible to estimate L andDPSSA for an infinite rod. These are useful as limiting con-ditions. For an infinite rod, the mean intercept length L is 1and DPSSA is 1.5.

5.4. Plate

Consider now the case of a square plate with thickness1/5 of the plate edge. The total number of intercepts shownis again about 900.

Once again the shape of the distribution is very differentshowing a maximum at reasonably low lengths – between0.1 and 0.2 of the maximum.

The maximum intercept length, used for the normaliza-tion, is now 7.14 times the plate thickness. On this normal-ized length scale the square edge of the plate measures 0.70and the plate thickness is 0.14. Thus the most likely inter-cept length about the plate thickness.

The mean intercept length is 0.20 on the normalizedscale or 1.43 times the plate thickness. The size as estimatedby DPSSA is 2.14, i.e. about double the thickness of theplate.

Size of Grain Volume %(micron)

5 515 1525 6035 1545 5

0

10

20

30

40

50

60

70

Per

cen

t in

Inte

rval

Fig. 6. Grain sizes distribut

Using Eqs. (1) and (2) again it is possible to estimate L

and DPSSA for an infinite slab. These are useful as limitingconditions. For an infinite slab, the mean intercept length L

is 2 and DPSSA is 3.

5.5. Summary

• Edges and corners have a strong effect on the shape ofthe intercept length distribution plot and lead to manysmall intercepts.

• Elongation of shapes in one or two dimensions leads tothe possibility of longer intercepts, but the mean inter-cept stays in the region of the rod or plate thickness.

• The size as estimated from the PSSA increases with elon-gation for particles of the same thickness.

Details are summarized in Table 1.

6. Effects of size distribution on intercept length distributions

Changes in the grain size distribution have a strongeffect on the shape of the intercept length distribution.Two cases will be considered both where the grains arecubic in shape. In one case the grain sizes are spread abouta mean size (Figs. 6 and 7), and in the other case there is abimodal distribution of grain size (Figs. 8 and 9).

It is clear that the distribution of grain size has a drasticeffect on the shape of the intercept length distribution plot.With even a small proportion of smaller particles, as indi-cated in the first case, the plot shows a predominance of

Grain Size Distribution

0 10 20 30 40 50

Grain Size (micron)

ed about a mean value.

Run Length Distribution

0

50

100

150

200

250

300

350

400

8 16 24 32 40 48 56 64 72 80

Intercept Length in microns

Nu

mb

er

of

Inte

rce

pts

Fig. 7. Intercept lengths of cube with size spread about a mean.

Run Length Distribution

0

100

200

300

400

500

600

700

800

900

1000

8 16 24 32 40 48 56 64 72 80

Intercept Length in microns

Nu

mb

er o

f In

terc

epts

Fig. 9. Intercept lengths of cube with bimodally distributed size.


small intercepts. A bimodal distribution is dominated bythe small grain size.

In practice there will always be a range of grain sizes sothat the observed intercept distribution found in practicemust be a combination of shape and size distributioneffects.

6.1. Summary

• The shape of the intercept distribution of real systems isstrongly affected by the presence of small grains.

• This suggests that all real systems have a range of grainsizes.

7. Effects of statistics on intercept length distributions

It is important to know how many intercepts should bemeasured in order to get a reasonable estimate of the grainsize. Plots of the effect of the number of intercepts mea-sured on the shape of the intercept length distribution areshown below (Figs. 10 and 11).

It can be seen that there is considerable variability in theshape when there are only 100 intercepts.

Table 2 is also shown giving an idea of the standarddeviation of estimates of the mean ‘‘size’’ based on 10 esti-mates each with about 100, 200, 500 and 1000 intercepts of

Size of Grain Volume %(micron)

5 5015 025 035 045 50

0

10

20

30

40

50

60

Per

cen

t in

Inte

rval

Fig. 8. Bimodal grain

a cube. Using 200 intercepts the standard deviation isabout 3%. This seems a reasonable basis for grain sizeestimates.

7.1. Summary

• The shape of the intercept length distribution can varysignificantly when the number of intercepts is small.

• At least 200 intercepts should be measured when esti-mating grain size.

• This is a minimum number since it is based on a uniformgrain size.

8. Estimation of the liberation size

Mineral grain size is measured because liberationbecomes easier as grain size increases. Consequently grainsize is used as a very important index for ore classification.

The size at which minerals liberate is not easy to define.It must be less than the size of the mineral grains, but willvary as their shape and size distribution changes. Experi-mental information is needed, but the following discussionindicates the trends that might be expected.

The PSSA can be readily estimated from the linear anal-ysis since both area and volume can be measured. As

Grain Size Distribution

0 10 20 30 40 50

Grain Size (micron)

size distribution.


0

5

10

15

20

25

30

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1


Nu

mb

er o

f in

terc

epts


0

5

10

15

20

25

30

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1


Nu

mb

er o

f in

terc

epts


0

5

10

15

20

25

30

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1


Nu

mb

er o

f in

terc

epts


0

5

10

15

20

25

30

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1


Nu

mb

er o

f in

terc

epts

Fig. 10. Intercept length distribution of cube (100 intercepts).


discussed above, this permits a ‘‘size’’ to be determinedbased on the assumption that the particles are sphericalor cubic. In each case a particle of unit volume has a sur-face area of 6 square units.

8.1. Experimental results

The PSSA size has been shown to be a useful guide toliberation size. In an AMIRA study of effects of ore char-acteristics on processing performance the PSSAs of differ-ent ores were plotted against the P80s of the finalconcentrates (Sutherland et al., 1991). The following plot(Fig. 12) shows results for lead-zinc ores from a varietyof Australian concentrators. The line drawn representsthe PSSA size, i.e. 6000/PSSA, where size is in micronsand PSSA is in mm2/mm3. The fact that this line is a rea-sonable representation of the experimental data indicatesthat the PSSA size for the feed ore is approximately thesame as the P80 in final concentrate.

This is strong evidence that the PSSA size is a usefulindex of ore characterization.

8.2. Comparing PSSA sizes for grains of different shapes

The effects of grain shape on PSSA were shown above forrods and plates. We would expect the ‘‘liberation’’ to be

related to the thickness of the plate or rod, i.e. the grainsshown below would require similar grind sizes for liberation.

Table 1 indicates that the PSSA size increases as thegrain becomes more elongated in one or two dimensions.

The fact that the PSSA size is always greater than the‘‘true grain size for liberation’’ may be the reason for thegeneral agreement noted above in experimental cases. Thisis because we must grind to less than the grain size to lib-erate the minerals.

Irrespective of the accuracy of the agreement betweenliberation size and PSSA size it remains a good way to rank

different ores where grain shapes are likely to be similar.Fig. 13 shows the shapes of the intercept distributions for

cube, rod and plate all having the same thickness. The inter-cept length distributions shown are not normalized, butshown as absolute values as they would be measured. Thisis a fairer comparison than the normalized plots shown ear-lier and indicates less obvious differences between the differ-ent shapes. Rather the differences are more from theexistence of some longer intercepts with elongated grains.

8.3. Summary

• PSSA size is recommended for ranking ores because:� It is a stereologically sound estimate of mean grain

size.


0

40

80

120

160

200

240

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1


Nu

mb

er o

f in

terc

epts


0

40

80

120

160

200

240

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1


Nu

mb

er o

f in

terc

epts


0

40

80

120

160

200

240

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1


Nu

mb

er o

f in

terc

epts


0

40

80

120

160

200

240

0-.1 .1-.2 .2-.3 .3-.4 .4-.5 .5-.6 .6-.7 .7-.8 .8-.9 .9-1


Nu

mb

er o

f in

terc

epts

Fig. 11. Intercept length distribution of cube (1000 intercepts).

Fineness of Grind

50

100

150

200

250

300

350

400

450

Ph

ase

Sp

ecif

ic S

urf

ace

Are

a (m

m2 /m

m3 )

Table 2Variability of mean intercept length based on 10 estimates

No. of inercept Ratio = meas mean/true mean SD ratio

104 0.977 0.07201 1.000 0.03496 0.987 0.02994 0.996 0.02


� Shape effects are less likely to be a problem for similarminerals.

� Experimental results from a wide range of ores sup-port its use.

• But predictions of liberation size still require experimen-tal work.

0

0 25 50 75 100 125 150

p80 in Concentrate (microns)

Fig. 12. Relationship between grind size and PSSA (Sutherland et al.1991).

9. Conclusions

The following suggestions are made for a sound methodto measure grain sizes:

Samples

• Sample coarse material, crush and sub-sample.• Measure particles at about 1 mm.

Image analysis measurements

• Base the grain size estimation on line scanmeasurements.


0

100

200

300

400

500

0 -.4 .4 - .8 .8 - 1.2 1.2 - 1.6 1.6 - 2 2 - 2.4 2.4 - 2.8 2.8 - 3.2 3.2 - 3.6 3.6 - 4 4 - 4.4 4.4 - 4.8 4.8 - 5.2 5.2 - 5.6 5.6 - 6 6 - 6.4 6.4 - 6.8 6.8 - 7.2

Intercept length (cube edge = 1)

Nu

mb

er o

f in

terc

epts

Intercept Length Distribution of Square Rod (5:1:1)

0

100

200

300

400

0 - .4 .4 - .8 .8 - 1.2 1.2 - 1.6 1.6 - 2 2 - 2.4 2.4 - 2.8 2.8 - 3.2 3.2 - 3.6 3.6 - 4 4 - 4.4 4.4 - 4.8 4.8 - 5.2 5.2 - 5.6 5.6 - 6 6 - 6.4 6.4 - 6.8 6.8 - 7.2

Intercept length (rod thickness = 1)

Nu

mb

er o

f in

terc

epts

Intercept Length Distribution of Square Plate (5:5:1)

0

100

200

300

400

0 -.4 .4 - .8 .8 - 1.2 1.2 - 1.6 1.6 - 2 2 - 2.4 2.4 - 2.8 2.8 - 3.2 3.2 - 3.6 3.6 - 4 4 - 4.4 4.4 - 4.8 4.8 - 5.2 5.2 - 5.6 5.6 - 6 6 - 6.4 6.4 - 6.8 6.8 - 7.2

Intercept length (plate thickness = 1)

Nu

mb

er o

f in

terc

epts

Fig. 13. Effect of grain shape on intercept length distribution.


• Aim for at least 200 intersections of the mineral ofinterest.

• Qualitatively assess grain shape and texture with anareal image.

Grain size estimation

• Rank grain sizes using the PSSA estimate of meansize.

• Experimental results from an AMIRA study stronglysupport this approach.

• The size distribution of ill-defined grains is very difficultto infer from images.

Refer grain size to liberation size with caution!

• Experimental investigations are still needed.• Build-up a data bank of results for ores of interest.


• Layered minerals can give size estimates up to 3 timesthe layer thickness.

• Rod or needle-like grains can give size estimates up to1.5 times the rod thickness.

Acknowledgement

Falconbridge limited kindly helped support this work.

References

Intellection, 2006. <http://www.intellection.com.au/index.asp?pgid = 9>.JKMLA, 2006. <http://www.jkmla.com/index.htm>.King, R.P., 1984. Measurement of particle size distribution by image

analyzer. Powder Technology 39, 279–289.Sutherland, D.N., Gottlieb, P., Wilkie, G., Johnson, C.R., 1991. Assess-

ment of ore processing characteristics using automated mineralogy. In:XVIIth International mineral processing congress, Dresden, vol. III,pp. 353–361.

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