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The influence of segregation of particulates on sampling variance — the question of distributional heterogeneity
ELSEVIER
Int. J. Miner. Process. 55 (1998) 95–112
The influence of segregation of particulates on sampling variance — the question of distributional heterogeneity G.J. Lyman * Julius Kruttschnitt Mineral Research Centre, University of Queensland, Isles Road, Indooroopilly, Qld. 4068, Australia Received 9 September 1997; accepted 16 June 1998
Abstract Gy’s theory of sampling of particulate materials is now known to be the most realistic and comprehensive theory that can be applied. His theory of distributional heterogeneity of a material, which accounts for segregation effects is seldom discussed and is usually ignored in sampling calculations. This discussion examines the question of distributional heterogeneity in detail, retracing Gy’s derivation and demonstrating that his derivation introduces some rather restrictive assumptions. An alternate approach is proposed and illustrated with an example, showing that the restrictive assumptions lead to unreasonable results when applied to a material of wide size distribution while the new theory provides reasonable results. The new means of dealing with distributional heterogeneity is consistent with all other aspects of Gy’s theory and can be characterised by one material-dependent constant and one mixing parameter. 1998 Elsevier Science B.V. All rights reserved. Keywords: Gy; sampling theory; distributional heterogeneity; particulates; segregation; coal; minerals
1. Introduction Sampling theory according to Gy (Gy, 1982; Pitard, 1989) deals very well with all that needs to be known about sampling from a particulate material that is not segregated in some way. Indeed the concept of ‘constitutional heterogeneity’ introduced by Gy seems to be universally applicable to particulate materials, encompassing materials showing narrow distributions of composition and size to those showing massive variation in particle size and composition from one particle to the next, Ł
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such as gold ores. The area of Gy’s work with which it is somewhat more difficult to come to terms is his development of ‘distributional heterogeneity’. Distributional heterogeneity seeks to describe that heterogeneity of a particulate material in excess of the constitutional heterogeneity. In sampling, the constitutional and distributional heterogeneities control all sampling variance associated with the particulate nature of the material. Gy’s ‘short-range quality fluctuation error’ (QE1 in the notation of his 1982 work — we will retain his notation defined in that work in reference to his theory in this discussion), is entirely dependent on these factors. We note Gy’s remark that the particulate structure of the material is irrelevant to the long-range periodic and non-periodic (continuous) quality fluctuation errors and his further remark that these latter quality fluctuation errors would be present if the material sampled were a liquid (p. 241, Gy, 1982). In carrying out sampling calculations, knowledge of the approximate distribution of composition over particles of different sizes (the grade distributions on a size by size basis), augmented by such information on particle density as is necessary, permits the determination of the constitutional heterogeneity. This is particularly true when we deal with coal ash content where float and sink fractionation of the coal on a size by size basis provides nearly perfect sorting of the coal particles according to their ash content. The calculation of distributional heterogeneity is more difficult as, according to Gy, we must know the ‘grouping factor’, , and the ‘segregation factor’, ¾ . The grouping factor is (generally) a large number which is approximately equal to the number of particles in a group and the segregation factor lies in the range 0 to 1, taking the value 0 when the particulate is as well-mixed as is physically or statistically possible and the value 1 in the opposite case when the particles are all perfectly segregated. Gy states that the distributional heterogeneity cannot exceed the constitutional heterogeneity. This discussion seeks to examine the relation between the distributional heterogeneity and constitutional heterogeneity and also to look at how one makes the distinction between the short-term and long-term errors. The tool applied in the domain of long-term error or material quality fluctuation is time series analysis or the variogram and that in the domain of short-term fluctuation is the statistical model of particulates. It is important to note that in order to determine the sampling variance with respect to a critical quantity or content of the particulate material, it is not necessary to invoke the idea of constitutional heterogeneity. The derivation given by Lyman (1986) makes no definition of heterogeneity per se. Lyman determined the variance of sampling from a well-mixed material (the same as Gy’s fundamental error) by considering that the number statistics of particles falling into a ‘correct’ increment followed a Poisson distribution. Gy uses a binomial process and then notes the connection between the variance formulae derived and the definitions of his constitutional or distributional heterogeneity. The development of distributional heterogeneity made herein as alternative to Gy’s formulation follows on from earlier work, and requires no specific definition of distributional heterogeneity; the results arise naturally from the statistics of correct sampling.
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2. Gy’s model for distributional heterogeneity Gy attacks the problem of determination of the distributional heterogeneity of a set of particles by considering some lot LF of discrete particles and the division of this set into disjoint subsets of particles Gn , n D 1; 2; : : :; NG , all of approximately the same magnitude. The measure of the magnitude of a subset is taken to be the number of particles in the subset. The variance of his defined heterogeneity between these subsets or groups is taken to be the measure of the distributional heterogeneity. This idea is consistent with putting all the particles into a rectangular box and then moving an imaginary ‘sampling box’, smaller than the first, to a discrete set of locations within the first box, such that the volumes occupied by the sampling box do not overlap and in fact encompass the sets Gn . We then compute the compositions (critical content) of the contents of the sampling box at each location and use this composition to calculate the heterogeneity of the subsets. If we invoke the simple model of the particulate material used by Gy (a mixture of quartz and magnetite grains which is conveniently black and white with each particle type of a different composition), the differences in composition are very easy to visualise in terms of colour or composition. If we were geostatisticians, we would compare this situation to the block to block grade variation in an ore deposit. We would then describe the composition variance (do not confuse this with heterogeneity, as heterogeneity is dimensionless and measured relative to an average particle) with a variogram and the relevant value of variance would be the sill of the variogram. In the case of heterogeneity, if all groups are very close in mass the distributional heterogeneity value would be proportional to this sill variance. The picture is then one of not-too-large deviation of the ‘block’ composition from the mean composition over the whole box that is defined as a discrete stationary random function within the whole box. Gy’s development of a relationship between the constitutional heterogeneity and distributional heterogeneity of a material is simple to understand when systematically followed. However, a careful scrutiny of the arguments and algebra demonstrates that his final results are a consequence of some very important assumptions regarding the composition or make-up of the particulate material. In particular, the assumptions limit the application of the results to a material of very narrow size distribution and very particular grade distribution. To understand how the assumptions arise, it is necessary to recapitulate Gy’s derivation. Gy’s notation will be used initially. The work appears in chapter 19 of his 1982 volume. 2.1. Heterogeneity of a particle within the lot L (composed of NF particles) Gy defines the heterogeneity of the ith particle as (see notation for explanation of variables) .ai aL /Mi hi D aiN MiN MiN D
ML NF
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aiN D aL so aL /Mi N F (1) aL ML The heterogeneity of a particle is thus proportional to a composition difference from the average value for the lot, and the particle mass. It becomes the ratio of the signed excess weight of critical component in the particle to the mass of the critical component in an average particle. hi D
.ai
2.2. Heterogeneity of a particle within a group Gn (composed of Nn particles) Gy defines, for the jth particle in the nth group (total of NG groups) .an j aL /Mn j .an j aL /Mn j N F hn j D D hi (2) aiN MiN aL ML This definition is equivalent to the first. We have simply considered the ith particle to now be the kth particle within the jth group so we have Mi Mn j , and ai D an j . 2.3. Heterogeneity of a group Gn Gy defines .an hn D
aL /Mn .an aL /Mn NG D (3) anN MnN aL ML While it is not vital to the argument, Gy also defines the heterogeneity of the average fragment in a group and then equates this heterogeneity to the heterogeneity of the group. In doing so, he must make the assumption that all groups contain essentially the same number of particles, Nn D N F =NG , for all n D 1; 2; : : :; NG . This assumption, which is used again later, precludes grouping a material containing a number of size fractions into groups that contain all particles of any one size as, in such a case, the groups containing the small particles will have many more particles than the groups containing the larger particles. Thus the important case of segregation according to size is eliminated from consideration by this assumption. Using more than one group per size fraction would introduce a weighting of the heterogeneity, with corresponding loss of generality. 2.4. Constitutional heterogeneity of the lot L, CHL Gy defines the constitutional heterogeneity of the lot L to be the variance of the heterogeneity carried by the particles within the lot, or Varfh i g. We note that the expected value of h i is zero (expectation taken over all particles in the lot). We have NF NF 1 X NF X .ai 2 hi D 2 CHL D N F iD1 M L iD1
aL /2 Mi2 aL2
(4)
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2.5. Distributional heterogeneity of the lot L, DHL Gy defines the distributional heterogeneity of the lot L to be the variance of the heterogeneity carried by the groups within the lot, or Varfh n g. We note that the expected value of h n is zero (expectation taken over all groups in the lot). We have DHL D
NG NG 1 X NG X .an h 2n D 2 NG nD1 M L nD1
aL /2 Mn2 aL2
(5)
2.6. Linking CHL and DHL To develop a relationship between constitutional heterogeneity and distributional heterogeneity, Gy proceeds as follows. Put h i D h n j D .h n j
hn / C hn
so h 2i D .h n j
h n /2 C 2h n .h n j
h n / C h 2n
(6)
Then, summing over all fragments in the lot NF X iD1
h 2i D
NG X Nn X nD1 jD1
.h n j
h n /2 C 2
NG X nD1
hn
Nn X
.h n j
jD1
hn / C
NG X Nn X
h 2n
(7)
nD1 jD1
We note here that Gy omits the middle summation on the right hand side above (while Pitard (1989) includes it). Expanding this middle term, we find ½ Nn NG NG X X X NF hn .h n j h n / D 2 hn Mn .an aL / h n Nn 2 aL ML nD1 jD1 nD1 (8) NG X .an aL / hn Mn [N F NG Nn ] D2 aL ML nD1 making it clear that the term vanishes or becomes very small only if the number of particles in all groups Nn , n D 1; 2; : : :; NG is equal, or nearly so, to the same value N F =NG . Having earlier made this assumption in dealing with the heterogeneity of the average fragment in a group, Gy does not consider the point further. Pitard, on the other hand, states that since the heterogeneity of the average fragment in a group is equal to the heterogeneity of the group, the term in question must vanish. In earlier dealing with this alleged equality, Pitard simply states that (see his eq. 6.28 and preceding remarks) “it has been found that the heterogeneity h n carried by a group of fragments Gn is nothing else than the heterogeneity carried by the average fragment in the group.”
Pitard does remark, however, in moving from Eq. 6 to Eq. 7 that Nn should be relatively constant. In fact, for the algebra to be valid, it must be exactly constant. Continuing Gy’s development and dividing by N F , we have NG X Nn NF 1 X 1 X h 2i D CHL D .h n j N F iD1 N F nD1 jD1
h n /2 C
NG 1 X Nn h 2n N F nD1
(9)
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The last term on the right is equated to DHL (see Eq. 5) and to do this we must again take Nn to be a constant. Thus this link between CHL and DHL , which allows Gy to state that CHL ½ DHL ½ 0
(10)
is very much dependent on the condition that all groups must carry almost exactly the same number of particles. 2.7. Natural distributional heterogeneity of the lot In his section 19.3.5.3, having previously concluded that DH L can never be precisely zero, Gy develops an estimate of the minimum value that DHL may take on. He does this by considering that in a particulate that is as well-mixed as can be, the groups Gn of particles cannot be more similar to each other than when all particles in each group are drawn at random from a large batch of particles having the same properties as the ones with which we wish to concern ourselves. This is quite correct and what we will see in such a case is a variation of the composition from one group to the next that is a consequence of the Poisson distribution of the numbers of particles (variance of the number of particles equals the expected number of particles) in each particle class. However, such variation from group to group is a consequence of constitutional heterogeneity of the group; one wonders whether classifying it as minimal distributional heterogeneity might not be confusing the issue. The rule that the particles be drawn at random is actually sufficient to estimate the magnitude of (DHL )min , if this is how (DHL )min is to be defined. ToP relate the magnitude DHL to CHL , Gy states that the quantity P Nof N G P Nn F 2 2 2 h D ¦ h ¦ 2 nD1 jD1 n j iD1 i can be treated as a chi-squared random variable with N F 1 degrees of freedom, if ¦ 2 is the variance of h i in a population of an effectively infinite number of particles. Whether or not this can be generally accepted as true can be examined by considering the value of h i for each particle as we draw a finite number N F of particles at random from the very large lot. h i is given by Eq. 1 .ai aL /Mi N F hi D aL ML and may be interpreted as a random variable, where its randomness derives from the fact that each particle has a different value of critical content, ai , and a different mass Mi . The values of aL and M L are taken to be material constants. For a given material, we can accumulate a histogram of the results for h i as we draw the particles. To simplify our thinking about the material, let us admit that we may divide the material into Ns size fractions and Nd density classes (or composition classes — these will be equivalent in a binary system of two minerals having different densities). We may denote particle composition as auv and particle mass as m uv D Vu ²v when the particle is drawn from the uth size class and the vth composition class in which the typical fragment volume is Vu and the typical fragment density is ²v . The particle compositions must be distributed to either side of aL so we get positive and negative values of h i . With our simplified (but very valid) view of the material, the histogram that builds up will be of the nature illustrated in Fig. 1.
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Fig. 1. Histogram of particle h values for a sample of particles.
Now, the definition of a chi-squared random variable with N degrees of freedom is the sum of the squared values of N independent standard normal random variables (the standard normal random variable has zero mean and unit variance). We may scale the h i values by their standard deviation, so that the variance of the distribution is unity, but the values of h i will not in general be normally distributed. They may in fact be far from normally distributed as in the quartz–magnetite mixture when Fe is the critical component; such a system will have a bimodal distribution on either side of the mean composition. They will in fact be normally distributed only if the fragment masses the ai D auv are normally distributed about Mi D m uv are all substantially constant andP NF h 2i become chi-squared. Note that we aL . Only then will the distribution of ¦ 2 iD1 cannot make appeal to the central limit theorem here as we are dealing with a sum of squares of random variables rather than a sum of random variables which is later squared. Thus Gy’s conclusions that follow from the above assertion are questionable, in general, and are true only for a very restricted condition. His important conclusion that the minimum expected value of DHL can be expressed as NG 1 EfDHL gmin D (11) CHL NF 1 can be expected to be valid only when his assumptions are fulfilled and in error in other circumstances. It may also be true that the upper limit placed on distributional heterogeneity as DHL CHL is also restricted to a particular case of equal particle numbers and a particular distribution of critical content over the particles of the lot. The final consequence of this formulation of the distributional heterogeneity appears when Gy links up his continuous and discrete models. The grouping and segregation error is defined in section 21.3.3 of Gy’s work and its variance is taken as the difference between the variance of critical content when sampling is carried out taking groups of particles (the ‘potential increments’ that can be formed from the lot) and the variance
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when sampling is carried out by selecting particles at random from the entire lot (fundamental error). It is an excess variance (which does seem to be somewhat at odds with his discussion of the minimal value of distributional heterogeneity). Gy writes
¾ CHL D ¾ ¦ 2 .FE/ (12) NF S where is the grouping factor (approximately equal to the number of fragments in a group, or the potential increment in this case), ¾ is the segregation factor (a value between 0 and 1 — the latter value indicates worst-case segregation) and N F S is the total number of fragments in a sample. ¦ 2 .FE/ is the variance due to fundamental error. We will now consider an example of estimated sampling error for a sampling situation of common interest. However, it is worthwhile to preface the example with a further consideration of the general nature of distributional heterogeneity. ¦ 2 .GE/ D
3. Logical definition of distributional heterogeneity The constitutional heterogeneity of a material is an inescapable heterogeneity which results from the particulate nature of the material. Gy’s definition is given by Eq. 4. Let us take that definition and write it in terms of sums over size and density classes for particles in the nominal or expected make-up of the material. NF NF X .ai CHL D 2 M L iD1
aL /2 Mi2 aL2
We replace Mi by m uv D Vu ²v and take the number of particles in a particle class within the lot of mass M L to be n uv . Then CHL D
Nd Ns X .auv NF X 2 M L uD1 vD1
aL /2 n uv m 2uv aL2
If we take X u to be the mass fraction of particles in the uth size fraction and Yuv to be the mass fraction of particles in the vth composition class within the uth size class, we have CHL D
Ns Nd X .auv aL /2 NF X X u Vu ²uv Yuv M L uD1 aL2 vD1
(13)
Note that M L =N F is the average fragment mass in the lot so that CHL is effectively a constant for the material and is not a function of the quantity of the material we consider. The quantity CHL M L =N F can be taken to be the ‘sampling constant’, K s , for the material. If we draw particles at random from a very large lot of material having a particular nominal composition and properties in order to form a smaller lot L, we can do it in two ways. We can either demand that the total number of particles in L be a fixed constant or we can relax our requirements slightly and simply demand that the expected value of the mass of the lot L be a value M L . In the latter case, we can allow the numbers
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of particles in each class to be independent random variables that follow Poisson distributions. In the former case, when we fix the total number of particles, we find that equiprobable sampling of the particles results in the numbers of particles following a multinomial distribution. We note that in both cases, the total lot mass becomes a random variable. In practice, in trying to make up two nominally identical subsamples of a material, we will never achieve splitting that gives equal numbers of particles; the best we can do is to achieve nominally equal masses that differ between subsamples in a random way. Thus the model of particle numbers as Poisson random variables is the more appropriate statistical model for nominally identical subsamples. There is a minor statistical convenience in this assumption as it removes the covariance between numbers of particles in all particle classes and makes the algebra a bit simpler. Using the Poisson model of the number statistics, it is a simple matter to demonstrate that for samples or lots of expected mass M L , the relative variance in the critical content between lots is exactly equal to CHL =N F (see Lyman, 1986, eq. 18). This variance between lots is the expression of the concept of constitutional heterogeneity. If we now consider the lot L to be composed of NG sublots or groups, each group having been constructed by drawing particles at random from a very large lot, we must find that the relative variance of critical content between the groups is CHG =Nn and since CHG is a material constant, CHG D CHL . If we now take Mn to be a constant in Eq. 5 and note that Mn ¾ D M L =NG , we can write " # NG 2 .an aL /2 NG Mn2 1 X (14) .DHL /min D NG nD1 M L2 aL2 and we see that the term in square brackets is just the relative variance of critical content over the groups and the factor outside the bracket is unity. Thus, regardless of the distribution of critical content, but provided the number of particles in each group is the same CHL NG .DHL /min D D CHL (15) Nn NF This result is similar to the one found by Gy, but it makes no appeal to the chi-squared distribution. However, it must be emphasised here that this minimal (apparent) distributional heterogeneity displayed by the groups is actually a consequence of constitutional heterogeneity and is, in our view, misnamed. It is preferable and less conceptually complex to define distributional heterogeneity as the increase in heterogeneity above the constitutional heterogeneity for a sample of the same mass and nominal composition that is observed for a material when it is sampled in a ‘correct’ manner. In simple terms, any heterogeneity in a sample that cannot be explained by the constitutional heterogeneity for the nominal composition of the material must be deemed to be distributional heterogeneity. In light of Eq. 12 and considering that ¾ D 0 when the material is as well-mixed as can be, such a definition of distributional heterogeneity is probably the one that Gy really intended to make. With this definition, the minimal distributional heterogeneity of the groups is zero and not the term related to the constitutional heterogeneity given in Eq. 11 or Eq. 15. When we deal with a sampling situation, we wish to be able to describe the ‘fundamental error’ or the inescapable variance component associated with the mass of
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sample we collect from the lot. To this error or variance component, we wish to add a component that accounts for distributional heterogeneity, which Gy calls the grouping and segregation error or variance. Again, translating to practical terms and thinking of sampling from a moving stream, we want to know what the variance between samples will be if we sample correctly from a material when it is as well-mixed as can be. And then we wish to know how much the variance between samples will increase in similar circumstances (same number and nominal mass of increments taken in the same way, using random sampling or stratified random sampling) if the material from which we are sampling is segregated in some way. Ideally, we wish to be able to contrive a circumstance of segregation which, even when sampling is correct, will induce a maximal increase in sampling variance, thereby establishing this scenario as the worst that can happen in the event that the material is segregated. Gy’s arguments that establish a maximal value for the distributional heterogeneity are an attempt to do just this. However, as has been established above, the manner in which the maximal distributional heterogeneity is established precludes consideration of a material with a wide size distribution. It is precisely the situation of wide size distribution that we commonly expect to lead to segregation and difficulties in sampling or increase in sampling variance. We will now show that an excess variance due to segregation does indeed exist, but that its description takes a different form from that given by Gy.
4. Example of variance increase due to segregation Let us contrive a situation which would seem to provide a worst-case result of sampling variance due to segregation. Consider a large batch of a particulate laid out on a conveyor belt at a constant mass per unit length but in a totally segregated state. Let us model the material as being composed of K particle classes, each having a distinct critical content ai , i D 1; 2; : : :; K . Let the mass fraction in the ith particle class be X i . Within each length L of the belt, let the particles be sorted into classes which are arranged in a random order. Then each class will cover a length L X i of the section. Further, let consecutive lengths L repeat this sorting by particle classes. We will then sample the belt by taking N increments of mass m I out of each length L according to random sampling. It will be assumed that any one increment is taken from a single class only. Within the set of N increments from the jth section of length L, let the number of times that the ith class is ‘hit’ be n i j . The values of the random variables n i j must then follow a multinomial distribution and we can write the critical content of a sample taken from M sections of belt as K M X X mI n i j ai M K 1 XX jD1 iD1 as D D n i j ai (16) K M X M N jD1 iD1 X mI ni j jD1 iD1
as is a random variable which is written in a way that suppresses the particulate nature of the material and focuses only on the segregation of the material. We will ignore
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the fact that as carries an additional component of variance that is due to the fact that each time we ‘hit’ the ith particle class, we do not collect exactly the same number of particles. We are therefore ignoring a stochastic element in m I and focussing only on the randomness of n i j . The properties of the random variable n i j are: Efn i j g K X ni j
D N Xi DN
any j
iD1
Varfn i j g
D N X i .1
Covfn i j ; n kl g D D0
N Xi Xk
(17)
Xi / j Dl j 6D l
and so the expected value of as is M K K X 1 XX ai Efn i j g D ai X i D aN s Efas g D M N jD1 iD1 iD1 With some simple algebra, we may show that the variance of as is given by K mI X X i .ai aN s /2 Varfas g D Ms iD1
(18)
(19)
where Ms is the mass of sample collected and is simply M N m I . We can also identify as with aL and note that our sampling is unbiased in the long run. In fact, the sampling scenario set up here is absolutely ‘correct’ by all of Gy’s rules of sampling (constant mass flow and random increments from the material). The variance of as is the second moment of the critical content distribution, weighted by the mass fraction in the particle class; note that particle masses do not enter into the calculation. The variance can be reduced only by increasing the number of increments taken and not the mass of the increments taken. This contrasts with the fundamental variance of a sample which depends only on the total mass of the sample collected and the basic properties of the material as expressed in the sampling constant, K s . If the material were not segregated according to particle class, it is a simple matter to demonstrate that the sample critical content variance determined in a manner similar to that used to take account of segregation (suppressing the particulate nature of the material) would be zero. Thus, as the material becomes progressively less well-mixed, Varfas g varies from zero to the value given above. Eq. 19 is the expression of distributional heterogeneity that we seek and it is a maximal value. This expression for distributional heterogeneity demonstrates that, for a given sample value of the variance due to distributional heterogeneity is mass P K Ms , the maximal 2 X .a a N / which corresponds to taking a single increment to form the sample. i i s iD1 This variance can easily exceed the variance due to constitutional heterogeneity. We will define a second quantity, which we will call the sample segregation constant, as K X .ai as /2 Xi (20) Ds D as2 iD1
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This value is dimensionless and is an (intensive) property of the material, in the same way as the constitutional heterogeneity, and may be defined for any critical content. In the practical notation of Eq. 13 Ds can be written as Ds D
Nd Ns X X
X u Yuv
.auv
uD1 vD1
a L /2
(21)
aL2
and the expression is clearly the same as that for the sampling constant, except for the omission of the particle mass weighting. The next section provides an example calculation of the distributional heterogeneity for a coal and compares the value with the constitutional heterogeneity. 4.1. Example of calculation of constitutional heterogeneity and distributional heterogeneity for a coal Let us take a coal sample with the mass and ash distribution as a function of particle density as in Table 1, which is assumed to be the same for all size fractions of the coal. Let the size distribution be as in Table 2. We now calculate the sampling and segregation constants for the material according to the calculations in Tables 3 and 4. Table 1 Density and ash distribution by density fraction Mean density ²uv
Mass fraction Yuv
Ash auv
1.325 1.375 1.425 1.475 1.525 1.575 1.625 1.675 1.750 1.900
0.117 0.604 0.085 0.019 0.009 0.005 0.004 0.003 0.007 0.147 1.000
7.50 9.50 13.50 19.20 27.80 30.30 35.60 40.20 45.90 82.00
Table 2 Size distribution du (cm)
duC1 (cm)
Mass fraction X u
2.64 1.32 0.56 0.28 0.05
1.32 0.56 0.28 0.05 0.01
0.231 0.294 0.139 0.189 0.147 1.000
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Table 3 Calculation of constitutional heterogeneity and distributional heterogeneity terms (density factors) Mean density ²uv
Mass fraction Yuv
Ash auv
Ash mass Yuv auv
Heterogeneity terms CHv
Heterogeneity terms DH
1.325 1.375 1.425 1.475 1.525 1.575 1.625 1.675 1.750 1.900
0.117 0.604 0.085 0.019 0.009 0.005 0.004 0.003 0.007 0.147 1.000
7.50 9.50 13.50 19.20 27.80 30.30 35.60 40.20 45.90 82.00
0.8775 5.738 1.1475 0.3648 0.2502 0.1515 0.1424 0.1206 0.3213 12.054 a L D 21.168
0.064632 0.252328 0.015894 0.000242 0.001347 0.001466 0.003022 0.004062 0.016723 2.306676 2.666
0.048779 0.183512 0.011153 0.000164 0.000883 0.000931 0.001859 0.002425 0.009556 1.21404 1.473
The sampling constant, K s , is found to have a value of 3.694 g and the segregation constant, Ds , is found to have a value 1.473 [-]; the lot ash, aL content is 21.168% ash. The variance of ash content due to constitution heterogeneity for a 25 kg sample of this coal is 3:694 ð 21:12 D 0:066 Varfas;CH g D 25000 The variance of ash content due to distributional heterogeneity is mI 1:473 ð 21:12 D 656m I Varfas;DH g D 25 where m I is the increment mass in kg. For 2.5 kg increments, the standard deviations of ash are 0.256% ash and 40.5% ash for constitutional heterogeneity and distributional heterogeneity, respectively, which clearly shows the large influence of the distributional heterogeneity. The condition for equality of the variances due to constitutional heterogeneity and distributional heterogeneity, when the material is fully segregated is ND
Ds Ms Ks
(22)
Table 4 Calculation of constitutional heterogeneity terms (size factors) du (cm)
duC1 (cm)
Mass fraction X u
Volume Vu
X u Vu
X u Vu CHv
2.64 1.32 0.56 0.28 0.05
1.32 0.56 0.28 0.05 0.01
0.231 0.294 0.139 0.189 0.147 1.000
5.174928 0.618896 0.049392 0.005519 3.15 ð 10
1.195408 0.181955 0.006865 0.001043 4.63 ð 10
3.187427 0.485164 0.018306 0.002781 1.23 ð 10 3.694 (g)
5
6
5
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In this case, the required number of increments is 9969. This figure may seem ridiculous, but it is valid for sampling from a fully segregated material. In practice, the material will never be so segregated and Ds may be reduced by a factor between 0 and 1 expressing the degree of segregation (in the same way as Gy invokes a factor ¾ to account for degree of segregation). Nonetheless, this development and example bears out the truth of Gy’s recommendation that one should take the smallest practical increments consistent with no loss of sample and correct increment extraction and as many increments as possible in the forming of a sample of a particulate material, in order to minimise the effects of material segregation. The next comparison to be made is between the variance given by Eq. 19 and that given by Eq. 12. To do this, we need an estimate of the number of fragments in an increment, . The conceptual sampling scheme we used above considered all particle sizes and ash contents to fall in a different class. It is clear from Eq. 20 or Eq. 21 that only the distribution with respect to ash content influences the value of Ds and thus all size classes with the same ash content can be taken as a single class. We used a 2.5 kg increment. The mean density of the coal can be determined as 1.439 g cm 3 . Using the fragment volumes defined in Table 4, we find that an increment containing all sizes has 8:17 ð 106 fragments so ³ 8:17 ð 106 . Taking the value of ¾ as 1 (total segregation), the estimate of ¦ 2 .GE/ ³ 0:066 ð 1 ð 8:17 ð 106 D 5:39 ð 105 making the corresponding standard deviation 734% ash, which is rather too large. This value would become even larger if the bottom size of the coal were decreased. In fact, because of the dependence of Gy’s grouping and segregation variance on particle numbers, its value is effectively arbitrary. The core difficulty here is that the magnitude of the sampling constant of 3.694 g is not influenced by the number of particles; it is controlled by the magnitude of the volume of the largest particles. Reducing the minimum particle size in Table 4, has no effect on K s , but will radically change the number of particles and hence the estimate of the grouping and segregation error, according to Eq. 12. This difficulty arises from the fact that Gy’s development of grouping and segregation error retains the link to the expression for constitutional heterogeneity which retains a weighting with respect to particle volume. Our development above does not have such a weighting. The retention of the particle mass weighting in Gy’s grouping and segregation error is a consequence of his assumption that all groups have essentially similar numbers of particles, which is an invalid assumption when dealing with wide size distributions. It is thus not surprising that we arrive at an incorrect result when we apply Gy’s expression in a case of wide size distribution. The present development corrects this problem. We may propose a new expression for the sampling variance (relative variance of critical content) as mI Ds ¦ 2 .GE/ D ¾ Ms (23) Ns X Nd X .auv aL /2 Ds D X u Yuv aL2 uD1 vD1 where Ds is a dimensionless segregation constant for the material, 0 ¾ 1 is a segregation factor which is zero when the material is well-mixed and m I and Ms
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are the increment mass and total sample mass, respectively. The fundamental error remains unchanged and related to the constitutional heterogeneity as defined by Gy or, equivalently, the critical content variance which is a consequence of Poisson particle number statistics when sampling correctly from a well-mixed material. We have ¦ 2 .FE/ D
1 Ms
½ Ns Nd X .auv aL /2 1 X X u Vu ²uv Yuv M L uD1 aL2 vD1
(24)
where we have allowed for extraction of a sample of mass Ms from a lot of mass M L . The only restriction placed on the application of Eq. 23 is that the increments taken must be taken over a lot that is genuinely representative of the material, that is, all particle classes must be represented in their proper proportions in the lot, even though it is segregated. The only difficulty with this means of expressing the variance due to distributional heterogeneity is that it is very difficult to make an a priori estimate of the value of ¾ .
5. Reconciling the new expression of distributional heterogeneity with Gy’s overall theory There remains only one aspect of Gy’s overall theory of sampling to be discussed, which is the separation of the segregation effect from the long-term quality fluctuation error. The above development treated segregation using the idea of a segregated material on a conveyor belt and the extraction of increments from such a belt. Such an example immediately raises the question of how to separate segregation effects from genuine long-term fluctuations of the coal properties. The practical answer is that such a separation can never be made unequivocally as both the long-term quality fluctuation and the segregation are random processes and it is never possible to exactly separate the characteristics of two random processes that simultaneously contribute to the randomness of a single measurement. The following argument shows that we may retain Gy’s concept of using the variogram to make the separation by assigning all variance found in the intercept of the variogram (the nugget) to short-term quality fluctuation. The variogram expresses the extent to which the properties of increments extracted a time k∆t, k D 1; 2; : : : apart are statistically correlated; the variogram measure is designed to have minimum sensitivity to trends or drifts of the mean of the process (but it is not immune from such effects). If we were to take increments from a perfectly mixed material and analyse them for critical content, the variogram calculated from the analyses would be flat, with the intercept on the y (variance axis), or the ‘nugget’ value, equal to the ‘sill’ (asymptotic value at long times) of the process. Such a variogram indicates that consecutive measurements of critical content are entirely uncorrelated in time (white noise). The nugget value is equal to the variance of an individual measurement and, in this case, should be equal to the increment variance due to constitutional heterogeneity plus the preparation and analysis variance. We will detect no long-term variance of the critical content in sampling from a large perfectly mixed lot.
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If we now sample from a lot that retains the same mean or expected critical content and critical content distribution over time, but is badly segregated in the sense contrived above, the variogram should not express any correlation between increment assays as the segregation has been set up in a random way. The variogram will be flat. However, the sill and nugget values will be much higher as they will now express the constitutional heterogeneity, distributional heterogeneity and preparation and analysis variance. The increase in the nugget value will be equal to the variance due to distributional heterogeneity. If we now consider sampling from a lot of material that does possess correlated time variation in critical content, but also suffers segregation, we will, in general, find a variogram with a sill larger than the nugget value. The nugget variance will be the sum of constitutional heterogeneity, distributional heterogeneity and preparation and analysis variance as before. If the preparation and analysis variance is established separately by simple (well-known) techniques, and the average composition of the material is known (distribution of critical content by particle class), the preparation and analysis and constitutional heterogeneity variance can be subtracted from the nugget variance to provide an estimate of the variance due to distributional heterogeneity. Using the value of Ds for the material, the value of ¾ can be estimated. It is, of course, important that the increment extraction be correct according to all of Gy’s principles. We note that because distributional heterogeneity is inherently a ‘spatial’ phenomenon (or one distributed in time when the spatial variations are presented on a conveyor belt), it is logical (and necessary) that a spatial or time technique such as the variogram be used to deduce the variance due to distributional heterogeneity. The same technique can be extended to sampling from stockpiles, but in such a case, a three-dimensional variogram must be used, in the same way as in geostatistical analyses. On the assumption that the stockpile under scrutiny consists of material that possesses distributional heterogeneity and is locally segregated to some significant extent, increments taken from throughout the volume of the pile can be analysed and the results used to construct a variogram. Depending on the geometry of the pile, we may expect the variograms to indicate spatial anisotropy. Each directional variogram will then have a different shape, nugget and sill variance. Again using a separate estimate of preparation and analysis variance and a value of constitutional heterogeneity calculated from the average properties of the material, the nugget values can be used to determine the distributional heterogeneity variance. This distributional heterogeneity variance will also be different for each direction, expressing differing degrees of segregation in the different directions. As an example of this effect, we would expect a long stockpile created by simple stacking on the centreline of the pile (no chevroning) to be badly segregated with respect to particle size in the plane normal to the pile axis, but to be much less segregated in the direction parallel to the axis. Such an analysis of a stockpile may be somewhat academic as, in practice, it is effectively impossible to correctly sample from point locations throughout the volume of a large stockpile. Nonetheless, it is interesting to consider how distributional heterogeneity will be expressed through variograms, as suggested above. The concept of using the nugget value of the variogram to determine the effective distributional heterogeneity of a material, when the preparation and analysis and con-
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stitutional heterogeneity variances have been estimated, is entirely consistent with Gy’s breakdown of sampling variance into constitutional heterogeneity and distributional heterogeneity components that depend only on the particulate nature of the material and long-term components that are independent of the particulate nature of the material.
6. Conclusions This work has brought to light an essential assumption within the derivation of Gy’s theory for the distributional heterogeneity of a particulate material that precludes the application of his results to materials of wide size distribution and=or arbitrary distribution of critical content over fragments. It has also been shown that Gy’s expression for the grouping and segregation variance can be in error by an arbitrary amount. An alternate theory and expression for distributional heterogeneity has been developed, which is shown to provide reasonable practical results. The new proposal is also consistent with the use of the variogram to estimate the sum of sampling variance due to constitutional and distributional heterogeneity. The theory is consistent with all of the remaining elements of Gy’s sampling theory which has proved over time to be a tool of inestimable value.
7. Notation ai aiN aL an an j auv anN as aN s CHL DHL (DHL )min Ds hi hn hn j K Ks m uv
D critical content of the ith particle or particles within the ith particle class D critical content of the ‘average particle’ D critical content of the lot D average critical content of the nth group of particles D critical content of the jth particle within the nth group of particles D critical content of particles in the vth composition (or density) class and the uth size class D critical content of the ‘average group’ D critical content of sample D expected value of critical content of sample D constitutional heterogeneity D distributional heterogeneity D minimal distributional heterogeneity (according to Gy) D segregation constant [–] D heterogeneity of the ith particle D heterogeneity of the nth group of particles D heterogeneity of the jth particle within the nth group of particles D number of particle classes D sampling constant (mass) D typical fragment mass within the vth composition (or density) class and the uth size class
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mI M MiN ML Mn Mn j MnN Ms n uv
D mass of increment D number of sections D mass of the ‘average particle’ D mass of the lot L D mass of the nth group of particles D typical fragment mass of the jth particle within the nth group of particles D mass of the ‘average group’ D mass of sample D number of particles within the vth composition (or density) class and the uth size class N D number of increments per section D number of density or composition classes Nd NF D number of particles in the lot L D number of particles in the sample NF S D number of groups in the lot L NG Nn D number of particles in the nth group D number of size classes Ns D short-term quality fluctuation error QE1 Vu D typical particle volume in the uth size class D mass fraction of material in the vth composition class within the uth size Yuv class D mass fraction of material in the ith particle class Xi D mass fraction of material in the uth size class Xu
D Gy’s grouping factor D density of a particle in the vth composition class ²v D density of a particle in the vth composition class within the uth size class ²uv ¦2 D variance of particle heterogeneity over a very large set of particles ¦ 2 .FE/ D relative variance due to fundamental error (or due to constitutional heterogeneity) ¦ 2 .GE/ D relative variance of critical content due to grouping and segregation ¾ D mixing parameter or Gy’s segregation parameter Efg D statistical expectation Covfx; yg D covariance operator D Ef[x Efxg][y Efyg]g Varfg D variance operator D Ef[z Efzg]2 g
References Gy, P., 1982. Sampling of Particulate Materials, Theory and Practice. Elsevier, Amsterdam. Lyman, G.J., 1986. Application of Gy’s sampling theory to coal: a simplified explanation and illustration of some basic aspects. Int. J. Miner. Process. 17, 1–22. Pitard, F.F., 1989. Pierre Gy’s Sampling Theory and Sampling Practice. CRC Press, Baton Rouge, LA.
Int. J. Miner. Process. 55 (1998) 95–112
The influence of segregation of particulates on sampling variance — the question of distributional heterogeneity G.J. Lyman * Julius Kruttschnitt Mineral Research Centre, University of Queensland, Isles Road, Indooroopilly, Qld. 4068, Australia Received 9 September 1997; accepted 16 June 1998
Abstract Gy’s theory of sampling of particulate materials is now known to be the most realistic and comprehensive theory that can be applied. His theory of distributional heterogeneity of a material, which accounts for segregation effects is seldom discussed and is usually ignored in sampling calculations. This discussion examines the question of distributional heterogeneity in detail, retracing Gy’s derivation and demonstrating that his derivation introduces some rather restrictive assumptions. An alternate approach is proposed and illustrated with an example, showing that the restrictive assumptions lead to unreasonable results when applied to a material of wide size distribution while the new theory provides reasonable results. The new means of dealing with distributional heterogeneity is consistent with all other aspects of Gy’s theory and can be characterised by one material-dependent constant and one mixing parameter. 1998 Elsevier Science B.V. All rights reserved. Keywords: Gy; sampling theory; distributional heterogeneity; particulates; segregation; coal; minerals
1. Introduction Sampling theory according to Gy (Gy, 1982; Pitard, 1989) deals very well with all that needs to be known about sampling from a particulate material that is not segregated in some way. Indeed the concept of ‘constitutional heterogeneity’ introduced by Gy seems to be universally applicable to particulate materials, encompassing materials showing narrow distributions of composition and size to those showing massive variation in particle size and composition from one particle to the next, Ł
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such as gold ores. The area of Gy’s work with which it is somewhat more difficult to come to terms is his development of ‘distributional heterogeneity’. Distributional heterogeneity seeks to describe that heterogeneity of a particulate material in excess of the constitutional heterogeneity. In sampling, the constitutional and distributional heterogeneities control all sampling variance associated with the particulate nature of the material. Gy’s ‘short-range quality fluctuation error’ (QE1 in the notation of his 1982 work — we will retain his notation defined in that work in reference to his theory in this discussion), is entirely dependent on these factors. We note Gy’s remark that the particulate structure of the material is irrelevant to the long-range periodic and non-periodic (continuous) quality fluctuation errors and his further remark that these latter quality fluctuation errors would be present if the material sampled were a liquid (p. 241, Gy, 1982). In carrying out sampling calculations, knowledge of the approximate distribution of composition over particles of different sizes (the grade distributions on a size by size basis), augmented by such information on particle density as is necessary, permits the determination of the constitutional heterogeneity. This is particularly true when we deal with coal ash content where float and sink fractionation of the coal on a size by size basis provides nearly perfect sorting of the coal particles according to their ash content. The calculation of distributional heterogeneity is more difficult as, according to Gy, we must know the ‘grouping factor’, , and the ‘segregation factor’, ¾ . The grouping factor is (generally) a large number which is approximately equal to the number of particles in a group and the segregation factor lies in the range 0 to 1, taking the value 0 when the particulate is as well-mixed as is physically or statistically possible and the value 1 in the opposite case when the particles are all perfectly segregated. Gy states that the distributional heterogeneity cannot exceed the constitutional heterogeneity. This discussion seeks to examine the relation between the distributional heterogeneity and constitutional heterogeneity and also to look at how one makes the distinction between the short-term and long-term errors. The tool applied in the domain of long-term error or material quality fluctuation is time series analysis or the variogram and that in the domain of short-term fluctuation is the statistical model of particulates. It is important to note that in order to determine the sampling variance with respect to a critical quantity or content of the particulate material, it is not necessary to invoke the idea of constitutional heterogeneity. The derivation given by Lyman (1986) makes no definition of heterogeneity per se. Lyman determined the variance of sampling from a well-mixed material (the same as Gy’s fundamental error) by considering that the number statistics of particles falling into a ‘correct’ increment followed a Poisson distribution. Gy uses a binomial process and then notes the connection between the variance formulae derived and the definitions of his constitutional or distributional heterogeneity. The development of distributional heterogeneity made herein as alternative to Gy’s formulation follows on from earlier work, and requires no specific definition of distributional heterogeneity; the results arise naturally from the statistics of correct sampling.
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2. Gy’s model for distributional heterogeneity Gy attacks the problem of determination of the distributional heterogeneity of a set of particles by considering some lot LF of discrete particles and the division of this set into disjoint subsets of particles Gn , n D 1; 2; : : :; NG , all of approximately the same magnitude. The measure of the magnitude of a subset is taken to be the number of particles in the subset. The variance of his defined heterogeneity between these subsets or groups is taken to be the measure of the distributional heterogeneity. This idea is consistent with putting all the particles into a rectangular box and then moving an imaginary ‘sampling box’, smaller than the first, to a discrete set of locations within the first box, such that the volumes occupied by the sampling box do not overlap and in fact encompass the sets Gn . We then compute the compositions (critical content) of the contents of the sampling box at each location and use this composition to calculate the heterogeneity of the subsets. If we invoke the simple model of the particulate material used by Gy (a mixture of quartz and magnetite grains which is conveniently black and white with each particle type of a different composition), the differences in composition are very easy to visualise in terms of colour or composition. If we were geostatisticians, we would compare this situation to the block to block grade variation in an ore deposit. We would then describe the composition variance (do not confuse this with heterogeneity, as heterogeneity is dimensionless and measured relative to an average particle) with a variogram and the relevant value of variance would be the sill of the variogram. In the case of heterogeneity, if all groups are very close in mass the distributional heterogeneity value would be proportional to this sill variance. The picture is then one of not-too-large deviation of the ‘block’ composition from the mean composition over the whole box that is defined as a discrete stationary random function within the whole box. Gy’s development of a relationship between the constitutional heterogeneity and distributional heterogeneity of a material is simple to understand when systematically followed. However, a careful scrutiny of the arguments and algebra demonstrates that his final results are a consequence of some very important assumptions regarding the composition or make-up of the particulate material. In particular, the assumptions limit the application of the results to a material of very narrow size distribution and very particular grade distribution. To understand how the assumptions arise, it is necessary to recapitulate Gy’s derivation. Gy’s notation will be used initially. The work appears in chapter 19 of his 1982 volume. 2.1. Heterogeneity of a particle within the lot L (composed of NF particles) Gy defines the heterogeneity of the ith particle as (see notation for explanation of variables) .ai aL /Mi hi D aiN MiN MiN D
ML NF
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aiN D aL so aL /Mi N F (1) aL ML The heterogeneity of a particle is thus proportional to a composition difference from the average value for the lot, and the particle mass. It becomes the ratio of the signed excess weight of critical component in the particle to the mass of the critical component in an average particle. hi D
.ai
2.2. Heterogeneity of a particle within a group Gn (composed of Nn particles) Gy defines, for the jth particle in the nth group (total of NG groups) .an j aL /Mn j .an j aL /Mn j N F hn j D D hi (2) aiN MiN aL ML This definition is equivalent to the first. We have simply considered the ith particle to now be the kth particle within the jth group so we have Mi Mn j , and ai D an j . 2.3. Heterogeneity of a group Gn Gy defines .an hn D
aL /Mn .an aL /Mn NG D (3) anN MnN aL ML While it is not vital to the argument, Gy also defines the heterogeneity of the average fragment in a group and then equates this heterogeneity to the heterogeneity of the group. In doing so, he must make the assumption that all groups contain essentially the same number of particles, Nn D N F =NG , for all n D 1; 2; : : :; NG . This assumption, which is used again later, precludes grouping a material containing a number of size fractions into groups that contain all particles of any one size as, in such a case, the groups containing the small particles will have many more particles than the groups containing the larger particles. Thus the important case of segregation according to size is eliminated from consideration by this assumption. Using more than one group per size fraction would introduce a weighting of the heterogeneity, with corresponding loss of generality. 2.4. Constitutional heterogeneity of the lot L, CHL Gy defines the constitutional heterogeneity of the lot L to be the variance of the heterogeneity carried by the particles within the lot, or Varfh i g. We note that the expected value of h i is zero (expectation taken over all particles in the lot). We have NF NF 1 X NF X .ai 2 hi D 2 CHL D N F iD1 M L iD1
aL /2 Mi2 aL2
(4)
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2.5. Distributional heterogeneity of the lot L, DHL Gy defines the distributional heterogeneity of the lot L to be the variance of the heterogeneity carried by the groups within the lot, or Varfh n g. We note that the expected value of h n is zero (expectation taken over all groups in the lot). We have DHL D
NG NG 1 X NG X .an h 2n D 2 NG nD1 M L nD1
aL /2 Mn2 aL2
(5)
2.6. Linking CHL and DHL To develop a relationship between constitutional heterogeneity and distributional heterogeneity, Gy proceeds as follows. Put h i D h n j D .h n j
hn / C hn
so h 2i D .h n j
h n /2 C 2h n .h n j
h n / C h 2n
(6)
Then, summing over all fragments in the lot NF X iD1
h 2i D
NG X Nn X nD1 jD1
.h n j
h n /2 C 2
NG X nD1
hn
Nn X
.h n j
jD1
hn / C
NG X Nn X
h 2n
(7)
nD1 jD1
We note here that Gy omits the middle summation on the right hand side above (while Pitard (1989) includes it). Expanding this middle term, we find ½ Nn NG NG X X X NF hn .h n j h n / D 2 hn Mn .an aL / h n Nn 2 aL ML nD1 jD1 nD1 (8) NG X .an aL / hn Mn [N F NG Nn ] D2 aL ML nD1 making it clear that the term vanishes or becomes very small only if the number of particles in all groups Nn , n D 1; 2; : : :; NG is equal, or nearly so, to the same value N F =NG . Having earlier made this assumption in dealing with the heterogeneity of the average fragment in a group, Gy does not consider the point further. Pitard, on the other hand, states that since the heterogeneity of the average fragment in a group is equal to the heterogeneity of the group, the term in question must vanish. In earlier dealing with this alleged equality, Pitard simply states that (see his eq. 6.28 and preceding remarks) “it has been found that the heterogeneity h n carried by a group of fragments Gn is nothing else than the heterogeneity carried by the average fragment in the group.”
Pitard does remark, however, in moving from Eq. 6 to Eq. 7 that Nn should be relatively constant. In fact, for the algebra to be valid, it must be exactly constant. Continuing Gy’s development and dividing by N F , we have NG X Nn NF 1 X 1 X h 2i D CHL D .h n j N F iD1 N F nD1 jD1
h n /2 C
NG 1 X Nn h 2n N F nD1
(9)
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The last term on the right is equated to DHL (see Eq. 5) and to do this we must again take Nn to be a constant. Thus this link between CHL and DHL , which allows Gy to state that CHL ½ DHL ½ 0
(10)
is very much dependent on the condition that all groups must carry almost exactly the same number of particles. 2.7. Natural distributional heterogeneity of the lot In his section 19.3.5.3, having previously concluded that DH L can never be precisely zero, Gy develops an estimate of the minimum value that DHL may take on. He does this by considering that in a particulate that is as well-mixed as can be, the groups Gn of particles cannot be more similar to each other than when all particles in each group are drawn at random from a large batch of particles having the same properties as the ones with which we wish to concern ourselves. This is quite correct and what we will see in such a case is a variation of the composition from one group to the next that is a consequence of the Poisson distribution of the numbers of particles (variance of the number of particles equals the expected number of particles) in each particle class. However, such variation from group to group is a consequence of constitutional heterogeneity of the group; one wonders whether classifying it as minimal distributional heterogeneity might not be confusing the issue. The rule that the particles be drawn at random is actually sufficient to estimate the magnitude of (DHL )min , if this is how (DHL )min is to be defined. ToP relate the magnitude DHL to CHL , Gy states that the quantity P Nof N G P Nn F 2 2 2 h D ¦ h ¦ 2 nD1 jD1 n j iD1 i can be treated as a chi-squared random variable with N F 1 degrees of freedom, if ¦ 2 is the variance of h i in a population of an effectively infinite number of particles. Whether or not this can be generally accepted as true can be examined by considering the value of h i for each particle as we draw a finite number N F of particles at random from the very large lot. h i is given by Eq. 1 .ai aL /Mi N F hi D aL ML and may be interpreted as a random variable, where its randomness derives from the fact that each particle has a different value of critical content, ai , and a different mass Mi . The values of aL and M L are taken to be material constants. For a given material, we can accumulate a histogram of the results for h i as we draw the particles. To simplify our thinking about the material, let us admit that we may divide the material into Ns size fractions and Nd density classes (or composition classes — these will be equivalent in a binary system of two minerals having different densities). We may denote particle composition as auv and particle mass as m uv D Vu ²v when the particle is drawn from the uth size class and the vth composition class in which the typical fragment volume is Vu and the typical fragment density is ²v . The particle compositions must be distributed to either side of aL so we get positive and negative values of h i . With our simplified (but very valid) view of the material, the histogram that builds up will be of the nature illustrated in Fig. 1.
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Fig. 1. Histogram of particle h values for a sample of particles.
Now, the definition of a chi-squared random variable with N degrees of freedom is the sum of the squared values of N independent standard normal random variables (the standard normal random variable has zero mean and unit variance). We may scale the h i values by their standard deviation, so that the variance of the distribution is unity, but the values of h i will not in general be normally distributed. They may in fact be far from normally distributed as in the quartz–magnetite mixture when Fe is the critical component; such a system will have a bimodal distribution on either side of the mean composition. They will in fact be normally distributed only if the fragment masses the ai D auv are normally distributed about Mi D m uv are all substantially constant andP NF h 2i become chi-squared. Note that we aL . Only then will the distribution of ¦ 2 iD1 cannot make appeal to the central limit theorem here as we are dealing with a sum of squares of random variables rather than a sum of random variables which is later squared. Thus Gy’s conclusions that follow from the above assertion are questionable, in general, and are true only for a very restricted condition. His important conclusion that the minimum expected value of DHL can be expressed as NG 1 EfDHL gmin D (11) CHL NF 1 can be expected to be valid only when his assumptions are fulfilled and in error in other circumstances. It may also be true that the upper limit placed on distributional heterogeneity as DHL CHL is also restricted to a particular case of equal particle numbers and a particular distribution of critical content over the particles of the lot. The final consequence of this formulation of the distributional heterogeneity appears when Gy links up his continuous and discrete models. The grouping and segregation error is defined in section 21.3.3 of Gy’s work and its variance is taken as the difference between the variance of critical content when sampling is carried out taking groups of particles (the ‘potential increments’ that can be formed from the lot) and the variance
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when sampling is carried out by selecting particles at random from the entire lot (fundamental error). It is an excess variance (which does seem to be somewhat at odds with his discussion of the minimal value of distributional heterogeneity). Gy writes
¾ CHL D ¾ ¦ 2 .FE/ (12) NF S where is the grouping factor (approximately equal to the number of fragments in a group, or the potential increment in this case), ¾ is the segregation factor (a value between 0 and 1 — the latter value indicates worst-case segregation) and N F S is the total number of fragments in a sample. ¦ 2 .FE/ is the variance due to fundamental error. We will now consider an example of estimated sampling error for a sampling situation of common interest. However, it is worthwhile to preface the example with a further consideration of the general nature of distributional heterogeneity. ¦ 2 .GE/ D
3. Logical definition of distributional heterogeneity The constitutional heterogeneity of a material is an inescapable heterogeneity which results from the particulate nature of the material. Gy’s definition is given by Eq. 4. Let us take that definition and write it in terms of sums over size and density classes for particles in the nominal or expected make-up of the material. NF NF X .ai CHL D 2 M L iD1
aL /2 Mi2 aL2
We replace Mi by m uv D Vu ²v and take the number of particles in a particle class within the lot of mass M L to be n uv . Then CHL D
Nd Ns X .auv NF X 2 M L uD1 vD1
aL /2 n uv m 2uv aL2
If we take X u to be the mass fraction of particles in the uth size fraction and Yuv to be the mass fraction of particles in the vth composition class within the uth size class, we have CHL D
Ns Nd X .auv aL /2 NF X X u Vu ²uv Yuv M L uD1 aL2 vD1
(13)
Note that M L =N F is the average fragment mass in the lot so that CHL is effectively a constant for the material and is not a function of the quantity of the material we consider. The quantity CHL M L =N F can be taken to be the ‘sampling constant’, K s , for the material. If we draw particles at random from a very large lot of material having a particular nominal composition and properties in order to form a smaller lot L, we can do it in two ways. We can either demand that the total number of particles in L be a fixed constant or we can relax our requirements slightly and simply demand that the expected value of the mass of the lot L be a value M L . In the latter case, we can allow the numbers
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of particles in each class to be independent random variables that follow Poisson distributions. In the former case, when we fix the total number of particles, we find that equiprobable sampling of the particles results in the numbers of particles following a multinomial distribution. We note that in both cases, the total lot mass becomes a random variable. In practice, in trying to make up two nominally identical subsamples of a material, we will never achieve splitting that gives equal numbers of particles; the best we can do is to achieve nominally equal masses that differ between subsamples in a random way. Thus the model of particle numbers as Poisson random variables is the more appropriate statistical model for nominally identical subsamples. There is a minor statistical convenience in this assumption as it removes the covariance between numbers of particles in all particle classes and makes the algebra a bit simpler. Using the Poisson model of the number statistics, it is a simple matter to demonstrate that for samples or lots of expected mass M L , the relative variance in the critical content between lots is exactly equal to CHL =N F (see Lyman, 1986, eq. 18). This variance between lots is the expression of the concept of constitutional heterogeneity. If we now consider the lot L to be composed of NG sublots or groups, each group having been constructed by drawing particles at random from a very large lot, we must find that the relative variance of critical content between the groups is CHG =Nn and since CHG is a material constant, CHG D CHL . If we now take Mn to be a constant in Eq. 5 and note that Mn ¾ D M L =NG , we can write " # NG 2 .an aL /2 NG Mn2 1 X (14) .DHL /min D NG nD1 M L2 aL2 and we see that the term in square brackets is just the relative variance of critical content over the groups and the factor outside the bracket is unity. Thus, regardless of the distribution of critical content, but provided the number of particles in each group is the same CHL NG .DHL /min D D CHL (15) Nn NF This result is similar to the one found by Gy, but it makes no appeal to the chi-squared distribution. However, it must be emphasised here that this minimal (apparent) distributional heterogeneity displayed by the groups is actually a consequence of constitutional heterogeneity and is, in our view, misnamed. It is preferable and less conceptually complex to define distributional heterogeneity as the increase in heterogeneity above the constitutional heterogeneity for a sample of the same mass and nominal composition that is observed for a material when it is sampled in a ‘correct’ manner. In simple terms, any heterogeneity in a sample that cannot be explained by the constitutional heterogeneity for the nominal composition of the material must be deemed to be distributional heterogeneity. In light of Eq. 12 and considering that ¾ D 0 when the material is as well-mixed as can be, such a definition of distributional heterogeneity is probably the one that Gy really intended to make. With this definition, the minimal distributional heterogeneity of the groups is zero and not the term related to the constitutional heterogeneity given in Eq. 11 or Eq. 15. When we deal with a sampling situation, we wish to be able to describe the ‘fundamental error’ or the inescapable variance component associated with the mass of
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sample we collect from the lot. To this error or variance component, we wish to add a component that accounts for distributional heterogeneity, which Gy calls the grouping and segregation error or variance. Again, translating to practical terms and thinking of sampling from a moving stream, we want to know what the variance between samples will be if we sample correctly from a material when it is as well-mixed as can be. And then we wish to know how much the variance between samples will increase in similar circumstances (same number and nominal mass of increments taken in the same way, using random sampling or stratified random sampling) if the material from which we are sampling is segregated in some way. Ideally, we wish to be able to contrive a circumstance of segregation which, even when sampling is correct, will induce a maximal increase in sampling variance, thereby establishing this scenario as the worst that can happen in the event that the material is segregated. Gy’s arguments that establish a maximal value for the distributional heterogeneity are an attempt to do just this. However, as has been established above, the manner in which the maximal distributional heterogeneity is established precludes consideration of a material with a wide size distribution. It is precisely the situation of wide size distribution that we commonly expect to lead to segregation and difficulties in sampling or increase in sampling variance. We will now show that an excess variance due to segregation does indeed exist, but that its description takes a different form from that given by Gy.
4. Example of variance increase due to segregation Let us contrive a situation which would seem to provide a worst-case result of sampling variance due to segregation. Consider a large batch of a particulate laid out on a conveyor belt at a constant mass per unit length but in a totally segregated state. Let us model the material as being composed of K particle classes, each having a distinct critical content ai , i D 1; 2; : : :; K . Let the mass fraction in the ith particle class be X i . Within each length L of the belt, let the particles be sorted into classes which are arranged in a random order. Then each class will cover a length L X i of the section. Further, let consecutive lengths L repeat this sorting by particle classes. We will then sample the belt by taking N increments of mass m I out of each length L according to random sampling. It will be assumed that any one increment is taken from a single class only. Within the set of N increments from the jth section of length L, let the number of times that the ith class is ‘hit’ be n i j . The values of the random variables n i j must then follow a multinomial distribution and we can write the critical content of a sample taken from M sections of belt as K M X X mI n i j ai M K 1 XX jD1 iD1 as D D n i j ai (16) K M X M N jD1 iD1 X mI ni j jD1 iD1
as is a random variable which is written in a way that suppresses the particulate nature of the material and focuses only on the segregation of the material. We will ignore
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the fact that as carries an additional component of variance that is due to the fact that each time we ‘hit’ the ith particle class, we do not collect exactly the same number of particles. We are therefore ignoring a stochastic element in m I and focussing only on the randomness of n i j . The properties of the random variable n i j are: Efn i j g K X ni j
D N Xi DN
any j
iD1
Varfn i j g
D N X i .1
Covfn i j ; n kl g D D0
N Xi Xk
(17)
Xi / j Dl j 6D l
and so the expected value of as is M K K X 1 XX ai Efn i j g D ai X i D aN s Efas g D M N jD1 iD1 iD1 With some simple algebra, we may show that the variance of as is given by K mI X X i .ai aN s /2 Varfas g D Ms iD1
(18)
(19)
where Ms is the mass of sample collected and is simply M N m I . We can also identify as with aL and note that our sampling is unbiased in the long run. In fact, the sampling scenario set up here is absolutely ‘correct’ by all of Gy’s rules of sampling (constant mass flow and random increments from the material). The variance of as is the second moment of the critical content distribution, weighted by the mass fraction in the particle class; note that particle masses do not enter into the calculation. The variance can be reduced only by increasing the number of increments taken and not the mass of the increments taken. This contrasts with the fundamental variance of a sample which depends only on the total mass of the sample collected and the basic properties of the material as expressed in the sampling constant, K s . If the material were not segregated according to particle class, it is a simple matter to demonstrate that the sample critical content variance determined in a manner similar to that used to take account of segregation (suppressing the particulate nature of the material) would be zero. Thus, as the material becomes progressively less well-mixed, Varfas g varies from zero to the value given above. Eq. 19 is the expression of distributional heterogeneity that we seek and it is a maximal value. This expression for distributional heterogeneity demonstrates that, for a given sample value of the variance due to distributional heterogeneity is mass P K Ms , the maximal 2 X .a a N / which corresponds to taking a single increment to form the sample. i i s iD1 This variance can easily exceed the variance due to constitutional heterogeneity. We will define a second quantity, which we will call the sample segregation constant, as K X .ai as /2 Xi (20) Ds D as2 iD1
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This value is dimensionless and is an (intensive) property of the material, in the same way as the constitutional heterogeneity, and may be defined for any critical content. In the practical notation of Eq. 13 Ds can be written as Ds D
Nd Ns X X
X u Yuv
.auv
uD1 vD1
a L /2
(21)
aL2
and the expression is clearly the same as that for the sampling constant, except for the omission of the particle mass weighting. The next section provides an example calculation of the distributional heterogeneity for a coal and compares the value with the constitutional heterogeneity. 4.1. Example of calculation of constitutional heterogeneity and distributional heterogeneity for a coal Let us take a coal sample with the mass and ash distribution as a function of particle density as in Table 1, which is assumed to be the same for all size fractions of the coal. Let the size distribution be as in Table 2. We now calculate the sampling and segregation constants for the material according to the calculations in Tables 3 and 4. Table 1 Density and ash distribution by density fraction Mean density ²uv
Mass fraction Yuv
Ash auv
1.325 1.375 1.425 1.475 1.525 1.575 1.625 1.675 1.750 1.900
0.117 0.604 0.085 0.019 0.009 0.005 0.004 0.003 0.007 0.147 1.000
7.50 9.50 13.50 19.20 27.80 30.30 35.60 40.20 45.90 82.00
Table 2 Size distribution du (cm)
duC1 (cm)
Mass fraction X u
2.64 1.32 0.56 0.28 0.05
1.32 0.56 0.28 0.05 0.01
0.231 0.294 0.139 0.189 0.147 1.000
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Table 3 Calculation of constitutional heterogeneity and distributional heterogeneity terms (density factors) Mean density ²uv
Mass fraction Yuv
Ash auv
Ash mass Yuv auv
Heterogeneity terms CHv
Heterogeneity terms DH
1.325 1.375 1.425 1.475 1.525 1.575 1.625 1.675 1.750 1.900
0.117 0.604 0.085 0.019 0.009 0.005 0.004 0.003 0.007 0.147 1.000
7.50 9.50 13.50 19.20 27.80 30.30 35.60 40.20 45.90 82.00
0.8775 5.738 1.1475 0.3648 0.2502 0.1515 0.1424 0.1206 0.3213 12.054 a L D 21.168
0.064632 0.252328 0.015894 0.000242 0.001347 0.001466 0.003022 0.004062 0.016723 2.306676 2.666
0.048779 0.183512 0.011153 0.000164 0.000883 0.000931 0.001859 0.002425 0.009556 1.21404 1.473
The sampling constant, K s , is found to have a value of 3.694 g and the segregation constant, Ds , is found to have a value 1.473 [-]; the lot ash, aL content is 21.168% ash. The variance of ash content due to constitution heterogeneity for a 25 kg sample of this coal is 3:694 ð 21:12 D 0:066 Varfas;CH g D 25000 The variance of ash content due to distributional heterogeneity is mI 1:473 ð 21:12 D 656m I Varfas;DH g D 25 where m I is the increment mass in kg. For 2.5 kg increments, the standard deviations of ash are 0.256% ash and 40.5% ash for constitutional heterogeneity and distributional heterogeneity, respectively, which clearly shows the large influence of the distributional heterogeneity. The condition for equality of the variances due to constitutional heterogeneity and distributional heterogeneity, when the material is fully segregated is ND
Ds Ms Ks
(22)
Table 4 Calculation of constitutional heterogeneity terms (size factors) du (cm)
duC1 (cm)
Mass fraction X u
Volume Vu
X u Vu
X u Vu CHv
2.64 1.32 0.56 0.28 0.05
1.32 0.56 0.28 0.05 0.01
0.231 0.294 0.139 0.189 0.147 1.000
5.174928 0.618896 0.049392 0.005519 3.15 ð 10
1.195408 0.181955 0.006865 0.001043 4.63 ð 10
3.187427 0.485164 0.018306 0.002781 1.23 ð 10 3.694 (g)
5
6
5
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In this case, the required number of increments is 9969. This figure may seem ridiculous, but it is valid for sampling from a fully segregated material. In practice, the material will never be so segregated and Ds may be reduced by a factor between 0 and 1 expressing the degree of segregation (in the same way as Gy invokes a factor ¾ to account for degree of segregation). Nonetheless, this development and example bears out the truth of Gy’s recommendation that one should take the smallest practical increments consistent with no loss of sample and correct increment extraction and as many increments as possible in the forming of a sample of a particulate material, in order to minimise the effects of material segregation. The next comparison to be made is between the variance given by Eq. 19 and that given by Eq. 12. To do this, we need an estimate of the number of fragments in an increment, . The conceptual sampling scheme we used above considered all particle sizes and ash contents to fall in a different class. It is clear from Eq. 20 or Eq. 21 that only the distribution with respect to ash content influences the value of Ds and thus all size classes with the same ash content can be taken as a single class. We used a 2.5 kg increment. The mean density of the coal can be determined as 1.439 g cm 3 . Using the fragment volumes defined in Table 4, we find that an increment containing all sizes has 8:17 ð 106 fragments so ³ 8:17 ð 106 . Taking the value of ¾ as 1 (total segregation), the estimate of ¦ 2 .GE/ ³ 0:066 ð 1 ð 8:17 ð 106 D 5:39 ð 105 making the corresponding standard deviation 734% ash, which is rather too large. This value would become even larger if the bottom size of the coal were decreased. In fact, because of the dependence of Gy’s grouping and segregation variance on particle numbers, its value is effectively arbitrary. The core difficulty here is that the magnitude of the sampling constant of 3.694 g is not influenced by the number of particles; it is controlled by the magnitude of the volume of the largest particles. Reducing the minimum particle size in Table 4, has no effect on K s , but will radically change the number of particles and hence the estimate of the grouping and segregation error, according to Eq. 12. This difficulty arises from the fact that Gy’s development of grouping and segregation error retains the link to the expression for constitutional heterogeneity which retains a weighting with respect to particle volume. Our development above does not have such a weighting. The retention of the particle mass weighting in Gy’s grouping and segregation error is a consequence of his assumption that all groups have essentially similar numbers of particles, which is an invalid assumption when dealing with wide size distributions. It is thus not surprising that we arrive at an incorrect result when we apply Gy’s expression in a case of wide size distribution. The present development corrects this problem. We may propose a new expression for the sampling variance (relative variance of critical content) as mI Ds ¦ 2 .GE/ D ¾ Ms (23) Ns X Nd X .auv aL /2 Ds D X u Yuv aL2 uD1 vD1 where Ds is a dimensionless segregation constant for the material, 0 ¾ 1 is a segregation factor which is zero when the material is well-mixed and m I and Ms
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are the increment mass and total sample mass, respectively. The fundamental error remains unchanged and related to the constitutional heterogeneity as defined by Gy or, equivalently, the critical content variance which is a consequence of Poisson particle number statistics when sampling correctly from a well-mixed material. We have ¦ 2 .FE/ D
1 Ms
½ Ns Nd X .auv aL /2 1 X X u Vu ²uv Yuv M L uD1 aL2 vD1
(24)
where we have allowed for extraction of a sample of mass Ms from a lot of mass M L . The only restriction placed on the application of Eq. 23 is that the increments taken must be taken over a lot that is genuinely representative of the material, that is, all particle classes must be represented in their proper proportions in the lot, even though it is segregated. The only difficulty with this means of expressing the variance due to distributional heterogeneity is that it is very difficult to make an a priori estimate of the value of ¾ .
5. Reconciling the new expression of distributional heterogeneity with Gy’s overall theory There remains only one aspect of Gy’s overall theory of sampling to be discussed, which is the separation of the segregation effect from the long-term quality fluctuation error. The above development treated segregation using the idea of a segregated material on a conveyor belt and the extraction of increments from such a belt. Such an example immediately raises the question of how to separate segregation effects from genuine long-term fluctuations of the coal properties. The practical answer is that such a separation can never be made unequivocally as both the long-term quality fluctuation and the segregation are random processes and it is never possible to exactly separate the characteristics of two random processes that simultaneously contribute to the randomness of a single measurement. The following argument shows that we may retain Gy’s concept of using the variogram to make the separation by assigning all variance found in the intercept of the variogram (the nugget) to short-term quality fluctuation. The variogram expresses the extent to which the properties of increments extracted a time k∆t, k D 1; 2; : : : apart are statistically correlated; the variogram measure is designed to have minimum sensitivity to trends or drifts of the mean of the process (but it is not immune from such effects). If we were to take increments from a perfectly mixed material and analyse them for critical content, the variogram calculated from the analyses would be flat, with the intercept on the y (variance axis), or the ‘nugget’ value, equal to the ‘sill’ (asymptotic value at long times) of the process. Such a variogram indicates that consecutive measurements of critical content are entirely uncorrelated in time (white noise). The nugget value is equal to the variance of an individual measurement and, in this case, should be equal to the increment variance due to constitutional heterogeneity plus the preparation and analysis variance. We will detect no long-term variance of the critical content in sampling from a large perfectly mixed lot.
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If we now sample from a lot that retains the same mean or expected critical content and critical content distribution over time, but is badly segregated in the sense contrived above, the variogram should not express any correlation between increment assays as the segregation has been set up in a random way. The variogram will be flat. However, the sill and nugget values will be much higher as they will now express the constitutional heterogeneity, distributional heterogeneity and preparation and analysis variance. The increase in the nugget value will be equal to the variance due to distributional heterogeneity. If we now consider sampling from a lot of material that does possess correlated time variation in critical content, but also suffers segregation, we will, in general, find a variogram with a sill larger than the nugget value. The nugget variance will be the sum of constitutional heterogeneity, distributional heterogeneity and preparation and analysis variance as before. If the preparation and analysis variance is established separately by simple (well-known) techniques, and the average composition of the material is known (distribution of critical content by particle class), the preparation and analysis and constitutional heterogeneity variance can be subtracted from the nugget variance to provide an estimate of the variance due to distributional heterogeneity. Using the value of Ds for the material, the value of ¾ can be estimated. It is, of course, important that the increment extraction be correct according to all of Gy’s principles. We note that because distributional heterogeneity is inherently a ‘spatial’ phenomenon (or one distributed in time when the spatial variations are presented on a conveyor belt), it is logical (and necessary) that a spatial or time technique such as the variogram be used to deduce the variance due to distributional heterogeneity. The same technique can be extended to sampling from stockpiles, but in such a case, a three-dimensional variogram must be used, in the same way as in geostatistical analyses. On the assumption that the stockpile under scrutiny consists of material that possesses distributional heterogeneity and is locally segregated to some significant extent, increments taken from throughout the volume of the pile can be analysed and the results used to construct a variogram. Depending on the geometry of the pile, we may expect the variograms to indicate spatial anisotropy. Each directional variogram will then have a different shape, nugget and sill variance. Again using a separate estimate of preparation and analysis variance and a value of constitutional heterogeneity calculated from the average properties of the material, the nugget values can be used to determine the distributional heterogeneity variance. This distributional heterogeneity variance will also be different for each direction, expressing differing degrees of segregation in the different directions. As an example of this effect, we would expect a long stockpile created by simple stacking on the centreline of the pile (no chevroning) to be badly segregated with respect to particle size in the plane normal to the pile axis, but to be much less segregated in the direction parallel to the axis. Such an analysis of a stockpile may be somewhat academic as, in practice, it is effectively impossible to correctly sample from point locations throughout the volume of a large stockpile. Nonetheless, it is interesting to consider how distributional heterogeneity will be expressed through variograms, as suggested above. The concept of using the nugget value of the variogram to determine the effective distributional heterogeneity of a material, when the preparation and analysis and con-
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stitutional heterogeneity variances have been estimated, is entirely consistent with Gy’s breakdown of sampling variance into constitutional heterogeneity and distributional heterogeneity components that depend only on the particulate nature of the material and long-term components that are independent of the particulate nature of the material.
6. Conclusions This work has brought to light an essential assumption within the derivation of Gy’s theory for the distributional heterogeneity of a particulate material that precludes the application of his results to materials of wide size distribution and=or arbitrary distribution of critical content over fragments. It has also been shown that Gy’s expression for the grouping and segregation variance can be in error by an arbitrary amount. An alternate theory and expression for distributional heterogeneity has been developed, which is shown to provide reasonable practical results. The new proposal is also consistent with the use of the variogram to estimate the sum of sampling variance due to constitutional and distributional heterogeneity. The theory is consistent with all of the remaining elements of Gy’s sampling theory which has proved over time to be a tool of inestimable value.
7. Notation ai aiN aL an an j auv anN as aN s CHL DHL (DHL )min Ds hi hn hn j K Ks m uv
D critical content of the ith particle or particles within the ith particle class D critical content of the ‘average particle’ D critical content of the lot D average critical content of the nth group of particles D critical content of the jth particle within the nth group of particles D critical content of particles in the vth composition (or density) class and the uth size class D critical content of the ‘average group’ D critical content of sample D expected value of critical content of sample D constitutional heterogeneity D distributional heterogeneity D minimal distributional heterogeneity (according to Gy) D segregation constant [–] D heterogeneity of the ith particle D heterogeneity of the nth group of particles D heterogeneity of the jth particle within the nth group of particles D number of particle classes D sampling constant (mass) D typical fragment mass within the vth composition (or density) class and the uth size class
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mI M MiN ML Mn Mn j MnN Ms n uv
D mass of increment D number of sections D mass of the ‘average particle’ D mass of the lot L D mass of the nth group of particles D typical fragment mass of the jth particle within the nth group of particles D mass of the ‘average group’ D mass of sample D number of particles within the vth composition (or density) class and the uth size class N D number of increments per section D number of density or composition classes Nd NF D number of particles in the lot L D number of particles in the sample NF S D number of groups in the lot L NG Nn D number of particles in the nth group D number of size classes Ns D short-term quality fluctuation error QE1 Vu D typical particle volume in the uth size class D mass fraction of material in the vth composition class within the uth size Yuv class D mass fraction of material in the ith particle class Xi D mass fraction of material in the uth size class Xu
D Gy’s grouping factor D density of a particle in the vth composition class ²v D density of a particle in the vth composition class within the uth size class ²uv ¦2 D variance of particle heterogeneity over a very large set of particles ¦ 2 .FE/ D relative variance due to fundamental error (or due to constitutional heterogeneity) ¦ 2 .GE/ D relative variance of critical content due to grouping and segregation ¾ D mixing parameter or Gy’s segregation parameter Efg D statistical expectation Covfx; yg D covariance operator D Ef[x Efxg][y Efyg]g Varfg D variance operator D Ef[z Efzg]2 g
References Gy, P., 1982. Sampling of Particulate Materials, Theory and Practice. Elsevier, Amsterdam. Lyman, G.J., 1986. Application of Gy’s sampling theory to coal: a simplified explanation and illustration of some basic aspects. Int. J. Miner. Process. 17, 1–22. Pitard, F.F., 1989. Pierre Gy’s Sampling Theory and Sampling Practice. CRC Press, Baton Rouge, LA.