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Statistical relationship between pyrite grain size distribution and pyritic sulfur reduction in Ohio coal
International Journal of Coal Geology, 9 (1988) 371-383
371
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
Statistical relationship between pyrite grain size distribution and pyritic sulfur reduction in Ohio coal MAINAK MAZUMDAR', RICHARD W. CARLTON 2 and GINO A. IRDI 3
1Industrial Engineering Department, University o[ Pittsburgh, Pittsburgh, PA 15261, U.S.A. 2Department of Natural Resources, Division of Geological Survey, Columbus, OH 43224, U.S.A. ~U.S. Department of Energy, Pittsburgh Energy Technology Center, Pittsburgh, PA 15236, U.S.A. (Received April 22, 1987; accepted for publication October 30, 1987)
ABSTRACT Mazumdar, M., Carlton, R.W. and Irdi, G.A., 1988. Statistical relationship between pyrite grain size distribution and pyritic sulfur reduction in Ohio coal. Int. Coal Geol., 9: 371-383. This paper presents a statistical relationship between the pyrite particle size distribution and the potential amount of pyritic sulfur reduction achieved by specific-gravity-based separation. This relationship is obtained from data on 26 Ohio coal samples crushed to 14 × 28 mesh. In this paper a prediction equation is developed that considers the complete statistical distribution of all the pyrite particle sizes in the coal sample. Assuming that pyrite particles occurring in coal have a lognormal distribution, the information about the particle size distribution can be encapsulated in terms of two parameters only, the mean and the standard deviation of the logarithms of the grain diameters. When the pyritic sulfur reductions of the 26 coal samples are related to these two parameters, a very satisfactory regression equation (R 2 = 0.91 ) results. This equation shows that information on both these parameters is needed for an accurate prediction of potential sulfur reduction, and that the mean and the standard deviation interact negatively insofar as their influence on pyritic sulfur reduction is concerned.
INTRODUCTION
Purpose The purpose of this paper is to present a relationship between the pyrite particle size distribution and the amount of pyritic sulfur reduction achieved by specific-gravity-based separation. This relationship is based on the data collected by Carlton (1985) on 26 coal samples crushed to 14 × 28 mesh. Carlton found that a good prediction for pyritic sulfur reduction is obtained when 0166-5162/88/$03.50
© 1988 Elsevier Science Publishers B.V.
372
one takes into account the pyrite volume percent below a certain size (i.e., 24 /lm ) in the coal sample. In this study, we attempt to relate the washability data to the distribution by volume of all the particle sizes. We explain below why this approach makes sense from a physical viewpoint. The numerical information presented suggests that the proposed prediction procedure is quite accurate also.
Pyrite size/pyritic sulfur reduction relationship The distribution of pyrite particle sizes has long been known to influence the amount of pyritic sulfur reduction that will take place when the coal is washed by gravimetric methods. One of the earlier studies showing this relationship was by McCartney et al. (1969). Their results showed that a significant relationship exists between the mean grain size of pyrite and the pyrite sulfur reduction of washed coal. They also found a very strong correlation between the proportion of coal particles containing more than 50 percent pyrite and pyritic sulfur reduction. Harvey and DeMaris (1985) made a similar investigation and observed that the characteristic most useful for evaluating the float-sink behavior of coat was the percentage of the pyrite grain diameters within the various maceral associations. Based on his data on Ohio coal samples, Carlton (1985) observed a strong negative correlation between cumulative percent pyrite in certain pyrite size ranges and pyritic sulfur reduction of' coal washed by float-sink methods. The reasons why the pyrite distribution plays an important role in pyritic sulfur washability have been explained by Carlton (1985). Pyrite removal by washing in heavy media or liquids is possible because of pyrite's relatively high specific gravity (~5.0) compared to that of coal macerals (~1.35). Pyrite is most easily removed by washing when pyrite grains are large and occupy a large portion of individual crushed coal fragments or are broken completely free of' the coal. On the other hand, a coal with most of its pyrite occurring as finely disseminated particles that are intimately associated with coal macerals, especially vitrinite, will prove to be more difficult to clean by mechanical grinding and gravity separation. Leaving aside the question of coal association for the time being, a depiction of the effect of pyrite size distribution on pyrite sulfur reduction is given in Fig. 1. This figure suggests that if the mean pyrite particle size is small, a corresponding small standard deviation is detrimental for the purpose of pyrite removal by gravimetric methods. On the other hand, it should be possible to liberate pyrite particles easily if the mean particle size is large and the standard deviation is small. Thus, it appears that for the purpose of predicting washability, more accurate estimates can be obtained by considering more than one parameter related to the pyrite size distribution. (Previous work reported in the literature has considered only a single parameter for these distributions. ) More accurate pre-
373 Broken coal fragment with coarse pyrite
Coal fragment with coarse pyrite
Crushed to finer size
% Pyrite AI: 9.45 (sink) % Pyrite A2:2.75 (float) % Pyrite A3:1.96 (float)
A % Pyrite: 3.85
Decreasing standard deviation for pyrite
.c_ I "~
Broken coal fragment with fine pyrite
Coal fragment with fine pyrite
== Crushed to finer size
B % Pyrite: 3.84
% Pyrite BI: 3.58 (float) % Pyrite B2:3.76 (float) % Pyrite B3:3.96 (float)
Fig. 1. In this idealized example finer crushing of A, which contains coarse pyrite, results in a 49 percent reduction of pyritic sulfur. Crushing an identical coal fragment (B) with about the same amount of total pyrite, but occurring as much finer pyrite grains, does not increase the pyritic sulfur reduction. dictions should result by providing as much descriptive information as possible about the pyrite size distribution.
Occurrence o/pyrite in coal and lognormal character of its size distribution Pyrite found in coal can occur either as primary or secondary grains exhibiting many different morphologies. These different forms of pyrite have been well described by Gray et al. ( 1963 ), Caruccio et al. (1977), Maxwell and Kneller (1982), and Wiese and Fyfe (1986). Primary pyrite generally ranges from submicron to about 600 microns in size (Caruccio et al., 1977 ), whereas secondary pyrite may range from very large masses tens of centimeters in thickness to very thin coatings microns thick. There have been only a few studies on the size distribution of pyrite in coal. Maxwell and Kneller (1982) found that the size distribution of pyrite in seven Ohio coals crushed to minus 20 mesh followed a lognormal distribution. ReyesNavarao and Davis (1976) also found logormal distributions, and McCartney et al. (1969) observed that pyrite size distributions in minus-14-mesh crushed
;]74
coal were well-fitted by a Rosin-Rammler distribution (see Herden, 1953). ~ data set that is well-fitted by a Rosin-Rammler distribution is frequently alsc~ well-described by a lognormal distribution. The latter distribution is analyti cally simpler to handle. It has also the mathematical property that il: the dis tribution of the sizes by count is lognormal, then their distribution by area or volume is also logormal (Allen, 1981 ). This property holds in the case of spher ical grains only. If it is assumed that the particle sizes have a lognormal distribution, then it means that the logarithms of the sizes have a normal distribution. It is well known that a normal distribution is fully characterized by two parameters: the mean (/~) and the standard deviation (c7). Therefore, if~ and a are specified for the logarithmic-size distribution, and if we assume that the distribution is lognormal, p nd a will completely describe the relative proportion of all sizes in the coal sample. Because of the mathematical property referred to above, it would not matter if the parameters are specified for the distribution of sizes by count or by volume. METHODS
Overview of statistical approach In this paper the distribution of the pyrite grain diameters is examined for the 26 Ohio coal samples considered by Carlton ( 1985 ). Each of Carlton's samples was stage crushed and sieved to 14 X 28 mesh and then float-sink tested using a liquid of 1.60 specific gravity. Table 1 reproduces some of the relevant information about the coal samples from Carlton's paper, which also provides details about the method of preparation of these samples. In the size analysis for the the individual coal samples, pyrite grains were converted to equivalent circular diameters (ECD). An ECD is the diameter of a circle with the same area as the pyrite grain measured. The area measurements of a randomly distributed pyrite on a polished surface are taken to be equal to the volume percent pyrite. An example of the size distribution data that were the object of the analysis of this paper is given in Table 2. This table gives the cumulative volume percent pyrite for all sizes between 2 ~m and 72 ttm at steps of 2/tm for five coal samples. The pyrite size distribution of each of Carlton's (1985) coal samples for his Study A was tested to determine whether the size distribution could be adequately described by means of a lognormal distribution. For almost all the coat samples, the lognormal distribution provided a good fit to the distribution of particle sizes. Next the mean (~) and the standard deviation (a) were estimated for the logarithms of the pyrite grain diameters using data as illustrated in Table 2 for the 26 Ohio coal samples. The estimated values are denoted by and &, respectively. Finally, the relationship was determined between the
375
TABLE 1 Sample location,coal thickness, and pyritic-sulfurreduction (Carlton, 1985 ) DGS file No.
1351 1352 1353 1354 1355 1356 1358 1359 1360 1361 1379 1380 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1395 1396
Coal
L. Kittanning M. Kittanning M. Kittanning L. Kittanning L. Kittanning M. Kittanning M. Kittanning U. Freeport M. Kittanning M. Kittanning M. Kittanning L. Kittanning L. Kittanning M. Kittanning L. Kittanning L. Kittanning L. Kittanning L. Kittanning M. Kittanning M. Kittanning M. Kittanning M. Kittanning L. Kittanning M. Kittanning M. Kittanning M. Kittanning
County
Holmes Holmes Coshocton Coshocton Coshocton Coshocton Muskingum Muskingum Perry Perry Vinton Vinton Vinton Jackson Jackson Jackson Stark Stark Stark Stark Stark Columbiana Stark Columbiana Columbiana Columbiana
Township
Walnut Creek Berlin Mill Creek Mill Creek Lafayette Lafayette Washington Wayne Pike Bearfield Swann Swann Clinton Madison Madison Bloomfield Pike Osnaburg Osnaburg Sugar Creek Paris Columbia, West Paris Franklin Center Hanover
Coal thickness (cm)
91.44 66.04 71.12 91.44 59.06 113.03 52.71 120.02 140.97 104.14 125.73 67.95 134.62 31.12 68.58 86.36 97.79 107.95 104.14 71.12 64.77 138.43 62.87 53.98 91.44 95.25
14 × 28 mesh coal Wt.% pyritic sulfur Unwashed
Float
2.97 7.23 2.86 5.81 1.88 3.03 4.71 3.58 2.51 1.82 3.28 5.94 3.40 2.10 1.60 2.36 3.24 5.79 1.58 2.82 1.54 1.98 6.94 3.65 5.54 3.43
1.53 1.42 0.58 2.50 0.96 0.96 1.92 1.30 1.63 1.06 0.61 1.28 0.96 0.56 0.37 1.10 0.46 2.69 0.64 0.66 0.51 0.42 1.03 0.18 0.64 0.76
% pyritic-sulfur reduction using 1.60 specific gravity liquid 48.5 80.4 79.7 57.0 48.9 68.3 59.2 63.7 35.1 41.8 81.4 78.5 71.8 73.3 76.9 53.4 85.8 53.5 59.5 76.6 66.9 78.8 85.2 95.1 88.4 77.8
pyritic sulfur reduction for the coal samples and the respective estimated parameters,/~ and &, by use of linear regression techniques. As the results in the following sections show, the agreement between the observed data and the corresponding least-squares fit is quite good. Estimation of pyrite mean and standard deviation The process of estimating the two parameters, # and a, for each of the coal samples was, however, not without complications. For his study A. Carlton (1985) used a 32 X oil immersion objective, which permitted him to measure maximum grain size up to 72 ttm only (see Table 2). Thus no direct measurements are available for grains whose ECD's exceed this particular value. This
376 TABLE 2 Cumulative volume percent pyrite at different size for five coal samples ECD* (Microns) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72
DGS Sample Number 1351
1352
1353
1354
1355
0.68 5.77 12.25 18.14 22.40 26.28 29.84 33.03 36.10 38.56 40.73 43.72 47.38 50.49 53.70 57.82 60.25 62.15 63.82 66.63 67.49 69.08 69.78 70.55 72.17 74.40 75.87 77.45 79.13 80.91 82.83 84.93 87.85 88.63 90.26 91.12
0.18 1.51 3.36 5.03 6.65 8.13 9.39 10.76 12.25 13.42 14.42 15.53 16.75 18.45 20.25 21.63 22.79 23.89 25.26 26.79 28.74 29.79 31.78 33.53 34.60 36.04 37.44 38.80 40.80 41.58 43.86 45.84 47.02 49.77 52.62 53.01
0.46 3.54 7.26 9.91 12.21 14.06 15.50 17.16 18.90 20.18 21.71 23.02 25.17 26.66 27.99 29.50 30.71 31.89 33.21 33.92 36.07 37.84 38.44 41.30 41.67 42.51 43.40 45.33 45.85 47.55 50.53 51.81 55.25 58.84 60.36 61.16
0.54 5.04 11.59 17.35 22.25 26.35 30.01 32.97 35.84 38.00 40.60 42.81 44.70 46.13 48.48 50.85 52.20 54.28 55.98 57.38 58.68 59.97 60.92 61.42 63.10 64.50 65.59 66.99 67.98 68.25 69.96 71.19 72.50 73.54 75.77 78.09
0.60 5.42 12.19 17.63 21.39 25.25 29.09 32.20 35.53 39.88 43.28 48.45 53.40 56.54 58.25 61.81 65.75 67.15 69.03 71.64 74.00 77.67 80.54 83.66 85.68 85.68 88.03 89.70 91.52 91.52 92.55 94.76 95.94 97.17 98.49 98.49
*ECD: Equivalent Circular Diameter. s i t u a t i o n is c o m m o n l y k n o w n i n s t a t i s t i c a l t e r m i n o l o g y as a d a t a s e t h a v i n g c e n s o r e d o b s e r v a t i o n s . S e v e r a l d i f f e r e n t a l g o r i t h m s (e.g., R a a b e , 1971; W o l y netz, 1979) could be a p p l i e d for o b t a i n i n g t h e m a x i m u m l i k e l i h o o d e s t i m a t e s of the two p a r a m e t e r s f r o m l o g n o r m a l l y d i s t r i b u t e d c e n s o r e d d a t a sets. In this
377 study the readily available SAS software (1985) on probit analysis (Finney, 1971 ) was used to derive these estimates. This computer package also has the built-in capacity for testing for the lognormality of~he underlying distribution. In the process of estimating the size distribution parameters for the coal samples, an analogy is drawn with the problem considered in probit analysis. This technique is predominantly used in statistical analysis of data pertaining to biological assays. These assays are methods for the estimation of the potency or toxicity of a given material as measured by the reaction that follows its application to living material. An interesting application of probit analysis in coal preparation was made by Luckie (1969). In the language of bio-assay, let x denote the level of the dose of a given material being tested and y the corresponding probability of response corresponding to this dosage level. For example, in experimental toxicity studies, this probability is estimated by the proportion of animals killed as a result of the application of the dose at the level x. Probit analysis assumes that y can be expressed as the cumulative normal distribution function (with parameters p and a) at the point x. In applications, the numerical values of H and a will be unknown. Denote by ¢(x; lt, a) the normal probability density function with mean H and standard deviation a and by ¢b(x; H,a) the corresponding cumulative distribution function. In algebraic notation, the above assumption is equivalent to the following:
y = (I)(x;H,a)
(1)
where
CP(x;H,a) = i f~(t;H,a)dt --
OO
and ¢(t;
H,a) = (1/2n)l/2a-lexp I - (t-p)2/2a2].
The probit analysis techniques provide the best estimates of the parameters H and a from an observed set of paired data (x,y), where x is the level of applied dose and y is the estimated probability of response. The statistical software (SAS, 1985) contains a procedure known as the PROC Probit for estimating H and a from such data pairs. This procedure also provides an option whereby it is possible to do the probit analysis using the logarithm of the data value x to the base e. For the purpose of estimating the parameters H and a from the pyrite size distribution data for the coal samples, a correspondence is set between x (the dose level) and the pyrite size ECD, and between y (the probability of response
378 at dose level, x) and the cumulative pyrite volume percent at x. For example~ Table 2 shows that for the Ohio Division of Geological Survey (DGS) sample 1351, y = 57.82 for x = 32 ~lm. In analogy with probit analysis, we can regard y as the probability of an animal being killed (--0.5782) when the level of the dose is x = 32. For each of the 26 coal samples, data are available for 36 pairs of x and y values. The SAS probit procedure using the LOG option for x is performed for these samples. This option has the effect of assuming that y can be expressed as the cumulative normal distribution function in terms of the natural logarithm of x; that is, the pyrite sizes (ECD) have a lognormal distribution for each of the coal samples. The output of the SAS procedure will now provide the "best" estimated values (maximum likelihood estimates ) forg and a, which we denote by/~ and ~. They can now be regarded as the estimates for the mean and the standard deviation of the volume distribution of the logarithms of the pyrite ECD's in a given coal sample, the logarithms being taken to the base e. The output also provides information on results of statistical tests carried out to determine whether a lognormal distribution provides a good fit to the observed data for each of the coal samples. RESULTS Table 3 gives the estimated values ofp and a for each of the coal samples as furnished by the SAS output. The fourth column of this table indicates whether a lognormal fit to the observed data for the particular coal sample is considered satisfactory. This is determined by the SAS package using a chi-square goodness-of-fit test. The symbol 'S' indicates that the calculated chi-square is insignificant at the 10% level in which case the distribution fit is considered satisfactory. The symbol 'U' indicates that the calculated chi-square statistic is significant, in which case the fit is not satisfactory. In only three cases out of a total twenty-six is the lognormal fit considered inadequate according to the chi-square test (Table 3 ). The SAS output also provides graphical plots of the observed data against the backdrop of the fitted normal distribution. From a visual inspection, however, each of the coat samples appeared to provide a reasonable fit to the lognormal distribution. Therefore, all 26 samples were used in this study. The lognormal assumption implies that all the information about the pyrite particle size distribution for a given coal sample is contained in these two parameters tt and a. From the data given in Table 3, a relationship was established between the sulfur reduction values and ~ and ~. This is done by using the techiques of multiple regression. A quadratic equation that describes this relationship was obtained by first using the stepwise regression procedure of SAS (1985) on a full quadratic model in # and a with two linear terms, two square terms, and one cross-product term. Subsequently, a least-squares regression was fitted
379 TABLE 3 Logarithmic mean and standard deviation values for the pyrite size distribution of 26 coal samples and the corresponding sulfur reduction data DGS sample No.
1351 1352 1353 1354 1355 1356 1358 1359 1360 1361 1379 1380 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1395 1396
Estimated parameters /~
~
3.2126 4.3701 4.1535 3.4180 3.0848 3.5864 3.5390 3.7991 3.0166 3.2238 4.3190 4.4086 3.9788 4.5342 4.3685 3.5575 4.2691 3.5188 3.5286 3.7532 4.6407 4.8047 5.1315 4.7118 4.8152 4.0224
1.0451 1.2627 1.4433 1.3343 0.8258 1.2627 1.3635 1.2615 1.5667 1.2633 1.2042 1.4428 1.0213 1.3025 1.2246 1.4586 1.0704 1.0051 1.3331 1.0190 1.6682 1.6188 1.4483 0.9615 1.1971 1.0225
Goodness of lognormal fit*
Sulfur reduction (% )
S S S S U S S S S S S S U S S S S S S S S S S U S S
48.5 80.4 79.7 57.0 48.9 68.3 59.2 63.7 35.1 41.8 81.4 78.5 71.8 73.3 76.9 53.4 85.8 53.5 59.5 76.6 66.9 78.8 85.2 95.1 88.4 77.8
*"s" indicates fit is considered satisfactory according to a statistical test. "U" indicates fit is considered unsatisfactory according to a statistical test.
w i t h t h e t e r m s j u d g e d s i g n i f i c a n t b y t h e s t e p w i s e r o u t i n e . T h e m o d e l is described by the following equation: S R -- - 184.613 + 110.243/~ - 5.172 ~ & - 9.871 ~2
(2)
w h e r e S R s t a n d s for s u l f u r r e d u c t i o n , fi r e p r e s e n t s t h e e s t i m a t e d m e a n p a r t i c l e size, a n d ~ r e p r e s e n t s t h e e s t i m a t e d s t a n d a r d d e v i a t i o n o f t h e p a r t i c l e size d i s t r i b u t i o n . T h e p o s i t i v e sign for t h e c o e f f i c i e n t of/~ s u g g e s t s t h a t as ~ in-
380
creases, SR increases up to a point. However, this relationship is not exactly linear. The negative coefficient of the/~l~ term tempers the rate of increase as increases. The most interesting facet of the equation is that the coefficient of' the/5# term is negative. In statistical terminology,/~ and # are said to have a negative interaction. That is, if 7l has a relatively small value, a relatively small value of ~ is detrimental for the purpose of sulfur reduction. On the other hand, if'~ is relatively large, then a small value of h is conducive to sulfur reduction. This equation is consistent with Figure 1 and our understanding of coal cleaning based on gravimetric separation. The regression equation (2) had a R 2 value of' 0.91. Table 4 compares the predicted values of sulfur reduction using eqn. (2) against the actual observed values. For the most part, the agreement between the observed and the preTABLE 4 Comparison of actual sulfur reduction values with those predicted by Equations (2) and (3) DGS Sample No.
Actual pyritic sulfur reduction
Predicted pyritic sulfur reduction ( This paper )
Predicted pyritic sulfur reduction ( Carlton's study A, 1985 )
1351 1352 1353 1354 1355 1356 1358 1359 1360 1361 1379 1380 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1395 1396
48.5 80.4 79.7 57.0 48.9 68.3 59.2 63.7 35.1 41.8 81.4 78.5 71.8 73.3 76.9 53.4 85.8 53.5 59.5 76.6 66.9 78.8 85.2 95.1 88.4 77.8
50.31 80.11 71.99 53.29 48.36 60.38 56.95 66.95 33.68 47.14 80.50 76.66 76.74 81.77 80.94 55.81 82.49 62.80 57.16 69.76 74.37 76.97 82.73 92.25 87.55 77.85
50.37 79.69 71.90 51.32 45.45 59.65 56.39 65.59 41.88 46.15 79.81 76.48 76.46 81.10 78.97 58.51 81.84 60.16 54.65 70.45 77.16 79.92 87.76 90.65 88.04 76.50
381 dicted values is quite satisfactory. Also shown in this table are predicted values given by the regression equation obtained by Carlton (1985), which is: SR = 95.84 - 1.04z
(3)
where z is the cumulative pyrite volume percent at size 24/~m. The predictions based on eqn. (2) are in general more accurate than those based on eqn. (3). The average absolute difference between the actual pyritic reduction and the prediction based on eqn. (2) is 3.62%; the corresponding figure for eqn. (3) is 4.19%. Independent estimates of # and a were also derived for the 26 Ohio coal samples using a modified version of the algorithm proposed by Wolynetz (1979). The results are similar to those given in Table 3. DISCUSSION The statistical analysis given in the preceding section has provided an algebraic relation between pyrite particle size distribution and pyritic sulfur reduction by gravimetric methods. Under the assumption that the size distribution under question is lognormal, it is possible to encapsulate all the information about the size distribution in terms of two constants. A prediction equation for sulfur reduction is then developed in terms of these two constants. For the particular set of 26 Ohio coal samples for which we had detailed information, the prediction equation performed very well. It is difficult to say whether this equation will hold for a much wider variety of coal samples. A much larger data base is needed before one can attempt to answer this question. It is hoped, nevertheless, that the methodology considered in this paper will continue to prove to be useful in the analysis of pyrite reduction studies carried out from a petrographic viewpoint. Although the prediction eqn. (3) did not perform too badly in comparison with eqn. (2), there is a statistical reason why the latter equation should be preferred in practice. The statistics,/~,~z and z, will be estimated from a limited number of pyrite particles. The standard errors associated with/~ and ~ are expected to be smaller than that of z. Thus, in general, the "repeatability" of /~ and ~zwill be better than of z. The analysis given here has considered the relationship between the pyrite particle size distribution and pyritic sulfur reduction. No doubt a stronger relation can be obtained if, additionally, information on the degree of liberation of pyrite particles from coal macerals is provided for each of the coal samples. This information is ideally given by one or two numerical indexes. McCartney et al. (1969) provided such an index as a measure of degree of liberation. They separately obtained the relationship between pyritic sulfur reduction and mean pyrite particle size, and between the former quantity and their index of degree of liberation, taken one at a time. An analysis of their data as given in Table 2
382 of their paper using both the size parameter and the index of liberation as two independent variables in one common linear regression equation results in a stronger regression equation with a R ~ approximately 89%. This leads us to believe that if information on the degree of liberation is available in addition to the size distribution, a still more accurate equation than eqn. (2) will result. Modern automated image analysis offers a potentially rapid, nontiring method of' determining information on the size characteristics and degree of liberation of pyrite particles associated with coal. The United States Department of Energy at the Pittsburgh Energy Techology Center is currently working on developing automated image analysis procedures to estimate pyrite particles size distributions and degree of liberation (Irdi and Rohar, 1986). It is developing a technique fbr automatically quantifying the degree of pyrite liberation in ultrafine coal using a Zeiss IBAS image analyzer. A preliminary program is being tested, and the results will be published soon. CONCLUSIONS Based on data on pyrite size distribution available for 26 Ohio coal samples, this paper has presented a relationship between the estimated mean and the standard deviation of the logarithm of the pyrite grain sizes and potential pyritic sulfur reduction. For the most part, the agreement between the observed and predicted values is quite satisfactory. The derived equation performs better than that provided by Carlton (1985), where he developed a similar prediction based on the cumulative volume percent distribution at size 24 microns. The numerical results of the present study appear to validate the argument given here, namely, that the distribution of all the pyrite particle sizes should be considered for the purpose of developing a prediction equation for potential pyritic sulfur reduction based on gravimetric tests. ACKNOWLEDGEMENT The research of M. Mazumdar was supported in part by an appointment to the U.S. Department of Energy Fossil Fuel Part-time Faculty Participation program administered by Oak Ridge Associated Universities at the Pittsburgh Energy Techology Center. Portions of this paper were published with permission of H.R. Collins, Chief of the Division of Geological Survey. DISCLAIMER Reference in this paper to any specific commercial product, process, or service is to facilitate understanding and does not necessarily imply its endorsement or favoring by the United States Department of Energy.
383 REFERENCES Allen, T., 1981. Particle Size Measurement, Third Edition, Chapman and Hall, New York and London, p. 678. Carlton, R.W., 1985. Image analysis of pyrite in coal: relation between pyrite grain-size distribution and sulfur reduction. In: Y.A. Attia (Editor), Processing and Utilization of High Sulfur Coals. Elsevier, Amsterdam, pp. 3-17. Caruccio, F.T., Ferm, J.C., Horne, J., Geidel, and Baganz, B., 1977. Paleoenvironment of coal and its relation to drainage quality. Interagency Energy Environment Research and Development Program Report, EPA-600/7-77-067, U.S. Environmental Protection Agency, Cincinnati, OH. Finney, D.J., 1971. Statistical Methods in Biological Assay, Second Edition. Hafner Press, New York, NY, p. 668. Gray, R.J., Schapiro, N. and Coe, G.D., 1963. Distribution and forms of sulfur in a high volatile Pittsburgh seam coal. Trans., Soc. Min. Eng., June: 113-121. Harvey, R.D. and DeMaris, P.J., 1985. Size and maceral association of sulfide grains in Illinois coals and their washed products. Ill. State Geol. Surv., Champaign, IL, 49 pp. Herden, G., 1953. Small Particle Statistics. Elsevier, Amsterdam, 520 pp. Irdi, G.A. and Rohar, P., 1986. An automated microscopic method for the determination of pyrite size distributions in crushed coal. Presented at the AIME-SME Fall Meeting, St. Louis, MO, September 7-10, 1986. Luckie, P.T., 1969. The application of probit analysis methodology to the determination of the nature of the partition curve. M. Sc. thesis (unpublished). Pennsylvania State University, University Park, PA. Maxwell, G.P. and Kneller, W.A., 1982. Size, shape, and distribution of microscopic pyrite in selected Ohio Coals. Final Report V for Ohio Coal Research Laboratory Association OCRLA1. University of Toledo Carbon Facility, 158 pp. McCartney, J.T., O'Donnell, H. and Ergun, S., 1969. Pyrite size distribution and coal-pyrite particle association in steam coals. U.S. Bureau of Mines Rept. of Invest. 7231, 18 pp. Raabe, G., 1971. Particle size analysis using grouped data and the lognormal distribution. Aerosol Sci., 2: 289-303. Reyes-Navaro, J. and Davis, A., 1976. Pyrite in coal: its forms and distributions as related to the environments of coal deposition in three selected coals from western Pennsylvania. Special Research Report No. SR-110, Coal Research Section, PA., 141. SAS User's Guide: Statistics, Version 5 Edition, 1985. SAS Institute, Inc., Box 8000, Cary, NC. Wiese, R.G., Jr., and Fyfe, W.S., 1986. Occurrences of iron sulfides in Ohio coals. Int. J. Coal Geol., 6: 251-276. Wolynetz, M.S., 1979. Maximum likelihood estimation from confined and censored normal data, Algorithm AS 138. Appl. Statist., pp. 185-195.
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Statistical relationship between pyrite grain size distribution and pyritic sulfur reduction in Ohio coal MAINAK MAZUMDAR', RICHARD W. CARLTON 2 and GINO A. IRDI 3
1Industrial Engineering Department, University o[ Pittsburgh, Pittsburgh, PA 15261, U.S.A. 2Department of Natural Resources, Division of Geological Survey, Columbus, OH 43224, U.S.A. ~U.S. Department of Energy, Pittsburgh Energy Technology Center, Pittsburgh, PA 15236, U.S.A. (Received April 22, 1987; accepted for publication October 30, 1987)
ABSTRACT Mazumdar, M., Carlton, R.W. and Irdi, G.A., 1988. Statistical relationship between pyrite grain size distribution and pyritic sulfur reduction in Ohio coal. Int. Coal Geol., 9: 371-383. This paper presents a statistical relationship between the pyrite particle size distribution and the potential amount of pyritic sulfur reduction achieved by specific-gravity-based separation. This relationship is obtained from data on 26 Ohio coal samples crushed to 14 × 28 mesh. In this paper a prediction equation is developed that considers the complete statistical distribution of all the pyrite particle sizes in the coal sample. Assuming that pyrite particles occurring in coal have a lognormal distribution, the information about the particle size distribution can be encapsulated in terms of two parameters only, the mean and the standard deviation of the logarithms of the grain diameters. When the pyritic sulfur reductions of the 26 coal samples are related to these two parameters, a very satisfactory regression equation (R 2 = 0.91 ) results. This equation shows that information on both these parameters is needed for an accurate prediction of potential sulfur reduction, and that the mean and the standard deviation interact negatively insofar as their influence on pyritic sulfur reduction is concerned.
INTRODUCTION
Purpose The purpose of this paper is to present a relationship between the pyrite particle size distribution and the amount of pyritic sulfur reduction achieved by specific-gravity-based separation. This relationship is based on the data collected by Carlton (1985) on 26 coal samples crushed to 14 × 28 mesh. Carlton found that a good prediction for pyritic sulfur reduction is obtained when 0166-5162/88/$03.50
© 1988 Elsevier Science Publishers B.V.
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one takes into account the pyrite volume percent below a certain size (i.e., 24 /lm ) in the coal sample. In this study, we attempt to relate the washability data to the distribution by volume of all the particle sizes. We explain below why this approach makes sense from a physical viewpoint. The numerical information presented suggests that the proposed prediction procedure is quite accurate also.
Pyrite size/pyritic sulfur reduction relationship The distribution of pyrite particle sizes has long been known to influence the amount of pyritic sulfur reduction that will take place when the coal is washed by gravimetric methods. One of the earlier studies showing this relationship was by McCartney et al. (1969). Their results showed that a significant relationship exists between the mean grain size of pyrite and the pyrite sulfur reduction of washed coal. They also found a very strong correlation between the proportion of coal particles containing more than 50 percent pyrite and pyritic sulfur reduction. Harvey and DeMaris (1985) made a similar investigation and observed that the characteristic most useful for evaluating the float-sink behavior of coat was the percentage of the pyrite grain diameters within the various maceral associations. Based on his data on Ohio coal samples, Carlton (1985) observed a strong negative correlation between cumulative percent pyrite in certain pyrite size ranges and pyritic sulfur reduction of' coal washed by float-sink methods. The reasons why the pyrite distribution plays an important role in pyritic sulfur washability have been explained by Carlton (1985). Pyrite removal by washing in heavy media or liquids is possible because of pyrite's relatively high specific gravity (~5.0) compared to that of coal macerals (~1.35). Pyrite is most easily removed by washing when pyrite grains are large and occupy a large portion of individual crushed coal fragments or are broken completely free of' the coal. On the other hand, a coal with most of its pyrite occurring as finely disseminated particles that are intimately associated with coal macerals, especially vitrinite, will prove to be more difficult to clean by mechanical grinding and gravity separation. Leaving aside the question of coal association for the time being, a depiction of the effect of pyrite size distribution on pyrite sulfur reduction is given in Fig. 1. This figure suggests that if the mean pyrite particle size is small, a corresponding small standard deviation is detrimental for the purpose of pyrite removal by gravimetric methods. On the other hand, it should be possible to liberate pyrite particles easily if the mean particle size is large and the standard deviation is small. Thus, it appears that for the purpose of predicting washability, more accurate estimates can be obtained by considering more than one parameter related to the pyrite size distribution. (Previous work reported in the literature has considered only a single parameter for these distributions. ) More accurate pre-
373 Broken coal fragment with coarse pyrite
Coal fragment with coarse pyrite
Crushed to finer size
% Pyrite AI: 9.45 (sink) % Pyrite A2:2.75 (float) % Pyrite A3:1.96 (float)
A % Pyrite: 3.85
Decreasing standard deviation for pyrite
.c_ I "~
Broken coal fragment with fine pyrite
Coal fragment with fine pyrite
== Crushed to finer size
B % Pyrite: 3.84
% Pyrite BI: 3.58 (float) % Pyrite B2:3.76 (float) % Pyrite B3:3.96 (float)
Fig. 1. In this idealized example finer crushing of A, which contains coarse pyrite, results in a 49 percent reduction of pyritic sulfur. Crushing an identical coal fragment (B) with about the same amount of total pyrite, but occurring as much finer pyrite grains, does not increase the pyritic sulfur reduction. dictions should result by providing as much descriptive information as possible about the pyrite size distribution.
Occurrence o/pyrite in coal and lognormal character of its size distribution Pyrite found in coal can occur either as primary or secondary grains exhibiting many different morphologies. These different forms of pyrite have been well described by Gray et al. ( 1963 ), Caruccio et al. (1977), Maxwell and Kneller (1982), and Wiese and Fyfe (1986). Primary pyrite generally ranges from submicron to about 600 microns in size (Caruccio et al., 1977 ), whereas secondary pyrite may range from very large masses tens of centimeters in thickness to very thin coatings microns thick. There have been only a few studies on the size distribution of pyrite in coal. Maxwell and Kneller (1982) found that the size distribution of pyrite in seven Ohio coals crushed to minus 20 mesh followed a lognormal distribution. ReyesNavarao and Davis (1976) also found logormal distributions, and McCartney et al. (1969) observed that pyrite size distributions in minus-14-mesh crushed
;]74
coal were well-fitted by a Rosin-Rammler distribution (see Herden, 1953). ~ data set that is well-fitted by a Rosin-Rammler distribution is frequently alsc~ well-described by a lognormal distribution. The latter distribution is analyti cally simpler to handle. It has also the mathematical property that il: the dis tribution of the sizes by count is lognormal, then their distribution by area or volume is also logormal (Allen, 1981 ). This property holds in the case of spher ical grains only. If it is assumed that the particle sizes have a lognormal distribution, then it means that the logarithms of the sizes have a normal distribution. It is well known that a normal distribution is fully characterized by two parameters: the mean (/~) and the standard deviation (c7). Therefore, if~ and a are specified for the logarithmic-size distribution, and if we assume that the distribution is lognormal, p nd a will completely describe the relative proportion of all sizes in the coal sample. Because of the mathematical property referred to above, it would not matter if the parameters are specified for the distribution of sizes by count or by volume. METHODS
Overview of statistical approach In this paper the distribution of the pyrite grain diameters is examined for the 26 Ohio coal samples considered by Carlton ( 1985 ). Each of Carlton's samples was stage crushed and sieved to 14 X 28 mesh and then float-sink tested using a liquid of 1.60 specific gravity. Table 1 reproduces some of the relevant information about the coal samples from Carlton's paper, which also provides details about the method of preparation of these samples. In the size analysis for the the individual coal samples, pyrite grains were converted to equivalent circular diameters (ECD). An ECD is the diameter of a circle with the same area as the pyrite grain measured. The area measurements of a randomly distributed pyrite on a polished surface are taken to be equal to the volume percent pyrite. An example of the size distribution data that were the object of the analysis of this paper is given in Table 2. This table gives the cumulative volume percent pyrite for all sizes between 2 ~m and 72 ttm at steps of 2/tm for five coal samples. The pyrite size distribution of each of Carlton's (1985) coal samples for his Study A was tested to determine whether the size distribution could be adequately described by means of a lognormal distribution. For almost all the coat samples, the lognormal distribution provided a good fit to the distribution of particle sizes. Next the mean (~) and the standard deviation (a) were estimated for the logarithms of the pyrite grain diameters using data as illustrated in Table 2 for the 26 Ohio coal samples. The estimated values are denoted by and &, respectively. Finally, the relationship was determined between the
375
TABLE 1 Sample location,coal thickness, and pyritic-sulfurreduction (Carlton, 1985 ) DGS file No.
1351 1352 1353 1354 1355 1356 1358 1359 1360 1361 1379 1380 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1395 1396
Coal
L. Kittanning M. Kittanning M. Kittanning L. Kittanning L. Kittanning M. Kittanning M. Kittanning U. Freeport M. Kittanning M. Kittanning M. Kittanning L. Kittanning L. Kittanning M. Kittanning L. Kittanning L. Kittanning L. Kittanning L. Kittanning M. Kittanning M. Kittanning M. Kittanning M. Kittanning L. Kittanning M. Kittanning M. Kittanning M. Kittanning
County
Holmes Holmes Coshocton Coshocton Coshocton Coshocton Muskingum Muskingum Perry Perry Vinton Vinton Vinton Jackson Jackson Jackson Stark Stark Stark Stark Stark Columbiana Stark Columbiana Columbiana Columbiana
Township
Walnut Creek Berlin Mill Creek Mill Creek Lafayette Lafayette Washington Wayne Pike Bearfield Swann Swann Clinton Madison Madison Bloomfield Pike Osnaburg Osnaburg Sugar Creek Paris Columbia, West Paris Franklin Center Hanover
Coal thickness (cm)
91.44 66.04 71.12 91.44 59.06 113.03 52.71 120.02 140.97 104.14 125.73 67.95 134.62 31.12 68.58 86.36 97.79 107.95 104.14 71.12 64.77 138.43 62.87 53.98 91.44 95.25
14 × 28 mesh coal Wt.% pyritic sulfur Unwashed
Float
2.97 7.23 2.86 5.81 1.88 3.03 4.71 3.58 2.51 1.82 3.28 5.94 3.40 2.10 1.60 2.36 3.24 5.79 1.58 2.82 1.54 1.98 6.94 3.65 5.54 3.43
1.53 1.42 0.58 2.50 0.96 0.96 1.92 1.30 1.63 1.06 0.61 1.28 0.96 0.56 0.37 1.10 0.46 2.69 0.64 0.66 0.51 0.42 1.03 0.18 0.64 0.76
% pyritic-sulfur reduction using 1.60 specific gravity liquid 48.5 80.4 79.7 57.0 48.9 68.3 59.2 63.7 35.1 41.8 81.4 78.5 71.8 73.3 76.9 53.4 85.8 53.5 59.5 76.6 66.9 78.8 85.2 95.1 88.4 77.8
pyritic sulfur reduction for the coal samples and the respective estimated parameters,/~ and &, by use of linear regression techniques. As the results in the following sections show, the agreement between the observed data and the corresponding least-squares fit is quite good. Estimation of pyrite mean and standard deviation The process of estimating the two parameters, # and a, for each of the coal samples was, however, not without complications. For his study A. Carlton (1985) used a 32 X oil immersion objective, which permitted him to measure maximum grain size up to 72 ttm only (see Table 2). Thus no direct measurements are available for grains whose ECD's exceed this particular value. This
376 TABLE 2 Cumulative volume percent pyrite at different size for five coal samples ECD* (Microns) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72
DGS Sample Number 1351
1352
1353
1354
1355
0.68 5.77 12.25 18.14 22.40 26.28 29.84 33.03 36.10 38.56 40.73 43.72 47.38 50.49 53.70 57.82 60.25 62.15 63.82 66.63 67.49 69.08 69.78 70.55 72.17 74.40 75.87 77.45 79.13 80.91 82.83 84.93 87.85 88.63 90.26 91.12
0.18 1.51 3.36 5.03 6.65 8.13 9.39 10.76 12.25 13.42 14.42 15.53 16.75 18.45 20.25 21.63 22.79 23.89 25.26 26.79 28.74 29.79 31.78 33.53 34.60 36.04 37.44 38.80 40.80 41.58 43.86 45.84 47.02 49.77 52.62 53.01
0.46 3.54 7.26 9.91 12.21 14.06 15.50 17.16 18.90 20.18 21.71 23.02 25.17 26.66 27.99 29.50 30.71 31.89 33.21 33.92 36.07 37.84 38.44 41.30 41.67 42.51 43.40 45.33 45.85 47.55 50.53 51.81 55.25 58.84 60.36 61.16
0.54 5.04 11.59 17.35 22.25 26.35 30.01 32.97 35.84 38.00 40.60 42.81 44.70 46.13 48.48 50.85 52.20 54.28 55.98 57.38 58.68 59.97 60.92 61.42 63.10 64.50 65.59 66.99 67.98 68.25 69.96 71.19 72.50 73.54 75.77 78.09
0.60 5.42 12.19 17.63 21.39 25.25 29.09 32.20 35.53 39.88 43.28 48.45 53.40 56.54 58.25 61.81 65.75 67.15 69.03 71.64 74.00 77.67 80.54 83.66 85.68 85.68 88.03 89.70 91.52 91.52 92.55 94.76 95.94 97.17 98.49 98.49
*ECD: Equivalent Circular Diameter. s i t u a t i o n is c o m m o n l y k n o w n i n s t a t i s t i c a l t e r m i n o l o g y as a d a t a s e t h a v i n g c e n s o r e d o b s e r v a t i o n s . S e v e r a l d i f f e r e n t a l g o r i t h m s (e.g., R a a b e , 1971; W o l y netz, 1979) could be a p p l i e d for o b t a i n i n g t h e m a x i m u m l i k e l i h o o d e s t i m a t e s of the two p a r a m e t e r s f r o m l o g n o r m a l l y d i s t r i b u t e d c e n s o r e d d a t a sets. In this
377 study the readily available SAS software (1985) on probit analysis (Finney, 1971 ) was used to derive these estimates. This computer package also has the built-in capacity for testing for the lognormality of~he underlying distribution. In the process of estimating the size distribution parameters for the coal samples, an analogy is drawn with the problem considered in probit analysis. This technique is predominantly used in statistical analysis of data pertaining to biological assays. These assays are methods for the estimation of the potency or toxicity of a given material as measured by the reaction that follows its application to living material. An interesting application of probit analysis in coal preparation was made by Luckie (1969). In the language of bio-assay, let x denote the level of the dose of a given material being tested and y the corresponding probability of response corresponding to this dosage level. For example, in experimental toxicity studies, this probability is estimated by the proportion of animals killed as a result of the application of the dose at the level x. Probit analysis assumes that y can be expressed as the cumulative normal distribution function (with parameters p and a) at the point x. In applications, the numerical values of H and a will be unknown. Denote by ¢(x; lt, a) the normal probability density function with mean H and standard deviation a and by ¢b(x; H,a) the corresponding cumulative distribution function. In algebraic notation, the above assumption is equivalent to the following:
y = (I)(x;H,a)
(1)
where
CP(x;H,a) = i f~(t;H,a)dt --
OO
and ¢(t;
H,a) = (1/2n)l/2a-lexp I - (t-p)2/2a2].
The probit analysis techniques provide the best estimates of the parameters H and a from an observed set of paired data (x,y), where x is the level of applied dose and y is the estimated probability of response. The statistical software (SAS, 1985) contains a procedure known as the PROC Probit for estimating H and a from such data pairs. This procedure also provides an option whereby it is possible to do the probit analysis using the logarithm of the data value x to the base e. For the purpose of estimating the parameters H and a from the pyrite size distribution data for the coal samples, a correspondence is set between x (the dose level) and the pyrite size ECD, and between y (the probability of response
378 at dose level, x) and the cumulative pyrite volume percent at x. For example~ Table 2 shows that for the Ohio Division of Geological Survey (DGS) sample 1351, y = 57.82 for x = 32 ~lm. In analogy with probit analysis, we can regard y as the probability of an animal being killed (--0.5782) when the level of the dose is x = 32. For each of the 26 coal samples, data are available for 36 pairs of x and y values. The SAS probit procedure using the LOG option for x is performed for these samples. This option has the effect of assuming that y can be expressed as the cumulative normal distribution function in terms of the natural logarithm of x; that is, the pyrite sizes (ECD) have a lognormal distribution for each of the coal samples. The output of the SAS procedure will now provide the "best" estimated values (maximum likelihood estimates ) forg and a, which we denote by/~ and ~. They can now be regarded as the estimates for the mean and the standard deviation of the volume distribution of the logarithms of the pyrite ECD's in a given coal sample, the logarithms being taken to the base e. The output also provides information on results of statistical tests carried out to determine whether a lognormal distribution provides a good fit to the observed data for each of the coal samples. RESULTS Table 3 gives the estimated values ofp and a for each of the coal samples as furnished by the SAS output. The fourth column of this table indicates whether a lognormal fit to the observed data for the particular coal sample is considered satisfactory. This is determined by the SAS package using a chi-square goodness-of-fit test. The symbol 'S' indicates that the calculated chi-square is insignificant at the 10% level in which case the distribution fit is considered satisfactory. The symbol 'U' indicates that the calculated chi-square statistic is significant, in which case the fit is not satisfactory. In only three cases out of a total twenty-six is the lognormal fit considered inadequate according to the chi-square test (Table 3 ). The SAS output also provides graphical plots of the observed data against the backdrop of the fitted normal distribution. From a visual inspection, however, each of the coat samples appeared to provide a reasonable fit to the lognormal distribution. Therefore, all 26 samples were used in this study. The lognormal assumption implies that all the information about the pyrite particle size distribution for a given coal sample is contained in these two parameters tt and a. From the data given in Table 3, a relationship was established between the sulfur reduction values and ~ and ~. This is done by using the techiques of multiple regression. A quadratic equation that describes this relationship was obtained by first using the stepwise regression procedure of SAS (1985) on a full quadratic model in # and a with two linear terms, two square terms, and one cross-product term. Subsequently, a least-squares regression was fitted
379 TABLE 3 Logarithmic mean and standard deviation values for the pyrite size distribution of 26 coal samples and the corresponding sulfur reduction data DGS sample No.
1351 1352 1353 1354 1355 1356 1358 1359 1360 1361 1379 1380 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1395 1396
Estimated parameters /~
~
3.2126 4.3701 4.1535 3.4180 3.0848 3.5864 3.5390 3.7991 3.0166 3.2238 4.3190 4.4086 3.9788 4.5342 4.3685 3.5575 4.2691 3.5188 3.5286 3.7532 4.6407 4.8047 5.1315 4.7118 4.8152 4.0224
1.0451 1.2627 1.4433 1.3343 0.8258 1.2627 1.3635 1.2615 1.5667 1.2633 1.2042 1.4428 1.0213 1.3025 1.2246 1.4586 1.0704 1.0051 1.3331 1.0190 1.6682 1.6188 1.4483 0.9615 1.1971 1.0225
Goodness of lognormal fit*
Sulfur reduction (% )
S S S S U S S S S S S S U S S S S S S S S S S U S S
48.5 80.4 79.7 57.0 48.9 68.3 59.2 63.7 35.1 41.8 81.4 78.5 71.8 73.3 76.9 53.4 85.8 53.5 59.5 76.6 66.9 78.8 85.2 95.1 88.4 77.8
*"s" indicates fit is considered satisfactory according to a statistical test. "U" indicates fit is considered unsatisfactory according to a statistical test.
w i t h t h e t e r m s j u d g e d s i g n i f i c a n t b y t h e s t e p w i s e r o u t i n e . T h e m o d e l is described by the following equation: S R -- - 184.613 + 110.243/~ - 5.172 ~ & - 9.871 ~2
(2)
w h e r e S R s t a n d s for s u l f u r r e d u c t i o n , fi r e p r e s e n t s t h e e s t i m a t e d m e a n p a r t i c l e size, a n d ~ r e p r e s e n t s t h e e s t i m a t e d s t a n d a r d d e v i a t i o n o f t h e p a r t i c l e size d i s t r i b u t i o n . T h e p o s i t i v e sign for t h e c o e f f i c i e n t of/~ s u g g e s t s t h a t as ~ in-
380
creases, SR increases up to a point. However, this relationship is not exactly linear. The negative coefficient of the/~l~ term tempers the rate of increase as increases. The most interesting facet of the equation is that the coefficient of' the/5# term is negative. In statistical terminology,/~ and # are said to have a negative interaction. That is, if 7l has a relatively small value, a relatively small value of ~ is detrimental for the purpose of sulfur reduction. On the other hand, if'~ is relatively large, then a small value of h is conducive to sulfur reduction. This equation is consistent with Figure 1 and our understanding of coal cleaning based on gravimetric separation. The regression equation (2) had a R 2 value of' 0.91. Table 4 compares the predicted values of sulfur reduction using eqn. (2) against the actual observed values. For the most part, the agreement between the observed and the preTABLE 4 Comparison of actual sulfur reduction values with those predicted by Equations (2) and (3) DGS Sample No.
Actual pyritic sulfur reduction
Predicted pyritic sulfur reduction ( This paper )
Predicted pyritic sulfur reduction ( Carlton's study A, 1985 )
1351 1352 1353 1354 1355 1356 1358 1359 1360 1361 1379 1380 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1395 1396
48.5 80.4 79.7 57.0 48.9 68.3 59.2 63.7 35.1 41.8 81.4 78.5 71.8 73.3 76.9 53.4 85.8 53.5 59.5 76.6 66.9 78.8 85.2 95.1 88.4 77.8
50.31 80.11 71.99 53.29 48.36 60.38 56.95 66.95 33.68 47.14 80.50 76.66 76.74 81.77 80.94 55.81 82.49 62.80 57.16 69.76 74.37 76.97 82.73 92.25 87.55 77.85
50.37 79.69 71.90 51.32 45.45 59.65 56.39 65.59 41.88 46.15 79.81 76.48 76.46 81.10 78.97 58.51 81.84 60.16 54.65 70.45 77.16 79.92 87.76 90.65 88.04 76.50
381 dicted values is quite satisfactory. Also shown in this table are predicted values given by the regression equation obtained by Carlton (1985), which is: SR = 95.84 - 1.04z
(3)
where z is the cumulative pyrite volume percent at size 24/~m. The predictions based on eqn. (2) are in general more accurate than those based on eqn. (3). The average absolute difference between the actual pyritic reduction and the prediction based on eqn. (2) is 3.62%; the corresponding figure for eqn. (3) is 4.19%. Independent estimates of # and a were also derived for the 26 Ohio coal samples using a modified version of the algorithm proposed by Wolynetz (1979). The results are similar to those given in Table 3. DISCUSSION The statistical analysis given in the preceding section has provided an algebraic relation between pyrite particle size distribution and pyritic sulfur reduction by gravimetric methods. Under the assumption that the size distribution under question is lognormal, it is possible to encapsulate all the information about the size distribution in terms of two constants. A prediction equation for sulfur reduction is then developed in terms of these two constants. For the particular set of 26 Ohio coal samples for which we had detailed information, the prediction equation performed very well. It is difficult to say whether this equation will hold for a much wider variety of coal samples. A much larger data base is needed before one can attempt to answer this question. It is hoped, nevertheless, that the methodology considered in this paper will continue to prove to be useful in the analysis of pyrite reduction studies carried out from a petrographic viewpoint. Although the prediction eqn. (3) did not perform too badly in comparison with eqn. (2), there is a statistical reason why the latter equation should be preferred in practice. The statistics,/~,~z and z, will be estimated from a limited number of pyrite particles. The standard errors associated with/~ and ~ are expected to be smaller than that of z. Thus, in general, the "repeatability" of /~ and ~zwill be better than of z. The analysis given here has considered the relationship between the pyrite particle size distribution and pyritic sulfur reduction. No doubt a stronger relation can be obtained if, additionally, information on the degree of liberation of pyrite particles from coal macerals is provided for each of the coal samples. This information is ideally given by one or two numerical indexes. McCartney et al. (1969) provided such an index as a measure of degree of liberation. They separately obtained the relationship between pyritic sulfur reduction and mean pyrite particle size, and between the former quantity and their index of degree of liberation, taken one at a time. An analysis of their data as given in Table 2
382 of their paper using both the size parameter and the index of liberation as two independent variables in one common linear regression equation results in a stronger regression equation with a R ~ approximately 89%. This leads us to believe that if information on the degree of liberation is available in addition to the size distribution, a still more accurate equation than eqn. (2) will result. Modern automated image analysis offers a potentially rapid, nontiring method of' determining information on the size characteristics and degree of liberation of pyrite particles associated with coal. The United States Department of Energy at the Pittsburgh Energy Techology Center is currently working on developing automated image analysis procedures to estimate pyrite particles size distributions and degree of liberation (Irdi and Rohar, 1986). It is developing a technique fbr automatically quantifying the degree of pyrite liberation in ultrafine coal using a Zeiss IBAS image analyzer. A preliminary program is being tested, and the results will be published soon. CONCLUSIONS Based on data on pyrite size distribution available for 26 Ohio coal samples, this paper has presented a relationship between the estimated mean and the standard deviation of the logarithm of the pyrite grain sizes and potential pyritic sulfur reduction. For the most part, the agreement between the observed and predicted values is quite satisfactory. The derived equation performs better than that provided by Carlton (1985), where he developed a similar prediction based on the cumulative volume percent distribution at size 24 microns. The numerical results of the present study appear to validate the argument given here, namely, that the distribution of all the pyrite particle sizes should be considered for the purpose of developing a prediction equation for potential pyritic sulfur reduction based on gravimetric tests. ACKNOWLEDGEMENT The research of M. Mazumdar was supported in part by an appointment to the U.S. Department of Energy Fossil Fuel Part-time Faculty Participation program administered by Oak Ridge Associated Universities at the Pittsburgh Energy Techology Center. Portions of this paper were published with permission of H.R. Collins, Chief of the Division of Geological Survey. DISCLAIMER Reference in this paper to any specific commercial product, process, or service is to facilitate understanding and does not necessarily imply its endorsement or favoring by the United States Department of Energy.
383 REFERENCES Allen, T., 1981. Particle Size Measurement, Third Edition, Chapman and Hall, New York and London, p. 678. Carlton, R.W., 1985. Image analysis of pyrite in coal: relation between pyrite grain-size distribution and sulfur reduction. In: Y.A. Attia (Editor), Processing and Utilization of High Sulfur Coals. Elsevier, Amsterdam, pp. 3-17. Caruccio, F.T., Ferm, J.C., Horne, J., Geidel, and Baganz, B., 1977. Paleoenvironment of coal and its relation to drainage quality. Interagency Energy Environment Research and Development Program Report, EPA-600/7-77-067, U.S. Environmental Protection Agency, Cincinnati, OH. Finney, D.J., 1971. Statistical Methods in Biological Assay, Second Edition. Hafner Press, New York, NY, p. 668. Gray, R.J., Schapiro, N. and Coe, G.D., 1963. Distribution and forms of sulfur in a high volatile Pittsburgh seam coal. Trans., Soc. Min. Eng., June: 113-121. Harvey, R.D. and DeMaris, P.J., 1985. Size and maceral association of sulfide grains in Illinois coals and their washed products. Ill. State Geol. Surv., Champaign, IL, 49 pp. Herden, G., 1953. Small Particle Statistics. Elsevier, Amsterdam, 520 pp. Irdi, G.A. and Rohar, P., 1986. An automated microscopic method for the determination of pyrite size distributions in crushed coal. Presented at the AIME-SME Fall Meeting, St. Louis, MO, September 7-10, 1986. Luckie, P.T., 1969. The application of probit analysis methodology to the determination of the nature of the partition curve. M. Sc. thesis (unpublished). Pennsylvania State University, University Park, PA. Maxwell, G.P. and Kneller, W.A., 1982. Size, shape, and distribution of microscopic pyrite in selected Ohio Coals. Final Report V for Ohio Coal Research Laboratory Association OCRLA1. University of Toledo Carbon Facility, 158 pp. McCartney, J.T., O'Donnell, H. and Ergun, S., 1969. Pyrite size distribution and coal-pyrite particle association in steam coals. U.S. Bureau of Mines Rept. of Invest. 7231, 18 pp. Raabe, G., 1971. Particle size analysis using grouped data and the lognormal distribution. Aerosol Sci., 2: 289-303. Reyes-Navaro, J. and Davis, A., 1976. Pyrite in coal: its forms and distributions as related to the environments of coal deposition in three selected coals from western Pennsylvania. Special Research Report No. SR-110, Coal Research Section, PA., 141. SAS User's Guide: Statistics, Version 5 Edition, 1985. SAS Institute, Inc., Box 8000, Cary, NC. Wiese, R.G., Jr., and Fyfe, W.S., 1986. Occurrences of iron sulfides in Ohio coals. Int. J. Coal Geol., 6: 251-276. Wolynetz, M.S., 1979. Maximum likelihood estimation from confined and censored normal data, Algorithm AS 138. Appl. Statist., pp. 185-195.