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Statistical representation of generalized distribution data for float-sink coal-cleaning devices: Baum jigs, Batac jigs, Dynawhirlpools
International Journal of Mineral Processing, 15 (1985) 231--236 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
231
Technical Note STATISTICAL REPRESENTATION OF GENERALIZED DISTRIBUTION D A T A F O R FLOAT-SINK C O A L - C L E A N I N G DEVICES: BAUM JIGS, BATAC JIGS, DYNAWHIRLPOOLS
NANCY E. FALLON1 and BYRON S. GOTTFRIED 2 ' Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, PA 15261, U.S.A. Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA 15261, U.S.A. (Received September 18, 1983; revised and accepted January 31, 1985)
ABSTRACT Fallon, N.E. and Gottfried, B.S., 1985. Statistical representation of generalized distribution data for float-sink coal-cleaning devices: Baum jigs, Batac Jigs, Dynawhirlpools. Int. J. Miner. Process., 15: 231--236. In an earlier paper (1978), Gottfried presented a method for combining distribution data for float-sink coal-cleaning devices into a single generalized distribution curve which, for a given device and feed size, is independent of specific gravity of separation. A nonlinear, exponential-type equation was utilized to represent the generalized distribution curve, along with the corresponding generalized probable error. Distribution data for six common coal-cleaning devices have previously been treated by this method. This paper is an extension of two previous studies (Gottfried, 1978, 1980). The method described above is applied to three different float-sink coal-cleaning devices: Baum jig (replacing previously reported results), Batac jig and Dynawhirlpool separator. Results for the Baum jig and Batac jig reflect a two-stage separation process, with a set of generalized distribution curves obtained for each stage and another set for the overall separation. Several different feed size fractions are given for each vessel.
INTRODUCTION Float-sink coal-cleaning devices separate a coal f r o m its impurities t h r o u g h d i f f e r e n c e s in specific gravity. T h e p e r f o r m a n c e o f this t y p e o f vessel is c h a r a c t e r i z e d b y a d i s t r i b u t i o n curve, such t h a t t h e w e i g h t p e r c e n t o f feed r e p o r t i n g t o clean coal is a f u n c t i o n o f specific gravity. T h e l o c a t i o n o f t h e curve f o r a given vessel and feed size d e p e n d s o n t h e c o n t r o l settings o f t h e coal cleaning device. A given vessel and feed size results in a f a m i l y o f dist r i b u t i o n curves, each curve w i t h its o w n specific gravity o f s e p a r a t i o n (i.e., t h e value o f specific gravity at w h i c h 50% o f t h e feed r e p o r t s t o clean coal). G o t t f r i e d ( 1 9 7 8 ) has s h o w n t h a t , given a vessel a n d a feed size, each f a m i l y o f curves can be r e p l a c e d b y a single generalized d i s t r i b u t i o n curve w h i c h is i n d e p e n d e n t o f specific gravity o f separation. F u r t h e r m o r e , he has
0301-7516/85/$03.30
© 1985 Elsevier Science Publishers B.V.
232
shown that the generalized distribution curve can be expressed by a nonlinear, exponential type of equation known as the Weibull function. Once the parameters are determined for a given vessel and feed, the generalized probable error corresponding to the generalized distribution curve can be calculated. This method has thus far been applied to six c o m m o n floatsink coal-cleaning devices: concentrating table, dense-medium vessel, densemedium cyclone, hydrocyclone, Baum jig and sand cone. In this paper the m e t h o d described above is applied to three float-sink coal-cleaning devices: Baum jig (replacing earlier results with more recent and more extensive performance data), Batac jig and Dynawhirlpool separator. Generalized distribution data for the Baum and Batac jigs reflect twostage separation processes, with results presented for the primary, secondary and overall separation of coal and its impurities. Several size fractions are analyzed for each device, along with overall feed composites. The jig data were taken from the values reported by Killmeyer (1980), and the Dynawhirlpool data from the work of Maronde et al. (1983). MATHEMATICAL REPRESENTATION OF THE G E N E R A L I Z E D DISTRIBUTION CURVE
Gottfried (1978) has shown that the generalized distribution curve can be expressed as a modified Weibull function o f the form: f ( x ) = 100 [f0 + c e x p { - (x - X o ) a / b ~ ]
(1)
where x is the reduced specific gravity, P/Ps (Ps is the specific gravity of separation); f(x) is the percent of feed reporting to clean coal; and f0, c, a, b and x0 are constants to be determined for a given vessel and a given feed size fraction. Of the five constants to be determined, four are independent and the fifth, x0, can be calculated from the condition that f ( x ) = 50 when x = 1. When this condition is imposed on eq. 1, the following expression is obtained: x0=l-
[
bloge
( c
0.5;fo
(2)
The values of f0 and c are determined directly from the performance data: f0 represents the minimum fraction of feed that will report to clean coal at high gravities (i.e., very large values of x); c represents the difference between the highest and lowest fraetions of feed reporting to clean coal, so that: C+fo<~ 1
For the float-sink coal-cleaning devices reported herein, 100% of the feed reports to clean coal at low x values so that c is taken to equal the quantity (1 - f0).
233
Two parameters remain to be determined in eq. 1: a and b. The values for a and b are obtained through nonlinear regression techniques. Once the values of f0, c, a, b and x0 are known, the generalized probable error corresponding to the curve can be calculated from the following equation: GPE=0.5
t [b loge ( 0 . 2 5 - - f 0 ) ]
c
-[bloge (0.75-f0)]
1
(3)
RESULTS
Values of a, b, c, f0, x0 and GPE are given in Table 1 for several size fractions and the overall feed composite for each coal cleaning device. Results TABLE 1 N u m e r i c a l v a l u e s t o be u s e d in e q u a t i o n 1 0 0 [f0 + c e x p ( - ( x - x o ) a / b ) ], x > x o F e e d size fraction (inches and mesh)
a
b
for the generalized distribution
curve, f(x) =
c
f0
xo
GPE
S t d . dev.
B a u m jig:
primary separation 6X 4 4x 2 2× 1 1 × '/2 composite
1.9067 2.3773 2.2817 2.5403 2.6929
9.7550× 1.0765X 1.2621x 6.1792× 8.3684X
1 0 -2 10 -~ 10-' 10-: 1 0 -2
0.998 0.964 0.897 0.700 0.845
0.002 0.036 0.103 0.300 0.155
0.75619 0.65674 0.63092 0.63477 0.61789
0.0986 0.1152 0.1381 0.1237 0.1312
7.55X 4.12X 6.56× 3.87× 4.18X
10 10 10 10 10
-3 -3 -3 -~ -~
2.0978X 4.0590X 5.7034× 6.0208 x 6.1881 X 8.4666X 1.3787X 1.5743 X
10 2 10 3 10 -2 10-: 10-: 1 0 -2 10 -' 10-'
1.000 1.000 0.982 0.975 0.995 0.999 0.980 0.980
0. 0. 0.018 0.025 0.005 0.001 0.020 0.020
0.95227 0.85861 0.89616 0.88339 0.77941 0.80105 0.81033 0.96565
0.0266 0.0363 0.0578 0.0631 0.0820 0.0875 0.1073 0.0468
2.65X 1.21× 1.52× 1.14× 1.29X 1.14X 6.36X 3.15X
10 -2 10 -2 1 0 -2 10-: 10 -2 10 2 1 0 -3 10 2
1.6554 × 2.8450x 1.8473× 7.5754X 1.0516X 9.2961X 1.7940x 1.0675x
10 -2 10 -2 1 0 -2 10-: 10-' 10 -= 10 ' 10-'
1.000 1.000 1.000 1.000 1.000 1.000 0.997 1.000
0. 0. 0. 0. 0. 0. 0.003 0.
0.95426 0.92069 0.79233 0.87465 0.74464 0.68246 0.65074 0.77582
0,0244 0,0396 0.0577 0.0684 0,1025 0.1022 0.1363 0.0992
1.85 × 1.79X 1.64x 2.85× 1.48X 1.52× 1.54X 1.10×
10 2 10 -2 10 2 10 2 10 -2 10 -2 10 2 10 -2
secondary separation 6× 4× 2X 1 X
4 2 1 '/2
1/2 X '/4 I/,X 8 8 X 14 composite
1.3907 3.0023 1.4148 1.4610 2.0786 1.7552 1.3947 0.6484
overall s e p a r a t i o n 6 × 4 4X 2 2× 1 1 × 1/: ' / : X '/4 '/,× 8 8 X 14 composite
1.4483 1.5491 2.7725 1.4190 1.9184 2.3904 1.9776 1.7413
(continued)
234 TABLE 1
(continued)
F e e d size fraction (inches and mesh)
a
b
c
f0
x0
GPE
Std. dev.
B a t a c jig
primary separation 3/4 x '/2 '/2 x 3/8 3/8 x '/4 '/~ x 8 8 x 14 1 4 × 28 28× 48 composite
6.3851 15.156 2.8959 6.9717 10.955 8.8426 2.9400 8.0514
2.2845 X 3.4350 2.7957 x 2.8782× 1.3948 x 4.3400x 2.9145 x 8.0574×
1 0 -2 10 -2 10 -2 10-' 10 -2 1 0 -2 10 -2
0.675 0.650 0.650 0.650 0.659 0.601 0.591 0.625
0.325 0.42007 0.350-0.11073 0.350 0.66814 0.350 0.36493 0.341 0.13730 0.399 0.25126 0.409 0.62822 0.375 0.22408
0.0678 0.0482 0.0672 0.0579 0.0542 0,0372 0.0422 0.0515
1.36X 1.89X 8.49X 1.08X 1.04X 1.74X 7.25X 8.60x
10 2 10 -2 1 0 -' 10 -2 10 -2 10 -2 10 -3 1 0 -~
4.3224 × 4.1216 X 3.3588 X 5.5967X 6.3789x 1.3802× 3.8426× 1.1232×
10 -2 1 0 -~ 10 -~ 10 -2 1 0 -~ 10~' 1 0 -~ 10 - '
1.000 1.000 1.000 1.000 1.000 1.000 0.800 1.000
0. 0. 0. 0. 0. 0. 0.200 0.
0.81326 0.81395 0.81161 0.82824 0.78203 0.89929 0.60499 0.84118
0.0688 0.0677 0.0644 0.0717 0.0820 0.0779 0.1147 0.0887
1.59x 1.80x 1.48X 1.42X 1.64× 2.60X 4.42X 1.39x
10 -2 10 -2 10 -2 1 0 -2 10 -2 10 -2 1 0 -3 10 -2
2.5542 2.5077 4.1062 2.0770 2.0707 1.7999 1.6415 7.3558 1.6870
2.7426 × 3.1968 × 1.0471 x 5.3424× 6.9885× 5.6753× 5.5162x 1.8014× 7.8245x
10 -~ 10 -2 10 -2 1 0 -~ 10 -2 10 -2 1 0 -2 10-' 1 0 .2
1.000 1.000 1.000 1.000 1.000 0.970 0.878 0.666 1.000
0. 0. 0. 0. 0. 0.030 0.122 0.334 0.
0.78807 0.78110 0.69867 0.79545 0.76823 0.83017 0.84578 0.17165 0.82228
0.0639 0.0672 0.0568 0.0758 0.0862 0.0746 0.0836 0.0798 0.0812
7.58x 8.54X 2.67X 2.01× 1.03 X 1.31X 1.60x 1.03× 2.39×
10 -3 10 -3 10 2 10 -2 1 0 -2 1 0 -2 10 -2 10 -2 10 -2
1.2870 1.7210 1.2139 1.6980 0.8691 1.2149 1.1677 1.2768
3.3656× 1.6772X ,4.4847 X 1.2037 X 1.0331X 7.7903 X 1.0680X 5.3836x
1 0 -2 10 -~ 1 0 -2 1 0 -2 10-' 1 0 -~ 10-' 10 -2
1.000 1.000 1~000 1.000 1.000 1.000 1.000 1.000
0. 0. 0. 0. 0. 0. 0. 0.
0.94607 0.92486 0.94270 0.94032 0.95186 0.90952 0.89242 0.92388
0.0326 0.0337 0.0368 0.0271 0.0447 0.0581 0.0721 0.0464
2,21X 2,49X 1.72× 2.04× 3.01 x 1.47X 1.67x 2.09X
10 : 10 -2 10 : 1 0 -2 10 -2 1 0 -2 10 -2 1 0 -2
secondary separation ~/~ x 1/~ '/2 X 3/~ 3/~ X '/, ~/, x 8 8 x 14 1 4 x 28 28x 48 composite
2.0904 2.1141 2.2526 1.8446 2.0472 1.0224 3.5295 1.3875
overall separation ~/~ x '/2 '/2 x 3/~ ~/~ x '/4 ~/,× 8 8 x 14 14x 28 28x 48 48× 100 composite Dynawhirlpool
separator l'/2X 1 1 X '/~ '/2 × 3/~ ~/~ X ~/4 '/,× 8 8 × 14 14× 28 composite
for the Baum jig and Batac jig include primary, secondary and overall separation steps. The appropriate coefficients can be used in conjunction with eq. 1 to describe the generalized distribution curve of each coal cleaning device and feed size fraction b y an analytical equation.
235
-••
100.00 -
80.00-
-- Weibull function ts
o 60.00u
g "5 o
40.00-
L
_m O
20.000.(30
).40
o16o
i
J
0-80 1.00 Reduced specific grafity
~
120
OT
1.40
Fig. 1. Generalized distribution curve representation using the Weibull function for Batac jigs (overall), 1/2 x 3/~ inch.
100¸00"
80.00-
°~°°
o \° oO~
-- Weibull function o Data points
~ 60.00
g
40.00o o o o°
°o o
2000 i 0.00 060
0.80
I O0 I.'20 140 Reduced specific gravity
n
180
i
Fig. 2. Generalized distribution curve representation using the Weibull function for Dynawhirlpool separators, 14 x 28 mesh.
236
The standard deviation (based upon ordinate values ranging from 0 to 1) for each device and feed size is also presented in Table 1 to provide an idea of the accuracy of each equation. As found in previous studies, the Weibull function is least accurate where the data spans the entire range of ordinate values (c = 1). Figure 1 shows a typical curve fit, for an overall (2-stage) separation using a Batac jig with a 1/2 by 3/8-inch size fraction. Table I indicates that the standard deviation is 8.54 X 10 -3 in this case. Figure 2 shows another fit, for a Dynawhirlpool with a 14 by 28 mesh size fraction. Notice the difficulty in fitting the upper "shoulder" of the curve. (The data scatter further contributes to the poor fit.) In this case, Table 1 shows that the standard deviation is 1.67 X 10 -2. This value is not greatly different than the value of 8.54 X 10 -3 obtained for the Batac jig; yet Figs. 1 and 2 indicate noticeable differences in the quality of the fits. These results are typical o f the better fits and the poorer fits, respectively. ACKNOWLEDGEMENTS
This work is a part of a project supported by the U.S. Department of Energy, contract no. DE-AC22-83PC62684. The assistance and support of Messrs. A.W. Deurbrouck and J.T. Wizzard, of the Pittsburgh Energy Technology Center, USDOE, are gratefully acknowledged. REFERENCES Gottfried, B.S., 1978. A generalization of distribution data for characterizing the performance of float-sink coal-cleaning devices. Int. J. Miner. Process., 5 : 1 -20. Gottfried, B.S., 1981. Statistical representation of generalized distribution data for float-sink coal cleaning devices: sand cones. Int. J. Miner. Process., 8: 89--91. Killmeyer, R.P., Jr,, 1980. Performance characteristics of coal-washing equipment: Baum and Batac jigs. Report of investigations. USDOE RI-PMTC-9(80), Oct. 1980. Maronde, C.P., Killmeyer, R.P. and Deurbrouck, A.W., 1983. Performance characteristics of coal-washing equipment: Dynawhirlpool separator. USDOE/PETC/TR-83/8, June 1983.
231
Technical Note STATISTICAL REPRESENTATION OF GENERALIZED DISTRIBUTION D A T A F O R FLOAT-SINK C O A L - C L E A N I N G DEVICES: BAUM JIGS, BATAC JIGS, DYNAWHIRLPOOLS
NANCY E. FALLON1 and BYRON S. GOTTFRIED 2 ' Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, PA 15261, U.S.A. Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA 15261, U.S.A. (Received September 18, 1983; revised and accepted January 31, 1985)
ABSTRACT Fallon, N.E. and Gottfried, B.S., 1985. Statistical representation of generalized distribution data for float-sink coal-cleaning devices: Baum jigs, Batac Jigs, Dynawhirlpools. Int. J. Miner. Process., 15: 231--236. In an earlier paper (1978), Gottfried presented a method for combining distribution data for float-sink coal-cleaning devices into a single generalized distribution curve which, for a given device and feed size, is independent of specific gravity of separation. A nonlinear, exponential-type equation was utilized to represent the generalized distribution curve, along with the corresponding generalized probable error. Distribution data for six common coal-cleaning devices have previously been treated by this method. This paper is an extension of two previous studies (Gottfried, 1978, 1980). The method described above is applied to three different float-sink coal-cleaning devices: Baum jig (replacing previously reported results), Batac jig and Dynawhirlpool separator. Results for the Baum jig and Batac jig reflect a two-stage separation process, with a set of generalized distribution curves obtained for each stage and another set for the overall separation. Several different feed size fractions are given for each vessel.
INTRODUCTION Float-sink coal-cleaning devices separate a coal f r o m its impurities t h r o u g h d i f f e r e n c e s in specific gravity. T h e p e r f o r m a n c e o f this t y p e o f vessel is c h a r a c t e r i z e d b y a d i s t r i b u t i o n curve, such t h a t t h e w e i g h t p e r c e n t o f feed r e p o r t i n g t o clean coal is a f u n c t i o n o f specific gravity. T h e l o c a t i o n o f t h e curve f o r a given vessel and feed size d e p e n d s o n t h e c o n t r o l settings o f t h e coal cleaning device. A given vessel and feed size results in a f a m i l y o f dist r i b u t i o n curves, each curve w i t h its o w n specific gravity o f s e p a r a t i o n (i.e., t h e value o f specific gravity at w h i c h 50% o f t h e feed r e p o r t s t o clean coal). G o t t f r i e d ( 1 9 7 8 ) has s h o w n t h a t , given a vessel a n d a feed size, each f a m i l y o f curves can be r e p l a c e d b y a single generalized d i s t r i b u t i o n curve w h i c h is i n d e p e n d e n t o f specific gravity o f separation. F u r t h e r m o r e , he has
0301-7516/85/$03.30
© 1985 Elsevier Science Publishers B.V.
232
shown that the generalized distribution curve can be expressed by a nonlinear, exponential type of equation known as the Weibull function. Once the parameters are determined for a given vessel and feed, the generalized probable error corresponding to the generalized distribution curve can be calculated. This method has thus far been applied to six c o m m o n floatsink coal-cleaning devices: concentrating table, dense-medium vessel, densemedium cyclone, hydrocyclone, Baum jig and sand cone. In this paper the m e t h o d described above is applied to three float-sink coal-cleaning devices: Baum jig (replacing earlier results with more recent and more extensive performance data), Batac jig and Dynawhirlpool separator. Generalized distribution data for the Baum and Batac jigs reflect twostage separation processes, with results presented for the primary, secondary and overall separation of coal and its impurities. Several size fractions are analyzed for each device, along with overall feed composites. The jig data were taken from the values reported by Killmeyer (1980), and the Dynawhirlpool data from the work of Maronde et al. (1983). MATHEMATICAL REPRESENTATION OF THE G E N E R A L I Z E D DISTRIBUTION CURVE
Gottfried (1978) has shown that the generalized distribution curve can be expressed as a modified Weibull function o f the form: f ( x ) = 100 [f0 + c e x p { - (x - X o ) a / b ~ ]
(1)
where x is the reduced specific gravity, P/Ps (Ps is the specific gravity of separation); f(x) is the percent of feed reporting to clean coal; and f0, c, a, b and x0 are constants to be determined for a given vessel and a given feed size fraction. Of the five constants to be determined, four are independent and the fifth, x0, can be calculated from the condition that f ( x ) = 50 when x = 1. When this condition is imposed on eq. 1, the following expression is obtained: x0=l-
[
bloge
( c
0.5;fo
(2)
The values of f0 and c are determined directly from the performance data: f0 represents the minimum fraction of feed that will report to clean coal at high gravities (i.e., very large values of x); c represents the difference between the highest and lowest fraetions of feed reporting to clean coal, so that: C+fo<~ 1
For the float-sink coal-cleaning devices reported herein, 100% of the feed reports to clean coal at low x values so that c is taken to equal the quantity (1 - f0).
233
Two parameters remain to be determined in eq. 1: a and b. The values for a and b are obtained through nonlinear regression techniques. Once the values of f0, c, a, b and x0 are known, the generalized probable error corresponding to the curve can be calculated from the following equation: GPE=0.5
t [b loge ( 0 . 2 5 - - f 0 ) ]
c
-[bloge (0.75-f0)]
1
(3)
RESULTS
Values of a, b, c, f0, x0 and GPE are given in Table 1 for several size fractions and the overall feed composite for each coal cleaning device. Results TABLE 1 N u m e r i c a l v a l u e s t o be u s e d in e q u a t i o n 1 0 0 [f0 + c e x p ( - ( x - x o ) a / b ) ], x > x o F e e d size fraction (inches and mesh)
a
b
for the generalized distribution
curve, f(x) =
c
f0
xo
GPE
S t d . dev.
B a u m jig:
primary separation 6X 4 4x 2 2× 1 1 × '/2 composite
1.9067 2.3773 2.2817 2.5403 2.6929
9.7550× 1.0765X 1.2621x 6.1792× 8.3684X
1 0 -2 10 -~ 10-' 10-: 1 0 -2
0.998 0.964 0.897 0.700 0.845
0.002 0.036 0.103 0.300 0.155
0.75619 0.65674 0.63092 0.63477 0.61789
0.0986 0.1152 0.1381 0.1237 0.1312
7.55X 4.12X 6.56× 3.87× 4.18X
10 10 10 10 10
-3 -3 -3 -~ -~
2.0978X 4.0590X 5.7034× 6.0208 x 6.1881 X 8.4666X 1.3787X 1.5743 X
10 2 10 3 10 -2 10-: 10-: 1 0 -2 10 -' 10-'
1.000 1.000 0.982 0.975 0.995 0.999 0.980 0.980
0. 0. 0.018 0.025 0.005 0.001 0.020 0.020
0.95227 0.85861 0.89616 0.88339 0.77941 0.80105 0.81033 0.96565
0.0266 0.0363 0.0578 0.0631 0.0820 0.0875 0.1073 0.0468
2.65X 1.21× 1.52× 1.14× 1.29X 1.14X 6.36X 3.15X
10 -2 10 -2 1 0 -2 10-: 10 -2 10 2 1 0 -3 10 2
1.6554 × 2.8450x 1.8473× 7.5754X 1.0516X 9.2961X 1.7940x 1.0675x
10 -2 10 -2 1 0 -2 10-: 10-' 10 -= 10 ' 10-'
1.000 1.000 1.000 1.000 1.000 1.000 0.997 1.000
0. 0. 0. 0. 0. 0. 0.003 0.
0.95426 0.92069 0.79233 0.87465 0.74464 0.68246 0.65074 0.77582
0,0244 0,0396 0.0577 0.0684 0,1025 0.1022 0.1363 0.0992
1.85 × 1.79X 1.64x 2.85× 1.48X 1.52× 1.54X 1.10×
10 2 10 -2 10 2 10 2 10 -2 10 -2 10 2 10 -2
secondary separation 6× 4× 2X 1 X
4 2 1 '/2
1/2 X '/4 I/,X 8 8 X 14 composite
1.3907 3.0023 1.4148 1.4610 2.0786 1.7552 1.3947 0.6484
overall s e p a r a t i o n 6 × 4 4X 2 2× 1 1 × 1/: ' / : X '/4 '/,× 8 8 X 14 composite
1.4483 1.5491 2.7725 1.4190 1.9184 2.3904 1.9776 1.7413
(continued)
234 TABLE 1
(continued)
F e e d size fraction (inches and mesh)
a
b
c
f0
x0
GPE
Std. dev.
B a t a c jig
primary separation 3/4 x '/2 '/2 x 3/8 3/8 x '/4 '/~ x 8 8 x 14 1 4 × 28 28× 48 composite
6.3851 15.156 2.8959 6.9717 10.955 8.8426 2.9400 8.0514
2.2845 X 3.4350 2.7957 x 2.8782× 1.3948 x 4.3400x 2.9145 x 8.0574×
1 0 -2 10 -2 10 -2 10-' 10 -2 1 0 -2 10 -2
0.675 0.650 0.650 0.650 0.659 0.601 0.591 0.625
0.325 0.42007 0.350-0.11073 0.350 0.66814 0.350 0.36493 0.341 0.13730 0.399 0.25126 0.409 0.62822 0.375 0.22408
0.0678 0.0482 0.0672 0.0579 0.0542 0,0372 0.0422 0.0515
1.36X 1.89X 8.49X 1.08X 1.04X 1.74X 7.25X 8.60x
10 2 10 -2 1 0 -' 10 -2 10 -2 10 -2 10 -3 1 0 -~
4.3224 × 4.1216 X 3.3588 X 5.5967X 6.3789x 1.3802× 3.8426× 1.1232×
10 -2 1 0 -~ 10 -~ 10 -2 1 0 -~ 10~' 1 0 -~ 10 - '
1.000 1.000 1.000 1.000 1.000 1.000 0.800 1.000
0. 0. 0. 0. 0. 0. 0.200 0.
0.81326 0.81395 0.81161 0.82824 0.78203 0.89929 0.60499 0.84118
0.0688 0.0677 0.0644 0.0717 0.0820 0.0779 0.1147 0.0887
1.59x 1.80x 1.48X 1.42X 1.64× 2.60X 4.42X 1.39x
10 -2 10 -2 10 -2 1 0 -2 10 -2 10 -2 1 0 -3 10 -2
2.5542 2.5077 4.1062 2.0770 2.0707 1.7999 1.6415 7.3558 1.6870
2.7426 × 3.1968 × 1.0471 x 5.3424× 6.9885× 5.6753× 5.5162x 1.8014× 7.8245x
10 -~ 10 -2 10 -2 1 0 -~ 10 -2 10 -2 1 0 -2 10-' 1 0 .2
1.000 1.000 1.000 1.000 1.000 0.970 0.878 0.666 1.000
0. 0. 0. 0. 0. 0.030 0.122 0.334 0.
0.78807 0.78110 0.69867 0.79545 0.76823 0.83017 0.84578 0.17165 0.82228
0.0639 0.0672 0.0568 0.0758 0.0862 0.0746 0.0836 0.0798 0.0812
7.58x 8.54X 2.67X 2.01× 1.03 X 1.31X 1.60x 1.03× 2.39×
10 -3 10 -3 10 2 10 -2 1 0 -2 1 0 -2 10 -2 10 -2 10 -2
1.2870 1.7210 1.2139 1.6980 0.8691 1.2149 1.1677 1.2768
3.3656× 1.6772X ,4.4847 X 1.2037 X 1.0331X 7.7903 X 1.0680X 5.3836x
1 0 -2 10 -~ 1 0 -2 1 0 -2 10-' 1 0 -~ 10-' 10 -2
1.000 1.000 1~000 1.000 1.000 1.000 1.000 1.000
0. 0. 0. 0. 0. 0. 0. 0.
0.94607 0.92486 0.94270 0.94032 0.95186 0.90952 0.89242 0.92388
0.0326 0.0337 0.0368 0.0271 0.0447 0.0581 0.0721 0.0464
2,21X 2,49X 1.72× 2.04× 3.01 x 1.47X 1.67x 2.09X
10 : 10 -2 10 : 1 0 -2 10 -2 1 0 -2 10 -2 1 0 -2
secondary separation ~/~ x 1/~ '/2 X 3/~ 3/~ X '/, ~/, x 8 8 x 14 1 4 x 28 28x 48 composite
2.0904 2.1141 2.2526 1.8446 2.0472 1.0224 3.5295 1.3875
overall separation ~/~ x '/2 '/2 x 3/~ ~/~ x '/4 ~/,× 8 8 x 14 14x 28 28x 48 48× 100 composite Dynawhirlpool
separator l'/2X 1 1 X '/~ '/2 × 3/~ ~/~ X ~/4 '/,× 8 8 × 14 14× 28 composite
for the Baum jig and Batac jig include primary, secondary and overall separation steps. The appropriate coefficients can be used in conjunction with eq. 1 to describe the generalized distribution curve of each coal cleaning device and feed size fraction b y an analytical equation.
235
-••
100.00 -
80.00-
-- Weibull function ts
o 60.00u
g "5 o
40.00-
L
_m O
20.000.(30
).40
o16o
i
J
0-80 1.00 Reduced specific grafity
~
120
OT
1.40
Fig. 1. Generalized distribution curve representation using the Weibull function for Batac jigs (overall), 1/2 x 3/~ inch.
100¸00"
80.00-
°~°°
o \° oO~
-- Weibull function o Data points
~ 60.00
g
40.00o o o o°
°o o
2000 i 0.00 060
0.80
I O0 I.'20 140 Reduced specific gravity
n
180
i
Fig. 2. Generalized distribution curve representation using the Weibull function for Dynawhirlpool separators, 14 x 28 mesh.
236
The standard deviation (based upon ordinate values ranging from 0 to 1) for each device and feed size is also presented in Table 1 to provide an idea of the accuracy of each equation. As found in previous studies, the Weibull function is least accurate where the data spans the entire range of ordinate values (c = 1). Figure 1 shows a typical curve fit, for an overall (2-stage) separation using a Batac jig with a 1/2 by 3/8-inch size fraction. Table I indicates that the standard deviation is 8.54 X 10 -3 in this case. Figure 2 shows another fit, for a Dynawhirlpool with a 14 by 28 mesh size fraction. Notice the difficulty in fitting the upper "shoulder" of the curve. (The data scatter further contributes to the poor fit.) In this case, Table 1 shows that the standard deviation is 1.67 X 10 -2. This value is not greatly different than the value of 8.54 X 10 -3 obtained for the Batac jig; yet Figs. 1 and 2 indicate noticeable differences in the quality of the fits. These results are typical o f the better fits and the poorer fits, respectively. ACKNOWLEDGEMENTS
This work is a part of a project supported by the U.S. Department of Energy, contract no. DE-AC22-83PC62684. The assistance and support of Messrs. A.W. Deurbrouck and J.T. Wizzard, of the Pittsburgh Energy Technology Center, USDOE, are gratefully acknowledged. REFERENCES Gottfried, B.S., 1978. A generalization of distribution data for characterizing the performance of float-sink coal-cleaning devices. Int. J. Miner. Process., 5 : 1 -20. Gottfried, B.S., 1981. Statistical representation of generalized distribution data for float-sink coal cleaning devices: sand cones. Int. J. Miner. Process., 8: 89--91. Killmeyer, R.P., Jr,, 1980. Performance characteristics of coal-washing equipment: Baum and Batac jigs. Report of investigations. USDOE RI-PMTC-9(80), Oct. 1980. Maronde, C.P., Killmeyer, R.P. and Deurbrouck, A.W., 1983. Performance characteristics of coal-washing equipment: Dynawhirlpool separator. USDOE/PETC/TR-83/8, June 1983.