Technical note a simple equation for sink-float data

~

Minerals Engineering, Vol. 7, No. 11, pp. 1441-1446, 1994 Copyright ©1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0892-6875/94 $7.00+0.00

Pergamon 0892-6875(94)00082-4

A SIMPLE

TECHNICAL NOTE EQUATION FOR SINK-FLOAT

B. G O V I N D A R A J A N

DATA

and T.C. RAO

Regional Research Laboratory, Bhopal - 462 026, India (Received 9 Februaly 1994; accepted 22 June 1994)

ABSTRACT

7he published results of the sink-float data on coals of different origins have been analysed for smoothing and intelpolation. The cumulative weight fi'action of material floating and the ash content of the float fraction were used to estimate the recoveries of "non-ash" attd "ash" material at each relative density level. A simple equation was then derived relating recovery to the relative density. Further, it has been shown that these equations can be used for the purpose of interpolation thereby reducing the need for canting out sink-float tests at several relative density levels. Keywords Sink-float data, washability, curve fitting, interpolation.

~TRODUCTION Sink-float tests are widely used to estimate the amenability of coal towards gravity concentration techniques. These tests are time-consuming and cumbersome hence the sink-floats are often carried out only at limited relative density levels. The discrete data thus obtained from the sink-float tests are used to make a set of curves known as "Washability curves" which are then used to assess the degree of difficulty of gravity separation of raw coal and to provide qualitative or quantitative data for the products of the separation at a selected relative density [1,2]. Apart from the washability curves, many attempts have been made to smooth and interpolate the discrete sink-float data. Klima and Luckie [3] describe an interpolation technique using IMSL routines to predict the complete sink-float data for different size fractions and specific gravities. Hughes [4] used classical optimisation theory for smoothing sink-float data. Abbot and Miles [5] have suggested equations for the Mayer curve for the same purpose. However, these techniques involve complicated mathematical equations and require computational facilities. In the present work a simple equation for smoothing the sink-float data has been proposed. It has also been shown that such equations can be used for interpolation to significantly reduce the number of relative density levels normally needed for the sink-float test.

FORM OF EQUATION The cumulative form of the sink float data, more suitable for fitting equations, has been used in the present work. This is supported by the discussions on the use of cumulative data for interpolation methods presented by Kiima and Luckie [3]. 1441

1442

Technical Note

The major obstacles for fitting a simple equation for sink-float data are i)

the cumulative ash content of the float fraction and the levels of relative density used vary over a short range.

ii)

a large increase in float material at the highest relative density level due to the presence of well liberated shale.

These problems can be solved by making simple modifications in the parameters "cumulative weight" , " ash of float material" and "relative density" before correlating them. The sink-float data has been resolved into two components namely recovery of non-ash (Rn) and ash (Ra) which are calculated as

i)

R.

W/(IO0 -,4 i) -

(IO0-A/)

% - WiAi AI

ii)

(1)

(2)

where W i is the cumulative weight percentage of the feed floated at i th relative density fraction, A i is the cumulative ash content of the float material at i th relative density fraction and Af is the percentage ash content of the feed. The values of the relative densities (Sg) used in the sink-float test have been represented by a parameter X which is defined as x -

¾-%

(3)

where Mg is the highest relative density level used (generally 2.20) and Lg is the relative density at which no material will float. The value of Lg has been taken as 1.22 which is normally considered as the lowest relative density in sink-float testing. The value of parameter X varies between 0 to ** for the changes in the values of the recoveries of non-ash and ash material in float fraction between 100 to 0 respectively. The published data available on washability of coals of different origin have been collected to develop a simple equation to smooth the sink-float data using the proposed modifications. The relationship between the parameter X and recovery of non-ash and ash material of a difficult to wash coal (1) and an easy to wash coal (2) is shown in Figures I and 2 respectively. It is interesting to note that the simple modifications made to the sink-float data resulted in a smooth curve which can be easily fitted to a simple equation. The curves shown in Figures 1 and 2 can be mathematically expressed as

R.

= lOOe -ax}

R a = lOOe-pX~

(4)

(5)

Technical

Note

1443

G)

t~J2mJ~non-ash

0

E

xxx:~

o~h

N

e0"i

.

L

"

O

_

O I

-

e,.

ol -

ES.

Ii

°o~o

i ii

fll

ii

iii

i ii

Ii

lo.oo

iIiii

ii

2o.oo

iii

ii

I

~o.oo

X Fig. 1 Relationship between parameter X and Rn or R a (Coal 1)

qR.gg.~ n o n - a s h x x x x x ash

O O

cb.~

10.00

20.00

30.00

× Fig.2 Relationship between parameter X and Rn or R a (Coal 2) The values of the constants in the above equations have been estimated for coals 1 and 2 using the least squares method and are presented in Table 1. Using these two equations the cumulative weight and ash content of float material at different relative densities have been calculated and compared with the actual values in Table 1. From this Table, it may be noted that the equations fit the data well. The published data on coals of different origins have been quantified using Eqs 4 and 5. The values of the constants in these equations were estimated and are presented in Table 2 together with the correlation coefficient (CC) values. Table 2 confirms the validity of the equations over widely varying coal washability characteristics.

1444

Technical Note

TABLE 1 Comparison between predicted and actual values of sink-float data for coals 1 and 2.

Type of coal

Sg

W actu.

Easy to wash (Coal i)

Difficult to w a s h (Coal 2)

A pred.

actu.

pred.

1.26 1.35 1.45 1.55 1.65 1.75 2.20

63.10 79.20 85.40 88.30 90.20 91.40 I00.00

62.78 79.86 85.27 88.12 90.05 91.55 i00.00

2.70 4.50 5.90 6.80 7.60 8.10 13.90

2.65 4.65 5.87 6.77 7.50 8.17 13.90

1.26 1.35 1.45 1.55 1.65 1.75 2.20

12.98 36.21 56.67 67.92 74.29 79.18 i00.00

10.59 41.33 57.49 67.05 73.75 78.97 i00.00

4.50 9.55 13.39 16.08 17.97 20.03 29.61

4.65 9.73 13.34 15.96 18.07 19.88 29.61

TABLE 2 The values o f c o n s t a n t s i n E q s 4 and 5 ~ r c o a l s o f d i f f e r e n t o r i g i n coal 1 2 3 4 5 6 7 8 9 i0 ii 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

a

b

0.i01 0.120 0.037 0.172 0.030 0.024 0.035 0.050 0.068 0.049 0.108 0.044 0.102 0.067 0.051 0.011 0.015 0.029 0.026 0.135 0.027 0.029 0.044 0.070 0.063 0.180

0.503 0.653 0.940 0.887 0.878 0.996 0.740 0.972 0.611 0.620 1.049 0.804 0.692 0.837 0.876 1.211 1.078 0.751 0.577 0.792 0.802 0.905 1.002 0.742 1.019 0.662

CC 0.976 0.997 0.997 0.987 0.997 0.996 0.994 0.989 0.974 0.986 0.974 0.989 0.997 0.995 0.980 1.000 0.998 0.988 0.984 1.000 1.000 0.992 0.990 1.000 0.998 0.982

p

q

2.008 1.202 0.801 0.711 0.749 0.701 0.818 1.150 1.105 0.997 0.732 1.009 1.809 0.733 0.700 0.266 0.348 0.887 1.017 1.328 0.659 0.501 0.439 0.749 0.514 1.136

0.201 0.377 0.411 0.634 0.278 0.320 0.194 0.307 0.233 0.215 0.613 0.296 0.223 0.443 0.455 0.558 0.446 0.260 0.192 0.385 0.371 0.378 0.627 0.416 0.628 0.437

CC 0.994 0.994 0.990 0.996 0.981 0.970 0 977 0 965 0 978 0 996 0 987 0 969 0.994 0 997 0 985 0 997 0 993 0 961 0 976 1 000 0 999 0.986 0.990 0.994 0.997 0.998

Ref.

[2] [5] [5] [7] [8] [8] [8] [8] [8] [8] [9] [9] [10] [11] [11] [11] [11] [12] [12] [12] [13] [13] [13] [13] [13] [13]

INTERPOLATION OF SINK-FLOAT DATA

One of the important benefits of fitting an equation to the sink-float data is the possibility of reducing the number of relative density levels needed in the testing. To verify the use of the proposed equations for the interpolation of the sink-float results, the data presented by Sokasi et.al [6] on eight size fractions at seven relative density levels has been considered. The same data was used by Klima and Luckie [3] for interpolating washability of different size fractions. However, in the present work the relationship between size and washability is not considered.

Technical Note

1445

For the purpose of interpolation, the values of R n and R a were calculated at average relative density levels 1.26, 1.45 and 1.7. Using these values the constants "a", "b", "p" and "q" in Eqs. 4 and 5 were evaluated. These equations were then used to describe the complete set of the washability data at seven relative density levels. The comparison between actual and interpolated values of cumulative weight and ash content of float material is shown in Figures 3 and 4 respectively. From these Figures, it is noted that the equations proposed have accurately interpolated the sink-float data using only three of the original data points.

8

8-

]. 1 O

~1

0

I

I

I

I /

I

I

I

I /

|

I

I

I i~1

I

!

I

1100

Ao~u~l ~a~u4a

Fig.3 Comparison between actual and interpolated values of weight percent float material

// O

Irl 0

i i I i 8

li

i i I i i i i i 5

i I i i i i i i

B

10

i i i i i i 18

15

Ao~zo.l ~ s

Fig.4 Comparison between actual and interpolated values of ash percent of float material

CONCLUSIONS Sink-float data have been resolved into two components i.e., recovery of non-ash and ash material in the float fraction, which are related to the average relative density. These equations can be used to produce

1446

Technical Note

continuous washability data from the sink-float tests. Further, it has also been shown that these equations can be used to reduce the number of relative density levels needed in the sink-float tests. The published sink-float data on coals of different origins have been used to check the validity of the proposed equations and no specific limitations are noted. The traditional polynomial smoothing methods sometimes give false results such as negative weight fraction float material or reduction in ash percent at higher relative density. The smoothing method suggested in this work will not give such results due to the form of the equations. However difficult sink-float data [4] may produce higher deviation between actual and smoothed values.

REFERENCES 1.

.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Corriveau, M.P. & Schapiro, N., Projecting data from samples. Coal Preparation Ed., J.W. Leonard, 4th Edition, The American Institute of Mining, Metallurgical, Petroleum Engineers Inc., chapter 4, 4-27 (1977). Osborne, D.G., Coal Preparation Technology, Volume 1, Graham & Trotman Limited, 179-188 (1988). Klima, M.S. & Luck,e, P.T., An interpolation methodology for washability data. Coal Prep., 2, 165-177 (1986). Hughes, D.M., A mathematical programming method for smoothing washability data, CoalPrep., 9, 13-26 (1991). Abbott, J. & Miles, N.J., Smoothing and interpolation of float-sink data for coals. Minerals Engng., 4, no 3/4, 511-524 (1991). Sokaski, M., Jacobsen, P.S. & Geer, M.R., Performance of Baum jigs in treating Rocky Mountain coals, U.S. Bureau of Mines, Report of Investigations, 6306 (1963). Vanangamudi, M., Rao, T.C. & Sharma R.N., Studies on the operational characteristics of an industrial heavy medium cyclone. Coal Prep., 6, no 1/2, 79-90 (1988). Butch, C.F., btdustrial Practice of Fine Coal Processing, Eds. R.R.Klimpel and P.T.Luckie, Chapter 7, 49-55 (1988). Cierpisz, S. & Gottfried, B.S., Theoretical aspects of coal washer performance. Int. J. of Min. Prec., 4, 261-278 (1977). King, R.P. & Juckes, A.H., Cleaning of fine coals by dense medium hydrocyclone. Pew. Tech., 40, 147-160 (1984). Maronde, C.P., Killmeyer, R.P. & Deurbrouck, A.W., Performance characteristics of coal washing equipment. The Dynawhirlpool separator, Report no DOE/PETC/TR 83 -8 (1983). Zeilinger, J.E. & Deurbrouck, A.W., Physical desulfurization of fine size coals on a spiral concentrator, Report of h~vestigation USBM 8152, (1976). Geer, M.R., et. al., Performance of dense medium cyclone in cleaning fine coal, Report of b~vestigation USBM 5 732, (1961).