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Some observations on the utility of separation measures in coal flotation tests
The Chemical
Engineering
Journal,
27 (1983)
113
- 115
113
Short Communication Some observations on the utility of separation measures in coal flotation tests
It is proposed, however, to use recovery as defined by Gaudin [l] R(t)
E. T.
Department of Chemical Engineering, University of Manchester Institute of Science and Technology, Sackville Street, Manchester M60 1QD (Gt. Britain) September
-Xc(t))
F(1
WOODBURN
(Received 1983)
C(t){1
=
‘7,1982;
in final form June
16,
(2)
-xF)
and grade defined here as G(t) = 1 -
xc(t)
-
(3)
XF
as the primary measures defining separation. Both recovery and grade are bounded, O< G(t)<
1.0
1. INTRODUCTION
0 f R(t) < 1.0
For a conventional semibatch flotation test an initial feed coal containing a finite amount of mineral matter has a low mineral content concentrate withdrawn as froth overflow, leaving behind a high mineral residual tailings. The performance of the test can be described in the usual manner by overall and component mass balances: the overall mass balance is
and their trajectories plotted with time as a parameter are called graderecovery curves. These plots provide a basis for comparing different test conditions and the upper boundary of the family of experimental curves is the locus of the best attainable recovery at a specific grade within the conditions investigated. Although the upper boundary of the attainable region in grade-recovery space is optimal in a given context it is desirable to have an additional basis for estimating the degree of separation achieved on the upper bound. Gaudin [l] introduced the selectivity index SI which for the binary coal mixture becomes
F = C(t) + T(t)
(la)
and the components mass balance is FXFi = C(t)X,i(t)
+ T(t)XTi(t)
(lb)
where F is the original feed mass of untreated coal, C(t) and T(t) are the masses of total concentrate removed and tailings remaining after time t, XFi is the component mass fraction of component i in the feed, and Xci(t) and Xri(t) are the component mass fractions in the total concentrate withdrawn and tailings remaining after time t. It is convenient to consider coal as a binary mixture of mineral-free coal and mineral. The mineral content of the streams is approximated by the gravimetric ash assays of the feed, concentrate and tailings, i.e. 3tF, xc(t) and z+(t) respectively. For a binary mixture with a measured feed ash the separation is completely defined by two independent variables. C(t) and xc(t) are conveniently obtained with a minimum of data processing and can be used for this purpose. 0300-9467/83/$3.00
1’2
(4)
where RM is the recovery of mineral in the concentrate: Cxc
R,=
-
FXF
The selectivity index may be defined in terms of grade and recovery by substituting eqns. (2) and (3) in eqn. (4): sI = ~-xF(~-G)-R(~-x~)(~-G) 1
(l-R)(l-xr)(l--G)
I’* t
(5) Figure 1 shows two typical experimental curves plotted in grade-recovery space with @ Elsevier
Sequoia/Printed
in The Netherlands
Subassemblies related to the original system will have uncertainty measures H(C) or H(T). Using a separation analogy, if no selection were used in producing C and T from A then the product uncertainties of C and T will equal that of the original assembly A: H(A) =W(C) +H(T) If, however, selection does take place in the separation process the uncertainty will be reduced and in the limit with perfect separation it will tend to zero: 0.0 < H(C) + EHc(T)<
Fig. 1. Loci of SI in grade-recovery feed (XF = 0.19).
space for coal ash
superimposed SI loci. Point P clearly represents perfect separation and it is also reasonable to consider that points lying on the horizontal (G = 0) or vertical (R = 0) axes represent zero separation. The SI loci satisfy the criterion that separations represented by high values of the index completely enclose those with a lower value. They do not, however, uniquely identify point P as the perfect separation and they permit a variation 1 .O < SI < 00 on the vertical (R = 0) axis. 2. PROPOSED
NEW MEASURE
OF SEPARATION
EFFICIENCY
It is proposed
H=-_(xlln~l+(l-~l)ln(l-~xl))
(6)
as a measure of the degree of mixing of the two components. Equation (6) is analogous to the information entropy defined by Shannon [2], the properties of which have been formally investigated by Khinchin [3] as a basis for formulating a measure of the uncertainty of a finite system. To illustrate the basis of Khinchin’s analysis, let us consider a finite system of two mutually exclusive events such that either one or the other will occur at each trial with a probability of P, or Pz respectively. Then
H(A) = -(PI In P, +P2 In Pz) subject to the constraint
PI + P2 = 1.0.
(7)
where E is the expectation of the conditional uncertainty of H,(T) based on a knowledge of H(C). The function H defined by eqn. (6) satisfies these criteria. Equations (6) and (7) provide the basis for proposing that a separation efficiency for the binary coal-mineral mixture be defined as Sx=lOO
i
~{xclnr,+(l-~c)ln(l-x,)}+ I
l-
+ 1-g i
x {xF
i ln
1
{Xrlnxr+(l--+)ln(l--r)}
xF
+
(I-xF)
ln(l-xF))-’
X
!
(8)
Loci of SE can be plotted in grade-recovery space by substituting eqns. (la), (lb), (2) and (3) into eqn. (8). It should be noted that X T=
to define a function
H(A)
XF - (GIF)xc
l-C/F
Figure 2 shows the same experimental data as Fig. 1 but with constant S, loci superimposed. Both measures satisfy the criterion that high value loci, representing more complete separations, should enclose those of lower value. The S, loci set is bounded between 0% and 100% while the SI loci have no upper bound. The SE loci identify P uniquely as representing a condition of perfect separation (S, = 100%) and further define both horizontal and vertical axes as representing a nil separation (S, = 0.0%). The SI loci more closely resemble the grade-recovery curves than do the S, loci, particularly with respect to the ordinate (R = 0) on which the SI loci have finite values which correspond to non-zero varia-
115 S.E.% 5
1.0
0.9
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3. CONCLUSIONS \ \
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On this basis it is asserted that while SE provides additional information distinct from that available from grade-recovery tests the selectivity index provides a useful singleparameter approximation to the more complex grade-recovery data.
\
1 \
0.9
1
\
'10
I
1.0
Recovery
Fig. 2. Experimental grade-recovery trajectories with separation efficiency loci: curve 1, size range -20 w; curve 2, size range, +45 -63 pm.
tions in grade in the range 0 - 1.0. The SI = 100 locus over the entire range of recoveries resembles a single test at which high grades were achieved. In contradistinction a high grade single test would show a very sensitive dependence of separation efficiency on recovery.
(i) The proposed measure of separation efficiency has a sound conceptual basis identifying perfect and nil separation conditions over a bounded region. (ii) It should be used preferably with a complete set of experimental grade-recovery data but can also be useful in conjunction with the selectivity index.
REFERENCES A.M. Gaudin, Flotation, McGraw-Hill, New York, 1st edn., 1932, p. 526. G. E. Shannon, The mathematical theory of communication, Bell Syst. Tech. J., 27 (1948) 379 -423;623 -656. A. I. Khinchin, Mathematical Foundation of Information Theory, Dover Publications, New York, 1957, pp. 2 - 13.
Engineering
Journal,
27 (1983)
113
- 115
113
Short Communication Some observations on the utility of separation measures in coal flotation tests
It is proposed, however, to use recovery as defined by Gaudin [l] R(t)
E. T.
Department of Chemical Engineering, University of Manchester Institute of Science and Technology, Sackville Street, Manchester M60 1QD (Gt. Britain) September
-Xc(t))
F(1
WOODBURN
(Received 1983)
C(t){1
=
‘7,1982;
in final form June
16,
(2)
-xF)
and grade defined here as G(t) = 1 -
xc(t)
-
(3)
XF
as the primary measures defining separation. Both recovery and grade are bounded, O< G(t)<
1.0
1. INTRODUCTION
0 f R(t) < 1.0
For a conventional semibatch flotation test an initial feed coal containing a finite amount of mineral matter has a low mineral content concentrate withdrawn as froth overflow, leaving behind a high mineral residual tailings. The performance of the test can be described in the usual manner by overall and component mass balances: the overall mass balance is
and their trajectories plotted with time as a parameter are called graderecovery curves. These plots provide a basis for comparing different test conditions and the upper boundary of the family of experimental curves is the locus of the best attainable recovery at a specific grade within the conditions investigated. Although the upper boundary of the attainable region in grade-recovery space is optimal in a given context it is desirable to have an additional basis for estimating the degree of separation achieved on the upper bound. Gaudin [l] introduced the selectivity index SI which for the binary coal mixture becomes
F = C(t) + T(t)
(la)
and the components mass balance is FXFi = C(t)X,i(t)
+ T(t)XTi(t)
(lb)
where F is the original feed mass of untreated coal, C(t) and T(t) are the masses of total concentrate removed and tailings remaining after time t, XFi is the component mass fraction of component i in the feed, and Xci(t) and Xri(t) are the component mass fractions in the total concentrate withdrawn and tailings remaining after time t. It is convenient to consider coal as a binary mixture of mineral-free coal and mineral. The mineral content of the streams is approximated by the gravimetric ash assays of the feed, concentrate and tailings, i.e. 3tF, xc(t) and z+(t) respectively. For a binary mixture with a measured feed ash the separation is completely defined by two independent variables. C(t) and xc(t) are conveniently obtained with a minimum of data processing and can be used for this purpose. 0300-9467/83/$3.00
1’2
(4)
where RM is the recovery of mineral in the concentrate: Cxc
R,=
-
FXF
The selectivity index may be defined in terms of grade and recovery by substituting eqns. (2) and (3) in eqn. (4): sI = ~-xF(~-G)-R(~-x~)(~-G) 1
(l-R)(l-xr)(l--G)
I’* t
(5) Figure 1 shows two typical experimental curves plotted in grade-recovery space with @ Elsevier
Sequoia/Printed
in The Netherlands
Subassemblies related to the original system will have uncertainty measures H(C) or H(T). Using a separation analogy, if no selection were used in producing C and T from A then the product uncertainties of C and T will equal that of the original assembly A: H(A) =W(C) +H(T) If, however, selection does take place in the separation process the uncertainty will be reduced and in the limit with perfect separation it will tend to zero: 0.0 < H(C) + EHc(T)<
Fig. 1. Loci of SI in grade-recovery feed (XF = 0.19).
space for coal ash
superimposed SI loci. Point P clearly represents perfect separation and it is also reasonable to consider that points lying on the horizontal (G = 0) or vertical (R = 0) axes represent zero separation. The SI loci satisfy the criterion that separations represented by high values of the index completely enclose those with a lower value. They do not, however, uniquely identify point P as the perfect separation and they permit a variation 1 .O < SI < 00 on the vertical (R = 0) axis. 2. PROPOSED
NEW MEASURE
OF SEPARATION
EFFICIENCY
It is proposed
H=-_(xlln~l+(l-~l)ln(l-~xl))
(6)
as a measure of the degree of mixing of the two components. Equation (6) is analogous to the information entropy defined by Shannon [2], the properties of which have been formally investigated by Khinchin [3] as a basis for formulating a measure of the uncertainty of a finite system. To illustrate the basis of Khinchin’s analysis, let us consider a finite system of two mutually exclusive events such that either one or the other will occur at each trial with a probability of P, or Pz respectively. Then
H(A) = -(PI In P, +P2 In Pz) subject to the constraint
PI + P2 = 1.0.
(7)
where E is the expectation of the conditional uncertainty of H,(T) based on a knowledge of H(C). The function H defined by eqn. (6) satisfies these criteria. Equations (6) and (7) provide the basis for proposing that a separation efficiency for the binary coal-mineral mixture be defined as Sx=lOO
i
~{xclnr,+(l-~c)ln(l-x,)}+ I
l-
+ 1-g i
x {xF
i ln
1
{Xrlnxr+(l--+)ln(l--r)}
xF
+
(I-xF)
ln(l-xF))-’
X
!
(8)
Loci of SE can be plotted in grade-recovery space by substituting eqns. (la), (lb), (2) and (3) into eqn. (8). It should be noted that X T=
to define a function
H(A)
XF - (GIF)xc
l-C/F
Figure 2 shows the same experimental data as Fig. 1 but with constant S, loci superimposed. Both measures satisfy the criterion that high value loci, representing more complete separations, should enclose those of lower value. The S, loci set is bounded between 0% and 100% while the SI loci have no upper bound. The SE loci identify P uniquely as representing a condition of perfect separation (S, = 100%) and further define both horizontal and vertical axes as representing a nil separation (S, = 0.0%). The SI loci more closely resemble the grade-recovery curves than do the S, loci, particularly with respect to the ordinate (R = 0) on which the SI loci have finite values which correspond to non-zero varia-
115 S.E.% 5
1.0
0.9
\ \
-
o.s-
\
‘1\
’\ -2
20 \
lu
’
\
0.7-
\
xl \
\\
A-2
\
\
\
0.6-
\
\
d 0.5si
\
\
\
\
\
\
'\
\
\
'-'-'-‘I,,
\
\\
\
' \
P mo1
+y2.2.‘\\,
\
\
O.L0.3_
\
\ \ \\ \\ \
‘1
‘\
\
50
\
\
\ \
\
\
\
\
\
\
1' \' \
3. CONCLUSIONS \ \
‘q\,,
0.3
04
0.5
0.6
0.7
0.8
'aI
\\2o \
0.2
50 \
\
0.2-
0.1
\ \
\
On this basis it is asserted that while SE provides additional information distinct from that available from grade-recovery tests the selectivity index provides a useful singleparameter approximation to the more complex grade-recovery data.
\
1 \
0.9
1
\
'10
I
1.0
Recovery
Fig. 2. Experimental grade-recovery trajectories with separation efficiency loci: curve 1, size range -20 w; curve 2, size range, +45 -63 pm.
tions in grade in the range 0 - 1.0. The SI = 100 locus over the entire range of recoveries resembles a single test at which high grades were achieved. In contradistinction a high grade single test would show a very sensitive dependence of separation efficiency on recovery.
(i) The proposed measure of separation efficiency has a sound conceptual basis identifying perfect and nil separation conditions over a bounded region. (ii) It should be used preferably with a complete set of experimental grade-recovery data but can also be useful in conjunction with the selectivity index.
REFERENCES A.M. Gaudin, Flotation, McGraw-Hill, New York, 1st edn., 1932, p. 526. G. E. Shannon, The mathematical theory of communication, Bell Syst. Tech. J., 27 (1948) 379 -423;623 -656. A. I. Khinchin, Mathematical Foundation of Information Theory, Dover Publications, New York, 1957, pp. 2 - 13.