EVALUATION EFFICIENCY WITH PARTITION CURVES IN MINERAL PROCESSING APPARATUS

EVALUATION EFFICIENCY WITH PARTITION CURVES IN MINERAL PROCESSING APPARATUS G. M. Kosoy, D.Sc. (Mineral Processing leader of Kroosh Technologies Ltd, Israel) Partition curves proposed by K. Tromp [1] were originally used to evaluation the efficiency of gravity concentration of minerals. It was further found that they may also be used to describe of other processes, including screening and classification processes where the feed and final products can be separated into narrow size fractions. Partition curves constructed by a large number of narrow fractions give the maximum information about the performance of a mineral processing apparatus. They are known to represent the function of recovery narrow fractions into the final products depending on their specific gravity or size. Most clearly this can be shown by the example of the screening process.Two basic parameters of the screening process are determined by the partition curves: the cut point and efficiency. In most cases, the cut point is understood as a narrow fraction d50, which is divided equally between the undersize and oversize. There is no unambiguous evaluation of screening efficiency by partition curves since an analytical form of this function is not known. K. Tromp proposed to characterize the separation efficiency by "the area of misplaced particles" of the final products [1], which is calculated by the equation: ds

dn

0

ds

FT=   i (us) d i    i ( os) d i ,

(1)

where εi(us) is the recovery of a ∆di narrow fraction into the undersize (US); εi(os) is the recovery of a ∆di narrow fraction into the oversize (OS); dn is the maximum size of particles in the undersize. According to equation (1), "the area of misplaced particles" is equal to the sum of the mass of particles larger than a predetermined cut size ds in the undersize and the mass of particles smaller than a predetermined cut size ds in the oversize, depicted on the graph of the partition curve against the narrow fractions size. The partition curve can be constructed for both the undersize and the oversize. Since the form of equation (1) is unknown, it is impossible to calculate the numerical value of efficiency by this equation. However, "the area of misplaced particles" can be calculated if the partition curve εi(us) is constructed against the cumulative passing of fines in the feed "α" [2]. In this case, "the area of misplaced particles" will be equal to the cumulative amount of outsize fractions contained in the screening products:

F=γus (1 – β) + γos,

(2)

2 where γus, γos are respectively the yields of the solid phase in the undersize and oversize; β,  are respectively the fines contents in the undersize and oversize. The basic property of function (2) is that it takes the minimum value for narrow fraction d50, divided equally between the undersize and oversize εi(us) = εi(os) = 0.5 (here and below, all values are in decimal fractions). To prove this, let us differentiate function (2) by α: dF/dα = γus[d(1 – β)/dα] + γos(d/dα) = – εi(us) + εi(os) = 0. Hence εi(us) = εi(os) = 0.5, as εi(us) + εi(os) = 1. In coordinates (εi(us), α), the partition curve does not depend on the nature of the physical characteristic by which the material is separated into narrow fractions (particle size, density, magnetic susceptibility, etc.), therefore it can be used for evaluation the efficiency of any mineral process. The general view of the US and OS partition curves are shown in Fig. 1, 2.

Differential Partition Curve of undersize 1

B

Fos

0.9

C

1

0.7

2

Pos

γos

∆Sus

0.6

3

0.5

M

T

N

0.4 0.3

∆Sos

Pus

0.2

γus

Partition coefficient Pc(i), %

0.8

L

0.1

A

K (α)

Fus

0

α=0

Cumulative percent passed in feed, α

1. Partition Curve. 2. BLKD - a broken line of perfect separation. 3. BMND - a broken line of dividing a feed without separation.

Fig. 1 Differential Partition Curve of undersize

D

α=1

3

Differential Partition Curve of oversize 100

L

B

Fus

C

90

1

∆Sos

γus

Pus

70

60

M

T

N

50

3

40

2

∆Sus

30

Pos

γos

Partition coefficient Pc (i), %

80

20

10

Fos

A 0

α=0

K (α)

Cumulative percent passed in feed , α

D

α=1

1. Partition Curve. 2. BLKD - a broken line of perfect separation. 3. BMND - a broken line of dividen a feed without separation.

Fig. 2 Differential Partition Curve of oversize

Let the total content of fines in the feed at point K(α) (Fig. 1) be equal to α. Broken curves BLKD and BMND correspond to the limiting form of partition curve εi(us) in the case of perfect separation and separation into parts with the same qualitative composition of narrow fractions in the undersize ∆βi = ∆αi respectively. Since fractional recovery εi (partition coefficient Pc(i) ) into the undersize εi(us) = γus(∆βi/∆αi), in the latter case εi(us) = γus = const (straight line MN). Let us define the physical meaning of the individual parts of square ABCD received as a result of its division by the partition curve and segments of straight lines LK and MN. Passing to the limit (∆βi/∆αi) = dβi/dαi and given that εi(us)dα = γusdβ, we obtain 



0

0

  i (us) d   usd  us .

Thus, the area under the partition curve to the left of straight line segment TK is equal to the cumulative amount of fines in the undersize Sus= γusβ. Part of the area of square ABCD below the 1

whole partition curve is equal to   usd  us , i.e. to the undersize yield. 0

Fig. 1 also shows that γus = ∆Sus + Pus + Fus. Consequently, the undersize can be represented as

4 consisting of three parts: ∆Sus = Sus – Pus = γus (β – α) – quantitative increment of fines in the undersize; Pus=γusα – part of the fines that have passed from the feed into the undersize without changes in the size composition, i.e. when β = α; finally, Fus=γus – γusβ=γus (1 – β) – the amount of the course particles in the undersize. Part of the area of square ABCD bounded below by the partition curve to the left of straight line 

segment LT is equal to Fos =  (1   i (us) )d , where (1 – εi(us))=εi(os). Given that εi(os)= γos(d/dα), we 0



obtain Fos =   osd = γos. 0

Part of the area of square ABCD bounded below by the entire partition curve is equal to 1



os

d = γos, i.e. to the oversize yield.

0

Part of the area of square ABCD, bounded below by the segment of the partition curve to the right of 1

1





straight line segment LT, is equal to Sos =   i ( os) d =   osd = γos(1 – ), i.e. to the cumulative amount of coarse fractions in the oversize. Fig. 1 shows that γos = ∆Sos + Pos + Fos, where ∆Sos = Sos – Pos = γos(1 – ) – γos(1 – α) = γos(α – ) – quantitative increment of coarse fractions in the oversize. The oversize can also be represented as consisting of three parts: ∆Sos = γos(α – ) – quantitative increment of coarse fractions in the oversize; Pos = γos(1 – α) – part of coarse fractions that have passed into the oversize without changes in its composition, i.e. when  = α; and, finally, Fos = γos – the amount of fines in the oversize. Let us note that the quantitative increment of fines in the undersize ∆Sus identical to the quantitative increment of coarse fractions in the oversize ∆Sos: ∆Sus=∆Sos. The same results can be obtained by analyzing the partition curve in Fig. 2. Fig. 1, 2 shows the differential form of the undersize and oversize partition curves. Only one position of the partition curves can be represented in the coordinates, in which ∆Sus = ∆Sos. However, this equality is not dependent on the cut poinds [2]. Below, Fig. 3 shows the combined dependency graph Sus = γusβ = f(α) and Sos = γos(1 – ) = f(1 – α). Values Sus = γusβ are plotted up the left ordinate, values Sos = γos(1 – ) are

5 plotted down the right ordinate. The values of α from 0 to 1 are plotted from left to right on the lower abscissa, and the values (1 – α) from 0 to 1 are plotted from right to left on the upper abscissa.

Integral Partition Curves

(1 - α) = 1

(1 - α) = 0 C

Cumulative percent of fine in oversize γos (1- )

γos (1-

Pos

(1- α)

γos

Cumulative percent passed to undersize , γusβ

γusβ= 1

) =0

M

2

) =1

3

ΔST

Pus

α

ΔSus

γus

γusβ= 0

A

D

γos(1-

T

B

1

Fos

Fus

ΔSos

4

α= 0

α= 1

N

1. Partition Curve of undersize: γus β =f (α ). 2. Partition Curve of oversize: γos(1- ) =f (1 - α ). 3. Dividing line of undersize product for: β = α. 4. Dividing line of oversize product for: = α. 5. ATD - Perfect Partition Curve of undersize . 6. BTC - Perfect PartitionCurve of oversize . Fig. 3 Integral Partition Curves

Curves 1 and 2 represent, respectively, the integral form of the undersize and oversize partition curves because their derivatives by α and (1 – α) give the values of fractional recovery: εi(us) = d(γusβ)/dα = γus(dβ/dα); εi(os) = d[γos(1 – )]/d(1 – α) = γos(d /dα) and are equal to the tangents of the slope angle at all points of curves 1, 2 with respect to their abscissas.

6 Straight line segment AD corresponds to the integral partition curve of the undersize without changes in its composition (γusα); εi(us) = γus = const along it. Straight line segment CB corresponds to the integral partition curve of the oversize without changes in its composition γos(1 – α); εi(os) = γos = const along it for all values of (1 – α) in the interval (0, 1). The ordinates of the points on segment AC are equal to α and to (1 – α) on segment CA. Broken line ATD corresponds to the integral curve of perfect separation of the undersize, whereas broked line CTB – to that of the oversize. Since the angle of segment AC slope is 45º, then εi(us) = 1 on straight line segment AT and εi(us) = 0 on straight line segment TD. On straight line segment CT, εi(os) = 1, and on segment TB, εi(os) = 0. Real partition curves are intermediate between εi = const and the broked lines of perfect separation of the corresponding products. Areas ∆Sus, Pus, Fus and ∆Sos, Pos, Fos in Fig. 1, 2 correspond to the ordinate segments of the same name in Fig. 3 for all values of cut poinds α in the interval (0, 1). Let us use the graphs of functions of differential and integral partition curves to derive technological criteria for efficiency of size screening and classification of a polydisperse material. Fig. 1, 2 shows that the initial state of screening is characterized by limiting position of the differential partition curve in the form of broked line BMND. In this case part of the fines Pus and coarse fractions Pos is already in the undersize and oversize. They are recovered into these products in perfect separation (broked line BLKD) as well. In Fig. 3, the initial position of the integral partition curve is characterized by segment AD for the undersize and CB for the oversize. In all intermediate cases, the separation work at an arbitrary cut poind of α can be characterized by the sum of the increments of the corresponding fractions into screening or classification products, i.e., by the sum of areas ∆Sus + ∆Sos in Fig. 1, 2 or segments ∆Sus + ∆Sos in Fig. 3 obtained only through the action of the force field of the apparatus regardless of their content in the feed. Since the total amount of fine and coarse fractions that pass into the final products without changes in their composition is (Pus + Pos), the total theoretical increment is ∆ST = 1- (Pus + Pos). In Fig. 3, it corresponds to the vertical distance between straight line segments AD and CB: ∆ST =1 – γusα – γos(1 – α). Thus, the technological efficiency of screening can be evaluated numerically by partition curves constructed in coordinates (εi(us), α) or (Pc(i), α) and (εi(os), (1 – α)), or (Pc(i),(1- α)) by the ratio of the total increment of the amount of fine and coarse fractions in the undersize and oversize to the theoretical increment:

7 E = (∆Sus + ∆Sos)/∆ST;

(3)

By definition (3) E= [γus(β – α) +γos(α – )]/ [1 – γusα – γos(1 – α)].

(4)

When separated into two products, ∆Sus = ∆Sos, therefore E= [2γus(β – α)]/ [γus(1 – 2α) + α];

(5)

In formula (5), to calculate the efficiency, all values are expressed in decimal fractions. Efficiency values by formulas (4) and (5) change from 0 to 1, i.e. they are normalized. Below, table 1 shows an example of processing and presentation of results of an ordinary experiment on wet screening of quartz sand pulp. The table shows the conditions of the experiment, as well as all measured and calculated data of screening, including the yield of final products, the total recovery of control fractions and screening efficiency by formula (5) obtained by analysis of the graph of partition curves in coordinates (Pc(i), α) and Pc(i),(1- α). Tables 2, 3 and 4 show the results of calculating the partitions coefficients Pc(i)us and Pc(i)os in the undersize and oversize, as well as screening efficiency Еi of each narrow fraction. Partition curves and fractional efficiency curves were constructed in Microsoft Excel by numerical values of these parameters. Fig. 4 and 5 show the graphs of the partitions coefficients Pc(i)us and Pc(i)os in the undersize and oversize, as well as the screening efficiency Еi for each narrow fraction in respect to its mean size di and cumulative passing of fines in the feed α. Graphs are constructed by program Chart Tools Microsoft Excel. They show that the screening efficiency Еi by formula (5) takes the maximum value at the cut size d50 where the fractional recovery is Pc(i)us = 50 %. Shift from d50 is due to errors in the measurement of the mean size of a narrow fraction and deviation of each narrow fraction yield from the average yield of the entire product. The shapes of the partition curves and fractional efficiency of screening are absolutely identical in coordinates (Pc(i), Еi, di) and coordinates (Pc(i), Еi, α).

8

Screening slurrry of sand

Table 1 o

Pilot screen 600 x 1500 mm, 1500 RPM, slope 15 . Polyurethane sieve, opening 0.2 x 2.59 mm.

Particle size Partition coefficient distribution, μm Pc (os), %

Feed

OS

+ 500

99.84

9.93

16.92

- 500 + 300

99.94

26.26

44.79

0.40

- 300 +200

85.16

10.72

15.58

3.52

- 200 +150

56.53

7.11

6.86

8.73

- 150 + 106

31.22

5.78

3.08

9.17

- 106 +75

23.67

5.99

2.42

10.36

- 75 +45

18.74

8.35

2.67

17.20

-45+ 0

17.40

25.86

7.68

50.62

Total amount, %

100.00

100.00

100.00

Total fractions +200 μm, %

46.91

77.29

3.92

% Solids

42.25

70.16

28.15

1.37

0.43

2.55

Slurry flow rate, m /h

7.57

2.12

5.45

Capacity of dry material, t/h

4.34

2.64

1.86

7.23

4.40

3.10

12.62

3.53

9.08

Ratio of water / solids 3

2

Specific capacity of dry material, t/m x h 3

2

Specific capacity of slurrry, m /m x h

Mass % US

Mass balanced data Weigtht % of retained fractions + 200 μm

100

58.59

41.41

Recovery of retained fractions + 200 μm

100

96.54

3.46

Screening efficiency by formula (5)

70.45

E= {[2γos (β-α)]/[γos (100-2α) + 100α]}*100, α , β, γos - %, α=46.91%; β=77.29%, γos =58.59%, E=70.45%.

(5)

9

Table 2 Calculation of Partition coefficients (d1 - d2), μm di Pc(i) us 500-300 400 0.63 300-200 250 13.75 200-150 175 50.84 150-106 128 65.69 106-75 90.5 71.61 75-45 60 85.29 -45 22.5 81.05

Table 3 Calculation of screening efficiency Pc(i) os 99.37 86.25 49.16 34.31 28.39 14.71 18.95

α

β

di

Ei

90.7 68.83 53.09 45.98 40.2 34.21 25.86

100 99.6 96.08 87.35 78.18 67.82 50.62

400 250 175 128 90.5 60 22.5

13.51 47.87 70.46 69.49 65.10 58.87 44.72

Screening efficiency, (Ei) and Partition coefficients, Pc (i),%

Screening efficiency and Partition Curves of US and OS 100 90 80

70 60 50 40 30 20 10

d50

0 0

50

100

150

200

250

300

350

400

Mean size of fractions, di , μm

Screening efficiency Ei Partiton coefficients of OS

Partition coefficients of US d50=176μm

Fig. 4 Screening efficiency E(i) and Partition Curves in coordinates Ei, Pc i =f(d)

10

Table 4 Screening efficiency Еi= f(α) and Partition coefficients Pc(i) = f(α) α Pc(i)us Pc(i)os Ei 90.7 0.63 99.37 13.51 68.83 13.75 86.25 47.87 53.09 50.84 49.16 70.46 45.98 65.69 34.31 69.49 40.2 71.61 28.39 65.10 34.21 85.29 14.71 58.87 25.86 81.05 18.95 45.24

Screening efficiency Еi and Partition coefficients, Pc(i) ,%

Screening efficiency E (i) and Partition coefficients Pc(us), Pc(os) in coordinates E i =f(α), Pci = f(α) 100 90

80 70 60 50 40 30

d50

20 10 0 20

30

40

50

60

70

80

90

100

Cumulative mass passed in feed, α, %

Partition coefficients of US

Partition coefficients of OS

Screening efficiency, E(i)

Fig. 5 Screening efficiency E(i) and Partition Curves in coordinates Ei, Pc i =f(α)

11

Conclusion. A grapho-analytical analysis of Partition Curves in coordinates (Pc(i), α) has been made resulting in a quantitative criteria of separation efficiency evaluation (5), applicable not only for screening and classification processes, but also for mineral processing in general. The shapes of the Partition Curves and mineral processing fractional efficiency are absolutely identical in coordinates (Pc(i), Ei, di) and (Pc(i), Еi, α), where di- expresses any physical feature of separating products. The proposed form of experimental results processing gives full information about all the qualitative and quantitative parameters of mineral processing apparatus. References. 1. Tromp K. F. Neue Wege für die Beurteilumg der Aufberechtung von Steinkohlen. Glcükauf, 1937, Nr. 73. - S. 125-131. 2. Dr. G. Kosoy. Analysis and Engineering Estimate of Mineral Concentration Processes. The Israeli association for the advancement of mineral Enginneering. The Eleventh Conference. Nahariya - 21-23 December 1992.