Analytical evaluation of locked cycle flotation tests

International Journal o[ Mineral Processing, 27 (1989) 39-50

39

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

Analytical Evaluation of Locked Cycle Flotation Tests S. N I S H I M U R A ,

H. H I R O S U E , K. S H O B U

and K. JINNAI

Government Industrial Research Institute, Kyushu Shuku-machi, Tosu, Saga 841 (Japan) (Received March 28, 1988; accepted after revision January 10, 1989)

ABSTRACT Nishimura, S., Hirosue, H., Shobu, K. and Jinnai, K., 1989. Analytical evaluation of locked cycle flotationtests.Int. J. Miner. Process.,27: 39-50. A locked cycle test is one of the testing methods used to develop the optimum mineral recovery process. In order to understand the behavior of locked cycle tests,material balance equations are set up at the nth cycle of locked cycle tests consisting of three and four stages, called rougher, scavenger, cleaner and recleaner,and analytical solutions are obtained. From the analyticalsolutions, itbecomes clearthat the iterationnumber of the locked cycletestsrequired to reach a steady state depends on the flotationrate constant, the flotationtime, the arrangement of the flotation stages and the total mass of recycled middlings which are discharged from the nth cycle of the locked cycle testsand fed into the subsequent (n + 1 )th cycle.In addition,flotationcharacteristics obtained from analytical solutions of a locked cycle test are compared with those of a continuous flotation circuiton the assumption that each flotationstage is composed of the same number of perfectlystirredcellsof the same volume. As a result,it is found that good agreement between the theoreticallocked cycle test and continuous flotationresultsis obtained when the number of cells in each stage is large.

INTRODUCTION

When mineral processing circuits are developed, many tests are conducted using a laboratory apparatus or a pilot plant to collect the data necessary for determining the optimum process circuit and operating condition. Traditionally, a locked cycle test or a locked test (Taggart, 1948; Macdonald et al., 1985) has been conducted to experimentally simulate a continuous process. A locked cycle test has been said to provide as much information as a bench-scale continuous test and to save the cost, sample and labor necessary to carry out the continuous test (Agar and Kipkie, 1978; Macdonald et al., 1985 ). A locked cycle test is a series of repetitive batch tests conducted on a small-scale laboratory apparatus, in which the middiings generated in the one test (nth cycle) are added to the subsequent test ( (n + 1 ) th cycle ). Fig. I shows 0301-7516/89/$03.50

© 1989 Elsevier Science Publishers B.V.

]

Ne:~-

I

hatcher

I

I

I

I

] ,

i:

I '

Fig. 1. Flowsheet for locked cycle test.

an example of a flowsheet for a locked cycle flotation test and each cycle consists of four batch tests(stages) such as rougher, scavenger, cleaner and recleaner. To obtain satisfactory information from locked cycle tests, a steady state must be reached. Although locked cycle tests have many advantages as mentioned above, the following disadvantages have also been pointed out through experimental work. ( 1 ) These tests are time-consuming because five cycles are required to reach a steady state even in a simple circuit and generally six or more cycles are required ( Macdonald et al., 1985 ). (2) It becomes very difficult to reach a steady state with increasing the a m o u n t of the recycled middlings (Agar and Kipkie, 1978; Agar, 1985 ). It is therefore indiapensable that analyt-

41 ical equations are developed theoretically in order to understand these experimental results more deeply. Agar (1985) observed good agreement between the locked cycle flotation test results and the continuous flotation results. However, the main reason why the locked cycle test results coincided with the continuous flotation results has not been made clear to date. Therefore, we try to analyze theoretically locked cycle tests and to compare simulation results on a locked cycle test with those on a continuous flotation circuit. ANALYSISOF LOCKEDCYCLETEST In analyzing locked cycle tests, the following assumptions are made. (1) The flotation process in each stage is described by a first order rate equation:

Y= W/Wo = e x p ( - k T )

(1)

where k is the flotation rate constant of the first order rate equation, T is the flotation time, Y is the residual mass fraction of a component with the flotation rate constant k at time T a n d Wo and Ware the weight of a component retained in the cell at time zero and T, respectively. (2) For each component, the flotation rate constant k is not affected by the recycled middling and is constant in each stage. Fig. 2 shows the flowsheet for the nth cycle of a locked cycle test. Considering that the mass of new feed is equal to 1 in this flowsheet, one gets the following material balance equations:

Ct(n-l)

RCt(n-1)

Sc(n-l) ~I R = S = C = RC =

New feed

¢

Rc(n)

c = concentrate

rRt(n) ~n~CL~

Cc(n)

~

S~

rougher scavenger cleaner recleaner t = tailing

Sc(n) J Ct(n)

St(n)

~I

Fig. 2. Flowsheetfor lockedcycletest at the nth cycle.

iteration number RCc(n:

(n)

Rt(n)= Y[1 + S c ( n - 1) + C t ( n - 1 I J R c ( n ) = t l - Y) Rt(n)/Y St(n) = YRt(n) Sc(n)= ( 1 - Y) S t ( n ) / Y

(3)

Ct(n ) = Y[Rc(n ) + R C t ( n - 1 )] Cc(n) = ( 1 - Y) Ct(n)/Y

(4)

RCt(n) = Y Cc(n) R C c ( n ) = ( 1 - Y) RCt(n)/Y

(5)

where n is the number of iterations (cycles), Rt, St, Ct and RCt are the mass of tailing in rougher, scavenger, cleaner and recleaner, respectively, and Rc, Sc, Cc and RCc are the mass of concentrate in rougher, scavenger, cleaner and recleaner, respectively. From eqs. 2, 3, 4 and 5, one gets:

Rt(n)= Y ( 1 - Y) R t ( n - 1 ) + Y C t ( n - 1 ) + Y

(6)

Ct(n)= Y ( 1 - y)2Rt(n- 1) + 2 Y ( 1 - Y) C t ( n - 1 ) + Y ( 1 - Y)

(7)

Eqs. 6 and 7 are rewritten in the matrix form: Xn=A Xn-l+Xl

(8)

where:

[ Y ( 1 - Y) A--

Y

]

(9)

Y ( 1 - Y) 2 2 Y ( 1 - Y)

Xn=

[ Rt(n) Ct(n)

Xl=

[ Y Y ( 1 - Y)

]

(10)

]

(11)

The matrix A is diagonalized (see Appendix I) and Xn is obtained as follows:

Xn

=

[ [(l+x/~)an-(1-x/-~)bn]Y/2x/~ k [(3+',/5)an-(3-'/5)bn]Y(1-Y)/2x/~

where:

an= (1-rl") / (1-r,), bn= ( 1 - r 2 " ) / ( 1 - r 2 ) r, = (3 + j 5 ) Y ( 1 - Y)/2, r2= ( 3 - x / ~ ) Y ( 1 - Y)/2

]

(12)

43

Thus the following analytical solutions in each stage are obtained: Rt(n ) = [(I + V/5 )an - (1-v/5)bn] Y/2V/5

Rc(n)=[(l+~//5)an-(1-v/5)bn](1-Y)/2v/~

I

(13)

St(n)= [(1 + x//5)an - (1-x/~)bn] Y2/2x/~ Sc(n)= [ (I + x / 5 ) a n - ( l - ~ v / 5 ) b n ] Y ( l - Y ) / 2 V ~

I

(14)

]

Ct(n) = [ (3 + x f S ) a n - (3-x/~5)bn] Y ( 1 - Y)/2V~5 Cc(n) = [ ( 3 + ~ 5 ) a n - (3-x/5)bn] ( 1 - Y)2/2~/5

(15)

RCt(n) = [ (3 + x/ZS_)an- (3-x/~)bn] Y ( 1 - Y)2/2U/5 RCc(n) = [ ( 3 + x / 5 ) a n - (3-x/5)bn] ( 1 - Y)3/2x/5

(16)

It is found that these analytical solutions of each stage approach asymptotically their own steady state values with increasing the number of iterations n because r, and r2 are less than 1. DISCUSSION

Influence of kT and the arrangement of separation stages on the number of iterations Fig. 3 shows the mass ratio of recleaner concentrate at any iteration number to the one at a steady state as a function of the number of iterations with kT as a parameter, which was calculated from the equations derived above. The equation estimating the mass of the concentrate at a steady state is:

RCc= (1- y ) 3 / [ I - 3 Y ( I - Y) + {Y(I- y)}2] 1.0

I

O

0 0

0.5.

4D

OkT

" 0.11

, 2.30

~)kT

= 0.29

, 1.39

•

e~

00

I

1

•

() •

-j

•

(17)

I

2

I

3

I

4

I

5

kT = 0.69

I

6

I

7

I

8

I

9

10

Number o f i t e r a t i o n

Fig. 3. The ratio o£ recleaner concentrate to its steady state value as a function of the number of iterations.

l-I

These calculated results indicate that the number of iterations required to reach more than 95% of a steady state, which is called the critical iteration number in this paper, depends on the producl of k and 7'. For example, when kT' is equal to 0.11 or 2.30. :~ cycles are required, while 8 cycles are required t,> reach the same condition when kT' is equal to 0.69. Incidentally, as found from eq. 12, 0.69 is the kT-value which gives the maximum values o f t , and r:, causing the maximum iteration number. Moreover, kT-values other than 0.69 (fi~r example, 0. ] 1, 2.30, 0.29 and 0.:~9 ) give the smaller iteration number to converge. Another expression is given in Fig. 4, which shows the ratio of recleaner concentrate to its steady state value as a function of k T w i t h n as a parameter. It indicates the above results more clearly. When the calculated results are applied to a real flotation circuit consisting of three kinds of particles such as the locked mineral, the values and the gangue, it is supposed that the completely liberated values and gangue with the highest and lowest flotation rate constants, respectively, reach a steady state more quickly than the locked mineral with the medium flotation rate constant. Thus the critical iteration number seems to increase with the fraction of locked mineral. In addition, to understand the influence of' the arrangement of the separation stage on the critical iteration number, other flowsheets are examined in the same manner as the above on the assumption t h a t k T is constant in each stage. Three flowsheets, respectively the R - C - R C , R - C - C S and R - S - C systems, are taken up because they are the most typical circuits and widely used. The fbllowing analytical solutions on the mass of final concentrate are obtained in these three flowsheets as a function of iteration number n and kT.

1.o ------i

'

~

+

i

\

-

e ~_

n

F

o, o.o1

Ct(n) IRC-t(n) I I !!!Ill I 0.1

iii

= 9

-

St(n) I

1 i t Itlil 1.o kT

l

i I i IIII I0

Fig. 4. T h e ratio o f recleaner c o n c e n t r a t e to its s t e a d y s t a t e value as a f u n c t i o n of kT.

45 Rougher-cleaner-recleaner ( R - C - R C system):

RCc(n) = [ 1 - { (1 - Yr) Yc+ ( 1 - Yc) Yrc}n]Rec

(18)

where:

Rec= {1 - Y r ) ( 1 - Y c ) ( 1 - Y r c ) / [ 1 - { ( 1 - Yr) Yc+ ( 1 - Yc) Yrc} ] Rougher-cleaner-cleanerscavenger ( R - C - C S system):

Cc(n) = [1 - { ( 1 -

Yr) Y c ( 1 - Ycs)}n]Cc

(19)

where:

Cc= ( 1 - Y r ) ( 1 - Y c ) / { 1 - ( 1 - Yr) Y c ( 1 - Ycs)} Rougher-scavenger-cleaner ( R - S - C system )

Cc(n) = [ 1 - { Y r ( 1 - Ys)+ ( 1 - Yr)Yc}n]Cc

(20)

where:

Cc= ( 1 - Y r ) ( 1 - Y c ) / [ 1 - { Y r ( 1 -

Y s ) ÷ ( 1 - Yr) Yc} ]

where Yr, Ys, Yc, Ycs and Yrc are the residual mass fraction of a component with the rate constant k at time T in rougher, scavenger, cleaner, cleanerscavenger and recleaner stages, respectively, and are all the same from the assumptions (1) and (2). RCc and Cc are the mass of final concentrate at a steady state in the three flowsheets. The calculated results of these equations are shown in Fig. 5, which shows the mass ratio of final concentrate to its steady state value as a function of k T with n as a parameter. As shown in Fig. 5, it is recognized that the critical iteration number depends on k T j u s t like the result in Fig. 4. Besides, it is clear that the arrangement of stages causes the difference of the critical iteration number even if the number of the stages is the same. For example, in the R C-RC and R - S - C systems, more than 5 cycles are required to reach 95% of a steady state at k T giving the maximum iteration number. However, in the RC-CS system, only 2 cycles are required to reach the same condition. The above analytical solutions indicate that the critical iteration number depends on the functions of the residual mass fractions, that is, ( 1 - Yr ) Yc ÷ ( 1 - Yc ) Yrc, ( 1 - Yr ) Yc ( 1 - Ycs ) and Y r ( 1 - Ys ) ÷ ( 1 - Yr ) Yc, which are always less than 1 (see Appendix II). The required number of iterations increases with increasing the values of the function. The critical iteration number in R - C - C S system is smaller than those in R - C - R C and R - S - C systems because the maximum value of the function of R - C - C S system is smaller than those of two other systems as shown in Appendix II. In the R - C - C S and R - S - C systems, it is found that the functions are exactly equal to the total mass of the middlings which are the same as the ones recycled from new feed ( = 1 ), as recognized from the flowsheets in Fig. 5. This means

(!t ; n- 1

~

".

!

,

,,)

1_ ~"

-LJ

~-

" - ' - w ~ , ), /

" ~

"

"I

\/'"G\~

--

\ f"\.~-

0.5

L_.. ,_j

i

-[

ol

,

,

,

Ct(n)

RCt(n)

I

I I I

Ill

0.01

/

"]

n

i

ii

6

-

n=4 \,k--n = 3

",

I

Rt(n)

/

"

i

J

!

l

i

~ I,lJlIl

0.i

i

I I,llllJ

1.0

I0

RT

I

I.(

I

l I 1 ll~7-'t-"l~,l;

~."i'~[llll

CSc(n-l) ~"'-"~

0.5-

5 ~j

~n

=2

Co(n)

--

1

0

I

[

Rt(n) lllllll

O.Ol

0.i

CSc(n) [ 1 1111111 1.0

I

I 111111 I0

kT

1.0

0.5

,

Rt(n) 1

0.01

I,

~c(,~) I

o

.-.

,

~ \\

n :

5

r~ :~ .k--,~ = 2 • -n

: 1

t

Jl_LL

! -I

5c(nl

II

!I

O. I

1.0

~n

k'I" Fig. 5. T h e r a t i o o f f i n a l c o n c e n t r a t e to its s t e a d y s t a t e v a l u e as a f u n c t i o n o f kT.

47 that the critical iteration number increases with increasing the mass of recycled middling, that is, it becomes more difficult to reach a steady state. Essentially, the same conclusion is also derived in the R-C-RC system although the function is not necessarily equal to the mass of recycled middling. These calculated results agree qualitatively with the empirical fact which was already obtained in the locked cycle tests (Agar and Kipkie, 1978; Agar, 1985 ).

Comparison of locked cycle test and continuous circuit A locked cycle test is a series of repetitive batch tests and its purpose is to experimentally simulate the behavior of a continuous process as described in the introduction. It is therefore necessary to understand the relation between the locked cycle test results and the continuous flotation results. In the analysis of the continuous circuit (R-S-C-RC system), the following assumptions are made in addition to the assumptions (1) and (2) set in the analysis of the locked cycle tests. (3) Each flotation stage is composed of N perfectly stirred and isolated cells in a series. When a continuous flotation is carried out, the residual mass fraction of a component with the rate constant k is:

Y = ( l + k t ) -N

{21)

where t is the retention time per cell and N is the number of cells. (4) Each stage has the same number and volume of cells; therefore the mean retention time is equal in each stage. In analyzing the continuous circuit (R-S-C-RC system), the mass of final concentrate at a steady state is given by substituting eq. 21 into eq. 17. To compare the locked cycle test results with the continuous flotation ones, flotation characteristic curve analogous to Tromp's partition curve, which was demonstrated by Imaizumi and Inoue (1963), is introduced. It has already been known that the shape of the curve shows the separation characteristic of circuits and that the separation efficiency is improved as the slope of the curve increases. Flotation characteristic curves of the locked cycle test and the continuous flotation circuit for R-S-C-RC system are shown in Fig. 6, which shows the partition factor to the final concentrate as a function of k T with the number of cells as a parameter. The partition factor is equal to the recovery of a component with rate constant k and was calculated from eqs. 21 and 17. The flotation characteristic curve of the continuous circuit approaches steadily that of the locked cycle test with increasing the number of cells. Thus, it may probably be concluded that the continuous flotation results approximate to the locked cycle test results when the number of cells in each stage is large enough.

1.0

f I I I l llll

i I lllll

;-T f I i llJl ~

c~,c~.

I//'/x~\,

o~u

....

Y/A\\\\ -_J

0.5--

N=4 H=2

//

N = nun~ber of cells

f /

I I II IIIII

o o.oi

in each stage

I~t~lllll

0.I

i 1.0

I I lllll[

[ i0

flili

IO0

kT

Fig. 6. Comparison of flotation characteristic curves of the rougher-scavenger-cleaner-recleaner system for locked cycle test and continuous circuit.

SUMMARY Locked cycle flotation tests were analyzed theoretically. The conclusions obtained in this study are summarized as follows. ( 1 ) Analytical solutions were obtained as a function of k T and n from material balance equations at the nth cycle of the locked cycle tests. (2) Analytical solutions indicated that the iteration number required to reach a steady state (the critical iteration number) depended on k T and the arrangement of separation stages. (3) Analytical solutions showed that the critical iteration number increased with increase of the total mass of the recycled middlings. (4) The flotation characteristics in the continuous circuit were derived to approach steadily that in the locked cycle test when increasing the number of cells. APPENDIX I - DERIVATION OF DIAGONAL MATRIX AND ANALYTICAL SOLUTION

Matrix A is: A=

Y ( 1 - Y) ~

2 Y ( 1 - Y)

(A1)

In the following similarity transformation, this matrix A is diagonalized by giving an appropriate matrix U:

49 Jr, 0

U-1AU=

0 r2

]

(A2)

r~ and r 2 are obtained by solving the following quadratic equation:

{Y(I- Y) -r}{2Y(l- Y) -r}-y2 ( i - Y)2=0

(A3)

r, = (3+ x/~)Y(l- Y] )/2, r2= (3-x/~)Y(l - Y)/2

(A4)

From eq. A2, U- ' and U are calculated: I - Y (i+x/~5)/2 J l - Y (I-~/5)/2

U-,=

U=

[

(5-v/5)/10Y(1 -Y)

( 5 + x / ~ ) / 1 0 Y ( 1 - Y)

1/X/~

-- 1/X/~

(A5)

]

(A6)

Eq. 8 is expressed using U - ' and U as follows: U-IXn= (U-1AU)U-'Xn_ I+U-'X1

(A7)

n--I

= ~ (U-1AU)'(U-'X1)

(AS)

i=0

and: Xn=U

an 0 ] 0 bn U-iX1

(A9)

where:

an= ( 1 - r , " ) / ( 1 - r l ) , bn= ( 1 - r 2 " ) / ( 1 - r 2 ) Eq. 12 is obtained by solving eq. A9. APPENDIX II - CALCULATIONOF THE MAXIMUMVALUESOF THE FUNCTIONS The maximum values of the functions are obtained using the relation between geometric mean and arithmetic mean as follows: R-C-RC system ( 1 - Yr) Yc+ ( 1 - Yr) Yrc~ [{ ( 1 - Yr) + Yrc}/2] 2 + [{ ( 1 - Yr) + Yc}/2] 2

= yc2+(1--Yc) 2 =2(Yc-1/2)2+1/2 <1 where O< Yc< 1 for t=O or t = ~ .

(AIO)

R ('-CS system ( 1 -- Yr) Yc(1 - Yc,s~ < [', I 1 - Y r t +

Y ~ + ( 1 - Yc,~ i / 3 ]'

---- ~P(:l

< I

(All)

R-C-S system T h e m a x i m u m v a l u e is also g i v e n by:

Y r ( 1 - Y s ) + (1 - Y r ) Y c < [,', Yr+ ( 1 - Y s ) } / 2 l ~ +

[{(1-

Y r ) + Y s } / 2 ]'-'

= Yr-'+ (1 - Yr)'-' = (1 - Yc)'2+ Yc'-' <1 Thus, the order of the maximum Yc:~< ( 1 -

(A12) v a l u e s in t h e t h r e e s y s t e m s is:

Yc)Z+ Yc'2= Yr'-)+ (1 - Yr)'2< 1

(A13)

REFERENCES Agar, G.E., 1985. The optimization of flotation circuit design from laboratory rate data. XV Int. Miner. Process. Congr., Cannes, II: 100-111. Agar, G.E. and Kipkie, W.B., 1978. Predicting locked cycle flotation test results from batch data. CIM Bull., November, pp. 119-125. Imaizumi, T. and Inoue, T., 1963. Kinetic consideration of froth flotation. Proc. 6th Int. Congr. Miner. Process., Cannes, pp. 581-593. Macdonald, R.D., Hellyer, W.C. and Harper, R.W., 1985. Process development testing. In: N.L. Weiss (Editor), SME Mineral Processing Handbook. American Institute of Mining, Metallurgical and Petroleum Engineers, Inc., New York, N.Y., Ch. 30, p. 116. Taggart, A.F., 1948. Handb(rok of Mineral Dressing. Wiley, New York, N.Y., Ch. 19, p. 181.