derivation of a basic efficiency formula for concentrating operations

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Derivation

o

a Basic Efficiency Formula for

Concentrating Operations

E. DOUGLAS,t B.Sc., A.M.I.Mech.E., ASSOCIATE MEMBER

622.7.012.5

SYNOPSIS

A formula is derived which expresses the basic efficiency of concentration

when

the constituents of a two-constituent

mixture

are separated.

which transforms to

or to

C (c

f

(c

f

E = 1oo ~

c).

.

io0::-7)

x

loo

E

E

Rj

100c-])·

c f

. o o - = ~ J x 1 o

where C is the weight of concentrate, expressed as a percentage of feed weight;

c the valuable mineral contained in the concentrate, expressed as a percentage

of

concentrate weight; j the valuable mineral contained in the feed, expressed as a

percentage of feed weight; and R, percentage recovery.

f

c

and/

represent chemical assays, the last

term must

be modified to

c ~ a x

)f)'

where

Cmax

is the assay grade

of

a 100

per cent

pure concentrate.

This

basic efficiency formula is logically derived, is applicable to all grades

of

feed and gives consistent assessments of any one operation, whether calculated on

the basis of the concentration

of

values or of gangue.

By contrast, the limitations of the formula for expressing concentration efficiency

referred to by

F l e ~ i n g and

dealt with

in more

detail by Stevens and Collins,

are

shown, particularly"' n its application to high-grade feeds.

Examples are given demonstrating the use of the basic efficiency formula

in

sizing

and

dewatering operations.

IN A RECENT

PUBLICATION

2

the

inadequacies

of

the various efficiency

factors

in

use at the present time for assessing the efficiencies

of

con

centrating operations are discussed.

In

an attempt to produce an expression

which would overcome many of these deficiencies, Stevens and Collins

defined the following six basic requirements for any expression designed

to assess concentration operations:

(1) the number

of

terms required to make a quantitative assessment

should be reduced to a minimum;

*This paper is

Crown

copyright; it was received by the

Institution

of Mining

and Metallurgy on 6th June, 1962, and published on

6th

September, 1962; for

discussion at a General Meeting on 15th November, 1962.

tD.S.I.R. Warren Spring Laboratory, Stevenage, Herts.

etc. See references at the end of the paper.

697

Yy



698

E. DOUGLAS:

(2) simplicity: it is considered that there can be no fundamental basis for

any single quantifying expression and therefore such an expression

should be reduced to the simplest terms and no attempt should be

made to stress some part of the data more than another;

(3) any quantitative expression should be applicable to all concentration

operations and should present no obvious anomalies between one set

of results and another;

(4) the limits of such an expression should be (a) a minimum for any

sampling operation (i.e. nil concentration should correspond to nil

percentage efficiency of concentration) and b) a maximum for an

operation resulting

in

100 per cent recovery at

100

per cent grade.

Variations in grade

of

head should also be accommodated and the

assessment should be applicable to all multi-component systems;

(5) any efficiency expression compares the results obtained with the

maximum theoretically possible and does not attempt to measure the

ease or difficulty of attainment; i.e. no attempt should be made to

measure the operating variables but only the results obtained;

6)

no attempt should be made to include economic considerations in the

technical assessment

of

a concentration operation.

On the basis

of

these requirements, they fashioned an expression for

Concentration Efficiency * and one for a Concentration Index t. (See

also Fleming.

1

The effectiveness with which these expressions can be applied to

various concentrating operations, and their superiority over existing

efficiency expressions, is cogently argued. They are, however, formulated

on a completely empirical basis and as a result produce adequate repre

sentations

of

the concentration achieved only when applied to low-grade

feeds.

A new efficiency expression put forward in the present paper is a

fundamental and logically derived formula which subscribes to

five of

the

six basic requirements already mentioned (excluding (2)) and appears to

be completely unrestricted in its application. It also satisfies a further

necessary requirement, namely that when a two-consti tuent feed

is

treated

in one operation, the efficiency with which the first constituent is separated

from the second should be equal to the efficiency with which the second

constituent is separated from the first. As one operation only

is

involved,

the two separations must be considered together; one cannot be effected

without achieving the other.

*

and

t

Be =

_jc ·

x R

(emax -f

e R

le = - 1) X

100

where

e

is the percentage mineral

or

element

in

the concentrate; j, the percentage

mineral

or

element

in

the feed;

emax, the maximum

possible percentage mineral

or

element

lOO per

cent

concentrate);

and R

percentage

recovery.

DERIVATION OF A BASIC EFFICIENCY FORMULA 699

DERIVATION OF

THE BASIC

EFFICIENCY FORMULA

. For

t ~ e

purpose

of

this analysis

~ h e

concentrating operation is con

sidered m

tw.o

stage.s, a zero-efficiency sampling operation giving a

percentage weight spht

of

C and

100

- C, followed by the simultaneous

movement

of

the values from the tailing fraction into the concentrate and

the gangue from the C units of concentrate into t he tailings. The move:Uent

of

values governs the final recovery, while both this and the movement of

the gangue influence the concentrate grade to a similar degree.

Both

of

these

~ r a n s f e r

.operations can be expressed in simple terms.

The

extent to which each

IS

effected can be related to its maximum value

thereby indicating its efficiency, the overall efficiency being the p r o d u ~

of the two individual operation efficiencies.

Normally, concentration is assessed either by chemical assay or by

reference t? the percentage content of a particular mineral. In this paper

the latter

IS

used

as

the initial basis for deriving the efficiency

of

con

centration, while in a later section modifications which account for its

conversion to chemical assay are indicated.

The block diagram shown in Fig. 1 illustrates the three fractions to be

considered; the feed (ABDE), the concentrate (AFGE), and the tailing

(FBDG).

t

be seen that the concentrate fraction has been up-graded

from f to

c,

with the resultant depletion

of

tailings from f to t

A F

8

c=:::J

GANGUE:

= VALUE:S

IN

HE:AD

VALUE S

IN CONCENTRATE

c

•

t l o l u o b / ~ minflrol contoinfld in

hfl concentrolfl (os o pflrcflnlogfl

of

concrntroffl Wflight.)

f • Vo uob " minrzrol

contoinfld

in thfl

f f l f ld (os o ~ r f l n t o g r z of

frzfld

IWig/11.)

C

•

Wrzight

of

concflntrotfl

os a

przrcvntogfl of f f l f ld Wflight.

I Voluob/fl

minflro/ contotnfld

in thfl

toi l (os a pflrcflnlogfl

of

toil Wfltght)

Fig 1.-Block diagram

The relative proportions

of

the two mineral constituents contained in

any one of these fractions is indicated on the vertical axis while the

i n d i ~ i d u a l fraction weight, as a percentage

of

the feed, is shown on the

abscissa.

The p r o ~ u t of e r c e n t ~ g e

mineral content and fraction weight

represents the weight

of

the particular mineral contained in the fraction

u n ~ e r consideration. For example, the feed comprises

100

per cent

of

the

wei.ght,fper cent ofwhich is valuable mineral, (lOO- f per cent gangue,

while the concentrate contains c per cent values in C per cent

of

the

feed weight.



700

E. DOUGLAS

With respect to a sampling operation, the percentage weight of values

transferr ed into concentrate = C .

c

Total percentage weight of values available for transfer =

(lOO-

C)

1

{m

. . c ~ n

. . Efficiency of transfer of values =

100

=

C). --

f

--

X

100

1)

Percentage weight of gangue discarded from concentrate = C c ~ j l

Total percentage weight of gangue available for discard = C 1 0 ~ 0 ~

n

. .

( c -

f

..

Efficiency of gangue transfer = lOO___:./) X 100

:. Overall concentration efficiency= eq. 1 x eq. 2

c

( c -

f

( c -

n

= c 00 =cy ·c 00 - / )

x

100

2)

3)

When the concentrate weight, C, its percentage valuable mineral con

tent,

c,

and the percentage mineral in

the

feed, f, are known, the formula

gives

an

exact account of the efficiency of concentration.

It is

common

practice, however, for concentrating operations to be expressed in terms of

R

where

R

= c7 ,

c

and f; therefore by representing C in terms of

R,f, and

c

in eq. 3

we

have:

. . M ~ n

~ n

Efficiency of concentratiOn = 100c -=-:]) .

-

1

- . Too x 100

An alternative form, in terms

of R,

C,

c

and/, can be obtained

by

replacing

c in the second term of eq. 3 by c

7§

so that

.

R

- C c -f

Efficiency = 100 _

C).

100

_ f

X 100

4)

Chemical assays

When products are assessed by chemical assay,

c and/

are used to refer

to

the

percentage assay figures.

The

formula requires only a minor

modification, namely that the 100

-f

denominator of the last

term

be

replaced by

cmax

-f , where cmax represents the chemical assay which

would result from a 100 per cent grade concentrate. In this case:

Efficiencyofconcentration = ___ _S:

.

( c - f . ~ - / _ x 100 5)

100 - C) f cmax - f

DERIVATION OF A BASIC EFFICIENCY FORMULA

701

COMPARISON

OF THE TWO

FORMULAE

To demonstrate the inherent differences in expressing efficiency between

the basic efficiency formula presen ted here and the formula

of

Stevens and

Collins, theoretical examples are cited from the hypothetical mineral

balance shown in Table I

TABLE

I.-Balance

for the separation

of

constituents A a nd B from a mixture

of

A and B

I

. I

Constituents I I

Product

I

Wt. ~ - - - - Grade Recovery

Grade

Recovery

f

A , B A [ A

A_c_o_n_c_·-/-B-ta_i_l_.

___

6_i_ 4 ~ 1 2 1 _ 2 s __ l

A

tail/

B cone.

84 _

~ ~ 1·19

I 20

Head 100 5 I 95 I 5 . 100

I

I I I

S ~ 6

B

B

o;

75

12·6

98·8 87·4

95

100·0

I n i t ~ a l l y , the results in_Tabl e I can be assessed from two aspects; a) the

efficiency

of

concentratmg A from a relatively low-grade feed, i.e. C = 16,

c.=

25,

f

= 5,

R

.= 80; and b) the efficiency of concentrating B from a

high-grade feed, I.e. C = 84, c = 98 · 8, f = 95, R == 87 ·4. These give

the efficiencies shown in Table II.

TABLE I.--Comparisons of calculated efficicncies

Basis of calculation:

Concentration of

Concentration of B

into A tail)

Efficiency

Stevens and Collins

formula)

c f

Ec = loo-=- f R

16·9

66·5

Efficiency

Basic efficiency

formula)

C c -f c -f

c10o

-=-c)·

~ f

·

000

_ n x 100

16·05

16·05

For

low-:grade feeds the two formulae are in close agreement; this is

borne out m a l ~ the examples quoted in the Stevens and Collins paper.

However, for high-grade feeds there is considerable divergence between

the two formulae, 66 · 5 per cent efficiency resulting from

the

'Be' criterion

compared with

16

per cent

of

the basic efficiency formula.

Further

con

siderati?n

is

necessary in ?rder to establish which of the two efficiency

figures IS more representative of the actual concentration achieved.

A sampling operation, having the same weight split as that of concen

trate 'B' in Table I, will be associated with a 95 per cent grade, an 84 per

cent recovery and a zero efficiency.

. In the 'B'

c o n c e n ~ r a t e ,

recovery has increased from 84 to 87 · 4 per cent,

I.e. 3·4 per cent_umts out

of

a possible 16 2? per cent), and the resulting

grade shows an mcrease of 3 · 8 out of a possible 5 per cent units enrich

men_t ratio 1· Such a result does not seem to warrant a 66 · 5 per ce nt

e _ f f i c i e n c ~ , p a r ~ I c u l a ~ l y when compared with the 16 per cent 'A' concentra

tiOn

efficiency m which the recovery, over a simple sampling operation, was



702

E. DOUGLAS:

increased by 64 per cent units out

of

a possible 84, with a grade increase

from 5 to

25

per cent (enrichment ratio 5).

It is estimated that a

66

· 5 per cent efficiency from the low-grade feed

concentration would require a concentrate grade of 66·1 (enrichment

ratio 13 coupled with a recovery of88 per cent (an increase of72 per cent

values out

of

a possible 84 per cent). This

is

considerably more impressive

than the 'B' concentration operation for which the lower efficiency of

16 per cent must be considered more realistic. Also, it is in exact agreement

with the calculated efficiency of the same operation based on the concentra

tion of

'A'

from 'B'.

Finallv, with the basic efficiency formula it

is

possible to operate on a

'concentrate' which has been depleted in values and still achieve a con

sistent result-for example, 'A' tail (Table I) is considered to be a

concentrate depleted in values, where,

c

= 1· 19, 5, C 84, R 20.

Efficiency, according to the basic efficiency formula, is

16

per. cent,

indicating that values have been rejected, or gangue concentrated, with an

efficiency which is in complete agreement with the two alternative assess

ments of the same operation.

The

formula discussed by Stevens and

Collins, however, would produce a third and different efficiency, of -0·8

for the same operation.

The difference between the two formulae lies in the use

of

recovery in

the 'Be' derivation of efficiency, as compared with the 'value concentration'

C c

-I

· h b ·

ffi

·

fi

1 R

term, (l

00

=-C)

~

, m t e

asiC

e ciency ormu a. ecovery,

r e p r ~

senting the ratio of the values in the concentrate and the total

valu.es

m

the feed, gives undue credit to the proportions of the values automancally

extracted with the concentrate weight by a zero-efficiency sampling

operation; the basic efficiency expression accounts only for the values

concentrated additional to those of a sampling operation. With low-grade

feeds the difference

is

small, but as the grades of feed increase, the use of

the standard recovery term introduces a progressively increasing bias

which results in inconsistent efficiency figures (see Fig. 2).

X

_ ._

x

L . - ~ [ J . . . , . 0 - 0 - - , 2 0 ~ 0 - . , . . , - - ~ / 0 - : 0 - D l 7 Z - - 5 ' : Z - - J ~ o - - r - - - - c c : : - - ~ / 6 855 lit No

50 200 500

/000 c p ~ r t u r r

microns

reqvir s i z ~ o s . ~ p a r o t i o n

Fig. 2. Efliciency of separation at various sizes.

DERIVATION OF A BASIC EFFICIENCY FORMULA

703

Acknowledgement. This paper

is

published by permission of the

Director, Warren Spring Laboratory.

REFERENCES

1.

FLEMING,

M. G. Selectivity factors

in

flotation.

Chem.

&

Ind.,

Oct. 3,

1959, 123Q-8.

. Discus_sion on

paper

by KUN

LI,

R. W. LIVINGSTON and L. K. LEMKE.

FlotatiOn conditioning of iron ore

with petroleum

sulphonate.

Trans. Instn Min.

2

M

0

etall., Land.,

70, 196061

(Bull. Instn Min. Metal ., Land.,

no. 650, Jan. 1961),

6-7.

2.

SrEVENS,

J.

R.,

and CoLLINS,

D. N. Technical

efficiency of concentration

operations.

Trans. Amer. Inst. Min. Engrs,

220, 1961, 409-19.

APPENDIX--APPLICATION

OF

FORMULA

The

b a ~ i c

efficiency formula can be applied to any form

of

physical

concentratmg process and examples of its application are demonstrated

below, using results from cycloning and sieving operations.

Cyclone: desliming

. T h ~ w e i g h ~ b a l ~ n c e shown in Table Ill represents a deslirning operation

m which a 1-m diameter cyclone was employed. A 24 JL cut was required.

TABLE IlL Results of

desliming by cyclone

Product

Overflow .

Underflow .

Feed

.

Wt.

-241'-

2 4 ~ t

feed

Wt.( ) Wt.( )

2·4 93·8 6·2

97 . 6 2 . 6 97 . 4

100·0 4·8

95·2

4 · 8 per cent of the feed was smaller than the required 24-p. size limit;

2 · per cent

of

the underflow, which comprised 97 · 6 per cent of the feed

weight, was finer than 24 L· Considering the + 4 JL material

as

the con-

centrate, C = 9,_ 6, c 97 · 4, 95 · 2, from which E 43 per cent.

Cyclone: classification

T ~ e

result? used

in

this particular example (shown

in

Table IV) were

obtamed durmg a cyclone classifying operation in which it was necessary

to make a size cut at 72 mesh.

TABLE IV.--Size

distribution

of

cycloned products

Size

Fraction,

B.S.S.

- 16

+25

+36

+52

lOO

150

+200

-r300

Feed

Underflow, Overflow,

Cumulative Cumulative

retained retained

0·2 0·0

21·0 0·0

34·1

0·3

48·8 1·1

63·5 3·6

80·2

10·4

91·10 23·3

95·28

36·1

100·0 58·9

62·7

37·3

Feed,

Cumulative ~ : ,

retained

0·18

13·13

21·57

31·0

41·16

54·12

65·81

73·22

84·69

100·0



704

E.

DOUGLAS

Either - 72 mesh or + 2 mesh can be designated

as

the concentrate.

With the latter, C = 62 · 7

c

= 63 · 5,

I=

41· 16 and 34· 6 per cent.

In a similar manner, the efficiencies at a number

of

sizes were calculated

and plotted

as

shown in Fig. 2. I t can be seen that the sizing obtained in

this particular operation is too fine

and

that

the most efficient cut has been

made at approximately 150 mesh.

For

comparison, the efficiency curve derived from the Stevens and

Collins formula has also been included in Fig. 2. I t demonstrates its

divergence from the basic efficiency formula as the grade of feed increases.

Sieving

The results shown in Table V indicate the effectiveness of an industrial

lOO-mesh screening operation.

TABLE

V . Industrial

si ·ving

using 1 00 mesh

B S

si ·v

Product

Oversize

Undersize

Feed

.

Wt.

-100

Wt.

( < ) ~ )

15

9 ·3

85

96·6

100

83·5

In

this case, C = 85,

c

= 96·6, I= 83 · 5 and = 70· 5 per cent.



281

DISCUSSIONS

AND

CONTRIBUTIONS

Derivation

of

a

Basic

Efficiencv Formula for

Concentrating

Operations

E. DOUGLAS, B.Sc.,A.N1.l.i\.1ech.8.,SSOCIAIE trIEIIBER

Report of disLttssion t Noacmber,

1962, General Meeting

(Chairnan:

,Vr.

J.

B. Sinpnn,

Prcsidcnt).

Paper

publihed

in Septufiet, 1962

('l:ans-

actions,?rol.71, 1961-62),

pp.

697-701

Mr.

E. Douglas, prcscntiilg his paper,

first apologized

for a con,

sidcrable

eror

which appearedr

the

sccond

ormula of

thc

synopsis

(p. 697). The lirst tcrn ofth,,- denominator rvasprinted as 100c

I

that

should have rcad l00c rt/. Thc samemodilicarion

shou)d

bc

made on

page

700.

Continuing, hc said his

paper,

originally intended

as

part

of

the

discus-

sion to

the

paper

b1' Stevens and

Collins,

published

by the Amcrican

Institutc of Mining, N{ctallurgical

and

Petroleum

Engineers, rva:

con-

scquently concerned

to somc extent with

comparisons betwecn the

formula

vhich

hoseuorkcrs had rccommendc.d

nd the one dcscribecin

thc

presentpaper

as

a

basic

ormula for

cxpressing oncentrationefiiciency.

In dcriving the formula, no attempt had been made to introduce

economicbiasc's

or had

probabilities

or intermcdiate ratcs

of

concentra-

tion bccn introduced. Thc sole

conccrn

had

been. bv exaninins the facrs

associated

vith

the

fcccl

and rr,ith the

oroducts.

to aisess he

edcctivenes

rvirhuhich

thc con.titucnt.ofa mixturehad

hccn

rcparated.

For

simplicity thc concentratingopcration had been considcrcd

n trvo

stages-a simplc u,eight splir

giving

two

products,

in each of which the

constiruent

atios

wcrc identical and equal o thosc n

the

feed. ollowecibv a

simulrancou' nlcrchangc fonc eon.titucnr

ronr one

ploduct

rrrt ir

t irc

othcr constituent rom the orher product.

A t\\ o-constituent

feed

was illustrated bv

the block diasram shorvn

n

Fig. I

r1:l.

99).

l 'hclc the constituents ai bccn de'ignatcd n common

telms

as

values

and

ganguc,

and the initial wcight split

was

represclted by

the

inrermediate roducts

AFGE and FBDG. cach

containing

per

cent

values.

At that

point

no collstirucnt

separation had takcn

placc

and

consequently hc conccntration efliciency would be

ratcd as

nil.

During

the

second stage,

values

*'crc

movcd fiom FBDG into thc conccntratc

AFGE, thc

ganguc

moving

in thc opposite

direcrion,

with

thc

resulting

product gradcs

of c

pcr

cent and I

pcr

ccnt. Thc eilicicncy rvith which the

valucs had

be 'n transferred

could

bc

exprcssedas the

wcight of valucs

which nroved nto rhe

concentlrtc

ditrdcd bl

thc

ueighr ofialues

shich

were availablc or movcment, .c.

( c

f ) e

iioo

c17

loo'



28 2

Similarly,

the

efficicncywith which

the

gangue

u as.rcjecred

would bc

thc

wcight

ofgangue

ejecied

ivided

bl

thc

weight

availablc

or re)ectrorl '

e'

-(,'

.L).9._

too.

1 0 0

J ) c

Thus

it was

possible o indicatc

the

individual

eliciencies

of the t ' 'o

oDerations

whiih

iDfluenced he

conccDtration

proccss' They

could

be

cimbincd

in a number

of ways,

each

of

rvhich

would

adequatcly

cprcsent

thc

olcrall

eliciency.

For eximllc.

it could

be

cxprct'ed

a' the.arirhmetic

-.""

ot rl.t"

geomeiric

m"an,

& et

en morc

simply

as

the

product

of the

i*no .t-t.

l-i"

ptoduct

combination

had been

uscd n

deri ing

the

formula

orcsented

n the

paper.

'

Scveral

variationi

could

hc applied

to

the

general heme

used

in that

dcrivation; for examplc, a sliglitly more logical, but considerably

more

t

= C o _ t . o

f o )+ c b ( c b

- t b ) , C g ( . s - r s )

x / o o

( /OO-

Ca)

a

+

(OO-Cb) fb +( /OO

Cg) ts

Z C ( c

' f

)

- - - x t v v

>:

f

(/oo,c)

FiE.

4.

complicatcd,

cpresentation

ould

be

had

by

introducing

a

actor expressing

ihe

iate

nf raiuc" conccntratio

$ilh r('pcct

to

gangue eicctiolr '

An

alternative

ut almost

dentical

reatment

ould

bc

appJied

o

producc.an

efrciency

formula

to

givc directly

rhe overall

efliclencl'

ot

an

operatron

from

which

sevcral conccntrates

vcre

produced. One

o[ the

two block

diagramr

n

Fig. A

abore

cpre.cntcd

hc

fccd anJ

the

olhcr hc

product'

ui inna"nt

rtio"n..

A\.uminAth(

eiicicner

of collcctltralloD

o

be

Total

weight

of

mincrals

coircenttatcd

vith respect

o a

\amnllng

qunlat'on

' .

- ,

'

l u \ '

"

l c ta l

r tc tg l r t

[m i t r t ta l '

a \a j l r t ' l c

or

conccn l ra t lo l l

t t tn

resPcct

o a

sanpling

oPcration

B^src

EFFrcrENcy

oRMULA_DlscussroN

293

thcn

r

Ca

i1

fa)

Cb

b

tb,

(

f

,..,

li ,

t00

C. )

/ , ,100

cb;

73-

oo L . ,1 '

t uu

2 C ( t

I l

r ^ ^

:

Tqtoo

cl

ruu

(l )

For

a two-constituent

eed,

or uhen

the

conccntration

f onlv

on e

mineral

of a mulri-mineral

fccd

u'as

considered,

he eliiciency

crpr"s.ron

became

.

2C(c

f )

- / i 0 0

C r

L r 0 0

I

"

The appendix to thc

.papcr

demonstrated he use of thc efficiency

lormula

by

cxamples

electerJ

rom

indcpcnrlcntpractical

opcration".

In

one

instance r

had bcen

used to

deternrinc

thc cur sizc

in'a

cvclone

operation

that

had

already

aroused

discussion.

.He

had.attemptcd

o

devise

a form

of equation

which

v,as

simple

and

wlich )ogically

and

con\istendy

rcprcscnted

he

concentration

efficiency

ol

an]-

phy\lcal

conccntr3ting

rocess.

t

gat.e

zero ering

or

a.ampling

operalron

nd a

maximum

100

pcr

ccnt for rhe

total

"eparation

i the

consutuents

n

a mlxture.

_ {r.

T. H. Hughes*

read

the following

contribution

from Dr.

A.

J.

KOOrn50n.

Dr.

A.

J.

Robinson:

Thc

imporrarlce

of an efficicncl,

ormula

rvhich

rs

acceptablc

to

englllcers

and

rcscarch

workers

can hardly

bc

ovcr_

emphasized.

t onc

rime

or

anothermost

enginccrs

vorking

n

the ielcl

f

mineral

tehnolog

cel rhe rccd

or

an cfficiencl

ormula

hiclt s

simplc

to use, vhich

requircs

only that

inforrnation \r'hi;h is nornrallv obtainecl ntcst work and yer

s

ba\ed upon

a sound

heoretical

concept.

i is mv

bclicf

that Mr.

Douglas\

formula

meets hesc

cquircments.

t^is

oftenuieful

o

have

a clear

mental

imagc

of

cause

and effect

u.hen consjderinq

a unrr

operation

but

in

some

cases

hc usc

of

an elliciency

brmula

cliurls

the

mental

p(ture

and

may, in

fact, make

intcrprctaiion

of

performance

orrrculf.

under these

clrcumstances

an eficiency

formula

docs

more

harm than

good.

^

Hort'cve

, thcrc

ma1'be_ o

alternative

o the usc

of an

emciency

ormula

tor

asses:ing

he rc:ults

ofan

a,J ior

series

ftests.

fhis

is

particularlv

ruc

in the,ficld.whcre

herema1,be

nsufllcicnt

metallurgical

taffto

cariy

ou t

carelutiv

derlgned

expenm(nts

becausc

of

thc

prcssur.e

f routine

metallurgical

control

and

accounting

duties.

Under

thcsc

circumstances

t

rs not

uncommon

for test \r'ork

to

be

donc on an,as

and

when,basis

and

for tcsts

to procecd

from ouc good

idca

to another.

Consequently,

t is

iscnior

exprr imcnr

rl

ofirucr ,

D.S.LR.

\\ urr(n

Snnng

Lrqof.,rory.

E. DOUGLAS:

DERIVATION

OF A



284

E.

DOUGLAS;

ERI\IATION

F

A

not

al$'ays

possible

to

detcrminc

precisell'

the cffect

of

changing

the

operatingvariables; nd

an eiicicncy

lbrmula which

is

gcncrally

ecognized

ollers a meansofassessment vhich would

be rvidely

acceptable.

oo early

emphasis

of

cconomic

considerationsnray

mislead the investigator who

attaches oo much importancc

o

either recovery

or

gradc,

and

mav mean

that

a

promjsing

routc is abandoncd.

By

using an

cfficicncy formula

incorporating both

rccovery and

grade

actors herc is less chancc of this

nappenrng.

In

my

orvn

cxpericnce

on

dilicult flotation problems. vhercselectivitv

s

poor

and

simultancous

cjcction of more than one constitucrlt s attcmptcd

in

one

operation,

o

be

followcd by

further operations or

the

rccovcry of

valuc, he asscssncnt f

pcrfornanceposes

crious

roblens.

By makjng

assumptions hat weight

/recovery*

curves should be straight

lines, the

assessment fthe

eficiency

ofvarious flotatjon

procedures

or

the various

constirucntsma) be madc using he slopeofthe respr-ctiveccovervcurves.

lJnfortunately

(his

assessmcnts not

neccssarily

rue

and

the slope of thc

various rveight rccovery curvcs for lhe various

minerals present

cannot

renrain

constantovcr

the rrhole recole$r ranse where collection occurs al

different collector conccntrations.

n

such cises rhc use

of

an

cfFciency

formula

can

be

extrcmely

valuablcand fufihelmore

may

saveexperimental

time in amenabilitv estins. The mathernatician nd

rcscarch

cienrist

would

correctly

reio-merld &ctorial design to

truly

assesshc

effccts of

variables,but bcfore his ma.v ven

bc

attemptcd some

preliminary

sorting

ofavailable routes,

particularll'

in flotation,

nrust

be

madc;

herc the use oi

an efficicncy ormula

is

invaluable,

Another

problem

arises n the statisticalanal-vsisffactorial experimentsl

rvhat s

to be uscd to assess

hc

results?Appropriatc use

of

the

cfliciency

lbrmula

rvhich combines

both

'rccovery'

and

'grade'

providcs

a

singlc

measurcof

performancc

which lends tself to matircmaticalanalysis.

Mr. J. R. Stevens said that the author suggested hat conccDtlatron

operations

could be

assesscd

y a fundamental

fbrmula.

Ho$'ever, his

derivation of such a

'fundamental

and

loeicallv

derivcd formula'

rvas

conspicuous

by the

absence

o[ fundarnenrri conirdcrarions.His

forrnula

rvasbased

on

the

concept

of

mass

ransfcr

across

an

imaginary boundary

between

he conccntratc ,\,'eightnd thc tailing weight;

gangue

ransferring

to

thc

tailing aud values

of

cqual weight transferring

to thc concentrate.

Such

a processhad

no

parallel

in

mincral

processingand

he

found it

difficult to apprcciate he logic of such a conccpt. Hts interprctatio[

of

the

author's conccpt was

try

nfcrcnce,

he

admitted, but he did not thinL that

anv

other interDretation

could bc

made.

ilefercnce hail been made o a recent

Daner

n which Mr.

Collins and

the

speaker

ad dciined

ncw

erprcssionr br-thc asser.mcnt f mineralcon-

centrating processes.

l'herc

rvas

another

publication

rvhich

antedatcd

he

*\\'eight

rccovcry vcrsus value rccovcry,

BAsrc

EFFIcIENcy

oRMtTLA_DrscussIoN

295

tcfcrcnccs

in thc

author's

list

and rvhich

might

uscfully

bc

includcd_

Harrvell.Re'earch

loup

Rcporr

\.8.R.F. R2022,

ared

Augu.r,

lq59'

rnar

d(art $rth

thc.arne

\ubject

matter n morc

detail.

.

Rcfcrring

to thc aurhor,s

six

basic rcquirements

(pp.

697-g),

hc

still

belicved_rhc

ccoIJ

requjrcmcnr

o be

.alid

in

facr he

only

mo'Jificarion

he r'ou.ld

r.ggest

r

a\ that t

shoulJ

be rnrierlincJ.

t

$.a,

bqqsu.q

hslg *r,

no

undane_nt,tl

aris

lor

any, nngle quantifling

erpres\ion

hat

tL" cUim

made

by A1r.

llougla\

Dcrired

consideration.

When

referriDg

to

dili_

culr]

requrrement

ir was

not

meant

o

imply

that

,work

in a closed

sy.stcm'should

ot

bc uscd

asa basis

or

any formula,

but

that

thc

practical

ditficulLies.

.e. he'exrernal

\.ork'.

hould

not be

nclud<d.

freouirements

5 anJ

6

(r'ele

mct, r.t.

thc Jiliieulry

of

attainmen(

n(l hc

econom,e

urr_

sidcrations

wcrc not

included

n a

formula,

then

t shoukl

not be

surprisrng

ifthat eltrciencyormula gave esuhs

rvhich

werenot alrvavsn accoi rvit[

onc's

mcnral

asscssmcnt

fthc

cflicicncv

fa

gircn

opcraiion.

hc rcason

for

such anomalies

vas undoubtedly

due

to ihe

unionscious

and incon_

slstent

use

of thc vardsticks

of

difhcultl'

and economics,

whcn

mcntally

comparirrg

sets of

iesults.

The c,tn6sp1f

conctntrattlou

ficicuci

iis

unat

detetmined

he deiaed

equ.rtion

td

fheri'uere

as

many

iissible'formulaeas

ther.c

ere dc.finitions.

e

felt that

N{r,

Douglas

gave

no adequate

verbal

basis or his

efficicncyerorcssion-

The

fact

that the proposed

basic

_efticiency

ormula

gave

numerically

equal

results

or

the conccntration

of valucs

n

a conccniratc

and for

the

conce,ntration

fgangue

in

the taiiing, for

a

giten

operation,

vg:

nor

proof

that fundamcntal

clarionship

ad

6ccn

dc.cribcd.

Orhcr formulac'rrcrc

po\\ible

rhat

coulLl

l.o do

that. For

exanrple.

he

ormula

of

Luyken

and

Bi<rbrauer'.t_n

ch

$a\

1.,r,.\

r

thc formula

ol

van I:bbcnhors't.

cng_

bergen

anJ

Rietenra.

lneidenralJl-rhose

ould

be

.hown

to

be identical

formulae.)

I('hen.the

components

of a t|o-component

mixtufe

were

complelell

separated

Jron

6.11

other

thcrc rl,as itilc

doubt

thar

thc efficicircv

oi

scparation

vas

equal

for both

coD)ponents,

nd

in that

respect

he ag;eed

with.Llr. Dauglas. The formula for conccntrationcdicicniv (Er) rias in

accord

l,ith

that vicw;

i.e.

rvhenan

operarion

achiered

100

per

cent

grarle

and

l00.per

ent

recoverv.

he concentration

licicncy ,rarihe

a,,

ii,u-

tnat

coutJ bc atlarncd.

nd

wa.

100

per

ccnr rvhichcver

omDonenl

as

considered.

The denominator

of

the formula

for

concentration

efficiency

(Ic)

rvas

}

(l

/r),

and

that

was

a constant or

any

givcn

mixturcl

rr

representcd

omplctc

scparation.

The proposed

basic

eliciency

formula, however,

u,as

basetl

upon

the

assumptlon

that ellicienc)

of concenlrction

u,as

also equal.

for

each

component,

rvhen

scpararionswcrc

incompletc.

There

wai no

basis

for

_ rS tF \ l Ns . . 1 .

R . . . r n d

Cu l r r r . r . D .

N .

N(w c x p rc . . ro l h

f o r

(onc c n r r r l ron

cmi lcn.y

rnJ

c.nc{Dtrr ' , ,n

jnJ(x

in

rhe i l \ \{. .m.nt

ot mincrrl cunL

nrr:rt ion

processcs.

U.K.A.E.A.

l lc\errch

croup

Reporr

R2922,

Harwerr.

tgtrj .J i

n.

-

,

tI-y]-]<I\

\v., an.l

BTERBR\LrIRj

-

Crlcul.rrions

in

orc dr(\sinq.

Trans.

Atner.

Inst.

Min. Enr:rs,8?,

lg30

(Mi l l ing

mcthods),

429

51 .



286

[,

DoucLAs:

DERtvAT]oN

F

A

such

an assumption. On the

contmryr

considcration

of the

cnergy of a

two-component

system cd to

the

conclusion

that

oDly when completc

separation

was achicved was

the useful

work done

on each componcnt

equaI.

'Thc

technical

efliciency

of a

concentration

operation could

be derived

using

gravitational

energy

as a[ analogueand

assuming

unit activity

for

the

comDonenlc

f

a two-component

irture.

Ijscful \t( ORK done

T1ASS

POTENTIAL

DIFFERENCE

Eiicicncy

=

usirut

wonr"*-

*

luass

>.

PofEN-iAi

DIFFERENCE).,A"

,Vlass fValues in Conccntrate >r :J[Values]

Ila.r o[

Value. n

Conecntrltc,,..'

J[\'aluc.]'

., r

C c .

( .

/ r ) .

; , . ; , ' ra '

r lcc imal

ract ion)

. . ^ t a t I )

or

al tcrnarr \c l ) (

{ lO0 , )

pcr

cent .

Thus concentrationefliciency

Et

had

bcen

derived

n

tcrms

ofuseful

work,

i.e. work

done

on thc useful

compolent) which could of course

bc

either

component by

choice.

(It

should bc

noted that for a two-component

system he

maximum useful

work

was

a fixcd

quantity,

br a

given

mixture.)

Thc author's

proposed

ormula for'basic cfficiency':

E .

t '

/ ' l

..1^

1

. roo

("q.

;

, t 00 , l 100 c

'

(.

t,,la. ar\

could

bc shorvn o coual

'.

.

"i

i.,

' (a'

dccimal raction).

'

( l

h ) l h

( n )

'1[Valucs]. :l Massof Values)

T h c r c f o r c ' b a . i c e f f i c i c n c ] ' , , , ;

-,

'

alue.l"

*

. /

(Ma:.

qf

Vslus\)'

''

There was surely

no

analogue

or

the

author's orm ofcfficiency

expression

and

he

suggested

hat

thc'basic

cmciency' brmula was empirical

while the

formula for

concentration mciency

Er)

was

fundamental'.

Morc seriously

still rhc denominator of thc

basic cfficiency ormula containcd

a

variable;

that

was

untcnable,The

denominatorhad dillerent

values or separations,

of

the

same

mixture, that

rcsulted

in

equal

grades

of

concentratesbut

dillering concentrateweights. Thus

it was implied that the

maximum

uscful work that

could bc done

n

completcly separatinga

gi\'€n

mixtur€

was

a variable. He l clt that

-\'Lr.

Douglas

should look

a littlc

more

closely

at the implicit assumption

n

the bald

statemcnt

given

for efliciency of

transfcr of valucs

on pagc 700,

inc

4.

For

all the

cxamplcs

given in

rhe

paper,

the formula for concentration

cfficiency,Ec,

gave

he

cor(cct cvaluation

n

terms

of u:eful work

achicved

comparcd o the

maximum

possible

useful work.

BASTC

EFFICIENCY

FORMULA_DTSCUSSION

287

M1. D.

N. Collins,

before

dealing with

the applicarion

of basic

and

technical

efficiency ormulae,

referred

to the

simllarity

betrveenvarious

formulae

commonly uscd

for

measuring

rcchnical

eficiency,

and their

significancc.

All such

formulac

could bc

presented

n the

form ofR

(per-

certage recovery

of

values) and

J6

(pJrcentage

rc;rc(ion

of

ganguc).

The relationships,

hough

n ditlireni

symboli, u.ere

unmistakibly'the

samc

for

a two-component

systcm.

Gaudin's Selecrivity

Index: the

- R . . 7 b

square oot

ot

/ r^^ -,,;" ,a\

appeared

o bc unnccessarily

ompli-

n,, I uu

-lt)

cated n that

it

gave

a double $'eighting

to

both recolery

and rejection

terms, but

seemed o

give

more reasonablc

csults than

cirlicr formulae

and was

also the f irst

aftcmpt at

a non-linear

relationship

bettveen

he

rwo rerms.

Thc tcchnical cfrcicncy

uas thc

first cfrciency

formula

to use a

non-

linear relationship

and

basically t

involved modifying

the

rejection erm

to

accommodate

sampling

operation-

R

:

(Jb

000

c-))

Lc

(J&,,,:.

(roo

c,

where C

-.

per

cent concentrate

weight.

The

rejection

erm nas

soconstructedthat

anv sampling

opcration

\rhcre

Jb

-

100

-

C) would

result n nil

per

cent eliciency

and

that for a

perfect

separation

he resultant

eliciency

uould

be

100

pei

ccnt. Ir also trad

he

distinct advantage

hat

for largc

values of

C and hence

low

grades

of

concentrate,

the

Jb

term would

be considerably reduccd.

Hincc

the

e-fficiency

crm lras }eavily pcnalizcd

for lort

concenhate grades

and a

differentialuas made

heu'een

a

good

scalping

operation

and a

good

conccntratton

operatron.

Ba.ic

tfitien*

c

Jb

loo

c

'

-

R, ,

,

C

.71'," ,

(

0o

Ci

The only

differencc

bctwccn E'r

and E was hat

n the technical

elliciency

for':rula R

was an absolute

erm rvhereas

n the

basic efticicncv cxoression

it

was also modified

to

account

or

marerial rhar

would be

reioveied

by a

direct sampling

operation.

Ho$'ever,

though

a mineral

dresse did not

evaluatehis

results n

term," of horv

much

gangue

materialhc had throu,n

away,

but rathcr on

the

grade

of

thc

fina1

product,

he did

evaluare

hem

directly on

his recovery

igures,a

d

from a

practical

standpoint

R x rc)ec-

tlon telm

was more realistic.

-The

pre.ent

author had

also

pointed

out, quire corccllv,

that the

basrc

efficicncv ormulc gavc

the samc eflicicncy for

the conceitration

of

both

valuable and gangue

minerals. For

rhat matter

so did

previous

formulae

and_it vas

with that point

in mind

rhat he

now turned

to thc

practical

applications

of the two formulae.

A

convenient

xample

was

har

givcn

by

thc author n

his evaluarion

f

cyclone

roducts

as shown

n Fig.

2 and

fable

IV on

pages

02 and 703.

'I

he speaker

at that

point

wished

to co[ect one

fieure in

Table IV.

The



28 8

E. DOUGLAS:DIRJ\' iTION

OF

A

o

B o s i c

e f f i c i e n c y c u r v e

6

+

Te c h n i c o l e f f i c i e n c y

f .

( f q r

o v e r f l o w

m o te r i o l )

x

6 c

( o r

u n d e r f l o w

m o r e r r o

o

4

l o r e ,o q e )

,-x-^-x

100

2 0 0 5 0

t 0 0 7 2 5 2

REOUIRED

IZE OF SEPARATION

(8.S .

me h)

BASIC EFFICIENCY

FORMULA-DISCUSSION

289

concentration.

t was highly

likcly that the

separation

mesh

was chosen

bccausc

matcrial liner

than rltat

size would

no1 respond

to

that form of

separation and

rvould also

affect

the

(efaration

oi the coarsc

matcrial,

Thus, the

prcsence

f nearly

20

pcr

ccnt incs,

vhich

was he

analvsis f

thc 100

me.h

frodrrcr.

$a\

un.ali.facror]..

Their optimum

cfficiency

for that separarion

was at 200

mcsh whcre

81.6

per

cent

ofthe 200

mcshmatcrial

was

ecovered t a 95.3

ner

cenr

gradc.

-The

basic

cfficicncy formu)a

appeared o

be a measureof

thc average

efficiency f

th(

separalion

hich

incidcnralh

could be obrainrd

by rhe

addition of rhc

technical

elliciency

for both

coarsc and {inc

sizcs. The

resultant curve

for that fbllowed

thc conrour

of the curve in

Fig. 2 very

closcly but

it

rvas

of slightly

higher magnitudc

than

his

(see

Fig.'B).

His secondcxamplc was of the treatment of a glasssand.The sand n

the

following

metallurgical balanceu'as

bcing

uscd for

bottle

making and

must not contain

more

than 0

25

per

cent FejL).J.

, , , , . t . , ' , ,

Fe )o , . ' , , , . r ( r ,

J : t " t ' i b . ; : i :

Conccntrxtc

5.9

32 2 6?

I 9i

O

4. 1

t 0 0

>_a;

i t i o

Q =

I Q

3 6

25

lo9

p

sccle)

Fis.

B. Re/atinshit

b.tlten

technicdl

,rnd bu:ic e.lftientic:

far

a

claxillcat"'t

r f.rdti

"1 .

I

300

mcsh retained

n thc c,vclonc

ndctflow should

have

read

97

8 and

not

100

pcr

cent as

had

been isted.

That

madc

a considerable

iffcrence

to the

Et

currr-

in

Fig. 2 (st-e

Fig. B abovc).

l'hc

100

pcr

cent

had bccu

assumed,

nd

he ha,l obtained

he 97'

3 from a log-log

plot

of the cl

clone

underflow material.

Furthcrmore)

the

E

curve

shown

in

Fig.

2 was

an

analysis

of cumulatilc

per

ccnt

coaner

material

and representcd

he

cfficiencv of

separation

tnto thc

underflow.

The elliciency

curve

for

seDarati;n

nto the

overflorv,

basedon

cumulative

pcr

cent finer material>

ni. ofa

difl irent

form

.cc

Fig.

Br .

He thought the

oplimum

cfliciency

or scpamtion

of ovcrflow

material

should

be

d-iffcrent

iom thc optimumellicicniy

for separatiotr funderflorv

material- Assuming

hat

the

cyclone

was

being

used

n cloied circuit

$itll

a mill

and

that

the overflow

eprescnted

he feed

o a separation

cchliquc,

e.g. lotation, the

size analvsis

ofthe overflowproduct could beexamined.

'Ihc

author had arrived-at

an optimum

scparation

sjzc of 150

mesh

for

that

product.

lhe

analysis fthat

materialshorvcd

hat

81

7

pcr

cent of-rhe

l5b

mesh material

had

bccn

recovcred

n a cyclone

ovcrflow

producc

which

contained

3'3

pcr

cent

+150

mesh.

Thus,

lbr

a flotation

ec d

material nominally

at

150

mcsh

herc

rvould be

prcseDt

23

3

per

ccnt

of

oversize.

fhat,

he thought

would

bc agrecd, las an

exccssive mount

of

ovcrsize.The

optimum

scparation

size

using

thc

technical eflicicncy

*'as

100 mcsh at

rvhich sizc 72

9

per

cent of thc

100

mesh material

was

recovcrcd n

the ovcrllou'

product which contajned

oDly 10 4

pcr

cent

oversize.

n that case,

t would bc

seen hat the ovcrflow

containcd

90

pcr

ccnt

passina

separation

size, which was normally

quite

acceptable

The

rc.ultant lois of

l0

per

cenr more fine

material n the

underflow would

be

taken

up quitc adequatel,\-in

he circulating

oad to

the

mill.

Alternativcly, thcy

could consider

thc cvclonc

as

prcparing a coarsc

oroduct

for tiblins

whcrc thc

underflow became

he feed material

for

Tc i l i nq

9 1 . 1

0 16

9 q

8 . 1 5 0

95.9

r 0 0

0

2 0 9 8 0

1 0 00

l o ( t

0

It could bc

sccn hat as a

sand-clcaninq

oeration

he seoaration

va s

cxtrcmely

gnod

but as an irotl

ore separati;n

t;l\4

avssuppo\iris

one would

treat an ore conraining

2

pcr

ccnr

Fe .Or),

rhe

resuli u eii nor

iomparablc.

The eliciencies

were:

l'c

(for

Fe"O.,

scpararion)

29.3

.Ec

for

sanJ

cleaning)

91. I

a

L

28.6 .

Mr. F. D.

L, Noakes

said

hat

he

v.ould ikc

ro

congrarulate

he author

on.haringdetclopcd

what rvas,

s he had

himsclf

unJerlinc.l,

ssentially

a simple fbrmula.

It was o[ rhat

very fact

of

simplicity thar

he rvanted

o

take thc

author

to task, because

e had not,

at thc samc

imc,

issuedany

warnings about

its use. It was

so

simple that

the non-technical

man

not

fully

conversant

with mineral

dressing

mighr

thin_k hat it could

bc uscd

for thc dircct

calculation

ofrhe

elliciencies fopcrating plants.

The obvious

formula to usc

for such

a

prlrposc

ra'ould,

however,

ppcar

to

be the one

basedon chemical

assays formula

5,

p.

700).

Immidiarcly,

the nr,rn-

expert might

get

into trouble

because e had

to work our rhi-maximum

theorctical

assay

of a

pure

concentrare

c,,.*).

Quire

aparr

from

the wide

variations in

mincralogical

composition,

dcpending

on th,:

actual ore

being mined

and treated n thc plant

day by

day, in many operations

he

true cma\ ould

not be established

y the

chcmical

assay lone.

For examplc,



290 [. DoucLAs:DERTVATIoNF

A

a

coppcr ore might be

predominantly

chalcopyritic, but would

contain

varying ratios of bornitc, chalcocite,etc.l or a wolfram

ore might

yield

concentratesof similar WO., assa-vs

rom orcs of rvidely dilTering

wol-

framite,/scheelite

atios and

hence dilferent cu'a' values. An cnlightened

metallurgist could overcome he at ter diiiculty by a study

ofthe complete

analyses,deducting only

gangue

constituents

o

givc

a reasonable .^*,

but

a

css

experienced

man might

bc

much misled nto

gaining

thc

impres-

sion that the

'efhciency'

of the

plant

concerned appearcd

to be, say,

about 30

per

cent,

Any formula of that nature should be used only in its

propcr

context,

namely,

for comparing a number of tests carried out

on similar samples.

Horvcvcr, n

spitc of thosc,

he suggcsted hat opcrating

plant

resultscould

be

compared rom day to day, or

preferably

lrom month

to monthJ by

using he formula in conjunction with a

planned

systcmoflaboratory tcst-

v-ork. Montbly compositcsamplcsof mill fccd would be batch

tcsted by a

standardizedmethod

as closely

parallel

o the

plant

opcratingconditions as

possible.

The samc or

prcciscly

similar samplcswould then be uscd

for a

series

of optimizing batch tcsts o establish he maximum'basic

efficiency'

obtainable with thc orc, thc calculationsbeing based on the

maximum

conccntratc

grade

by assay r mineral count.

By

working

back hrough the

standard

laboratory tcsts to thc actual rcsults obtained in the

plant,

suitable actors

could be developed vherebyany variations

n

performance

would

show up and, frcqucntly, an indication lvould be obtained

as to

whethcr

the

variations

were due to changes n the orc

fccd or to faulty

plant

operation.Although that might sccm o bc a laborious

procedurc, t

was his opinion

that any short cut would only lead to

rvrong conclusions

being

reachedand ccrtainly any dircct application of the

'basic

efficicncy

formula' to

plant

results would be completely misleading.

The

speaker had uscd thc cxprcssion

'basic

cfliciency' during

the

developmcnt

of

his

argument, but he was not in favour of

thc use of the

rvord

eliciency'

in that contcxt for

thc

rcasons

utlincd.

It was his opinion

that the v'ord

'eliciency'

was very much overworked n

many branchesof

technologyand ndustry. Hc suggestedhat a ar lessmislcading mpression

would bc

created f the fiactions in the formulae developed

n

thc

paper

\rere

not multiplicd by 100

and that the functions dcrived

should be

termed

'basic

performance

actors' or

'indiccs'.

In spite of

Dr. Robinson's earlier remarks, the eliciency

that should

basically

be of interest

q'as

hc emciencyof converting

the muck that thc

miners

produced

into money for thc sharcholders' benefit, Any

real

efficiency

ormula would

introduce such actorsaswere sometimes

gnored

by

the mill operators hcmselyes

in

particular,

the actual

value of the

concentratcs.

Generally, the mill man *'as madc vcry consciousof

hrs

opcrating

costsr

but he was very often

quite

unawareofthe effect of varia-

tions

in t he comDosition

ofhis

products

on their actualvalue as

evenue o

the company,

which involvcd

iuch

factors

as freight,

insurance,smelter

contract terms

and so on, or, on thc biggcr mincs rvith smelters, etc.,

factors

relatcd

to the recovery and sale of the final metal. For a useful

ovemll

emciency formula hc l 'elt

all t hose factors would have to bc

in-

corporated.

Fonunately, that rvould

involve so many different

'variable

BAsrc

EFFrcrENcy oRMULA-DIscusstoN

291

constants'that

t $'ould necd

a metallurgist or mining cngineer

o interpret

it. 1'hat

being so, it uould

gcncrally

sccm preferablc

to

provide

the

operating staff with

thc essential inancial data,

cnabling hem to calculate

their economiceficicncics on

the basisoftheir knowlcdge

ofthe technology.

Any

attempt to take a short cut by making

some sort of overall

eficiency

slide-rule, suitable

or the non-technical

uscr, seemeddoomed

to failure.

Mr.

C. C.

Dell

said that one

of the fundamental

problems

of mineral

dressingwas cconciling quantity

with quality,

i,e. the recoverywith

grade,

That

problem

madc itsclf felt

every time a test

was carried out, whether

the test

was on a laboratory scalcor full

scale n the

plant.

S(hatever he

test, an unambiguous

result could only bc obtained

if all the relcvant

information

were condenscd nto a singlc critcrion of cfficiency, Many

formulac and

graphical

studies had bccn

put

forward for solving

that

problem,

the new onc bcing, it

was claimed, rnorc ogical

than the others.

1'he requirements

which wcrc set out in

the

first

two

pages

of thc

paper

werc by way of bcing a statement

of the

premiscs

rom

rvhich the basic

eficiency formula

was then dcrivcd. Thc.speaker's

criticism s'as not

on

the logic of the dcrivation of

the formula but on

the original require-

ments

on which

it was based. Hc suggested

hat an eficicncy

formula

should satisfy wo rcquirements

only:

(t)

It should

be directly

proportional

to the economic

gain

from

the

separationJ eing zero lr'hen

there was neither

gain

nor loss and 100 for

maximum

possiblegain.

(2)

-fhe

expression hould

be dimensionless.

It

might

be argued hat to bring economics

nto accountwould inevitably

make he criterion of cfliciency

sensitive o daily

market fluctuations.That

was not

the case, owever,as t r'as

only the relative economic mportance

of the

grade

and rccovery

which had to bc assessed,

nd that did not

fluctuatc.with

marSt condition: as

did the absolute

pricc

of

a

valuable

mineral. If the rclatlve imlortance

of thc

gradc

aad recovery were fixed

implicitly, as in the basic cfficiency formula, then too much emphasrs

would be

placed

on rccovery

n an examplcsuch

as luorspar beneficiation

and

too much

emphasison

grade

n

an examplesuch as

thc concentration

of

galena.

f

it were

impossible o avoid

such a compromise, hcn

such

criticisms would havc little

point.

However, an cconomic,

dimensionless

efficicncy cxpression

did exist,* and he

I'ould be intercsrcd

to hear the

author's reasons or ruling

out economics

at the start of his argument.

He would criticize thc basic

efficiencv ormula in

vet another resDect:

that it

took

no

account of the ratc

of sipararion. Raie of separation

was

clearly the factor ryhich

influenced he siic

of

machine,

or rhc number of

machines

equired o trcat

a

giyen

rate of tfuoughput. I f

rate were not taken

*D LL,

C. C. The anal,vsis of flotatirln

test data. Cold. Sch. lllines

Qltart.,

56,

no. 3,

July

1961, l l 27.

(50th

Annivcrsary of Froth Florarion

in rhe U.S.A.)

- Thc cvaluetion of

flotation

pcrformance.

Cala. Sch. Mines

Q

art.,

58,

no. 3,

July

1961J129

40.

(50th

Annivcrsary of

Frorh Florat ion in the

U.S.A.)

'.

A comprellcnsivc

critcrion of

coal-cleaning cfficicncy.

.f.

Inst.

F el, 35,

June

1962J .10

5.



292

T.

DOUGLAS:

DFRIVAITON OF A

BASIC EFFICIENCY

FORMULA-DISCUSSION

293

into accountand accuracy fscparation

orlly \\'ere

considcred, hen thc

best

cconomic esults u'ould

always

bc obtaincd with the biggcst

plants.

.l'hat

was Dotr howcver, n accordanccwith common sensc.

1'he

problcm

of choosing hc right criterion

was

particularly

acute vhen

studying the

dcsign

of

plant.

Thel' had found in rheir

uork

at Leeds

University, on flotation

ccll

design,

hat

choice

ofthe rvrong

critcrion could

give

cxtremcly misleading

csults, as to

punue

'rate'

alone rcsultcd in

a

l-ast lotation

ccll

rnaking a very

poor

separation,but to

pursue

accuracy'

alone

lcd

to an

inrpossibly arge

and

costly flotatiol

system. He would

g'elcome

he

author's

commcntson

that.

Finally,

hc

would like to takc up the

question

of the nccd for

simplicity

u'hich

secmed o entcr largch'

into

thc

discussion.Hc lelt strongly that

when one had taken a lot oftrouble to sarnple ome

orc,

split it down, mix

it, do the flotation tcst, thcn thc fi1tcriug, dlying, rvcighing and then

assaying, he actual ellbrt in calculation

vas

quite

in:ignificant compared

*'ith the rcst

ofthe

effort,

and thc

only thing that rcallv mattcrcd 1\as hat

thc answer

vas

ight, and that it told *'hat was

*antcd.

Mr. A. G. Moncrieff

said hat any

conccntrating pelarion cquired

two figures

to define hc results achieved:a

figure

to shou' he degrec of

concentration normally

given

b

thc ratio of

conccntration and a

ligure

to show thc elicicncy of achieving hat degrcc of

concentration

normally

it 'en

1'thc ecol'cr\ '.

fcourse.

n mn:t conccnrratingpcrations

there

was someco-relationbctween hose wo

lisures.

-l'he

onlv

co-relation

bL-tweenwo sctsoffigures that could be repre.Jntedhv a

ingie

figure

rvas

a suaight

line rclationship

utich

passcd

through the

origin-

l'hus it

rvould

appear hat any attcmpt to define

he

el]iciencv of

a

concentrating

operatiol

b_y

a

single ligurc

rr€s doomcd to fajlurc. Essentially, all the

attempts to do that, including Mr. Douglas's, had

produced

a formula

uhich

defined hc t1'pcof relationship

bctu'een

atio rrf concentrationand

rccovcry.'fhat

relationship varied from

process

o

process

and fiom ore

to ore, and t rvascertainlynot going o bc ofmuch value n comparing he

operation between wo different opcrations. t

coul,l also

vary in the same

ore,

and the

same

processJ

epending on

how

thc extra recovely was

obtaincd

(e.g.

extra reagcnt

cqnsuniption

or

longel

rctention

time).

Il

might even vary dorvn

a

singlc bank of

flotation

cells

producing

the same

material.

Presumably he urge to

produce:uch

an

index was due to the diliculty

of

mcntally

co-rclating the

figures

of recoveryand

ratio of

concentration.

A

general

managerorr reading thc monthly

rcport from the concentrator

superintendcnt might scc that the rccovcry had increascd rom 94 to

95

pcr

cent, and the concentate

gradc

had dropped

rom 47

o 45, and head

grade

changed

rom 3

2 to

3.

3,

but

that did not

cnablc him to

assess t

a

glancc

f onc month's

opcration

rvasbetter than another.

Thc speakcrmaintaincd that it $as irnpossiblc o

compare

such results

in tcrms of a single igure othcr than

profitabilitv.

A1l atternpts

o produce

an

efficicncy indcx only lcd

to an extra and unnccessary

igule being

reportcd, and then such statementswould appcar

as

'Efrciency

index

decreased

his month

lrom

43 to

40, but rccovcry

increaseel

rom

9.r to

95

per

ccnr.and

concenrrarerade

ropped

rom

47

to

15 pcr

ccnt.,

That

woulo

not

fct

thtm verv

ar .

One

of.the

most hclpful

ways

to compare

wo

concetrffating

csults

particularly

fthc

head

grade

vereconsrant

rvas

o

calculatc he

difference

berrlccn rhem,

rnd so

frnd

thc

grade

oi

the extra

concentrate ro<luced

t

tnc

l)rfner

ecoterv.

He

;ishcd ro

r.aise

he following particular

points;

(l)

In the synopsis

he

author siid

that rhcrc

was an

example

of

de-

rvatering

operarions

but

he

had been

unable

o fiDd

thar.

(2)

Vould thc

author

clarify thc

logic

of muhiplyrng

roecrhcr

eouarions

I

and 2 to

get

the

overall concentrarion

efficicncy

lequari6n

on

i.

ZOO;.

It

\rasonly

bccau'c hc)-uere

rukiplicd

ogethei

hai hc

hadobtaincd

he

undoubtcd adlantageoi thc sameeficiency for ganguceliminationsand

mlnelal

recoyerv.

(3)

Thc

author's formula

coultl bc expressecl

n

thc

form

" ( RC) ' f

(100

c)c 100

J

Thus for

thc same

recovery (R)

and

rutio

of conccntation

(1./C),

the

elliciency

ndcx

was

propo.tional

to

tOO

/

i,

,o

gir

rng a

highcr

eHiciency

index to a

higher

grade

eed. This

upp""."'d

to be illogical.

.

(4)^

f a plant

werc opcrared

t ughiecovery

and ori ratio

oI

concenrra-

tron lor onc pcriod,

and

then low

recovcry

and

high

ratio

of conccntratron

for anothcr^period,

lrc cllicicncy

index

foi each

piriod

rnight

be

the

same.

trowever,

r rc

t\\.o

penods

$'ere

aken ogcther

a verv

differcnt

ejliciencvlliciency

ror anouler

pe rod,

tJlc cthclcncy

tndex

lor each

pcriod

However,

ifthc

two

periods

rveie

taken togcther

a very

index

migbt

be

obtaincd. Thar

also

appeired

illogical,

an.l

coutd

leal

iir

difficultics

n intcrpreting

results.

Tltat

uras

best liusrrarcd

[.v the

follorving

cxamplc:

rR.foz./y Ratio of EfJiciency

''.,

.otl.cntr,ttiln

ozLlex

Fccd

Conccnrlilte

Ta i ls

2nd

t

etiad

Feed

Conc€ntrutc

Tai ls

Fccd

Cor:ccntrlttc

Tr i is

2 6 4

2 6 4

2 1

6

What

u'ondcrl'ul

opporrunitics

$ould

be

ollered

to

the concentr.a.or

supeontendcnt

sing

^rhat

t'r6u1o

o explain

away

a low

eficiency ndext

Also.

or courscr

he ettlctencl

tndex

was

no help

in

comparing

opcrations

for the

tvo periods.

Irt. Grale

1 0 0

1 5

35

40

0 5 1 5 , 1

l (10

1 5

7 . 3 8

8 0

9 2

6 2

1 l

)o0

t5

? 1 . 2 1 7

7 6 8

6 . 1

9 3 5

3 9 2

2 8 6

l t 5

1 7



Mr.

D. N. Moir said that in

vicq' of what had

bccn said by

prcvious

speakers

'ho wishcd to introducc economics

nto the formula,

hc fclt

that

it was

an appropriate moment to defcnd

the researchand

dcvclopment

mincral

engineer against a common

criticism rvhich

rvas evelled against

him whenevcr

he attcmptcd to bring out a

formula rvhich rvould enable

him to

assess is test results accuratcly.

That criticism usually took

the

form:'It

is all very well

for

you

chaps o invent fancy

ormulae o

give

you

an

idea of

what

you

arc doing, but

you

are

neglccting he economicsofthc

job;

on

the

plant

we have to set

grade

and recovery according

to the

economics

of the

proccss

and

your

formula does

not hclp us.' He thought

that confusion

aroseout of a misunderstanding

about the real

purposc of

such

a formula,

He was surc thc

good

research nd developmentcnginccr

t'as quite rvell

aware that

his

process

must be economically

sound and also,

while

mctallurgicall.v ccrtain rcsult rvashighly desira ble, he plant targct might

be different

and was adjusted o suit the

pafiicular

conditions

pertaining

at

any

given plant

undcr thc

prcvailing

market conditions.

At the bench

csting stage, he rcsearch nd developmcnt

cngineermust

work

up his

process

o the

highest

peak

of

performance

o show the vcry

best metallurgical

rcsult Ehich could be expected. t

rvasat that stageof

the opcration

that the

formulac lirst madc their appearan ce nd they

did

so in order to

cnablc the engineer o assess ach

part

of

thc circuit and

bring

it to its

peak

of

performance.

Subsequently, n small or

large

pilot-scale

testing a rangc

of rcsults

would be

indicatcd and sulicient data collected

to cnable a series of

calculations o be made

which would indicate the approximatc

level of

performancc

nccessaryo achieve he correcteconomics

ating into account

such factors as nature and

size of ore deposit,

grade,

mining

ratc, capital,

transport and

crection costs of equipment,

local labour costs, rate of

amorrization, etc.

(assuming

throughout that the

process appcared

economically

sound at the start).

Final economics would

bc scttlcd $,hcn a detailed design

study vas

undertaken br thc l'ull-scale lant. Ilven when hc plant was uDning, small

changcs

n

pcrlormance

might

have

o bc

madc from time to time to mect

changes, or examplc,

n

plant

input or in the market

pricc

ofthc

product,

the result

ofrhich would be that the

finallevcl of

pcrformance

might well

be diferent from those ndicatcd

in thc initial studies.

The formulae rvhich

had

been discusscd,

hose of Stevensand Colhns

and ofDouglas, and

othcn,

'"r'crc

ot in any u'ay ntended o solvecconomic

problcm'

and

t \\a\ unrearonableo expect hcm to do

so .

The economic

problcms

were individual to a

particular plant

and

orebody and which

gcneral

ormulae could not bc cx pcctcd

o solve.

In addition to

thcir use n developmentstagcsof the rvork thc

applica-

tion ofthosc

formulac n

gauging

he day-to-day

pcrformance

ofindividual

parts

of thc circuit should not bc overlookcd;

sornc of those applications

had been discussed v thc abovc

authors.

BASIC EFFICIENCY FORMULA-CONTRIBUTED

REMARKS

IiIRITTIN

CONTRIBUTION

Mr, K. C. G, Heath: The basic

cficiency

formula may

be of value

rn

researchwork undertaken

s''ithout a spccifi.c

pplication

n view. It would

be dangerous

or use n an operating

plantr

as would any

ndex ofefficiency

that conforms

with the author's basic

requirement

6

-'no

attempt

should

bc made o include economicconsiderations

n the technical

assessment

f

a concentrationopcration.'

If maximum tcchnical

efficiency

produces

maximum

ultimate

profit,

then we want

maximum technical

efliciency. f maximum

ultimate

profit

is

produced

by some esser echnical

elliciency,

hen that is the

cfficiency

we want, /rol the

ma-\imum. Far from

cxcluding economic

considerations,

we need o

hamrner hem home

at cvery

possible

opportunity.

Let us examine a

hlpothctical operation, n which

a metallic

ore with

a mill head grade of 5 pcr cent is being concentrat€d or a latcr smelting

process

that will

produce

a

salcablc metal.

-Ihe

assay

grade

of

the

chemically

pure

concentratc

s 50

per

cent.

Expcrimental work shorvs

hat

recoverv

and conccntmte srade

can

bc related as follorvs:

295

294

E.

DOUGLAS:

DERIVATION OF

A

Recovery

91,

Emciency indcx

'1,,

Recovery

ui

Concentratc

grade

",,

Mr. Douelas'seflciencv

index calculatcsas follows:

70

7 5 8 0 8 5 9 0 9 5

975

15 0 42.5 40 0

3'7 5 32 5 25 0

15.O

70 i5 80 85 90

95 97.5

6 0 0 6 0 5

6 0 5 6 0 0 5 1 0 4 1 7 2 t . 4

5 4 0

4 2 . 5 1 0 0

3 7 5 3 2 5 2 5 . 0 1 50

9 8 . 0 9 7 5 9 7 . 0 9 6 5 9 5 5

9 4 0 9 0 0

indicating that the most eficient operation is one in which

rccovcry rs

between75 and 80

per

cent with a

conccntrate

gradc

of betrveen 2 5 and

40

pcr

ccnt.

In order to examine he rcsults cconomically,

ct us assumc:

Cost of

rnining, milling

and

general

charges-30s.

pcr

ton

at 70

per

ccnt rccovcry

incrcasing by 6d.

per

ton for cach step in incr eased

recovery, o allorv or finer grinding, additional reagentsor whatever s

necessaryo obtain the higher

recovery.

Cost

of smclting o mctal*{3

pcr

ton of containcdmctal

plus

{2

per

ton of concentrate.Smelter recoverv. established v test work-

Concentrate

grade

oo

Smcltcr rccovery

";

Costs iom smelter o market

.

d20

per

ton of metal.

Market valuc of mctal

-f100

pcr

ton.

The

profit

to bc

gaincd

from

mining and treating 100

tons of ore ts

calculated as in thc following table, indicating

that the most

profitable

operation s one in which recovery s bcniccn 90

and

95 per

cent

rvith a

concentmte

grade

between32.5 and 25

per

ccnt.



296

( - o n c . n t r a t o r r c c o r c r ' ,

( l i n cen t . r te

s rade , ,

'I

on \ meta l n r co r .c r , r rc

T .ns

concent .a re

Tons mctal

produceJ

E. DOUGLAS: DERIVATION

OF A

t0

7-

s0 1 l -

1 5 : r . i l 0 0 3 i a

I 5 0 1 j j

1 . 0 0 4 1 .

l i o

7 7 U

3 S :

l 0 0 0 1 1 l

:l

,110

-l

656

3 l l l J

.1

lo l

trom sale

oi

.

113 0 165 6

3 l l l 0

l lo I

BASIC EFFICIINCY

IORMULA_AUTIIOR'S

REPLY

297

remarks

based

on the correct

interpretation

of concentrating

cmciency

lnd rhe

se(onddcal:

wirh

the cconomic

specl.

Apparently,

,\4r. Stevens

and

I

disagree

on almost

everv

DoiDt

and

I doubr

f

much reconciliarion

-.i l l

c

aihicved

rom

nrirren

dircursion.

However,

I would like

to record

my rcplies

to his numerous

points

and

opinions

in order to

prescnt

the

'other

side'

and, in

one instance,

ro

correcr

whar

I consider

a

'slcight

of hand'

technique, which

is

used to

acnlete

a known Oblectrve-

I,lr.

Stercns has

difficulty

iu appreciating

hc

logic of my

derivarron,

particularly

as he transfer

of

mass

actoss

one

maginary boundarv

has no

parallcl

in mineral processing

(I

assume

he

meins mincral processrng

operalrons).

If

one attempted

to formulate

an

cfficiency

expression

based

on

the

mechanisms f the concentrating cchniques

employed

hen obviously

no

one

form

t'ould suflicc.

mainiain rhai.

wheni

concerrrration

a:

bccn

made,

and

ccd

aDd

product

data are available,

t is

Dot necessarv

o know

the route_employed

n

order

o as5c\she. l l icienc)'

ith

uhich rhe

con-

stituents

have

been separated,

To give

a verbal

description

of my dcrivation;

I have

asscssed

he

products

n

relarion to feed,

I have dcicribed

the effcctswhich

thc intcr-

change

of

constitucnts avc

on

products

and

thcn, bascd

on the

actLlal

eight plir

achicved

have

ndicated hc

mass

ransfcrs vhich

areneces-

sary

to

produc '

the

final

rcsults.

The efficiencies

f lhesc

mass ranslirs

hase

been

cxprcssed

n simplc

mathcmatical

terms and combincct

rn

p:oduct

form

ro

ti{c

the basi-

efliciency ormula.

I

do

nor agree

hat this

rs

an cmpirical

approach.

Further, I

do not understand

whv.

evcn if

Nlr-

Stevens'

suggestion

hat

"fhcre

\yas

surely

no analogue

or

the

author's

orm

of

cliciency

expression'is

ruc,

this shouldbi

a critcnon

of

aliditv.

-

,\1r.

Stevcns

mpha.izcc i.

olinion

rhar rhcrc .

,no

funLlamcntal

a.i,

lor any

singls

1l2n1iff

ng

cxpresrion',

nd arcr.

sa rcsulrofh r

emplql.1,,t

a

'rrork'

analogl he has.uggc.tedrhat'rhc formula for concenLraii,,n

ethciency l:-

)

u'as

fundamcltal"'.'l'hesc

two claims are contradictory

at least

one

must be invalid

on trIr. Stevens'reckoninq.

I consider

b-oth

to

be_

ncorrcctl

thc first on

the basis of

my ou'n ethlicncy

Jcrir.ation.

and rhe

sccond

on rhc

basis of

the irregulariiics

n his appliiatior.r

on

thc

'rvork-'

analog_r'. feel

rhe

lattcr. is extremely

misleading;

surcly

some

qua)rfication

hould

be

givcn

to indicarc

rliar

.uch

an-appliciriol

i, ,

permissiblc

and furrher,

e,ien assuming t

to bc

acceptable,i ii

necessary

to apply

it in

a consistent

mamrcr.

InMr .

s rcvens ' rpp l ica t ion l r ranr la rch i r

. lu*

I rorcn

r l

d i r fe tunce

to

bc

analosous

o

rre

roduct

r,h"

n'r.r")lt'i,.,

irllilli$l'ilt'"1;';;

grade,

relative

to

the

ma-ximum

of this

funcrion,

Howcvcr. follou,i[e

rhis

analogy, e

ntroduces'mass

fvalues n concentrate'to

cpresent

iuss'

and t}is ha'

bccn

multiplicd

by JfValue,l.

'l

hi. stcp ,. nor

lalid;

raluc"

cannot change

n

gradc;

r is

the mass of conccntrata

uhich undergoes

a

J ) a

t i o

l: l

l 5

i )

.t

165 4

3rl?

129 7

'1 ,16 .5

. l I ' . 2

5 2 3

43A.1

1 6 5

0

I tn r ins , mi l l ins

xn J

*eneral

expenses

{

snc l r ins

cos(

mrrkc r

a

1500 152 2

r a a 0 1 5 ?5

r : 0

1 5 4

7 1 6 E : r r

TDIIL .osi ol mebl

r,.r.-

d u ct i o nd . 2 1 1 t - 241 a

26.1

6 271 9

9 a 3 l l t I

1 2 )

I | ) 4 2

:S7

I 10, . 33t

:l

t . t :

6 r12 I

r06

r

Even this does

not

tell the

u,holc storv.

If a leaching

proccss

rvcre

available

as an alternativc o sDrelting,

t might well bc

profitable

o accept

a concentratc

grade

of 15

per

ccnt

in ordcr

to achiele

the high rccovcry

of97.5

per

cenr.

This rvould

depend

n leaching ccovcry nd costs, nd

would rcquire

a

further set of calculations.

Although

the costs

and recovelies hat I havc uscd are

hypothetical,

I

submit that

they

are typical of actual

practice.

Even

then, thcy arc

an

over-simplif ication,ince o

obtain

the higher rccovc|y$ould

probably

involve

highl-r

capital

cost of

plant, and alloq'ance

ould

havc to bc-madc

for

rhis

in assessing

hc cconomicsof the operation.

To

the engineer

or

manager,

statistics,costs and emcicncy

igures have a value only if thcy

provide

reliable nformation on which

to basedccisions.

Anyone

basing

a

decision on an index rather than on

thc

economics

would bc dccidine

blind instead

f in

thc icht of

the

facts.

AUTHOR'S

REPLY

Mr. E. Douglas: As indicarcd

in

my

presentationl,

his

papcr

was

prepared

nitiallf

in

the

form

of a

discussion n the

pape(

bv Stevens

and Collins which

I had assumed,

apparently

erroncously,

would

be read

by

members n

conjunction

with mr

paper.

The most

common misunder-

standing

has ariscn

rom

the readers'

pcrsonal

nterpretation of'elicienry

formula for concentrating operations'.

In this

papcr it refers

to the

eliciency

v'ith rvhich a mixturc is separated nto its

individual

com-

ponents.

A numbcr of contributors

havc rccognized

his

but others havc

interpreted it as an

cconomic

eiiciencl'.

This lattcr interpretation is not

corrccl; the formula

I havc

derived caznol

be used or

complete

economic

guidancc n

relation

to thc

products from

a milling opcration.

My

reply to the

discussion s in

tu

o sectionsl he first

is

concernedwith



298

E. DoucLAS:

DERTVATIoN

F

A

change

n

grade rom

/

to c. Therefore'

if this

analogy

s continued

and

applicd

strictly

it results

n

thc

following

expresslon:

,

C ' ( c

f )

. . ,

n o

/

\

( i . r x x

- / ,

This,

I suggest,

s tbe corrcct

product

of Mr.

Stcvens'

analogy

but

I daubi

if it;;n

be considcred

scriousll'

as a satisfactory

expression

or

c0ncentration

elliciencY.

trIr,

Stcvens

qucries

the

presence

of rhe opcrational

variablc C

in the

denominator

oftle

basic

cliciency

formula. I agrec,

at

first sight' that

this

might appcar

untcnable.

Hoqever,

his

implication

that

'the

maximum

usiful

*ort

that could

bc donc

in separating

a

given mixture

lt'as a

variable'

is not a

complete onc.

Thc basic formula was derived on the basis of a sampling operation

which

gives

a weight

split identical

to

lhat

of thc

products under

con-

sideration.

fthc

sampling

operation

varies hen the

basevaries.

A more

correctconclusion

o

bc dr;wn

from the

fact

hat C

occum

n the denomirra-

tor

is that the

maximum

cxtent of

scparatjQn llich can

be achicved,

afl€r

a rright

split has

beenmade,

varies according

to the wcight

split.

This

prclinrinarv

lvci'rht splir

supelrmposesa

limit on

thc secondary

weight

1r'alues

ni

gan-guc)

ran.fcrs.

For example,

f the setting

on a tabling

oDeration

s such as

to

produce

a concentrate

vhich $'eighs

l0

per

ccnt

oi th"

f"ed, the

fced

grade

bcing 5

pcr

ccnt,

thcn

it is

viltually

impossible

to achicvc

a 100

per

ccnt eflicicirt

separation, he

maximum

grade

which

can

be expected

being only

50

per

ccnt.

Mr. Stcvens cites

the instancc

lhen diffcrent

$'eights of conccntrates

which have he

same

gradc

will result

n diffe rent values n

thc denominator.

I would

also

point out that

such changcsare automatically

and

correctly

comoensated

bv the C

factors

in th c rlumerator. Becausc

of the

intcr-

dependcnce

ofb

with other

variables

(c,

R), the

100

per

cent

efliciency

can only be

achieved

whcn C

f.

I wish to stress hat m;' proposed efficicncy forrnula was not based

upon the assumption

hat

the efhcicncy of conccn trates

hould

be equal

s'hen calculated-from

either

of the two

products.

Tbis

is a result

of the

form

pre'entcd

and one

uhich.

in ml

opirtion.

houlJ

be expccled

Fo r

example,

he separation

achieved

when

ihrcc

uhite and

two black

balls

are removed

rom a mixturc

of, say,

l0

black and

10 r'hite is

idcntical to

tlrat whcn

7 rvhitesand

8 blacksarc

rcmoved rom the same

mixture.

Repl,ving

briefly to

I'lr. Collins's

contribution I v'ish to

do no mole

tlran outline onc

or two discrepancies

u'hich occur

in his comments

First I re fer

to requiremcnts

and

6lor

formulating

an expression

or

concentrating

elicicncl,

set out

in his paper

with

Mr. Stevens:

.

'no

attcmpt should

bc made o strcss

ome

part

of the data more

han another',

and'no attempt

should

bc made to

include

economic

considerations'.

n

contradiction

of these

equirements

hc

explains

his dire ct use of

recovery

(instcad

ofpercentage

ncreasc

n recovery)

by

the argument hat

a mineral

dresser evaluates

his results

directly on recovery

figures. By using the

BAsrc EFFrcrENcy

oRMUrA-AUTHoR'sR[pLy 299

recovery erm as it stands Mr. Collins is stessing

this

part

of the data

in

relation

to

pcrcentage

ncrease

n

gradc.

If

bv

'minclal

dresscr' he is

referring to the

plant

engincer

hcn

his argument

takesau economicbias.

Altemativel.v, f his reference o mineral dresser

ndicatesexperimenters,

such

as himscl{,workrng

n flou'sheet evclopmentsr

hen this s'bcgging

thc

argumcnt'.

I ma intain that a conccntrating

opcratiou

s

achicvcd by a combination

of valu e concentrationand

gangue

ejection.

l'he

former influencesboth

recovery and

grade

and, consequentl ',changes

n both of thesc factors

should be represcntcd

wit}I respect o a sampling

operation.

Mr, Collins s in disagrcementwith me and,

ashe

points

out, with other

autho$

he has

cited, who consider hat the same

eficiency of concentra-

tion should rcsult whether

considcrcd

from

the aspect of concentrating

values or of concentratingganguc.He maintains that it is reasonable o

cxpcct

two

efliciency atings

for a ny one operation,

He usesmy examplc

(Fig.

2 and Table I\r,

pp.

702,7O3) to dcrnonsrratc

his ar gument. He

shows

(Fig.

B),

using

his

expression, har rwo

cut

points

are

produccd,

olle at 100mesh bascdon cyclon c ovcrflow

and anothcr at 200 mesh based

on cyclone underflow. Surely, a cyclonc operation

has only one cut

point.

The basic cfficicncy formula indicates

only onc, i.c. 150 mesh,

and

this

is in

exact agreementwith t he result produced

by the commonly

' . .an rp , l , l h . rL^ ,1

,\,tr.

Collins, assumingccrtain intcnrions lor

the c)'clo[e

products,

r.e.

ovcrflow

to

flotation,

underflorv ro ta bling,

claims that by assessinghe

overflow on the basisof his cfficicnq, formula

r bccomcsa bctrer lotation

fccd than if it

were assessed sing rhe basic

efticiency ormula.

A

similar

conclusion s reac hed with rcspcct to thc

undcrflow

product.

I cannot

acccpt the

suggestion hat the type of expression

used in assessing

product

can inllucncc its

composition

or

its rcacdon to subsequent

processcs.

From Mr.

Collins's

graph,

both

his

efiiciency

characteristics rc

highu

than

the single basic efficiencycharactcristic,

and the latter cannot here-

forc bc 'a mcasure of the average efficiency

of the separation which

incidentally

could bc obtaincd

by thc addirion of thc

tcchnical

cfficiency

for

both coarseand fine sizes'.

I find it dilicult to understand

the comme[ts

contained n the first

paragraph

of Mr. Moncricff's

remarks.Surel-v,

9 :;J

i, 11.r"

"o--6n1y

rccognized

co-relation bctwecn rccovcry and

ratio of conccntration. Only

/

is

constantbut, for any one value of c, the

R

1C

elationship s a straight

line

and for

different

values

of c we have

straightJine

palamctcrs, all

passing

hrough zcro, but rvith

clillering slopes. ntroducing

a

bias, eirher

by

specifying a constant

gradc

or a

constant

rccovcry, considerabll'

simplifies

the

problem.

In this case, as .l\1r.

Moncrieff

points

out, the

cffectivcness

of separation can be express ed

n terms of the remaining

variable,

.e, recovery

or

grade.

My

answer o the fust

of

Mr. Moncrieff's

four specific

questions s to



300 E. DouGr-AS:

ERTYATTON

F

A

admit

my mistakcand apologizc

or mislcading

iml 'dcrvatcring'should

rcad'dcsl iming'. 'his rct 'crs

o thc irst

cxample

n thc appcndir.

In

ansrvcr

o his sccondqucry,

I

gavc

a bricf descriptiou

of someof thc

simplcr choicesavailable or

combining

the individual cfficicncicsof two

related and

simultancousopcrations.

My selcction

of thc product

com-

bination is favourcd by simplicity

and

thc kno*ledge that any

alternatrvc

rvould

give

the same

relative positions

between thc

cfficicncy

ratings

produced.

It is not true that

tlle

product

combination s tlre

ouly reason

that the resulting expression

ives

dcntical cfficiencies,wheth .r

calculatcd

from

r-alues

conccntration

or

from

gangue

rejccrion.

For

instance, the

arithmctic mea n of the nvo e{}iciencics

vould

be similarll'advantageous.

As

gangue

and values

havc bccn given

equal considcration n formulating

thc cflicicn cy cxprcssion.unless

undue

bias s introduced the

result is not

aflectedby designatingcithcr onc or the other as values.

The third

question

concerns ccdsofincrcasing

grade:

fthese

are reated

to

give

constant

ccovcrics and

constant

ratios of conccntration

hen the

application ofthe eficiency

folmula

indicatcs ncreasingefliciency

atings.

,\'1r.Moncrieff suggcsts hat rhis

is illogical but

I cannot undcrstand

-hy.

I1, n a series fo perations)er.,'v othcr factor remaining constant, ncreases

in fr--ed

rade produccd

no clrange

n recovery,

.c. thc wcight

of values

recovele,.l n th e concentrate ncrcascd n rela tion to feed

grade,

then

I think it is rr--asonablco attach

he sameelhciencv

ating to

cachoDcration.

Hoivever, n the casc i teci

y

Mr. , \ loncricf i .hi

'peci-t ies

ncreai ing

te d

gradc

in combination ivith

constant

recovery and a c onstant

ratio of

concentrat ion.

I

he only

condit ions ndcr rvhich his combination

an be

attaincd is in conjunction lvith

an

increase n conccntratcgradc.

Surely,

if

two operations

result in

idcntical

pcrccntage

recoveries

but one

rs

associatcdwith a higher

product grade

thcn

this onc rvould

rcprescnt a

mole elhclent opefatlon.

Mr. Moncrieff's fourth

qucstion

poses

an interesting state vhich, from

a practical point of vierv, appearc o be inconsistcnt, duc lalgclv to the

extreme conditions which

hc

has sclcctcd, although, n spite

of this, one

must still cxpcct a rational result.

Examining thc rcsults

quotcd,

iaithoi i t

considcring hcir origin,

rv c

have:

Clcarly,

although the recoverv

n

(iii) follorvs

an arithrnctic

mean, by

virtue

of the interdependence

of

R,

C

and c, thc conccnrrate- radc

is

biased owards he lowest

valuc-,

.c. t improvesby only seven

nits out

of

fort-v

over thc casc

(i)

ligurc.

'l'his

additional b)ending

operation has

produccd

a

worsc

result it iras

associatedhc dcficicncies fconcentrates

(i)

and

(ii)

to a

greater

degree

han thc othcr

irctors.

Ditlcrcnt blcnds,

BASICEFFICIENCYFoRIItILA-,1I irHoR'sREpLy

30 1

obtained by running

periods (i)

and

(ii)

for di ll'erent times,

rvill

result rn

furthcl changes

n the clliciency

ofthc conccntrating

blcnding

scquence,

Fig. C is

prcsented

n order

o

give

a

risual

l lustrarion

fihcri argu-

ments and to ind icate

the

characteristics

f thc cllicicncy

formula.

Hire,

cflicicncy

has

becn

plotted

in relation

to

recovery

or vaiious

concentrate

grades.

n rhir pre.entarion.

he conccntrarc

rade

detcrmincs

hc

par-

ticular operativc

parameter

and

points

r,

y

and : reprcsent

the

quotcd

lst, 2nd

and combincd period

condit ions.

l

he blendinq

curve, which

shorvs hc cfflciencies

o

be

achieved by various

cornbinations

of the

l\t

and 2nJ

pcriod

operarior:.

as

al.o brtn addcd.

E : bos;c e tic i.ac)/

c

=

coltccht/ole groda

c R c l

(i)

35

q3

5

40 15

( i i )

i . ) a

3 9 2 8 0 1 5

i ; i )

21

2 66 47 15

R7 "

):ig.

C.

F.fliciorcl harauerixics.

I am indebted

to -N{r.

Noakes

or cmphasizing

hat thc

application

of

such

a formula

should bc

carried

out by opcrators

who

undiistand

such

'tools'.

Also, I am

gratel'ul

o him for

clarifying

the dillerences

bets,cen

the

subjecr

matrer of rhis paper

and the

economlcassessments

f existino

milling

opcrations.

fhese

a'rJtwo

otally diffcrcnt

factors.

.i;;;;p.ifiiy:

Mr,

Noakesconsiders

he latter

to be most important

ancl

agreeentiiiy

with

this- aftitudc.

However,

thc

concentration cfliciency,

6r index, ii

rnost

useful

at the srage

before

a workable

lowsheercxistj.

For

example,

one

may hate

o a"'e\\ hc

comlaral ive

mcnabil i r ie.

fa

parl icular

cp;.ir

to

a numbcr

of

po.. ib)c

t:.ci tmcut merhod.,

e.g. f lotat iou,

macncric

sepatatio[

or

tabling; ar rhis

stage

t rs muc-h oo

earlt

to con:idjr the

complicated

and

specific economL

factors nvolved

at ci

ery level of each

;az-

-,_/



302

[ .

DOUCLAS

of these

processes.

The obvious necd

of a reliable assessmcnt f con-

centration

s reflcctcd by

the numbcr

of

papcrs, publishcd by well-known

authors, on this subject.

A reolv to Mr, Dell's contribution will scrvc to demonstrate he con-

siderab-le-

ifferences betrveen economic

and

physical

assessments f a

concentratingopcration. At thc same ime, it will

present

a

general

com-

ment in answer o Nlr. Heath and others, vho

havc ntroduced economrcs

into

this discussion.

Mr. Dell askswhy economicshave been ruled out. The simple answer

is that I have not attempted o

produce

an economic

asscssment; y sole

concern was to

derive

an cxpression o indicate the effectiveness

:r'ith

which the constituents both valueless or this exercise)of a mixture have

been scoaratcd. Immcdiatclv suitable economic factors are introduced

the resuiting expressionbec'omes pecific to one operation, at one

par-

ticular

sitc,

in

one

particular

country-it

is

not universal.

Commenting, at Mr. Dcll's request, on his

papers,

may I refer briefly

to

his

most recent one

'A

comprehensive critcrion

of coal-cleaning

efficiency'. Knowing Mr. Dell's experience n the coal industry

I havc

confidence n

the adequacy of

his

assessments

f

coal-cleaning

opera-

tions, but I consider thc title misleading. As with combustion efficiency

which relates

he

heat

extracted and the total

heat

contcnt

of the coal.

'coal-cleaning

efficiency' suggests he elficiency with which the

non-

combustibles have

been removed, and should

thcrcfore not include

monetary considerations. n fact, tlle

paper presents

an cconomic ccoi)rt

of

a

cleaning

operation. As an example of this

I rcfcr to Mr. Dell's term

(specific

profit)

in which the cost of cleaning n the denominator s

zero.

This

only obtains

when no cleaning

s

carried out and thcrcfore,

such an

expressiondoes not assess

he

cleaningefficiency.

The facts that the expressions evelopedby Mr, Dell are indcpendent

of changes n

concentrateweight

(a

constant

s rccommended),and that

t}te influence of feed

grade

is not directly accounted for, demonstratc

clearly the spccific nature of

his

economicefficiency.

A similar degreeof

specificity, am surc,

will

apply

to almost any

other worthwhile economrc

assessment.

Again, if I may be allowed to stress he

point,

I agreewith

Mr.

Dcll,

Mr. Heath and others that an economicstandard s of utmost and ol'er-

riding importance.

Horvever, other emcicncy

facto$ arc required in the

initial developmentstagcsof

a

proccss

and, as I have already ndicated,

tlis was my objecrive or the basic elliciency formula.

derivation of a basic efficiency formula for concentrating operations

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