Sizing a Till a statistical approach H. Patnaik *

Introduction

In an ore-dressing plant, the energy con-

sumption (kwh/t) in grinding constitutes about

70% of the total processing cost. Thus the

energy consumption of a grinding mill becomes

an important factor in cost economics of the

process. Of the several factors affecting the

energy consumption of grinding mill, the size

(dia) of the mill is more important. It is also

seen that maximum energy consumption(1) is

attributed to the ball load as it constitutes 45-

50% of the mill volume.

In this paper, a log-log relationship is

obtained between power consumption (kwh/t)

and i) ball load and ii) mill-diameter. A know-

ledge of Bond's work index and further the total

power for the mill throughput are known a priori,

this power consumption is then utilized to find

ball load and/or mill size. The calculation is for

a closed circuit wet grinding in 0.F discharge

ball mill and a simple rod mill. For grate dis-

charge ball mill the appropriate correction

factors are utilized(2).

Description

Several complex empirical equations are

developed(2) relating to mill dia, speed, ball/rod

load (`)/01 mill volume), ball size. But speed varies

in a narrow range viz. 75 to 80% of Nc for ball

mill and 65 to 75% of Nc for rod mill. Nc is the

critical rpm of the mill. Maximum energy con-

sumption is found to be due to media load. So

for a fixed media load (% of mill volume) and

a constant speed, power consumed is related to

mill diameter. A log-log plot of the data(2)

shows a straight line relationship.

Regression equations are determined for

both the mills. The correlation co-efficient and

chi-square ( K2 )- test confirms the accuracy of

equation.

TABLE-1

The values of n and a in y=--a xn for mills.

Mill % mill volume filling

Ball mill 40 45 35

n 3.1597 3.1721 3.1379

a 12.2761 12.5846 11.90924

Rod mill

n 3.1296 3.15232 3.08302

a 15.40636 15.8016 14.928

Application :

For an ore the work index is determined

by Bond's method. With the knowledge of mill

throughput and with the help of several correc-

tion factors—dry, open circuit, high/low reduc-

tion ratio, the mill power (kw at pinion shaft) is

calculated. From the equations developed in this paper, the mill size is calculated.

EXAMPLE

Size an O'F ballmill with 40% ball load

with the use of data :

Mill throughput (tph)

500.0

Feed size ( p m )

1200

Product size ( p m )

175

Work index, Wi

11.7

* Scientist, National Metallurgical Laboratory, Jamshedpur-831007.

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Co E C

)- W = 10 Wi (_ v/-

P 1/175 Ni1 200 = 10 x 11.7 ( 1 = 5.47 kwh/t

So total power consumption = W x throughput 5.47 x 500 = 2735 kw

y = a xn, y = kw, x = dia (m) inside liner, n = 3.1296 and a = 12.2761

Substituting in equation : 2735 = 12.2761 (X)3.1597

So X = 5.23 (m)

Use a mill 5.53 x 5.53 inside liner or a 5.73 x 5.73 outside liner ballmill.

RODMILL REGRESSION EQUATION :

y = a xn or log y= log a + n log x. Here y = w, x n

or Y = ao + a1 X, Y = log y, ao = log a

Exy N ZXY- EX .EY = and = ao ai )1- or ao N T oxx N x2 - (zx)2

( 14 x 15.73675 - 5.5626 x 34.0364) 30.9836 = a 1

3.1296 = (1) (•4 x 2.9173311 -- ( 5.5626 )2 ) 9.9001

same for all al's

( 14 x 15.8644 - 5.5626 x 34.3175 ) 31.2083 a1 ___ = _ 3.15232 (2) Dr 9.9001

( 14 x 1 5.5724 - 5.5626 x 33.5839 ) 31.20 al = _ = 3.08302

(3) Dr 9.9001

a0 ---

(

- al g + V - - ( (

3.1 2960 )

3.15232 ) x

3.08302)

( .397331

(

) -I- (

(

2.43120 )

2.45125 )

2.39900)

(

= (

(

1.1877 )

1.1987 )

1.1740 ) = log a

( 15.406359 )

So a = ( 15.801560 )

( 14 927944 )

Thus in equation y = a x" the values of a and n for different % mill volume filling are :

% mill volume filling

40 45 35

n 3.12960 3.15232 3.08302

a 15.40636 1 5.80160 14.92800

35

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52

BALLMILL REGRESSION EQUATION :

ai - ( 14 x 19.75063 - 6.6925 x 36.3922) 32.954021 3.1597

(1) [ 14 x 3.94421 - ( 6.6925 )2 10.4294 Denominator same for all al's

(14 x 19.87124 - 6.6925 x 36.6252) 33.08321 a1 = _ 3.1721 (2) Dr 10.4294

( 14 x 19.57637 - 6.6925 x 36.0616) 32.7269 al - = 3.1379

(3) Dr 10.4294

( 2.5994) ( 3.1597) ( 1.08906 )

ao = Y-a1 X = ( 2.6161) - 0.4780 ( 3.1721) = ( 1.09984 ) = log a

( 2.5758) ( 3.1379) ( 1.07588 )

( 12.276088 ) So a = ( 12.584617 )

( 11.909230 )

Thus in equation y = a xn the values of a and n for different % mill volume filling are :

fiNGC■I■IOMMIC■

% mill volume filling

40

n 3.1597 3.1721 3.1379

a 12.2761 12.58462 11.90924

45 35

CORRELATION COEFFICIENT : (r)

15x r = a

1 SY

, where S S are sample - x Y - n-1 Ix or y

standard deviations and a1' are from the regression equations already determined i.e.

y = ao + al x

Sxx E( X - 30 ( X -T. ) x )-T )2

sx2 E( Sxx I ( n 1 ) ( n-1)

or ( n 1 ) Sx2 = Sxx

37

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RODIV1ILL

a1 — 3.1296 (1)

r = 3.1296 x 0.2330929 0.7309081 — 0.9980564

a1 0 3.15232 , r = 3.15232 x —

.2330929 .7352962- = 0.9993026

(2) 0

a1 — 3 08302 r = 3.08302 x 012330929 = 0.9765066 1 — 07359193 (3)

BALLMALL

a1 = 3.1597 , r 0.239392

3.1597 x _ 0.9998889 (1)

239392 a1 — 3.1721 , r = 3.1721 x 0.

— 0.9999757

0.7593938

a1 = 3 1379

(3)

0.239392 r = 3.1379 x = 0.9999468

0.7512281

0.7564909

(2)

Conclusions :

— Energy consumption in milling (rod and / or ball mill) constitutes about 70% of the total processing cost of an ore-dressing plant. A proper sizing of a mill is important.

— Regression equations are developed for rod and ball mills. It obviates the complicated

empirical equation developed earlier (2)

— A knowledge of Bond's work index and the plant throughput is required to find the total power required for the mill which in turn

establishes the size of mill through the reg-ression equations.

— The correlation coefficients and Chi-square

K2 test are made to establish the adequacy of fit of the equations.

Acknowledgement

The author wishes to acknowledge the encouragement given by the Director, National Metallurgical Laboratoy, Jamshedput for this work.

References :

(1) Hand Book of Mineral Dressing — Taggart A.F. pp

5-49, Power consumption, John Wilsey B. Sons

Inc 1927.

(2) Mineral Processing Plant Design — Mular A. L. and

Bhappu R.B.

Mineral Processing Division of Society of Mining,

Engineers of AIME, 2nd Edition, 1980. (American

Institute of Mining, Metallurgical and Petroleum

Engineers Inc)

i ) pp 248, equation 3, 3a and Table V pp 250

Rod mill

ii) pp 256, equation 4, 4a and Table VII pp 258

Ball mill

40

APPENDIX

EMPIRICAL EQUATIONS ( for power calculation ) :

Kwr = 1.752 Di ( 6.3 — 5.4 Vp ) fl\lc

3.2 — 3V ) fl\lc ( 1 0.1 Kwb 4.879 D°.3( 29 — Offic 4 SS

B --- 12.5D

\ 50.8

Kwr or Kwb , the power at mill pinion shaft ( Kw per tonne of rods or balls )

D, Diameter (m) inside liner of rod or ball mill

Vp , fraction of mill volume loaded with rods or balls

fNIc , fraction of critical mill speed

Ss , Kw per tonne of balls

B , ball size (mm)

WORK INDEX ( WI )

Work index Wi is kwh/ts required to break from infinite size, F =

Rallmill : Wi = 44.5/A, A = P9.23 k 10/v 10/Vr G(b). 82 --

Wi = 62/B , B = P( G0.625 )*23 10/1/ P — 10/T/

Wi , work-index ( kwh/ts for ball — or rod mill

P1 , mesh of grind (fz m )

P,F Product and feed size (u m ), 80% passing size

Gb Gr net gram produced per revolution

cc to P ------ 100 /.4 m

Power ( kw ) at mill pinionshaft is given by

W = Wi ( 10/1/V — 10/1/T), W ( kwh is ). Converting to tm by multiplying

a factor 1.1 we have ;

W=11 Wi (1/1/1.— 1/1,

EFFICIENCY / CORRECTION FACTORS :

Work-index ( Wi is caluclated for an overflow ball rod mill 2.44 m (id), closed

circuit wet grinding 250% circulating load ( C. L. ) with feed, F coarser than 4000 pm and

Ss = 1.102

41

1.3

1.2

( P + 10.3 ) / 1 .145 P

0.13

Rr 1.35

( 250 / C )0.1

( 2.44 I D )0.2

1 -i--

product, P not finer than 74 m. So for an industrial practice, efficiency/correction

factors are needed.

( . ) ball mill, ( — ) rod mill, ( —) ball and rod mill

ki corrections due to

1 ( )

2 ( )

3 ( . )

4 ( )

5 ( • )

6 ( )

7 ( — )

8 ( — )

—)

DISCUSSION

S. T. Kulkarni

NULL, Bombay

dry

open circuit

finer grinding than 74 m

low reduction ratio ( Rr <6 ) particulary

in regrinding circuit

C. L., Circulating load. C > 50 O/0

mill dia

F—F0 P oversize feed factor coarser than ( Wi — 7 )1

F0 F optimum feed (related to Wi )

F0 =16000 (13/Wi ) = x for rod mill

F0 = x for ball mill

1 ( Rr Rro )2 /150 , Rro 8 -f- 5 L/D high or low Rr , reduction rate

No equation for Rro 2 = Rr otherwise use it for low Rr and also for

high Rr ( when Wi ) 7 )

Rod mill only 1.4 0CC crushing feed OCC, Open circuit crushing

1.2 CCC crushing feed CCC, Closed circut crushing

rod and ball mill combined :

1.2 0CC crushing feed

No factor for CCC crushing feed

(a)

(b)

Question 1 : Please elaborate the change on

power consumption by conventional method

and by your method and how the saving is

achieved ?

Author : I have not mentioned any saving in

power consumption. The paper relates to the

statistical analysis of a set of data and forma-

tion of generalised equation which can be used

to size a mill.

M. S. Prakasa Rao

Manager (OD), R D, NMDC.

Question : Any attempt to design autogenous

grinding mill ?

Author . The equations are developed only

for bail mill and rod mill. We have not studied

the design aspects for autogenous grinding mill.

V. Venkatachalam

Maharashtra Minerals Ltd , Bombay.

Question : What is the definition of unit 150

to 200.

Author : These figures relate to the mesh size.

Ex: 200 mesh is equal to 74 micron (.074 mm).

These are some standard screen/sieve sizes.

There is nothing like 120 and 80 mesh in this series.

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