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Engineering Science, 1974, Vol. 29, pp. 5E9-599.

Pergamon Press.

Printed in Great Britain

A MATRIX THEORY OF COMMINUTION MACHINES W. J. WHITEN Julius Kruttschnitt Mineral Research Centre, Isles Road, Indooroopilly, Brisbane, Australia 4068 (Received 27 January

1973)

Abstract-Matrix differential equations are used to develop a theory of comminution machines that includes the effects of classification and predicts machine contents. Certain equations are found to be independent of classification effects. These equations indicate the importance of machine contents in experimental work. The relation between models of this form and energy consumption laws is investigated. A partial differential equation and boundary conditions are derived for a tumbling mill and a practical method for calculating the steady-state solution is given.

INTRODUCTION

Models of comminution machines have generally been one of three types. The first type describes in terms of the events which occur to the material being broken as it moves through the comminution machine. Usually models of this type have used discrete size distributions and matrix techniques. Selection for breakage, classification according to size and breakage product distribution matrices together with stages of breakage have been used in these models. Normally the effect of a distributed residence time is not included and the contents of the machine are not calculated. Typical examples of this approach are given in Broadbent and Callcott [ 11and in Lynch, Whiten and Draper [1 11. The second type of model uses the solution of the batch grinding equation and a residence time distribution to determine the products from the machine. The solution of the batch grinding equation either follows Kelsall and Reid[6] or uses direct integration (Luckie and Austin[9]). Models of this type describe the comminution machine product as the combination of sections of the feed which have been broken by an amount corresponding to their residence time. The third type of model describes in terms of the contents of fixed regions of the comminution machine. In the simplest case the model consists of a well mixed volume where breakage occurs and the discharge from this volume is not classified by size or type. Sometimes, particularly in analogue computer studies, a small number of these mixers are connected in series possibly with intermixing (e.g. Kelsall et a1.[7]). Classification effects are usually omitted (e.g. Jowett [5] and Luckie and

Austin [9]) however Stewart and Restarick[ 131 added classification to the last stage of a three stage analogue computer model, and Loveday and Tolmay [8] obtained (using continuous size distributions) special cases of some of the results in the first section of this paper by introducing classification of the discharge of perfect mixers. Horst and Freeh [4] describe a diffusion mixing model with classification but do not derive the boundary conditions. This model was considerably simplified before being solved for mill contents and products. Models of the second and third types generally use a rate of breakage per time function and both continuous and discrete size distributions have been used. Models of the third type can be converted to the second type provided classification is not present. Conversion to the first type of model can become coniplex and differences in definitions often preclude an exact conversion. This paper takes models of the third type and develops a general theory of comminution machines. The first attempt to obtain such a theory is reported in Lynch and Whiten[lO]. Matrix notation is used because of its compactness, however, the correspondence with continuous size distribution notation is given. The notation which follows that of the earlier paper, was chosen to avoid possible misinterpretation and conflict with existing definitions which have in the past been responsible for many arguments. PERFECT The

MIXING

MILLS

contents of the mill are described by a vector s, each element of which gives the amount in the mill of one component of the mill contents. Nor589

W. J.

590

mally, the elements of s contain the mass or volume in each size fraction and the water content of the mill, however if required, s may contain the size distribution of several types of particles, possibly even including particles of composite composition. The rate of change of the contents of the mill is as/at. This can be equated to the sum of the effects which are causing the contents of the mill to change. These effects are: 1. Removal of material for breakage. Some of this material may reappear in the same size fraction after breakage. The rate of removal for breakage is Rs where R is a diagonal matrix giving the breakage rate of each component of s. The actual values in R depend on the assumptions made about the appearance matrix A. This is discussed later. 2. Increase in the components of the mill contents due to receiving the products of breakage. The rate at which material appears from breakage is ARs. Each column of the matrix A gives the relative distribution after the breakage of the corresponding component of s. The values in A may be derived analytically, derived experimentally under various assumptions, or alternatively, approximate values are often assumed. 3. Flow of material into the mill. The rate of new material entering the mill is fi 4. Flow of material out of the mill. This material is obtained from the contents of the mill, however a classification of the mill contents can occur. Hence the product from the mill can be written as p = Ds where D is the diagonal discharge matrix which gives the fractional rates at which the components in the mill are discharged. These effects are added to obtain the equations of the perfect mixing mill: $=(AR-R-D)s+f

(1)

and p=Ds.

(2)

In obtaining these equations, no assumptions have been made about the quantities A, R, D and f. They may be functions of time, the mill feed rate or the mill contents as appropriate. An integral equivalent of Eq. (1) can be obtained, when s is a single distribution, by changing s to a continuous distribution function of the particle size x i.e. s(x). The summations in Eq. (1) become integrals using corresponding continuous functions for A, R, D and f. The integral form is:

WHITEN

as(x) -=

I -4x,

Y)~(Y)s(Y) dy - r(x)s(x)

at

I

-

d(x)s(x)+f(x)

(3)

The relationship to the matrix elements is follows: Let xi, ,, xi be the upper and lower size limits of the ith element of s. Then

1

aii= -Xi21-xi

*i+ta@, dy dx, I I ‘,+a

Y)

xi

I]

I,+, rii= - 1 r(x) dx, Xii1- xi I :, and d(x) dx

where the left sides are the elements of s, A, R and The functions a(x, y), r(x) and d(x) can be regarded as the more basic values describing the mill however the matrix notation generally provides a more convenient notation and a more direct relation to computational procedures. An integral form exists either explicitly or implicitly for all of the equations derived from Eq. (1). In the case of constant coefficients, Eq. (1) can be solved to give D respectively.

, e-‘“-R-D” f(T) d7) (4) where so is the contents of the mill at time zero. Appendix 1 indicates the definition of functions of matrices. If A, R and D are functions of time, the expression (AR -R - D)t in Eq. (4) should be replaced by ’ (AR-R-D)dt. I0 The second type of model mentioned in the introduction calculates the product from a continuous mill by taking the solution of the batch mill equation and applying a residence time function u(t). This gives a mill product of the form p(t)=

I’0

m(T)q(t, 7) d7

(5)

where q(t, 7) is the description of the contents of a corresponding batch mill which has been run for

591

A matrix theory of comminution machines time r after being filled with material described by f(t - T). The solution of the batch mill equation is Eq. (4) with f = 0 and D = 0. Hence (5) becomes

where the function v is the volume or volume flow of the argument as appropriate. The equation D = (v(f)/(0.25 T d*1))D’

I’

P(t) = 0 U(T) ecAn-n)Tf(t -7) dr.

(6)

Equations (2) and (4) give, when the initial contents can be neglected, p(t) = D

Equations

I0

dAR-R-D’Tf(t- 7) dr.

(7)

(6) and (7) are the same provided

where D’ is a constant diagonal matrix, is often useful for simulation. If D varies in a more complex manner with the operating conditions (e.g. s) an iterative solution to Eq. (9) and the relations definining D may be required to obtain the steady state behaviour. The parameters of the perfect mixing mill can be determined from steady state data as follows. DR-’ may be obtained from a mill feed and product from the relation p =DR-‘(I-A)-‘(f-p)

= D{I-

DT +fD2~2- f(DAR - ARD)T’+O(T*)}.

(8) This equation can only hold if D is a multiple of the unit matrix, in which case Eq. (8) reduces to CT(T) =

ke-“.

Hence a solution of the form of Eq. (5) holds when the initial contents of the mill can be neglected and there is no classification of the mill discharge. For steady state conditions Eq. (1) becomes (D+R-AR)s=f.

(9)

In the case of constant A, R and D, this is a set of triangular simultaneous equations for s and hence s is easily calculated. Equations (2) and (9) may be combined to give

(13)

(14)

provided the value of A is known or assumed and the submesh material has been omitted from the matrices and vectors. Additional information is required to separate the values of D and R. If the mill contents s are also measured D can be calculated from Eq. (2) and then R from DR-’ or the following equation Rs = (I- A)-‘(f -p)

(15)

where the submesh has again been omitted from the matrices and vectors. If sufficient independent feed and product size distributions are available, we can define the square matrices F= [fl, fi . . . . f”1,

S=[s,, sz.. . . s.1. p =D(D+R-AR)-‘f

(10)

which may be written p=DR-‘(DR-‘+I-A)-‘f

P=[PhPz....P.1. (11)

(16)

Using Eq. (9) then Eq. (2) gives R-AR=FS’-D

and if D is non-singular p =(I+RD-‘-ARF’)m’f.

and

(12)

Equations (11) and (12) show the interdependence of D and R for steady state feed-product relations. In the case of a constant volume mill D various with the feed rate. If D = PI, p may be calculated as

=(F’F-I)D

(17)

provided the inverse matrices exist. Equation (24) derived under more general conditions eliminates the matrix D from Eq. (17). In performing these calculations, it is desirable to choose feed conditions that allow accurate inverse matrices to be obtained. The use of a tracer component in the feed is one way of doing this, which also ensures a minimum

592

W. 3.

change in the operating conditions. A degree of freedom exists in each column when the components of R-AR are separated for we have

WHITEN

$=(AiR,-RikT

Q+T

Ckisi

(19)

for the feed (1 - ajj)ci = (R-AR), and

f =7

(18) - a;,ri,= (R-AR)ii

C”J”,

for i#j and for the discharge

where (R-AIQii is the ij element of R-AR. It can be seen that once u,~ is chosen, r,, and aii are determined. A convenient choice is ai,= 0, which corresponds directly with data from screening analysis. Another useful approach is to choose uii so that the more basic continuous function a(x, y) is smooth. Equation (18) provides the method of converting between appearance matrices derived under these different assumptions. When insufficient data is available to calculate the complete A matrix, it is usual to assume that the appearance distribution is similar for all particle sizes. This distribution may be determined experimentally or is often assumed to be of a standard form (Broadbent and Callcott [ 11).

(21) In the general case of steady state conditions have asj_O

at-

and A, R and C are not functions of time. Now, summing Eqs. (19) for i = 1 to II gives T (I-Ai)Rist=f-p

and if Ai and Ri are constant MULTIPLE

SEGMENT COMMINUTION MACHINES

Any comminution machine may be assumed to be divided into segments, each of which is perfectly mixed. However in some cases a large number of small segments may be required to obtain an accurate approximation. Using this assumption a model of the comminution machine may be constructed using the perfect mixing mill model of the previous section. If the comminution machine has n segments and the content of the ith segment is si, then the transfer of material from the ith segment of the jth segment is Ciisi where Cii is a diagonal matrix giving the fraction of s, to move to the jth segment in unit time. The C, may, of course, be a function of several variables e.g. s, or t. Two additional segments are introduced for the feed and the product to provide a uniform notation. The contents of these segments are calculated from f=IsO

and

p=Is,+,

where I is the unit matrix having units l/time. Now, the equation for the ith segment can be derived in the same manner as the perfect mixing mill equation was derived. Hence the equation for the ith segment is

we

R 2

(22) over i then

si=(I-A)-‘(f-p)

where the submesh material has been removed from the matrices and vectors. Equation (23) relates the contents of the comminution machine (Es,) to the feed and product. It assumes only steady state and constant A and R and hence applies to many comminution machines. Equation (23) provides a method of calculating R which is independent of classification effects. If a notation similar to Eq. (16) is introduced we have R-AR=(F-P)S-’

which provides a method of calculating dependent of classification effects.

(24) R - AR in-

RELATION BETWEEN ENERGY CONSUMFI’ION MODELS AND BREAKAGE MODELS The energy consumption ‘laws’ of comminution assume that the energy required to produce a certain size x from a standard size is some function of the size say w(x). Hence, an energy vector

w’ = [w(x,), w(xz) . . w(xJ1

may be defined.

593

A matrix theory of comminution machines

Hence, the energy relations can in this case be reexpressed in terms of the rate function. The power for an overflow ball mill is closely proportional to d*“i hence Eq. (26) implies

For the three laws of comminution Kick’s

law gives w(x) = - hs (x),

R a tiR’

(28)

Bond’s law gives where R’ is a constant diagonal matrix. However, for ball and rod mills even though approximate power-breakage relations are known to exist, it may be more appropriate to calculate power consumption directly from mill dimensions and the mill contents.

w(x)= l/G, and Rittinger’s

law gives w(x) Q l/x.

The energy required for comminution ‘&= w’p - w’f.

RELAXED CONDITIONS FOR PERFECT MIXING MILL EQUATION

is (25)

Summing Eq. (19) we find that the perfect mixing mill equation holds provided:

Hence, from Eq. (23) c % = w’(G A)Rs.

x [l, 1 . . . 11.

(26)

This equation provides a link between energy relationships and breakage relationships. If we can now assume the appearance function is similar for all sizes [i.e. a(x, y) = a(xly)l the product w’(l- A) can be simplified provided sufficient size fractions are introduced below the size being considered to make the effect of the submesh material negligible. This is done to overcome the problem of determining the energy vector component for the submesh material. From Kick’s law we obtain w’(l-A)

m [l, 1.. l]

and hence r,, = constant.

From Bond’s law we obtain

c

s,

(29)

and

If R 0: R’ this equation can be used to calculate the magnitude of R. Alternatively, for a given solids and water content in a tumbling mill, the power can be expected to remain almost constant, regardless of the size distribution of the mill contents. For this to happen, we require w’(l-A)R

(I-Ai)R,si=(I-A)R

(27)

c Ct.. tlsk = D 2 k

si.

(30)

Hence, we require from (29), either Ai and/or Ri independent of i or the mill contents to be well mixed, and from (30), discharge with equal probability from all segments of the mill. These relaxed conditions are often satisfied approximately and hence the perfect mixing mill can serve as an accurate approximation to a more complex model over a limited range. BATCH MILLS

For a batch mill f = p = 0. Hence, summing Eqs. (19) for i = 1 to n gives %

= Z(A;R, - R~s;.

(31)

As above, this equation reduces to the following, provided either A, and Ri are independent of i or the mill contents are well mixed. $=(AR-R)s

(32)

where s = Zsi. The solution to Eq. (32) for A and R functions of time only can be written

rii m fi, and from Rittinger’s law rii m xi.

CES

Vol. 29 No. 2-R

s(t) = exp U’ (AR-R)dts(O) I where s(O) is the distribution

(33)

of the total mill con-

594

W.

J.

tents at time 0. A form of this equation usually with constant A and R and one size distribution has been considered by many authors. If we put f = s(O) and p = s(t) and use the notation of Eqs. (16)

I’ (AR 0

-R) dt = log (RF-‘)

(35)

If the batch mill contains material with a range of breakage properties, it will be shown that average breakage properties do not always adequately describe the overall behaviour. For simplicity, a material with two components will be considered. If S=

4 [Iu

and A and R are partitioned nents of s, i.e.

A = [;::

negative exponentials the fitting of which is an illconditioned problem, the use of two components should give a close fit in almost all cases.

TUMBLING

(34)

and if A and R constant AR-R=+log(PF-‘).

WHITEN

to match the compo-

[

R’ o

MILLS

$1 $=

R=

MILLS AND VIBRATING

Tumbling and vibrating mills can be regarded as a single path from the feed to the discharge with breakage occurring at all points along this path. The material in the mill may move in both directions along this path. The equations for multiple segment machines can be applied and allow the following simplifications to be made. 1. If the mill is divided into n segments so that each segment is contained between two crosssections perpendicular to the mill axis, each segment can exchange material with only the two adjacent segments. 2. Since there is a continuous variation along the mill axis, the n segments can be converted to a continuous representation along the mill. The n segments are each of length Sa = l/n where 1 is the length of the mill and a the distance from the mill feed. Then from Eq. (19) for i = 2 to n-l

0 Rz

(AR - R)s;+ Ci_,,g-, - i&,si - CLi+,si + ci+,.isi+ I.

1

(37)

For i=l

then Eq. (32) becomes, after summing the two components of s,

%=(AR-R)s,+f-Cl,zs,+Cz.,s2.

(38)

and for i=n

v=(A”+A”-I)R’q+(A2’+A”-I)R’u. (36) The right side of Eq. (36) cannot be put into the form (A*R*-R*)(q+u) unless A* or R* is allowed to vary with q and K. Since in general the ratios of the elements of q, u and q + u change with the amount of grinding, no mean matrices can be found to describe the breakage properties of the combined material for both large and small amounts of grinding. However a mean matrix can give a good fit over a limited range. Since the amount of material in a size fraction during breakage is typically described by a sum of

g=(AR-R)s.+CL.wC

n.n_,S.-P

(39)

where p = & Ds.. The interchange of material between and the ith segment of the mill is

the (i- 1)th

ci_,.isi_, - ci.i_,si = (CiM.i - c,,i_,)S,_, - ci.i-!(Si_ S,?l)

(40)

The first term, as it is the component derived from the contents of the preceding segment, is the flow of material through the mill and can be written

595

A matrix theory of comminution machines

& Gsi-,

$=(AR-R)u--&Gu+$H$

where G is the diagonal classification matrix for the flow of material through the mill. The second term in Eq. (40) is a multiple of the difference between the contents of the two segments and is tending to bring the contents of the two segments closer together. This is the mixing term and can be written as --!-HZ

if terms of order &Y*are neglected. The diagonal matrix I-I is the mixing classification matrix. If we put II(u;)=&

(41)

f= Gu(O)- Hz, and p = D*u(l) = Gu(l)-

H$$?

Note that A, D*, R, G and H may be functions of t, u or (Y as may be appropriate. Steady state conditions are obtained when au/at =0 and the coefficients are independent of time. In this case making the substitution q=Ru

s,

allows Eqs. (41)-(43) to be expressed so that u(a) is the distribution of material along the mill and oi the distance of the ith segment from the mill feed, Eq. (40) becomes C,-,.isi-,-C_1si

= Gu(u,-J-H?.

When this term is regarded as the flow rates through a cross-section of the mill it is clear that the right side is independent of &X.Equations (37)-(39) can now be written ~~~~i)_(AR_R)u(n3_G~(~i))~(ai-~)

+

H -au(W) sol

each of which is a classification over a breakage rate constant term and hence, the mill product can be expressed without separating the effects of classification and breakage rate constant. This is the same as was found for the perfect mixing mill. Also, under steady state conditions, an equation corresponding to Eqs. (15) and (22) may be obtained by integrating Eq. (41) from 0 to /, i.e.

For constant (41) becomes

’

in terms of

GR-‘, HR-’ and DR.’

- “’ (AR-R)uda I

H ___ WC%) aa Sa

(43)

=f-p.

coefficients

and steady state, Eq.

Hg-Gg+(Al-R)u=O.

(44)

and Eqs. (42) and (43) remain as the boundary conditions. If there is no mixing, i.e. H = 0, Eq. (34) has the solution and u((y) = G-1 e&nR@f sa -= au@“) at

So(AR -R)u(cY.)+

(45)

Gu((Y.-,)

provided D* = G and then we have p=e

where D* = ID. Letting &Ygo to zero gives the partial differential equations and boundary conditions for tumbling and vibrating mills, i.e.

I(A-WfG-~ f.

(446)

Note that when G = H Eq. (46) is the same at the solution of the corresponding batch equation. Otherwise an apparent breakage rate constant RG-’ exists that will predict the same results in the batch

W.J. WHITEN

596

mill equation. Using this apparent breakage rate an equation corresponding to Eq. (35) can be obtained. An explicit solution to Eqs. (42)-(44) has been obtained when H is a multiple of G (Whiten [ 141) but is too complex to be of practical use. The following obtains a similar solution using factored matrices and is suitable for numeric computation. We put H=mG’,G=gG’,(I-A)RG’-‘=LEL-’

L-‘G’u

(48)

in Eq. (44) gives * m$-$--gg-Ev=O

(55) vi = ci+vi++ c,- vy

(56)

i-l pi = bi c lijv,+ b,v,. j=l

(57)

(47)

where E is a diagonal matrix and L is lower triangular with unit main diagonal. Putting v =

and the values vizi and pi are obtained from

(49)

The subscripts indicate the elements of the corresponding vector or matrix. Using Eqs. (53)-(57) v+v- and p can be determined starting from the coarsest element. The mill content can then be determined from Eqs. (48) and (50). The solution of Eqs. (42)-(44) can be shown to reduce to Eq. (46) when the mixing goes to zero and to Eq. (10) when the mixing goes to infinity. Numerical calculation has confirmed this. Corresponding to Eq. (13), a mill of approximately constant volume can be obtained by putting

which has the solution g = v(f)/(0.25rd2) v =e~Q+v++e"Q-v-

GOI

where

(58)

and D” = gD’

(59)

EXAMPLE

(51) Q-

=

-A

E(Q+)-’

and v+ and v- are constant vectors which must be chosen to satisfy the boundary conditions (42) and (43), i.e. L-‘f=gv(O)-rn?

L-‘D*G’-‘Lv(l)

= gv(l)-rn$$.

Taking the ith component

(52)

CONCLUSIONS

of these (53)

i-l

i-l

bi C li++- C liizi = (g - mq; j=! i=,

bi)cTvi

+(g - mqii - bi)ci-viwhere bi = dtJg{, ,-: = elq:i, ci_= e14T’

A rod mill is considered to have internal classification and a different discharge classification (Meyers and Lewis[ 121). The simplest model to include these effects is Eq. (44). Table 1 shows how Equations (42)-(44) together with Eqs. (58) and (59) can reproduce rod-mill data extracted from Meyers and Lewis [ 121. To obtain a complete model of rodmill behaviour it is necessary to relate the model parameters to the operating conditions using techniques similar to those described in Whiten[lS]. The classification effects in a rod mill have been shown by Draper and Lynch[2] to vary with the feed rate to the mill.

(54)

Most of the equations presented in this paper are suitable for simulation of comminution plants and hence will find use in the optimisation of existing plants and the design of new plants. The perfect mixing model with discharge function is as simple as any breakage model and yet the relaxed conditions for its application show that it applies over a limited range to many types of comminution machines. For example Wickham [ 171used it to predict the product from an industrial pebble mill and Whiten [lo] used it (expressed in different terms)

(%I

48.5 27.4 7.9

3.3 2.0 2.0 2.4

4.8 1.7

9400 4699 2362

1168 589 295 147

0 74

12.6 6.8

11.9 11.5 13.4 12.4

7.0 13.0 11-4

12.6 I.4

11.6 13.9 13.8 11.2

9.3 10.5 9.7

talc.

Section 1

Data

19.1 9.3

11.3 13.8 16.1 18.1

0.0 5.4 7-o

18.2 10.9

9.7 15.8 18.5 16.0

1.8 3.6 5.5

Calc.

Section 2 Data

24.5 11.7

5.7 12.6 21.3 20.2

0.0 1.7 2.2

- .-

-

22.8 13.4

14.5 20.3 18.8

0.4 1.1 ;::

Calc.

Section 3 Data

Measured data is abstracted from Meyers and Lewis[l2].

Feed

Size

(w)

24,3 12.0

13.3 22.5 20.8

0.0 1.0 :.;:

Data

25.0 14.4

5-o 13.5 20.6 19.7

0.1 0.4 1.4

Calc.

Section 4

28.3 13.7

28.4 16.0

2.1 11.2 20.8 21.4

0.0 0.0 0.1

Calc.

Product

1.6 10.3 22.5 23.6

0.0 0.0 0.0

Data

Table I. Comparison of measured and calculated rod mill data (m = 5.8) A

0.0086

0.1225 0.0654 0.0338 0.0172

0.1980 0.3308 0.2148

column

D

5.0 4.9

1.4 3.3 4.3 4.8

0.0 0.0 0.2

diag.

G’

4.2

4.0 4.0 4.1 4.1

3.8 3.9 3.9

diag.

5.0

86.0 37.0 15.0 8.0

273.0 215.0 155.0

difg.

P

; p: &

/$ _. 3 c 5

0,

s

!Z

.g.

> 3

W. J.

WHITEN

for industrial cone crushers. The solution for the tumbling mill, Eqs. (42 - 44) require less computation than many less general models and if the matrix factors (L and E) can be kept the computation involved is of the same order as the matrix models of Lynch et a/.[1 11. It is not always easy to obtain samples of the contents of an industrial comminution machine, however such samples provide very useful information. Residence time data (which may be equally difficult to obtain) provide similar information in the absence of classification effects but otherwise, are not so easily related to the model parameters. When the contents are not available, the models can be expressed in terms of the ratio of classification and breakage effects. The general equation for multiple segment comminution machines can be used to derive a partial differential equation in three dimensional space and boundary conditions in a similar manner to the equation for tumbling mills. At the current state of grinding theory it appears that this partial differential equation is of little practical use unless it is considerably simplified and, in this case, the resulting equation could equally well be derived directly from the results in this paper.

E f

598

F :

G’ H

k 1 L m P

P R

Acknowledgements--Several persons helped in the development of this work. Dr. A. J. Lynch and Dr. T. G. Callcott provided many useful discussions on this work and Dr. T. G. Callcott assisted with the choice of notation. Dr. Austin provided useful criticism of earlier work. Financial assistance was provided by Mount Isa Mines Limited and the Australian Mineral Industries Research Association.

R’ s

NOTATION

;

Ri

A

appearance distribution matrix (or breakage matrix). The ith column gives the average distribution, in terms of the components of the material, of the ith component when it reappears after it has been selected for breakage Ai appearance distribution matrix for the ith segment c, transfer factor matrix. A diagonal matrix giving the fractional rate of transfer of material from the ith segment to the jth segment d diameter of tumbling mill D discharge factor matrix. A diagonal matrix with dii giving the fraction of the mill contents going to the mill product D* the discharge matrix for a tumbling mill (= ID) D a proposed constant part of the D matrix $ the power used in comminution

t u(a)

V

w(x)

W a V

m

a factor of (I - A)RG’-’ ( = LEL-‘) the feed flow rate vector. Each element gives the flow rate of one component of the feed into the mill a matrix with f vectors as columns scalar flow factor (G = gG’) the flow factor matrix. A diagonal matrix containing the flow factors for each component for the calculation of the flow along vibrating and tumbling mills a proposed constant part of the G matrix the mixing factor matrix. A diagonal containing mixing factors for each component for the calculation of mixing in vibrating and tumbling mills a constant the length of the vibrating or tumbling mill a factor of (I - A)RG’-’ ( = LEL-‘) scalar mixing factor (H= mG’) the product flow rate vector. Each element gives the flow rate of one component out of the mill a matrix with p vectors as columns breakage rate constant matrix. A diagonal matrix with riigiving the fractional rate at which the ith component in the mill is being broken breakage rate constant matrix for the ith segment a proposed constant part of the R matrix the mill contents vector. Each element gives the amount of one component in the mill the mill contents vector for the ith segment a matrix with independent s vectors as columns time the distribution of mill contents vector. Each element gives the distribution along the mill of one component of the contents of a vibrating or tumbling mill transformed contents of tumbling mill = L-‘G’u the weighting factor for size in power calculation the weighting vector for power calculation the distance from the feed end of a vibrating or tumbling mill volume or volume flow (as appropriate) of the vector argument the residence time distribution function REFERENCES

Ill BROADBENT S. R. and CALLCOTT T. G., J. Inst. Fuel 1956 29 524, 1957, 30 13.

A matrix theory of comminution PI DRAPER N. and LYNCH A. J., Proc. Aust. Inst. Min. Met. 1965 213 89. ]31 GANTMACHER F. R., The Theory of Matrices. Chelsea, New York l%O. ]41 HORST W. E. and FREEH E. J., Trans. Am. Inst. Min. Engrs 1972 252 160. ]51 JOWET? A., Mineral Sci. Engng 1971 3 33. ]61 KELSALL D. F. and REID K. J.. Inst. Chem. Ennrs: Symp. Ser. No. 4, paper 4.2. Inst.‘Chem. Engng Lendon 1965. r71 KELSALL D. F., REID K. J. and STEWART P. S. B., ZFAC Symposium, Sydney pp. 205-218, 1968. @I LOVEDAY B. K. and TOLMAY A. J., Chemica 70 141-152, Melbourne 1970. r91 LUCKIE P. T. and AUSTIN A. J., Minerals Sci. Engng 1972 4. 24. HOI LYNCH A. J. and WHITEN W. J., 34th Annual Meeting Am. Chem. Sot., 1%7. 1111 LYNCH A. J.. WHITEN W. J. and DRAPER N.. Trans. Znst Min. Met. 1967 76 C 169. 1121 MEYERS J. F. and LEWIS F. M., Mining Technol. 1946 T.P. 2041. ]I31 STEWART P. S. B. and RESTARICK C. J., Proc. Aust. Inst. Min. Met. 1971 239 81. [14] WHITEN W. J., Theory of comminution machines. Internal report Julius Kruttschnitt Mineral Research Centre (unpublished) 1971. [15] WHITEN W. J., Symposium on Automatic Cintrol Systems in Minerals Processing Plants. pp. 129-148, 1971 (Aust. Inst. Min. Met. Southern Queensland Branch: Brisbane). [16] WHITEN W. J., .Z.S.A. Inst. Min. Met. 1972 72 257. [17] WICKHAM P., Julius Kruttschnitt Mineral Research Centre Technical Report Jan.-June 1972 (unpublished).

APPENDIX 1 FUNCTIONS OF MATRICES

If a power series expansion of the function exists, a matrix may be substituted in that series provided the resulting series converges.

599

machines

e.g. e” =Z+A+hA’... holds for all A. If A = LEL-’ where E is a diagonal matrix (see Appendix 2) then f(A) = Lf(E)L-‘. A function of a diagonal matrix can be obtained by taking that function of the diagonal elements i.e.

L

f(e.3.l.

Further details are given in Gantmacher[3]. APPENJMX 2 FACTORING A LOWER TRIANGULAR MATRIX

If X= LEL-’ where X and L are lower triangular matrices, L has ones on the principal diagonal, and E is a diagonal matrix then L and E are calculated from

provided eii# e,#.For comminution calculations, if ei,= eii, it seems sufficient to make a slight symmetric adjustment to the breakage rates of the offending rows. Alternatively, the limit, as this adjustment goes to zero, of the calculated vectors can be found numerically.