Chapter 10. Mathematical Modelling in Comminution 10. INTRODUCTION The process of comminution has been observed and studied over the years. Statistical correlations of variables have been used to develop mathematical models describing unit and integrated operations. The approach has necessarily been mechanistic, [1]. With a better understanding of the processes and application of basic laws of physics, mathematical models have been developed to describe operations more fully. The developed models help to simulate the process. With the aid of computers a range of options can be easily simulated and optimum conditions of operations and circuit designs determined with relative ease and accuracy. This technique provides rapid answers on plant designs and optimum conditions of operation under specific plant conditions. The basic approach in the modelling of comminution systems is to recognise the fact that all the comminution processes accept ore and imparting energy, disrupt the binding forces between particles constituting the ore. Depending on the process used, either a single impact or multiple impacts are applied till disintegration and size reduction takes place to acceptable values. In the metallurgical industry, size reduction commences as soon as rocks are mined or quarried. By the very nature of mining and quarrying operations, the material consists of a range of sizes. We have seen in the case of crushers that a fraction of mined material could be too small and pass through a breaking device without further size reduction. The remainder has to be subjected to a comminution process. Each individual particle larger than the size required has to be broken down. The fragments from each particle that appear after first breakage again may consist of a fraction that requires further breakage. The probability of further breakage would depend on the particle size distribution in the breakage product and also would be machine specific. Comminution is a repetitive process. It is continued till all the particles in a certain size fraction have been broken down to an acceptable size. Thus the design of the equipment used to break particles and the duration the material remains in the breaking zone controls the ultimate size of the product. Standard sieves are used to obtain quantitative estimates of the extent of size reduction. Usually the Tyler standard sieve series, with a ratio of 1/V2 between each sieve, is used. This nest of sieves has been accepted as the international standard. In modelling comminution systems, the basic idea is to obtain a mathematical relation between the feed and product size. It is necessary to take into account all the variables involved in the operation including the machine characteristics. The process of comminution is considered to be represented by two processes: 1. a particle is selected for breakage, 2. a broken particle produces a given distribution of fragment sizes. The distribution of sizes produced from a single breakage step is known as the breakage or appearance function. It denotes the relative distribution of each size fraction after breakage.
Transcript
Chapter 10. Mathematical Modelling in Comminution
10. INTRODUCTION
The process of comminution has been observed and studied over the years. Statisticalcorrelations of variables have been used to develop mathematical models describing unit andintegrated operations. The approach has necessarily been mechanistic, [1]. With a betterunderstanding of the processes and application of basic laws of physics, mathematical modelshave been developed to describe operations more fully. The developed models help tosimulate the process. With the aid of computers a range of options can be easily simulated andoptimum conditions of operations and circuit designs determined with relative ease andaccuracy. This technique provides rapid answers on plant designs and optimum conditions ofoperation under specific plant conditions.
The basic approach in the modelling of comminution systems is to recognise the fact thatall the comminution processes accept ore and imparting energy, disrupt the binding forcesbetween particles constituting the ore. Depending on the process used, either a single impactor multiple impacts are applied till disintegration and size reduction takes place to acceptablevalues.
In the metallurgical industry, size reduction commences as soon as rocks are mined orquarried. By the very nature of mining and quarrying operations, the material consists of arange of sizes. We have seen in the case of crushers that a fraction of mined material could betoo small and pass through a breaking device without further size reduction. The remainderhas to be subjected to a comminution process. Each individual particle larger than the sizerequired has to be broken down. The fragments from each particle that appear after firstbreakage again may consist of a fraction that requires further breakage. The probability offurther breakage would depend on the particle size distribution in the breakage product andalso would be machine specific.
Comminution is a repetitive process. It is continued till all the particles in a certain sizefraction have been broken down to an acceptable size. Thus the design of the equipment usedto break particles and the duration the material remains in the breaking zone controls theultimate size of the product. Standard sieves are used to obtain quantitative estimates of theextent of size reduction. Usually the Tyler standard sieve series, with a ratio of 1/V2 betweeneach sieve, is used. This nest of sieves has been accepted as the international standard.
In modelling comminution systems, the basic idea is to obtain a mathematical relationbetween the feed and product size. It is necessary to take into account all the variablesinvolved in the operation including the machine characteristics.
The process of comminution is considered to be represented by two processes:
1. a particle is selected for breakage,2. a broken particle produces a given distribution of fragment sizes.
The distribution of sizes produced from a single breakage step is known as the breakage orappearance function. It denotes the relative distribution of each size fraction after breakage.
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The breakage function is often found to be independent of the initial size but is not necessarilyso. In the matrix form it is written as a lower triangular matrix.
The probability of breakage of certain particle sizes will be greater than others as they passthrough the breakage stage. Thus selective breakage action takes place and the resultingproportion of particles broken from a size interval is known as the selection function orprobablility of breakage.
Using these concepts, mathematical relations between feed size and product size, aftercomminution, have been developed. The development of such relations for specific machinesis described in this chapter.
10.1. Basis for Modelling Comminution SystemsAll comminution processes impart forces to break and reduce ore size. Provided the total
breaking energy imparted is greater than the bonding energy between individual particles, theparticles disintegrate producing a distribution of smaller sizes. The breakage usually startsfrom a point (or area) of stress concentration and propagates within the particle along planesof weakness. The disintegration breakage could be either along cleavage planes or inter-granular. Forces responsible for abrasion and chipping also play an important part as alreadydescribed for tumbling machines.
An attempt to illustrate the mechanism of breakage is made in Fig. 10.1 where the sizeanalysis of a single breakage event is given in column 3 (this is the breakage function). Forsingle particles present in each size fractions 1, 2, 3, N , the application of force is shownby solid arrows and the conceptual stress lines are indicated by the dotted lines. Themovement of fragments to the same or lower sizes is indicated by the dotted arrows. Column1 also shows the size distribution of the feed, with particles of smaller sizes represented ingrey. Column 4 shows the product from breakage after n number of size reductions. Rows 1,2, ..., N show the progenies from the single size fraction of size 1. It can be seen that the massof size 1, when broken, distributes itself into other size fractions. The distribution becomesmore complicated if fragments of breakage from other sizes in the feed are included. At somestage the size fraction in size 1 will disappear, as the particles sizes are distributed to smallersizes. The process continues to n number of breakages. It should be noted that the total massremains the same.
To identify the products in different size fractions two conventions have been adopted.Firstly, the mass fraction of particles remaining in size 1, after breakage of particles of size 1is designated as bii. Similarly for material broken from size 1 into size 2, the mass fraction isb21 and so on. Thus b3,i, b^i etc. to DN,I represents those particles present in the 3rd, 4th... ,Nl sieve size intervals obtained from breakage of particles of size 1. Similarly, the breakagefrom size 2 of the feed will be recognised as b2,2, b32 ... bN,2 and so on. In the general case,when one particle in size j of the feed is broken into the ith size fraction of the product it isdesignated as by.
The second convention advocated by Austin et al. [2] is to record the cumulative amountpassing each sieve instead of that retained. This is represented as By where i and j have thesame convention
The form of the breakage function is shown in Fig. 10.2 for two types of materials.
10.1.1. Estimation of the Breakage FunctionSeveral workers like Broadbent and Callcott [3], Gaudin and Meloy [4], Kelsall and Reid [5],Klimpel and Austin [6], Lynch [1], Austin et al. [2] have attempted to describe the breakage
257
0.01 0.1 1.0
100
10
g nissap % evital u
muC
Relative Size
hard ore
soft ore
257
breakage of singleparticle along cleavageplanes
product after singlebreakage.the breakage function
product after N breakageactions
Fig. 10.1. Representation of the distribution of particles after breakage. Solid arrows represent theapplied force for breakage and dotted arrows indicate the distribution of fragments ofbreakage to the same or lower sizes. The fragments shown represent breakage of anoriginal single size particle.
100
E
d10
0.01 0.1
Relative Size
1.0
Fig. 10.2. Breakage distribution function of hard and soft ore.
258258
function mathematically. Klimpel and Austin's formula encompasses most of the others and isgiven below:
B ( d , ) = l - Htl i-K~"2
(10.1)
where dj = is the original size being broken,di = size of the progeny fragment of breakage,nj-n3 = constants depending on the particle shape and flaw density within
the particles andB(di) = the cumulative mass fraction finer than di where dj > d; > 0.
The expression used by Broadbent and Callcott [3] is much easier to use to determine thedistribution of particles after breakage. In this case, if dj is the size of the original particle,which is subjected to size reduction and B(dj) the fraction of particles less than size dj thenthe size distribution of the breakage products is given by the breakage matrix which allowsthe determination of the individual elements:
B(d,) =1-e1
1-e"= 1.58 1-e (10.2)
This expression is material independent and hence can only be an approximation. Ingeneral, a standard breakage function of this kind have proved to give reasonable results forrod and ball mills where breakage is primarily due to impact and shatter but is not adequatewhere attrition is important such as in autogenous mills [7].
hi defining the matrix an implicit assumption is that the particles of different sizes arebroken in a similar manner (normalised breakage) and that no agglomeration takes place. B isan N x N matrix where the elements of B denotes the proportion of material that occurs in thatparticular size range after breakage. As no agglomeration is assumed, it is obvious that theelements above the diagonal will be zero. Thus B may be written as a lower triangular matrix:
(10.3)
If we consider the standard sieves in a 1/V2 series then the numerical values of thebreakage function would be:
0.1004, 0.1906, 0.1661, 0.1361, 0.1069, 0.0814, 0.0607 etc.
Example 10.1 illustrates the use of Eq. (10.2) for estimating the breakage function.
B,B2
B3
BN
0B,B2
BN_i
00
B,
BN.
259259
An alternate method of determining the breakage function has been advocated by Napier-Munn et al. [8] and already discussed in Chapter 3. In this method the relative sizedistribution after breakage is plotted against the cumulative percent passing and the Tio indexdetermined. A plot of the breakage index TJO (fraction or %) against TN (fraction or % passing1/N* of the parent size) gives the material breakage function. This relationship is expressedas [9]:
TN = l - ( l - T 1 0 ) ^ l J (10.4)
where a = a material specific parameter.
Another simple method to determine B values in batch grinding is to take an appropriateamount of a sample of ore of one size fraction, grind for a short time and determine its sizedistribution by sieve analysis. The sample is then returned to the mill and the tumbling actionrepeated for different lengths of time. After each time interval, samples are taken out of themill and a size distribution determined. A compromise is made between too short a grindtime to provide insufficient mass for accurate sieving and too long a grind time to giveexcessive re-breakage of fragments. A correction is applied to account for any re-breakagethat does occur. For example, the BI or BII methods of Austin and Luckie [10] may be used.B values may be normalisable and reduced to a vector.
In the BII method, re-breakage of primary progeny fragments are compensated for byassuming that the product of the breakage rate function and the breakage distribution function,SjBij, is approximately constant. Under these conditions, the breakage distribution function isobtained from:
B,,= ) ^ 4 (10.5)log l ^ i
U-P2(t)
where Pj(O) = cumulative mass fraction less than size dj at time 0,Pi(t) = cumulative mass fraction less than size dj at time t, andBjj = cumulative mass fraction of particles passing the top size of interval i from
breakage of particles of size 1.
The recent trend is to use back-calculation methods to obtain the B values. This method ofcalculations is beyond the scope of this book, the interested reader is directed to the originalwork by Austin et al. [2].
Example 10.1This is a numerical example of the methods of calculating the breakage function from a sieveanalysis of single particle breakage in a laboratory mill, ground for 2 minutes. For breakage ofsingle size, -2400 + 1200 microns, the sieve analysis is:
260260
Interval -Size, nm 2400Feed % retained
120072.3
60016.5
3008.5
1501.5
750.3
-750.9
1. Broadbent and Callcott methodThe breakage function is described in Eq. (10.2) as:
B(dj) is the proportion of material of initial size dj that is less than di in the product. The sizedi refers to the top size of the interval. That is, &.% is the top size of the interval -1200 +600jim in this example. The initial size di (i.e. j =1) is the actual average particle size for the topfraction (-2400 +1200 urn) and is the geometric mean of the two sizes. That is:
di = (2400 x 1200)a5 = 1697 pm
From Eq. (10.2),
fori = l,di = dj, B(di) = 1
for i = 2, d2 = 1200, tfe/di = 1200/1697 = 0.70713
then B(d2) = V ^ = K ' = 0.80197l-exp(-l) 1-0.36788
Similarly for other values of i:
Size interval dj B(dj)123456
1697120060030015075
1.00.801970.471130.256330.133830.06839
Since B(dj) is the proportion less than size dj, the proportion broken into size dj is obtained bysubtracting B(di) from B(dj_i) or from 1 in the case of the top size. That is:
B(U) = 1 - 0.80197 = 0.1980
B(2,l) = 0.80197-0.47113 = 0.3308
B(3,l) = 0.47113-0.25633 =0.2148
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B(4,l) = 0.25633-0.13383 =0.1225
B(5,l) = 0.13383-0.6839 =0.0654
B(6,l) is the material broken into size interval 6, that is, less than 75 microns. This remainderis the 'pan' of the size distribution and is given by 1 - sum(B(i,l)) = 0.0685.
Size interval123456
d,1697120060030015075
B(dO1.0
0.801970.471130.256330.133830.06839
B(ij)0.19800.33080.21480.12250.06540.0685
The B matrix can then be written in the following form ('pan' fraction not included):
0.19800.3308
0.2148
0.1225
0.0654
00.1980
0.3308
0.2148
0.1225
00
0.1980
0.3308
0.2148
00
0
0.1980
0.3308
00
0
0
0.1980
2. JKMRC methodIf values of Ki and K.2 are 45.4 and 1.15 respectively and the specific comminution energyfrom a drop weight test was 2.8 kWh/t then from Eq. (3.45):
T1 0=45.4(l-e-1 1 5 < 2 8 ))= 43.6%
From Eq. (10.4) for a value of a = 0.75 the following set of TN values can be calculated fromthis Tio value for Y = 1.697 mm:
Size(mm)1.20
0.6000.300
N1.42.85.7
TN
0.9970.8490.609
bi j
0.0030.1480.240
bijcorrected
00.1480.241
Size(mm)0.1500.0750.038
N11.322.645.3
TN
0.4040.2570.159
bi,i0.2050.1470.097
bycorrected
0.2060.1470.098
Column 4, the breakage function (by), is obtained by subtracting the cumulative % passingmass fraction (TN) from the previous row (TN-0- The value of bi i should be zero as noproducts of breakage should report to size 1 from the breakage of any size. If this value is notzero (0.003 in this case) then bii is made equal to zero and the subsequent values adjusted bydividing each term by (1-bii) = (1-0.003) = 0.997 (the total mass minus the mass in the firstinterval).
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3. Austin and Luckie methodIn this procedure, a single size ball mill feed in ground for a short time (1-3 minutes). Theproduct is sized and provided no more than about 30% of the top size mass is broken, then thebreakage distribution function can be estimated from Eq. (10.5) (BII method), as shown in thetable below.
Interval
123456
Passing size|xm
2400120060030015075
Retainedsize um
1200600300150750
Feed %retained
72.316.58.51.50.30.9
Cumulative %passing
10027.711.22.71.20.9
Bu
11
0.3780.0870.0380.029
K.0.0000.6220.2910.0490.0100.029
For the data in the table above, P2(0) is the fraction of material in interval 2 before grinding(screening only) and should be small. This is the sieving error and in this example, we'llassume it is equal to 0.01. All other values of P;(0) should be zero. P2(2) is the cumulativefraction passing the top size of interval 2 after 2 minutes of grinding and is equal to 0.277 inthis example. Similarly for interval 3, P3(2) is equal to 0.112. The first term Bi,i is equal to 1by definition, since when a particle breaks it is assumed to fall to smaller sizes. Similarly,B24 is also equal to 1 as particles broken from size interval 1 must be less than the top size ofinterval 2 (which is the bottom size of interval 1). The first calculation then is for B3>i:
log
loi
B3l, = > ( = 0.378
1-0.277.
1-0log
and B41 = ) : (- = 0.087 and so on.J-0.0277
log,\ 1-0.277,
The retained form of the breakage function, by is calculated as before by:
b u = B u - B l + u
For example: b31 = 0.378 - 0.087 = 0.291
10.1.2. Estimation of the Selection FunctionWhen an ore or rock sample is charged to a breakage system, it contains particles in severalsize ranges. During breakage, the probability of breaking the larger sizes within a size
263263
fraction is considerably greater compared to the smaller sizes. That is, a certain proportion ofparticles within each size range is preferentially reduced in size. Thus selective breakageoccurs within a size range. The proportion of particles within each size range that is broken isrepresented by S. Thus Si, S2, S3 ... SN will be the fraction of material in each size fractionthat would be selected for size reduction with the remaining particles passing through withoutany change in size. This is known as the selection function (or the specific rate of breakage)which can be mathematically expressed as a diagonal matrix where each element of thematrix represents the proportion of particles that has the probability of breakage.
hi a batch grinding process, if the total mass load in the mill is designated as M, the massfraction of size i in the mill load is mi and the specific rate of breakage (or the fractional rateof breakage or the mass of size i broken per unit time per unit mass of size i present) is S;,then for a first order breakage process:
Rate of breakage of size i oc mass of size i = mj(t) M (10.6)
or _dk(t)M] = _ S m ( t ) M ( 1 0 7 )dt
where Sj = the proportionality constant andnii(t) = the mass fraction of size i after a grind time, t.
Since the total mass is constant in a batch mill and if Sj is independent of time, then onintegration, Eq. (10.7) becomes:
m;(t) = mj(0) exp (-Sj t) (10.8)
or taking logs:
logm1(t) = l o g m 1 ( 0 ) - ^ (10.9)
A plot of log mj(t) against grind time, t, should give a straight line of slope (-Sj/2.303) asshown in Fig. 10.3.
10.2. Mathematical models of comminution processesThe approaches more commonly accepted for the modelling of comminution processes are:
1. The matrix model,2. The kinetic model,3. The energy model
The technique adopted for developing a model is to establish a material balance ofcomponents and an energy balance of the comminution system. The material balance of acomminution system in operation may be stated as:
[Feed in + Breakage] = [Total Product out] (10.10)
264
0.01
0.1
1
0 1 2 3 4 5 6
Grind time, t (mins)
)0(im/)t(i
m
-Si2.303
264
0.1 -
0.01 -
- = * - —
j ~~-\
I ;si[_ 2.303 -
0 1 2 3 4Grind time, t (mins)
Fig. 10.3. First order plot for breakage rate determination from Eq. (10.9).
and the energy balance as:
Energy input
(for breakage)
["Energy transmittd to the
[particles for breakage
[Energy transformed as 1
[heat and sound energies](10.11)
The transformation of input energy to produce heat and sound energies are often very smalland therefore always neglected in the energy balance equation.
A fundamental assumption in the approach is that the residence time of particles in the millis the same as if the entire charge is mixed thoroughly and is uniform. Hence the approach isknown as the perfectly mixed model.
The mass balance and the energy balances give similar results.
10.2.1. Matrix ModelLynch [1] expressed the relation between the selection function S and feed analysis using amatrix model representing the feed and product size distributions as N size ranges. The matrixwas developed by assuming Si to be the proportion of particles within a sieve fraction, i, thatwould break preferentially (the others being too small). Representing the feed sizedistribution by the matrix F, the fraction that would selectively break would be S.F. Thus ifFi, F2, F3...FN are the masses of material in each size fraction and Si, S2, S3...SN are theproportion of particles that have the probability of breaking in the corresponding size intervalsthen according to Lynch, the breakage process can be written in the matrix form as:
265265
Size Feed
1
2
3
4
N
to"
F2
F3
F4
FN
9
's,0
0
0
0
Selection function
0
S2
0
0
0
0
0
S3
0
0
0
0
0
s4
0
... 0
... 0
... 0
... 0
••• S r
Mass of particles broken
F,
F2
F3
F4
s,s2S3
s4
(10.12)
Thus we see that the mass of the product from breakage of selected particles will be:
Mass of product = S . F (10.13)
and the mass of unbroken particles will be (I - S • F) where I represents the identity matrix.The total product of breakage will be the sum of the broken and the unbroken particles.
The broken particles will have a distribution, B, the breakage function. The breakage functionB is for all the particles actually broken, and therefore the product fragments from breakagecan be represented by B«S«F. The entire breakage operation being a sum of broken andunbroken particles can now be expressed by the general equation:
= B . S . F (10.14)
Eq. (10.14) is the mathematical relation between the feed, breakage and product. It servesas the basis of mathematical models that describes the process of comminution of a particle.This equation gives a relation between the feed and the product from known matrices ofbreakage and selection functions. In actual breakage operations however, the product issubject to some classification (either internal to the breakage unit or external) and theclassifier oversize is combined with the original feed to form a new feed for the next breakagestage. The character and composition of the input feed changes and so does the B and Svalues. The situation is easily understood by examining Fig. 10.4 where the contribution ofthe oversize fraction from the classifier to the feed process is clearly illustrated.
The oversize fraction is recycled for further comminution and the undersize forms theproduct from the breakage unit. If we used the conventional symbols of F and P for feed andproduct size distributions, q the size distribution to the classifier, B, S and C the breakage,selection and classification functions respectively and all terms are considered as vectors, thenfor the breakage process the mass balances may be written as:
At the feed end:
F2 =F ,+Cq (10.15)
where F2 is the size distribution of the feed plus the oversize from the classifier.
At the product end:
P = ( I - C ) - q (10.16)
266266
Feed"1 ' F,
Cq
breakage
classification, C
,, Product P
q
1
Fig. 10.4. Schematic representation of a breakage-classification sequence of events.
According to Eq. (10.14) breakage is given by:
q = (B.S + I - S ) . F 2 (10.17)
Substituting the values F2 and q in Eq. (10.16) and simplifying, Lynch [1] derived thematrix model for comminution as:
P = (i-C)-(B-S + I - S)- [l-C-(B-S + I -S)]"1 F, (10.18)
This model is a quantitative relation between feed size distribution and product sizedistribution in comminution systems. It has been widely accepted.
10.2.2. Kinetic ModelThe matrix model considers the size reduction, especially in grinding process, as a number ofdiscrete steps, which consists of a repeated selection-breakage-classification cycle.Researchers like Kelsall and Reid [11], Whiten [12], Lynch [1], Austin et al. [2], have treatedsize reduction as a continuous process and Gault [13] has elaborated on the time-dependantprocess characteristics.
Loveday [14] and Austin et al, [2] have found experimentally that in the case of batchgrinding the breakage rate obeyed a first order law as in a chemical reaction, though there isno valid reason for it. The breakage rate constant was a function of particle size. The basicassumption was that the entire charge was mixed thoroughly and was therefore uniformduring the grinding process. Following their work, the kinetics of the breakage process maybe described as:
1. the rate of disappearance of particles in size range j by breakage to any smaller sizerange, i,
2. the rate of appearance of size i from breakage of particles of size j ,3. the rate of disappearance of size i by breakage to smaller sizes.
Referring to Eq. (10.7) and using the same symbols we can write:
267267
Rate of disappearance of size j = Sjirij(t)M
Rate of appearance of size i = bjjSjini(t)M
Rate of disappearance of size i = Sjmj(t)M
For a constant mass, M, the material size-mass balance would be:
for N>i>j>l
(10.19)
(10.20)
(10.21)
(10.22)
Eq. (10.22) is the basic size-mass rate balance model for batch grinding of rocks and ores.The rate of production of material less than size Xi is the sum of the rates of production ofmaterial less than size Xj by breakage of all larger sizes and is given by:
and = 0
..«-£•where P ( x , , t ) = > mk(t)k=N
For the modelling of continuous steady state mills, the breakage equation is combined withthe material residence time distribution. The two extremes of residence time distributions areplug flow and fully mixed. For plug flow, all material have the same residence time and hencethe batch grinding equation is applicable when integrated from time zero to the residencetime. The solution to this integration was originally proposed by Reid [15] and is known asthe Reid solution.
The general form of the solution is:
i
m, (t) = y a,^1' for N > i > 1 (10.24)
wherea.J = i
0
rrijCO
1
Si-S
forN>i
for
)"Zait for
k=l
r i£Skb i kak j for
> 1
i<J
i=j
i>j
268268
For i = 1 this gives:
m,(t) = m1(0)e-Slt (10.26)
and for i = 2:
ma(t) = | ^ - m i ( 0 ) e - s - ' + m 2 (0 )e^ - A ^ _ m i ( 0 ) e ^ ' (10.27)
For i > 2 the number of terms in the expression expand rapidly.
For a fully mixed mill at steady state the equation becomes:
i-l
p. = Fj + T^bijSjinj - SiiriiT forN>i>j>l (10.28)j=ii>l
where T = mean residence time.
To estimate P; using Eq. (10.28) it is necessary to determine Sj, the breakage rate constant.This is determined experimentally by a log-log plot of the fraction of particle size j remainingafter different grind times. From the slope of the plot, the value of S can be obtained.
10.3. Modelling Crushing and Grinding SystemsThe general principles and techniques described for mathematical modelling of
comminution systems are directly applicable to conventional crushers and grinding mills usedin mining and metallurgical operations. Little work has been done on their application to otherforms of grinding mills like the Roller Mills, Fluid energy mills, vibratory or attrition mills.These mills are essentially pulverisers. In the mineral industry, pulverisation is seldomrequired to liberate a mineral from its gangue content. In fact, over grinding is usually notencouraged. Hence in this book the discussions are confined to conventional industrialcrushing and grinding systems and not to grinding by pulverisers.
10.3.1. Modelling Jaw and Gyratory CrushersIn Chapter 4 we have already seen the mechanism of crushing in a jaw crusher. Considering itfurther we can see that when a single particle, marked 1 in Fig. 10.5A, is nipped between thejaws of a jaw crusher the particle breaks producing fragments, marked 2 and 3 in Fig. 10.5B.Particles marked 2 are larger than the open set on the crusher and are retained for crushing onthe next cycle. Particles of size 3, smaller than the open set of the crusher, can travel downfaster and occupy or pass through the lower portion of the crusher while the jaw swings away.In the next cycle the probability of the larger particles (size 2) breaking is greater than thesmaller sized particle 3. In the following cycle therefore, particle size 2 is likely to disappearpreferentially and the progeny joins the rest of the smaller size particles indicated as 3 in Fig.10.5C. In the figure, the position of the crushed particles that do not exist after comminutionis shaded white (merely to indicate the positions they had occupied before comminution).Particles that have been crushed and travelled down are shown in grey. The figure clearlyillustrates the mechanism of crushing and the classification that takes place within the
269
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14
Size, di
C ,ytiliba
bor
Pi
K1 K2
269
Fig. 10.5 Classification within a jaw crusher.
breaking zone during the process, as also illustrated in Fig. 10.4. This type of breakageprocess occurs within a jaw crusher, gyratory crusher, roll crusher and rod mills. Eq. (10.18)then is a description of the crusher model.
hi practice however, instead of a single particle, the feed consists of a combination ofparticles present in several size fractions. The probability of breakage of some relativelylarger sized particles in preference to smaller particles has already been mentioned. Forcompleteness, the curve for the probability of breakage of different particle sizes is againshown in Fig. 10.6. It can be seen that for particle sizes ranging between 0 - Ki, theprobability of breakage is zero as the particles are too small. Sizes between Ki and K2 areassumed to break according a parabolic curve. Particle sizes greater than K2 would always bebroken.
0.8
o>; 0.6'5CD
a.
0.2 J11
/
Size, d,
Fig. 10.6 Classification function, Q, in a crusher [1,16].
270270
According to Whiten [16] this classification function Q, representing the probability of aparticle of size dj entering the breakage stage of the crusher, may be expressed as:
Q =0
Q =1K , - K 2
for d; < Ki
forKi <di<K2,and
for dj > K2 (10.29)
Where the upper and lower sieve sizes for the ith size interval is known, Q can be obtainedfrom:
c, = -dd (10.30)
j and dj+i are the upper and lower sieve sizes of the i* size fraction.
For jaw, gyratory and cone crushers, Ki is the value of the set and K.2 the size above whichall particles will be broken. The recommended value of the exponent in Eq. (10.29) is 2.3 [8].
The classification function can be readily expressed as a lower triangular matrix [1,16]where the elements represent the proportion of particles in each size interval that would break.To construct a mathematical model to relate product and feed sizes where the crusher feedcontains a proportion of particles which are smaller than the closed set and hence will passthrough the crusher with little or no breakage, Whiten [16] advocated a crusher model asshown in Fig. 10.7.
The considerations in Fig. 10.7 are similar to the general model for size reductionillustrated in Fig. 10.4 except in this case the feed is initially directed to a classifier, whicheliminates particle sizes less than Ki The coarse classifier product then enters the crushingzone. Thus only the crushable larger size material enters the crusher zone. The crusherproduct is combined with the main feed and the process repeated. The undersize from theclassifier is the product.
B.C.X
Feed, F
C.X
Fig. 10.7. Crusher model [16].
271271
hi order to describe the operation mathematically, Whiten [16] and Lynch [1] consideredthe mass balances at the feed and product ends and derived the relation between crusher feedand product as:
P = ( I -C)-(I -B-C)" ' -F (10.31)
where P = vector for product size distribution (mass),I = unit diagonal matrix having all elements = 1 (remaining are = 0),C = classification function written as a diagonal matrix,F = feed size distribution (mass).
Eq. (10.31) is the widely accepted model of crushers that is used to predict the sizedistribution of products in different types of crushers.
While considering the above aspects of a model of crushers it is important to rememberthat the size reduction process in commercial operations is continuous over long periods oftime. In actual practice therefore the same operation is repeated over long periods so thegeneral expression for product size must take this factor into account. Hence a parameter v isintroduced to represent the number of cycles of operation. As all cycles are assumed identicalthe general model given in Eq. (10.31) should therefore be modified as:
P = XV .F (10.32)
where X = ( I -C) . ( I - B • C)'1
The method of evaluating a crusher model is illustrated in Example 10.2.
Example 10.2The size distribution of an ore to be fed to a cone crusher was:
Size, mmMass %
-100 +5010
-50+2533
-25 +12.532
-12.6 +620
-65
The breakage and classification matrices, B and C have been determined as:
B =
0.58 0 0 0
0.2 0.6 0 0
0.12 0.18 0.61 0
0.04 0.09 0.2 0.57
C =
1.0 0 0 0
0 0.7 0 0
0 0 0.45 0
0 0 0 0
Estimate the size distribution of the crusher product.
272272
Solution
Using Eq. (10.31):
SteplMultiple vectors B • C written in matrix form:
This gives the size distribution of the products from the given feed after comminution.
10.3.2. Modelling Ball Mills
Perfect mixing modelIn metallurgical operations, ball mills operate continuously and the process is repeated untilsuch time as the required liberation size of the mineral is achieved. Whether the operation isbatch or continuous the grinding mill model has been developed on the assumption that thecontents are perfectly mixed and the mill can be represented by one perfectly mixed segmentor a number of perfectly mixed segments in series [17]. In the steady state operationtherefore, a mass balance forms the basis of the model. That is:
Feed + material broken from =larger particles
material selected forbreakage in the mill
+ material discharged.
These parameters can be evaluated in the following way:
1. Feed rate: Denoted by the matrix F.2. Removal rate for breakage: R.s where R is a diagonal matrix giving the breakage rate of
each component of the mill contents, s, (for any component i, the rate of breakage is R;.Si). The mill contents matrix, s, represents the mass of the mill contents retained in eachsize fraction.
3. Appearance rate: Denoted by A-R-s where A is the appearance matrix (i.e., it is the sameas the B matrix and can similarly be obtained experimentally, analytically orapproximate values assumed).
4. Discharge rate: Denoted by P = D-s, where D is a triangular matrix giving the fractionalrates at which the components of the mill are discharged. For a perfectly mixed mill theproduct will be the same as the mill contents. However classification within the millmay occur, particularly for grate discharge mills, hence the product from the millincludes D and s.
These factors are combined to obtain the rate of change of mill contents:
— = A-R-s -R-s + F - P9t
= (A-R-R-£>)-s + F
(10.33)
(10.34)
275275
At steady state the rate of change of mill contents is zero, or— = 0
Hence Eq. (10.34) would be;
( A R - R - D ) S + F= 0 (10.35)
Since P = D-$ or s = ZX'-P, we can substitute for s in Eq. (10.35). Simplifying the resultingequation the mathematical model for ball mills may be written as:
1-A)'l-F (10.36)
This perfect mixing model has been successfully used to simulate ball mill and rod milloperations. It can be used for steady state and dynamic simulations of milling processes.
It is possible to evaluate Eq. (10.35) providing the model parameters A, R and D areknown. The parameters R and D however are interdependent and can only be determined ifdata on the mill contents are available. This is not always known in industrial mills. This maybe overcome by lumping the two together as a combined parameter £>-R"'.
To evaluate the model parameters, assuming a steady state condition: Rearrange Eq.(10.35) and substituting s = D~l -P:
0 = F + (A-R - R - D)-s = F + ARD'K? - RD'1? - P
or F - P = (I-AJR-Z)"1-?
or P = DRl(l-AYi-(F-P)
From Eq. (10.33):
R-s = (I-A)"'-(F-P)
and hence P = D-R"'-R-s
(10.37)
(10.38)
(10.39)
(10.40)
(10.41)
Eq. (10.40) can be solved if the inverse of (1 - A) is known, but this is seldom available.However, Eqs. (10.40) and (10.41) are two simultaneous equation in R-s. Both theseequations are triangular matrix equations and can be solved using the back substitutiontechnique. To use this technique the equations have to be arranged in the matrix form:
(R-s)i can then be substituted in Eq. (10.41) to calculate (D-R"1). If the size distribution ofthe mill contents is known then the discharge matrix can be calculated from P = D-s and hencethe rate of breakage matrix, R, can be determined from D-R-l.
While considering the mass balance in a breakage system, Lynch and Whiten [18] and laterGault [13] related the discharge rate from the mill with the volumetric feed rate and thedimensions of the mill as:
(10.48)
where D* = standard discharge rate function,V = volume feed rateD = the mill diameterL = the length of mill.
D is a diagonal matrix which is independent of size unless classification at the milldischarge occurs. D* has diagonal values near unity. The combined parameter D-R"1 varieswith the feed rate and:
D-R'1 = / (DR 4 )* (10.49)
where / = a function of the feed rate to the mill and(D-R'1)* = a constant
For example, the breakage rate parameter could be defined as:
4VD-R~l = ^-(D-R-1)" (10.50)
277277
Example 10.3 illustrates the use of Eqs. (10.47)-(10.50).
Example 10.3The feed and product size distribution of a ball mill is given below. The feed rate was 200 t/h.The feed rate was increased to 285 t/h. Estimate the product size distribution at the higherfeed rate.
Data: Appearance matrix A has the following values:
0.188, 0.321, 0.208, 0.105, and 0.021
Size,Microns
120060030015075-75total
Feed% Retained
6.418.422.829.122.0
1.3100
FeedMass, t/h
12.836.845.658.244.02.6200
Product% Retained
0.36.59.5
15.428.939.4
100
ProductMass, t/h
0.613.019.030.857.878.8200
SolutionWriting a simple spreadsheet can easily solve the problem. But for those using a calculator thefollowing steps for calculations would be useful.
Stepl
A is a lower triangular matrix, therefore the elements in the diagonal, AJ; are all 0.188.
Step 3Assuming the breakage rate is constant over the range of 200 - 285 t/h, then from Eqs.(10.48) and (10.50):
279279
) 2 O O t / h -200 (D-R"1)* ,and
D 2L
then (D - R - ) 2 8 5 t / h = I ^V
For constant % solids in the mill at the two flow rates, the volume flow rates are proportionalto the solid flow rates, hence:
(Z).R-')285t/h = g | j (/>-R-')2OOt/h and
CD-R"1),, =(D-R~l)22 =
(D-R-')33 =
CD-R~')44 =
(Z)-R"')
1.425x0.03993 = 0.0569
1.425x0.3688 = 0.5255
1.425x0.3759 = 0.5357
1.425x0.4761 = 0.6784
1.425x2.1832 = 3.1111
The mill product is then given by the Eq. (10.36):
P = D-R~liD-Kl+l-A)~l-F
The matrix (Z>R~' + I - A) may now be calculated using the normal rules of matrix additionand subtraction:
0.0569 0 0 0 0
0 0.5255 0 0 0
0 0 0.5357 0 0
0 0.6784
0 0
0
3.1111
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
o"0
0
0
1
-
0.188 0 0 0 0
0.321 0.188 0 0 0
0.208 0.321 0.188 0 0
0.105 0.208 0.321 0.188 0
0.021 0.105 0.208 0.321 0.188
0.8689 0
-0.321 1.3375
-0.208 -0.321
0
0
1.3477
-0.105 -0.208 -0.321 1.4904 0
-0.021 -0.105 -0.208 -0.321 3.9231
280280
and taking the inverse:
[(D-R-'+I -A)]~' =
1.15090.2762
0.2434
00.7476
0.1781
00
0.7420
00
0
00
0
0.1721 0.1427 0.1598 0.6710 0
0.0405 0.0411 0.0524 0.0549 0.2549
By matrix multiplication:
0.0569 0 0 0 0
0 0.5255 0 0 0
0 0 0.5357 0 0
0 0 0 0.6784 0
0 0 0 0 3.1111
1.1509 0 0 0 0
0.2762 0.7476 0 0 0
0.2434 0.1781 0.7420 0 0
0.1721 0.1427 0.1598 0.6710 0
0.0405 0.0411 0.0524 0.0549 0.2549
0.0655 0 0 0 0
0.1452 0.3929 0 0 0
0.1304 0.0954 0.3975 0 0
0.1167 0.0968 0.1084 0.4552 0
0.1261 0.1279 0.1631 0.1708 0.7930
and the product is calculated by:
-' -[(D-R'+I -
0.06550.1452
0.1304
0.1167
0.1261
00.3929
0.0954
0.0968
0.1280
00
0.3975
0.1084
0.1631
00
0
0.4552
0.1708
00
0
0
0.7930
•
"18.24
52.44
64.98
82.94
62.70
=
1.19"23.25
33.21
52.00
83.49
Thus at the feed tonnage of 285 t/h the ball mill product size distribution is tabulated in thetable below. For comparison, product size distribution at 200 t/h throughput is also given.The results show that the effect on the product size distribution of increasing the feed ratefrom 200 t/h to 285 t/h, in this particular example is not great.
281281
Simulated product size distribution
Size,microns
120060030015075-75
Total
Pat 285 t/h
1.1923.2533.2152.0083.4991.85
285
P % at285 t/h
0.48.2
11.718.229.332.2100
P % at200 t/h
0.36.59.5
15.428.939.4100
10.3.3. Modelling Rod millsA study of the movement of materials in a rod mill indicates that at the feed end the largerparticles are first caught between the rods and reduced in size gradually towards the dischargeend. Lynch [1] contended that the next lower size would break after the sizes above it hadcompletely broken. He described this as stage breakage, the stages being in steps of V2. Thesize difference between the particles at the two ends of the mill would depend on:
1. mill length,2. speed of grinding, and3. feed rate.
The presence of this size difference indicates that a screening effect was generated within arod mill and that the movement of material in the mill was a combination of breakage andscreening effects. The breaking process was obviously repetitive and involved breakagefunction, classification function and selection functions. Therefore for rod mills an extensionof the general model for breakage within each stage applies, where the feed to stage (i+1) isthe product from stage i. That is, within a single stage i, the general model defined by Eq.(10.18) applies.
P = (I-C)-(B-S + I - S)-[l-C-(B-S + I - S)]"'-F = X j-F (10.51)
The number of stages, v, is the number of elements taken in the feed vector. A stage ofbreakage is defined as the interval taken to eliminate the largest sieve fraction from the millfeed or the feed to each stage of breakage. The very fine undersize is not included as a stage.
The breakage function described by Eq. (10.2) could be used. For the classification matrix,which gives the proportion of each size that enters the next stage of breakage, the value of theelement in the first stage Cn equals 1. That is, all of size fraction 1 is completely reduced to alower size and all the particles of the classification underflow are the feed to the second stageof breakage and so on. Hence the classification matrix is a descending series. If we take theV2 series, then the classification matrix C can be written as:
282282
y"i
10
0
0
0
0
00.50
0
0
0
00
0,25
0
0
0
00
0
0.125
0
0
00
0
0
0.0625
0
00
0
0
0
0.032
(10.52)
The selection matrix S is machine dependant. It is affected by machine characteristics,such as length (including length of rods) and the speed of operation. Both B and C have to beconstant to determine the selection function S within a stage.
In Whiten's model [16] for the probability of breakage against the size of particles,considering a single stage operation, we see that:
1. The region between K] and K.2 is almost linear (actually it is a part of a parabola, Fig.10.6)
2. K.2 decreases with time as the material travels down the mill towards the discharge end3. Ki = S.C
Thus for each stage a similar matrix can be developed resulting in a step matrix whichprovides a solution of the rod mill model. Calculations are similar to that shown previouslyfor grinding mill models.
10.3,4. Modelling AG ISAG MillsWhile in conventional tumbling mills the size reduction is primarily by impact of grindingmedia (steel balls or rods) in AG or SAG mills the size reduction is mainly by abrasion andchipping off of particles and less so by impact. The process is complicated by:
1. the grinding medium itself disintegrating with time,2. the transport of the material through the mill is constrained by the size, shape and
apertures in the grate at the discharge end, and3. the slurry characteristics during wet milling, such as the viscosity and density.
Stanley [19] and Duckworth and Lynch [20] in attempting to describe the process havequantified these variables and derived mathematical models to suit the operation of thesemills. Additional considerations were advocated by Leung et al. [21] and later by Morrison etal. [22] who introduced the concept of high and low energy breakage of particles in theirmodels. The high energy was defined as breakage due to crushing and impact and the lowenergy breakage due to abrasion.
Some evidence of the different kinds of breakage within SAG mills has been reported(Austin et al. [23], Amestica et al. [24], Napier-Munn et al. [8]). Working with experimentalSAG mills the breakage rate was plotted against the particle size. The curves showed that atthe initial stages the breakage rate increased proportionally with particle size, subsequentlythe rate decreased but again increased (Fig. 9.7). These inflections in the curve have beenattributed to the change in the mechanism of breakage from surface abrasion effects tocrushing effect.
283
PF F* P*
Ci
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
10 100 1000 10000 100000
Size (microns)
C ,n
oitcn
uf n
oita cifissalC
i
Xg
283
In an earlier work, Austin et al. [23] identifies three mechanisms of breakage, namely,normal fracture caused by balls and large pebbles that formed the grinding medium, chippingoff of particles rendering them roundish and spherical, and finally abrasion. However,Loveday and Whiten [25] have questioned the interpretations of Austin's data.
To develop a mathematical model of the process, Amestica et al. [24] considered thedischarge grate as a classifier. Their conceptual model is illustrated in Fig. 10.8. The dottedlines in the figure indicate the conceptual movement of material in the mill where the grateacting as classifier returned its underflow to the main feed stream for further size reductionand its overflow formed the final product from the mill. The measured classification functionis shown in Fig. 10.9.
O O
Fig. 10.8. Conceptual diagram of material flow in a SAG mill [23,24]
1.0
0.9
°_ 0.8
•2 0.7o= 0.6
30.5
0.4
M 0 3
(0iS 0.2
0.1
0.0
/
/////
/ ^ \
' \ 1\J
Xg
10 100 1000 10000
Size (microns)
100000
Fig. 10.9. Internal classification in a SAG mill with grate size Xg [24]
284284
In Fig. 10.9, Cj represents the fraction of size i which is returned to the mill. The drop in Ciclose to the grate size was observed on a number of occasions. The mill product, P; wascalculated from the expression:
P, =P,*(l+C)(l-c,) (10.53)
where P* = the product from the mill grind (pre-internal classification), andC = an apparent internal circulation ratio.
The feed to the internal mill is given by:
(1 + C)F,* ^F.+a+QP. 'cj and (10.54)
(10.55)
The relationship between Fi and Pi is given by the size-mass balance or population balancemodel for a fully mixed mill at steady state:
Sjm. - ( ^ . I s m (10.56)
where MH = hold up mass,MF = feed mass,m; = mass fraction in the ith size interval,Fi , Pj = mass fraction of size i in the internal mill feed and product.
Leung et al. [21], Morrell [26] and Napier-Munn et al. [8] made a similar and simplerapproach by considering that the classification at the grate allowed a certain fraction of themill hold-up to always pass through. This fraction contained particles smaller than a certainsize, Xm which in turn is less than the grate aperture and had fluid characteristics similar towater. Further they considered that the solids in the slurry containing particles of size greaterthan the grate aperture would never pass through. Slurries containing particle sizes in-betweenwould pass depending on the probability of classification function. This concept is somewhatsimilar to that advocated for the crusher model by Whiten [16]. According to Leung et al [21]the discharge containing only small size solids is considered to behave like water passingthrough orifices in the grate and can be represented by AB in the log-size-classification curve(Fig. 10.10). The remaining slurry was classified at the grate, the classification followed lineBC representing sizes associated with the discharge. Napier-Munn et al. [8] recognised thatthe characteristics of flow of products, (solids and water) through the grate were different.They assumed that the slurry passing through obeyed the laws of fluid flow through orifices,as propounded by the well-known Bernoulli theorem. Thus the fluid flow through the grateopening was considered as:
285
0
1
0 2 4 6 8 10 12
Log Size
noitc
nuf
noitacifissal
C
Xm Xg
A B
C
285
5
o
A B
\ ^ C
(10.57)
Log Size
Fig. 10.10. Classification function C, against particle size [8].
where Qy = the volumetric flow rate through an aperture in the grate,Ao = the cross section of the grate aperture,H = head of fluid,g = acceleration due to gravity.
The problem in Eq. (10.57) was to establish the value of H, the head of fluid, as it woulddepend on the rate of rotation, the solid-liquid ratio, the porosity of the bed and the inclinationof the liquid level inside the mill. Morrell et al. [27] therefore considered an average radialposition defined as y, which was the ratio of the open areas of all apertures at a radial positionrj from the centre to the radius of the mill inside diameter. Symbolically this was written as:
y =RSA,
(10.58)
where Aj = total area of apertures at radial position r*, andR = mill radius.
Morrell [26] also considered that the product flow was made up of flow of the fluidthrough the zone of grinding medium (i.e. between balls and some solids), and flow from thepool zone created by separation of excess slurry from the tumbling charge (slurry in excess ofthe charge volume and the interstices in the charge). The empirical equation representing theflow rate through the grinding media (m3/h) was determined as:
QM = 6100 il J H < J ,MAX (10.59)
286286
and the equation representing the flow through the pole zone as:
Q, = 935 Js y2 AD 0 5 Js = J P - JMAx, Jp > JMAX (10.60)
where JH = net fraction of slurry hold-up within the interstitial spaces of the grindingmedia,
Js = net fractional volume of slurry in the slurry pool,JMAX = maximum net fraction of slurry in the grinding zone,Jp = net fraction of the mill volume occupied by pulp/slurry,y = mean relative radial position of the grate apertures (as defined by Eq.
(10.58)),A = total area of all apertures, m2 ,<|>c = fraction of the critical speed of the mill,D = mill diameter, m,QM = volume flow rate through the grinding media zone, m3/hQt = volume flow rate of slurry through the pool zone, m3/h.
If the flow through the mill is assumed to follow the perfect mixing model, then thevolumetric flow of material of size i would be:
Pi = A»Si (10.61)
where A = discharge rate of particles of size i,Pi = the volumetric flow rate of size i in the mill product, andSj = the volume of size i in the mill content.
pThe discharge rate is thus A = -1- (10.62)
Substituting Pi = QM, in Eq. (10.62), the discharge rate A would be:
"6100JA = H ' ' r (10.63)
TTD2
From simple geometry, s( = .. P
Hence the discharge rate model for AG/SAG mills may be written as:
A =6100J2 y25 A(|)C
138D
| 0 . 2 5 T I D 2 JP
>c L 3 8
D 1 5 J p
(10.64)
In addition to the flow rate and discharge considerations within a SAG mill, the highenergy required for breakage of larger particles by impact was attributed to the top 20% of the
287287
feed. The size reduction of the remainder was assumed to be due to abrasion requiring lowenergy of breakage.
The high-energy estimate required for size reduction was attributed to the potential energy(PE) transmitted to particles falling from a height equal to the diameter of the mill. Thisenergy value was determined by the twin-pendulum test and the Tio parameter estimated. Theenergy E was then calculated by using Eq. (3.45).
For low energy breakage determination, a tumbling abrasion test is used to generate thelow energy Tio parameter. Combining the high and low energy Tio parameters, a combinedappearance function can be calculated as:
A J T L E A L E ) + ( T A H E )
where A = combined appearance function,ALE, AHE = low and high energy appearance functions respectively,TLE, THE = low and high energy Tio parameters, respectively.
From the equations describing the combined appearance function, the discharge rate andbreakage rate, the material balance of SAG mills for the feed, mill holdup and dischargeresulted in the perfect mixing model for AG/SAG mills given by:
P£ =F, + ^ R ^ A ^ - R . s , (10.66)
and Pi =A«Si (Eq. (10.61))
where Pi = rate of production of particles of size i,Fj = mass rate of particles of size i in the feed,Ajj = appearance function or breakage distribution function of particles of size i
from particles of size j ,Rj = breakage rate of particles of size i,Sj = amount of particles of size i in the mill contents andA = rate of discharge of particles of size i.
10.3.5. High Pressure Grinding Rolls (HPGR)A satisfactory mathematical model of HPGR has not yet been established [28]. The forces ofcompression and size reduction are complex and some light on its complexity has beenindicated by Schonert and Sander [29] and Lubjuhn and Schonert [30]. Discussion of thistopic is beyond the scope of this book. The interested reader is directed to the original papers.
10.3.6. Stirrer Ball MillsWhile in a ball mill the body of the mill rotates and the charge made to rotate by the help oflifters (provided by the shape of mill liners), in stirrer mills the body is stationary and thecharged moved by stirrers rotating inside the mill. These mills have been introduced over thelast two decades for grinding down to very fines sizes that is required to liberate finelydisseminated minerals in some host rock. Several types of such mills are now available such
288288
as the vertical stirred mills (Tower mill, Vertimill, Metroprotech mill) and horizontally stirredmills (ISAmill). The difference between them is in the stirring mechanism The Tower Milluses a helical screw while the ISAmill uses a horizontal shaft with various shapes and sizes ofdiscs. The grinding media used are hardened steel balls, river sand or ceramics.
The feed size (Fgo) to the mills ranges between 45 and 100 microns and the product sizearound 3 microns and less.
The mechanism of size reduction is entirely different from the ball mills. In this case thesize reduction is mostly by attrition. Morrell et al.[31] examined the applicability of the rate-size mass balance model and the perfect mixing Whiten model to ultra-fine grinding in aTower Mill. Further work in this direction is in progress.
10.4. ProblemsMost problems arising out of this chapter need the use of computer programs. A few that canbe solved by hand held calculators are included here.
10.1In numerical example 10.3, the product from the first stage of breakage was calculated.Continue calculations to determine the products from the second and third stages.
10.2Using the particle size distribution obtained from a batch grind at short grind time, determinethe breakage distribution function using the BII method for an incomplete sieving error of0.01.
Interval12345678
Size, um5600400028002000140010007100
Feed % retained78.79.045.163.121.760.870.410.94
10.3A ball mill was designed to operate at the rate of 1 million tonnes of iron ore per year working360 days in the year. The characteristics of the grind are given in the table below.
Size microns2000100050025012575-75
Feed % retained1.38.5
16.025.020.018.011.2
Product01.23.4
11.718.112.253.4
289289
It was desired to increase the plant throughput to 1.2 m t/year. If all other conditions remainthe same, estimate the product size expected.
10.4Ore samples were taken from a grinding mill operating as a batch process. The feed sizedistribution, breakage functions and size analysis of samples taken at intervals of 10 minutesup to 30 minutes are given. Determine:
1. The breakage rate of the top two feed sizes2. The % passing 710 urn after 1 hour of operation.
Hint: Assume breakage was constant within a size range. May need to use Solver in Excel(or similar) for part of the calculation.
10.5A copper ore was charged in a rod mill for size reduction. Three different ore sources are tobe treated. At steady state conditions the following results were obtained:
Size,range
12345678
FeedSize (1)
mm10.121.225.413.46.54.1
15.34.0
FeedSize (2)
mm3.25.8
14.318.221.0
8.26.0
21.3
FeedSize (3)
mm23.018.018.612.56.58.64.75.1
Selectionfunction
1.00.50.250.1250.0620.0320.0160.008
BreakageFunction
0.19800.33080.21480.12250.06540.03380.01720.0087
Classificationfunction
1.00000000
Determine the product sizes after each change of feed size.
10.6A drop weight test on a single size fraction (-5.6+4.75 mm) at different comminutionenergies, EG, gave the following product sizes:
290290
Size(mm)4.02.01.00.50.250.1250.0630
Test
Mass%0
11.025.721.313.910.0
7.810.5
1, EG = 5.11kWh/t
C u m %passing100.0
89.063.342.028.218.210.5
Test
Mass%08.1
27.523.516.010.06.78.2
2, EG = 3.87kWh/t
Cum %passing100.091.964.540.924.914.9
8.2
Test
Mass%
09.6
30.921.713.1
8.66.69.4
3, EG = 2.80kWh/t
Cum %passing100.090.459.537.824.716.19.4
Test 4., EG = 2.39kWh/t
Mass%0
10.030.623.314.4
8.65.87.3
Cum %passing100.090.059.526.121.713.17.3
1. Determine the Tio value for each set of test results2. Determine the relationship between Tio and EG3. Estimate the value of a in the relationship between TN and Tio (Eq. (10.4))4. For a comminution energy of 25 kWh/t determine the Tio value and hence estimate the
appearance function
10.7In a ball mill experiment, the feed size and classification function is assumed constant. If thebreakage function changes, e.g. as a result of a change in ore type, estimate the change inproduct size from the following data:
Sizerange
12345
Ratefunction,
DR."1
0.040.3690.3760.4762.18
Feedtonnes
19.937.844.827.120.3
Breakage
Function 10.1890.3310.2150.1230.065
Breakage
Function 20.1380.2480.2290.1600.090
Breakage
Function 30.0330.3500.1700.0840.038
10.8A crusher treats ROM ore at a rate of 100 t/h.below, estimate the product size distribution.
From the crusher/ore characteristics given
Sizefraction
Classificationfunction
FeedMass %
BreakageFunction
1234
10.50.10
15283522
0.1950.270.180.09
291291
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design and control, Elsevier Scientific Publishing Company, Amsterdam, 1977.[2] L.G. Austin, R.R. Klimpel and P.T. Luckie, Process Engineering of Size Reduction:
Ball Milling, SME/AIME, New York, 1984.[3] S.R. Broadbent and T.G. Callcott, Philosophical Transactions of the Royal Society of
Symposium Series, No. 4,1965, pp. 14 - 20.[6] R.R. Klimpel and L.G. Austin, Transactions of the Society of Mining Engineers, 232
(1965) 88.[7] A.J. Lynch and M.J. Lees, in Mineral Processing Handbook, NX. Weiss (ed),
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CIM, Toronto, session III, paper 1,1982.[21] K. Leung, R.D, Morrison and W.J. Whiten, Copper '87, Santiago, 1987.[22] R.D. Morrison, T. Kojovic and S. Morrell, Proceedings of SAGSEM '89, SAG Milling
Seminar, Australasian Institute of Mining and Metallurgy, Perth, 1989, pp. 254-271.[23] L.G. Austin, C.A. Barahona and J.M. Menacho, Powder Technology, 51 (1987) 283.[24] R. Ame'stica, G.D. Gonzalez, J. Barria, L. Magne, J. Menacho and O. Castro,
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[26] S. Morrell, in Comminution: Theory and Practice, S.K. Kawatra (ed), AIME, Chapter27,1992, pp. 369-380.
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[28] L.G. Austin, K.R. Weller and I.L. Lim, Proceedings, XVIII International MineralProcessing Congress, Sydney, 1993, pp. 87-95.
[29] K. Schonert and U. Sander, Powder Technology, 122 (2002) 130.[30] U. Lubjuhn and K. SehSnert, Proceedings, XVIII International Mineral Processing
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