Modelo Chancador Ouchterlony

A new three-parameter fragment size distribution functionhas been found that links rock fragmentation by blasting andcrushing. The new Swebrec function gives excellent fits tohundreds of sets of sieved fragmentation data withcorrelation coefficients of 0997 or better over a range offragment sizes of 23 orders of magnitude. A five-parameterversion reproduces sieved fragmentation curves all the wayinto the 100 m range and also handles ball mill grindingdata. In addition, the Swebrec function: (i) can be used inthe KuzRam model and removes two of its drawbacks thepoor predictive capacity in the fines range and the upperlimit cut-off to block sizes; (ii) reduces the JKMRC one-family description of crusher breakage functions based onthe t10 concept to a minimum; and (iii) establishes a newfamily of natural breakage characteristic (NBC) functionswith a realistic shape that connects blast fragmentation andmechanical comminution and offers new insight into theworking of the Steiners OCS sub-circuits of mechanicalcomminution. It is suggested that the extended KuzRammodel, with the Swebrec function replacing theRosinRammler function, be called the KCO model.

Finn Ouchterlony is at the Swedish Blasting Research Centre,Lule University of Technology, Box 47047, S-10074Stockholm, Sweden (Tel: +46 8 6922293; Fax: +46 86511364; E-mail: [email protected]).

2005 Institute of Materials, Minerals and Mining andAustralasian Institute of Mining and Metallurgy. Publishedby Maney on behalf of the Institutes. Manuscript received 7September 2004; accepted in final form 25 February 2005.

Keywords: Fragment size distribution, rock fragmentation,blasting, KuzRam model, CZM, TCM, crushing, breakagefunction, NBC, natural breakage characteristics, t10

BLAST FRAGMENTATIONIn an area as complicated as rock fragmentation byblasting, Cunninghams KuzRam model4,5,13 has for 20years done an invaluable job in fostering a structuredapproach to what can be done to change thefragmentation pattern. Much experience has gone intothe equations that describe the defining parameters, themedian or 50% passing size x50 and the uniformityexponent n of the underlying RosinRammler cumulativefragment size distribution in Figure 1:

( )P x e1 1 2( / ) ( / )ln x x x x2n2

50 50= - = -- - (1)

The KuzRam model does not cover all aspects ofblasting and was never meant to do so. Timing is onesuch area, the fines part of the muckpile another. On amore detailed level, the effect of single row blasts isincluded by a constant prefactor (11) and the finiteboulder limit is taken care of by a cut-off in theinfinite distribution.

The defining equations x50 = h1 (rock mass, scale,explosive and specific charge) and n = h2 (geometry)decouple geometry from the other quantities thatdescribe the blast, which is pedagogically attractive. Arecent finding24 suggests, however, that the expressionx50 should involve the prefactor (ln2)

1/n/(1 + 1/n),where is the gamma function. Its effect is to raise thepredicted amount of fine material in well-gradedmuckpiles, i.e. when n is small.

It has gradually become clear that very few sievedfragment size distributions follow Equation (1),especially in the fines range. The JKMRC has comeup with two models that address this problem thecrush zone model9 (CZM) and the two-componentmodel7 (TCM).

In the CZM, the fragment size distribution is madeup of two parts, a coarse one given by the KuzRammodel that corresponds to tensile fracturing and afines part that is derived from the compressivecrushing around a blast-hole. The fines distribution isalso of RosinRammler type but with different valuesprimarily for n and the characteristic size xc (x50 nowrefers to the combined fine and coarse parts).

The two parts are mutually exclusive so the resultingfragment size distribution has a knee at the graftingpoint (see Fig. 2) and the CZM refers all 1 mm fines tothe blast-hole region. In this way, the effects ofquantities like the rocks compressive strength, blast-hole pressure and VOD enter the prediction equations.

In the TCM,7 the fragment size distribution is alsomade up of two RosinRammler distributions thatrepresent the tensile and compressive fracture modes.The two populations co-exist over the whole range sothe result is a smooth curve (see Fig. 2). Again, theKuzRam model gives the coarse part but the finespart is obtained, for example, from scaled-up modelblasts.

The effect is that the TCM is a five-parameter model;two sets of xc and n and one parameter that determinesthe ratio of the two populations. The CZM is a four-parameter model. This makes their predictions morerealistic than the KuzRam model at the price of

Mining Technology (Trans. Inst. Min. Metall. A) March 2005 Vol. 114 A1DOI 10.1179/037178405X44539

The Swebrec function: linking fragmentation by blasting andcrushing

Finn Ouchterlony

complexity. The JKMRC prefers the CZM and theyhave used it successfully in a number of fragmentationprojects related to their Mine-to-mill concept.8,23

Meanwhile, experimental evidence has emerged22 thatclearly contradicts the idea that in massive rock all but anegligible amount of fines are generated in a crushed zonearound a blast-hole. In one case,27 300-mm diameter,100-kg mortar specimens with concentric coloured layerswere shot to produce 2 kg of 1 mm fines were produced,1kg from the 120-mm diameter inner layer and asmuch from the outer layer. The percentage of fines is, ofcourse, higher in the inner layer but what matters in

practice is the total amount. More evidence to this effect isprovided in the work of Moser et al.17 and Micklautsch.14

CRUSHING FRAGMENTATIONTraditionally, a quarry or mine produces rock on theground for a plant to process. An eye is kept onhauling and oversize with much less focus on the restof the blasted fragment-size distribution. This ischanged with the Mine-to-mill or drill-to-mill approaches.Still, what blasting and crushing fragmentation orcomminution have in common has been neglected.

A2 Mining Technology (Trans. Inst. Min. Metall. A) March 2005 Vol. 114

Ouchterlony The Swebrec function: linking fragmentation by blasting and crushing

1 Fragment size distribution of blasted hornfels from Mount Coot-tha quarry10 with RosinRammler curve fit.21 Datarange = 0352000 mm. Curve fit parameters x50 = 1158 mm, n = 0572. Coefficient of correlation r

2 = 09958

2 Comparison of fragment size distributions in CZM9 and TCM.7 Note different characters of curves but samelinear behaviour in logP versus logx space in fines range

Steiners25,26 approach to mechanical comminutionis based on the concept that a material, which isfractured under pure conditions exhibits a material-specific natural breakage characteristic (NBC).Some of the tenets are that:

(i) Rock broken in the crushing and grinding sub-circuits of an optimum comminution sequence(OCS) has the steepest possible cumulativefragmentation curve PNBC(x).

(ii) When the sub-circuit product streams are classified,the fragmentation curves are shifted verticallyupward as the comminution progresses.

(iii) When plotted in loglog space, this basicallybecomes a parallel shift so that the local slopedepends only on the fragment size (Fig. 3).

Steiners approach also contains the energy registerconcept, which says that when the specific surface (m2

kg1) created by an OCS is plotted versus the energyconsumed (J kg1), the points fall more or less on amaterial specific straight line. The slope R (m2 J1) is

equal to the Rittinger coefficient of comminution(Fig. 4). Points that represent practical comminutioncircuits tend to fall below this line.

This concept was recently expanded to cover modelscale blasting with different specific charges,16 withfurther support coming from the work in the Less Finesproject15 (Fig. 5). It may even carry over to boulderblasting and full-scale bench blasts (Fig. 6).18,20

The JKMRC19 engineering-oriented approach tocrushing fragmentation describes comminution circuitsand associated individual crushing and grindingbreakage functions. The approach uses a matrixdescription of the product flow through the system and aone-family description of breakage functions based onthe t10 concept. t10 is that part of a given size fraction ofmaterial subjected to a drop weight crushing test, whichafterwards is smaller than a tenth of the original feedmaterials size.

An exponential curve with two material parametersrelates the measure t10 to the drop energy. This is, in

Mining Technology (Trans. Inst. Min. Metall. A) March 2005 Vol. 114 A3


3 Fragmentation curves from crushing and grinding of amphibolite in OCS sub-circuits3,17

4 Energy register curves3,15 for limestones and the Hengl amphibolite of Figure 3

principle, equivalent to the KuzRam model equation,which equates x50 with the explosive charge as a sourceof fragmentation energy. The family of breakagefunctions is usually plotted as in Figure 7; but, fromvertical lines of constant t10 values, the associatedfragment size distributions may be extracted (Fig. 8).

The shape of the curves in Figure 8 looks similar tothe curves for model blasting tests and full-scale blastingin Figures 5 and 6, apart from the dips of the latter in the500 m range. The NBC crushing curves in Figures 3and 5 have a different character in that they do notapproach the 100% level smoothly at a tangent.

How does all this tie together?

THE SWEBREC FUNCTIONWhen analysing the data from the Less Finesproject,15 it was realised that the following fragmentsize distribution does a very good job of fitting sievedfragmentation data.21 The transformation:

P(x) = ( )f x1 1 +7 Awith f(xmax) = 0, f(x50) = 1 (2a)

ensures that xmax and x50 are fixed points on the curveand a suitable choice for f(x) is:

f(x) = ln lnx x x xmax maxb

50_ _i i8 B (2b)



5 Comparison of fragmentation curves from OCS comminution in Figure 3 and model-scale blasts on sameamphibolite15

6 Comparison of fragmentation curves from model- and full-scale blasts of Brarp granitic gneiss18

Like the RosinRammler function, it uses the median or50% passing value x50 as the central parameter but it alsointroduces an upper limit to the fragment size, xmax. Thethird parameter, b, is a curve-undulation parameter.Unlike the RosinRammler or the CZM/TCMfunctions, the asymptotic properties of f(x) for smallfragments is logarithmic, not a simple power of x.

Figure 9 shows sieved data from a 500-t bench blastwith 51-mm diameter blast-holes on a 18 22-mpattern and a specific charge of about 055 kg m3. Atthe Brarp18,20 dimensional stone quarry, 7 single-rowtest rounds with constant specific charge and anaccurate EPD inter-hole delay of 25 ms were shot. Thehole diameters ranged from 3876 mm. Themuckpiles were sieved in three steps, all of the 25500

mm material and quartered laboratory samples(0063224 mm). Figure 9 shows round 4. The 1000-mm value is a boulder counting estimate.

The Swebrec function fit is excellent in the range05500 mm. The average goodness of fit for the sevenrounds is r2 = 0997 0001 (mean SD). Theparameter statistics became x50 = 490 70 mm, xmax =1720 440 mm and b = 246 045 (Table 1). (So a highx50-value would probably give hard digging in anaggregate quarry but this was a 500-t test blast.) Thecorresponding RosinRammler fits have a goodness offit of about 098 and the curves start to deviate from thedata from 20 mm fragments.

Interestingly, the coarse fractions seem to containinformation about the fines. Using the +90 mm data



7 Family of breakage functions for crusher based onthe t10 concept.

19 8 Fragment size distributions extracted from thefamily of breakage functions in Figure 7

9 Fragment size distribution for Brarp round 4 with best fit Swebrec function.21 Data range 05500 mm. Curve fitparameters: x50 = 459 mm, xmax = 1497 mm and b = 2238 (Table 1). x = 1000 mm is based on oversize counting,not a sieved value

and fitting a RosinRammler function gives entirelydifferent results than fitting the Swebrec function (Fig.10). The filled symbols denote the data used for thefitting. For a typical Swedish aggregate quarry withmarketing problems for 4 mm material, theRosinRammler fit to the +90 mm material predicts0304% of fines whereas the Swebrec function predicts2%, which is much closer to the measured value 25%.

The Swebrec and Rosin-Rammler curves are verysimilar for fragment sizes around x50. Equating the slopesat x50 makes it possible to compare the parameter values:

ln lnn

x x

b

2 2 maxequiv

50$ $. _ i8 B (3)

Furthermore, the Swebrec function has an inflectionpoint in logP versus logx space at:

x x x xmax maxb

50

1b1

=-_ ]i g

or

x x x xmaxb

50 50

1b1 1

=-

-_ ]i g (4)When b 1, the inflection point tends to x = xmax. Forincreasing values, it moves towards x = x50, which isreached when b = 2. When b increases further, the

inflection point moves to smaller values of x and thenmoves back towards x = x50. The inflection point andhence the undulating character of the Swebrec functionis always there and this makes it possible to pick up thefines behaviour from the coarse fraction data.

Start instead with a sieved sample with fragmentsin the range 1224 mm from Brarp round 4. Thisdata set was obtained after quartering of the 25 mmfraction from an Extec sizer, which sieved all 200-mmmaterial. If we know the percentage of the 22 4-mmfraction and make the guess that xmax B = 1800 mmbecause the rock is massive, then a curve fit with theSwebrec function yields the result in Figure 11. Thefilled symbols again denote the data used for thefitting.

The curve runs remarkably well through themissing coarse fraction data and provides an excellentestimate of x50. It seems that limited portions of thefragment size distribution contain relatively accurateinformation about the missing mass fractions.

Taking samples from a crusher product streamwhere the percentage of say 224-mm fines is betterknown than in a muckpile and using the closed-sidesetting and fragments shape to estimate xmax is another



Table 1 Curve fit data for Brarp rounds18,20

Round Blast-hole x50 xmax b Range r2 Residuals

no. diameter (mm) (mm) (mm) (mm) (mm) (% of scale)

1 51 468 1090 1778 05500 09966 < 242 51/76 629 2011 2735 05500 09976 < 093 76 529 2346 3189 05500 09969 < 184 51 459 1497 2238 05500 09973 < 185 38 414 1517 2398 05500 09977 < 226 64 422 2076 2651 05500 09977 < 227 76 511 1509 2261 05500 09968 < 19

10 Comparison of Swebrec and RosinRammler fits to coarse fraction data +90 mm and extrapolation to finesrange

example of where missing mass fractions might besuccessfully determined. Figure 12 gives an examplefrom a granite quarry where the fines percentage wasknown to be 1820% and the largest crushed pieceswere 250300 mm.

The final dip in the Brarp round 4 fragment sizedistribution in the 500 m range in Figure 9 may betaken care of by adding a second term to f(x) in theSwebrec function (Fig. 13):

f(x) = ln lna x x x xmax maxb

50 +_ _i i8 Ba x x x x1 1 1max max

c

50- - -_ _ _i i i8 B (5)

This extended Swebrec function has 5 parameters andis able to fit most fragment size distributions withextreme accuracy.

The Swebrec function has been fitted to hundredsof sets of sieved blasting, crushing and grinding datafrom a large number of sources,21 including:

(i) Brarp full-scale and model blasts.18,20

(ii) Less Fines project model blasts on 5 types oflimestone and an amphibolite.15

(iii) Norwegian model and full-scale bench blasts indifferent rocks with different explosives.11

(iv) Bench blast samples before and after crushing ofgneiss and dolerite.



11 Using Brarp sample data in range 1224 mm plus estimate of xmax to make Swebrec prediction of x50 and coarsefractions

12 Using 05224 mm fraction data from jaw crusher sample of granite plus closed-side setting estimate of xmax =~300 mm to predict missing fractions. Curve fit parameters: x50 = 77 mm and b = 233

(v) Blasting of iron ore oversize.(vi) South African reef and quarry blasts.

(vii) US bench blasts in dolomite.(viii) Blasting of magnetite concrete models.(ix) Blasting of layered mortar models.27

(x) Feed and product streams from gyratory, coneand impact crushing of andesite.

(xi) Product stream samples from roller crushing oflimestone.

(xii) Single particle roll mill crushing.(xiii) Ball mill grinding of limestone.(xiv) High energy crushing test (HECT) crushing of

seven rock types.(xv) Drill cuttings from the Christmas mine, etc.



13 Brarp round 4 data with best fit extended Swebrec function.21 Data range 0075500 mm. Curve fit parameters:x50 = 459 mm, xmax = 1480 mm, b = 2224, a = 099999812 and c = 20. r

2 = 09976. Note magnitude of prefactor(1a) in Equation (3)

Table 2 Curve fit data for crusher product samples at Nordkalk,15 xmax free or fixed

Sample x50 xmax b Range r2 Residuals

number (mm) (mm) (mm) (% of scale)

Free xmax2-01 759 300 2558 8300 09990 < 192-02 649 508 2479 8300 09950 < 332-03 867 300 2211 8300 09997 < 092-04 485 376 2712 8300 09995 < 122-05 828 300 2178 8300 09992 < 162-06 573 488 2681 8300 09987 < 192-07 847 320 2195 8300 09998 < 062-08 784 300 2297 8300 09998 < 062-09 757 377 2218 8300 09985 < 152-10 868 345 2518 8300 09995 < 132-11 959 300 2198 8300 09986 < 202-12 763 360 2142 8300 09991 < 112-13 649 300 2333 8300 09995 < 10Mean SD 753 132 352 72 236 020 09989

Fixed xmax2-01 759 315 2625 8300 09989 < 192-02 660 315 1984 8300 09930 < 322-03 868 315 2270 8300 09997 < 092-04 489 315 2497 8300 09993 < 172-05 828 315 2234 8300 09992 < 162-06 582 315 2200 8300 09977 < 282-07 847 315 2175 8300 09998 < 062-08 784 315 2356 8300 09997 < 072-09 759 315 2027 8300 09982 < 142-10 867 315 2400 8300 09994 < 142-11 961 315 2254 8300 09985 < 202-12 764 315 2002 8300 09989 < 132-13 649 315 2386 8300 09994 < 11Mean SD 755 130 226 019 09986

Two examples of crusher product size distributionsare shown in Figures 14 and 15. For nearly all thesesets of sieved data, the 3-parameter Swebrec functiongives a better fit that the RosinRammler function.

The experience21 is that good-to-excellent fit todifferent kinds of fragmentation data is obtained withcorrelation coefficients of usually r2 = 0995 or betterover a range of fragment sizes of 23 orders ofmagnitude. This range is at least one order ofmagnitude larger than the range covered by the Rosin-Rammler function.

Of the three parameters, the central medianmeasure (i.e. the size of 50% passing x50) shows themost stable behaviour. The maximum fragment size,xmax, will be physically related to the block size in situin blasting; however, as a fitting parameter it varieswidely. On the other hand, in crushing it is more orless given by the closed-side setting. Fixing the valueof xmax has little effect on the goodness of fit and thevalues of x50 or b (Table 2).

The parameter b normally accepts values in therange 14. Only in special cases, like model blasting



14 Fragment size distribution for gyratory crusher product 2 (CSS = 15) of andesite with Swebrec fit.21 Data range042563 mm. Curve fit parameters: x50 = 356 mm, xmax = 68 mm and b = 1531. r

2 = 09961.

15 Fragment size distribution for belt sample 2-04 of Nordkalk limestone.21 Swebrec function fit with fixed xmax =315 mm. Data range 025300 mm. Parameters: x50 = 488 mm and b = 2451. r

2 = 09994 (Table 2)

near the critical burden, does the fitting give a value ofb < 1. When the sieved fragmentation curve becomesRosinRammler like, both b and xmax tend to driftduring the fitting procedure and can becomeunnaturally large. This tendency may be suppressedby choosing a pair of coupled b and xmax values thatare related through Equation (3) for nequiv. In thesecases, the value of nequiv tends to vary less than the b-value. Otherwise, in the majority of cases, b is moreconstant than nequiv.

Often, b remains constant for a given material evenwhen the fragmentation conditions change. A coupleof data sets show, however, that b also depends on theexplosive used (Table 3), on the charge concentration,

and on the size of blast (model scale or full-scale, forexample). There is thus no basis for considering b as amaterial property or as depending only on thespecimen geometry as the KuzRam model suggestsbe the case for the uniformity index n.

The Swebrec function is also an improvement overthe CZM and TCM. Both JKMRC models showlinear behaviour in the fines range in loglog spacewhere a vast majority of the data sets are clearly non-linear. The Swebrec function does not rest on anassumption of the origin of the blasting fines that hasbeen disproved by tests.

The extended Swebrec function with 5 parametersshows the capacity of reproducing sieved fragmentation



Table 3 Curve fit data for block samples blasted with different explosives11

Explosive x50 (mm) xmax (mm) b Range (mm) r2 Residuals (% of scale)

SyeniteExtra dynamite 329 113 2441 05100 09976 < 20Extra dynamite 332 135 2635 05100 09969 < 20Extra dynamite 338 124 2437 05100 09970 < 20Dynamite 629 167 2053 05150 09955 < 31Dynamite 761 168 1813 05150 09938 < 28Dynamite 721 184 1976 05150 09936 < 46

GraniteExtra dynamite 499 143 2233 1125 09984 < 08Extra dynamite 505 136 2102 1125 09994 < 15Extra dynamite 466 142 2198 1100 09996 < 06Dynamite 661 127 1657 1125 09996 < 08Dynamite 742 131 1607 1125 09994 < 16Dynamite 781 133 1592 1125 09988 < 20

GneissExtra dynamite 608 161 2465 1125 09990 < 21Extra dynamite 625 146 2279 1125 09988 < 16Extra dynamite 668 168 2537 1125 09958 < 51Dynamite 953 330 2939 1100 09985 < 09Dynamite 103 464 3559 1100 09995 < 04Dynamite 106 408 3261 1100 09989 < 07

16 Laboratory ball mill data for limestone after 6 min grinding with extended Swebrec function fit.21 Data range0063336 mm. Curve fit parameters: x50 = 100 mm, xmax = 336 mm, b = 1010, a = 09911 and c = 1753. r

2 =09995

curves all the way into the super fines range (x < 01 mm)and also of reproducing laboratory ball mill grindingdata (Fig. 16).

It could thus be said that the Swebrec functiongives the fragment size distributions from blasting andcrushing a common form.

CRUSHER BREAKAGE AND NBCCONNECTIONSA comparison of Figure 8 and the presented sizedistributions from blasting and crushing shows that

the general curve-forms look alike. Some mass passingversus non-dimensional fragment size data for crusherand AG/SAG mill breakage functions from theJKMRC19 were matched against the non-dimensionalversion of the Swebrec function:

( ) ln lnP 1 1b

50= +x x x_ _i i8 B' 1with = x/t and max = 1 (6)

One example from the data sets behind Figures 7 and8 is given in Figure 17. In fact, the whole t10 family ofcrusher curves given by Napier-Munn et al.19 (their



17 JKMRC breakage function for crusher19 at t10 = 30%, with Swebrec function.21 Data range 00251. Curve fit

parameters: 50 = 0197, max = 10 and b = 2431. r2 = 09985

18 Mass passing isolines for constant degrees of size reduction, plotted as function of the breakage index t10. Fulllines denote predictions from Equations (5) and (6)

Table 6:1) may be reduced to the following equation:

logt t n100 1 100 1b t

10n

10

= + -_ _ ]i i g8 B* 4 (7)where, in this case, n is the size reduction ratio. When n= 10 (e.g. log(n) = 1), Equation (7) reduces to thestraight line tn = t10. An approximate expression forb(t10) was obtained from the data fitting:

21

b = 1616 + 002735t10 (8)

and the results are plotted in Figure 18. The full lineswere obtained using Equation (8), the dashed ones forn = 2 and 75 using the value b = 2174 valid for t10 = 20%.It is seen that the variations in the b-value do make adifference. The data are well represented by theisolines, differing by at most 34%.

Excellent fits were also obtained to the data inTables 4:7 and 4:9 of Napier-Munn et al.19 This is anindication that the whole set of spline functions usedto describe the breakage functions could be replacedby two simple equations Equation (7) and a form ofEquation (8).

The connection with the NBC theory rests on anobservation of the asymptotic behaviour of the Swebrecfunction. The parallel shift property in logP versus logxspace reduces to the statement that P(x)/P(x) =constant, independent of some parameter that describesthe shift. This is not met by the Swebrec function itselfbut the behaviour when x 0:

P(x) 1/f(x) yields

P(x)/P(x) f(x)/f(x) = lnb x x xmax_ iwhich is independent of x50. To retain the meaning ofxmax as the maximum allowable and x50 as the medianfragment size, make the substitutions xmax xc andx50 xmax. Now xc denotes a characteristic size valuefor the distribution, which lies outside the acceptablerange of x-values, 0 xmax.

Then the following function has NBC properties:

ln lnP x x x x xmaxNBC c cb

=_ _ _i i i8 B (9)with x x x xmax maxc50

2 b1 1

=-_ i

Equation 9 describes a set of parallel shifted curveswhen the value of xmax is changed but xc and b are keptconstant. PNBC(x) is always concave upwards when x< xmax, which is the behaviour of the OCS sub-circuitcurves in Figure 3 except in the super fines range.

The simplest description of an OCS sub-circuit is a sharpsieve that lets the fines bypass the comminution chamber,which in turn processes the coarse material retained by thesieve. When the derivative of PNBC(x) is used to describe thebreakage function of the chamber, the following resultsemerge for the combined product stream:21

(i) If the entire feed stream passes the chamber,then the product stream has NBC propertiesirrespective of the feed stream properties.

(ii) If parts of the feed stream bypass chamber then ifthe bypass stream has NBC properties then so doesthe combined product stream.

Otherwise the combined product stream does not haveNBC properties.

This has a direct bearing on the interpretation ofFigures 3 and 5. As long as the size reduction ratio in anOCS sub-circuit is large enough to make the bypass flowhave NBC properties, then the product stream has NBCproperties no matter what the size distribution of theprocessed part looked like beforehand.

The fragment size distributions for OCS crushed ormilled rock and rock models blasted with differentlevels of specific charge look similar in the 20-mmrange in Figure 5. Hence, we may conclude that if theNBC curve from mechanical comminution indeed ismaterial specific, then the model blasting testsproduce material specific results too in that range. Therelation between the blasting curve and the NBC



19 Comparison of three model predictions of the fragment size distribution of the Brarp round 4 muckpile withactual data in logP versus logx space

curve is:

P x P x1 1 1 NBC= +_ _i i8 B (10)Similarly, Equation (10) contains a potentialcorrespondence between the OCS sub-circuits and theJKMRC crusher models.

THE KUZRAM CONNECTIONIn connection with the KuzRam model, Equation (3)offers the possibility of simply replacing the originalRosinRammler function in Equation (1) with theSwebrec function of Equation (2a,b). Thus, we arriveat an extended KuzRam model (or KCO model, seebelow) based on the prediction formulae shown asboxed text (see equation 11ad).

The factor lng n n2 1 1n1

= +C_ _ _i i i essentiallyshifts the fragment size distribution to smaller valuesof x50 or to predicting more fines.

24 For expedience,call the original KuzRam model with g(n) added,the shifted KuzRam model. The shifting factor g(n)could be incorporated into the extended model too, ifexperience proves that this is an advantage.

Further, the rock mass factor A is given by:

A = 006 (RMD+JF+RDI+HF) (11e)

where: RMD = rock mass description = 10 (powdery/ friable)JF (if vertical joints) or 50 (massive)JF = joint factor = JPS + JPA

(joint plane spacing + joint plane angle)JPS = 10 (average joint spacing SJ < 01 m), 20 (01 m-

oversize xO) or 50 (> oversize)JPA = 20 (dip out of face), 30 (strike vertical face) or

40 (dip into face)RDI = rock density influence = 0025 (kg m3) 50HF = hardness factor, uses compressive strength c (MPa)

and Youngs modulus E (GPa) HF = E/3 if E < 50 and c/5 if E > 50.

CZM9 and TCM7 have slightly different algorithmsfor determining A and n and Cunningham6 has a newversion of A as well. As the Swebrec function has abuilt-in fines bias, it remains to be seen whether thefactor g(n) in the x50 expression is really needed. Theexpression for xmax is tentative and would be replacedwhen a better description of blasting in a fracturedrock mass becomes available.

Despite these uncertainties, the extended KuzRamor KCO model overcomes two important drawbacksof the previous version the poor predictive capacityin the fines range and the infinite upper limit to blocksizes.

Use Brarp round 4 as an illustration.18,20 The rockis hard and weakly fissured so we may try A = 1312 tomake things simple. For the determination of x50, wefurther need Q = 924 kg, q = 055 kg m3 and sANFO =70 4/45 = 622%. Insertion into Equation (11b)yields x50 = 448 cm or 448 mm, which is very close tothe 459 mm value given in Table 1.

For the determination of n, we need the geometryof the blast: = 51 mm, B = 18 m, S = 22 m, H = 52m, Ltot = 42 m or 39 m above grade, Lb = Ltot and Lc= 0 and SD 025 m. Insertion into Equation (11c)yields n = 117. The shifting factor g(n) = 0659.Using the estimate xmax = (BS) 20 m = 2000 mmyields b = 2431, which is also close to the value 2238given in Table 1.

Figure 19 shows the sieved data together with thethree prediction equations in logP versus logx space.Figure 20 does the same in P versus logx space. Sincethe original x50 prediction incidentally was nearlyexact, the shifting adds nothing in this case. It maywell in other cases but neither the original nor theshifted KuzRam model captures the real behaviourof the fragment size distribution in the fines range aswell as the extended KuzRam or KCO model does.

Figure 20 focuses the perspective more on thecoarse fractions. The slope equivalence between the



ln lnP x x x x x1 1 max maxb

50= +_ _ _i i i8 B( 2 (11a)x g n A Q s q115 .ANFO50

1 619 30

0 8$ $ $= _ _i i; E with g(n) = 1 or (ln2)1/n/(1+1/n) (11b)lnlnb x x n2 2 max 50$ $= _ i8 B

lnln x x B SD B2 2 2 2 0 014 1max h50$ $ $ $ $ $Q= - -_ _ _i i i S B L L L L H1 2 0 1b c tot tot0 1$ $ $+ - + $_ _i i8 8B B (11c)xmax = min (in situ block size, S or B) (11d)

where: x50 = median or size of 50% passing (cm)Q = charge weight per hole (kg); q = specific charge (kg m3)sANFO = explosives weight strength relative to ANFO (%)B = blast-hole burden (m); S = spacing (m)h = drill-hole diameter (m)Lb = length of bottom charge (m)Lc = length of column charge (m)Ltot = total charge length, possibly above grade level (m)H = bench height or hole depth (m)SD = standard deviation of drilling accuracy (m).

RosinRammler function and the Swebrec function atx50, which is expressed by Equation (3), is clear. Thereis only one data point for x > x50 in Figure 20. Figure9 contains a value for x = 1000 mm. The value P(1000mm) = 983% obtained from boulder counting is notvery accurate but it lies closer to the Swebrec functioncurve than to the RosinRammler one. The P(1000mm) values for the other Brarp rounds range from75% to 99%.

The final judgement as to whether the Swebrecfunction or the RosinRammler function does a betterfitting job for the coarse fractions is left open untilsufficient sieved muckpile data from full-scale blastshave been studied. The general experience in fittingthe Swebrec function to sieved data sets is that it isfully capable of reproducing both the fines and thecoarse fractions.

It is hoped that the incorporation of the Swebrecfunction in the KuzRam model will enhance thetools available to blasting engineers and researchers.Since the underlying size distribution is no longer theRosinRammler function the name might be changedto the KCO (KuznetsovCunninghamOuchterlony)model.

The description of the effects of initiation delaybetween blast-holes on fragmentation remains. To dothis properly and to account for systematic variations inrock mass properties, for example, good numerical modelsare required. Model complexity and computation speedare two factors that, for the time being, limit developmentof such models. Until this is overcome, the KCO modelhas a role to play.

CONCLUSIONSA new fragment size distribution function has beenfound, which ties together rock fragmentation byblasting and crushing, the KuzRam, CZM and TCMmodels of rock blasting, the JKMRC approach to

describing crusher circuits and the original NBCconcept of material specific size distribution curves incomminution.

The new Swebrec function has three parametersand gives good-to-excellent fits to different kinds offragmentation data with correlation coefficients ofleast 0997 or better over a range of fragment sizes oftwo to three orders of magnitude. Hundreds of sets ofsieved data from crushing and blasting have beenanalysed with excellent results.

The inherent curvature of sieved fragment sizedistributions is captured by the Swebrec function.Using coarse fractions data to extrapolate into thefines range has the potential of giving accurate finespredictions. Similarly, using samples of fine materialfrom a muckpile or crusher products on a belt and anestimate of xmax has the potential to give accurateestimates of the coarser fractions.

The extended Swebrec function with 5 parametersshows capacity of reproducing sieved fragmentationcurves all the way into the super fines range (x < 01 mm)and also of reproducing laboratory ball mill grinding data.

The Swebrec function further gives the followingadvantages:

(i) It can be used in the KuzRam model andremoves two of its drawbacks the poorpredictive capacity in the fines range and theupper limit cut-off of block sizes.

(ii) It reduces the JKMRC one-family description ofbreakage functions based on the t10 concept to aminimum.

(iii) It establishes a new family of NBC functionswith a realistic shape that connects mechanicaland blasting comminution and offers newinsight into the working of Steiners OCS sub-circuits of mechanical comminution.

By analogy with the origin of the KuzRam name, it issuggested that the extended KuzRam model be called theKCO (KuznetsovCunninghamOuchterlony) model.



20 Comparison of three model predictions of the fragment size distribution of the Brarp round 4 muckpile withactual data in P versus logx space

Summing up, it may be said that the Swebrec functionhelps give a much less fragmented description of rockfragmentation than present models.

On a more philosophical note, it would be useful toexplore why such a simple function as the Swebrecfunction is so successful. Although the three-parameter Swebrec function does a better job thanother functions with four or five parameters this iscannot, a priori, be taken as evidence that these threeparameters are physically more relevant. It should,however, motivate further investigation and analysis.

Tests reported here show that geometricallyseparate regions can not be assigned separatefragment sizes. Assigning such regions separatefracturing modes is misleading too. It is probably alsomisleading to take the fact that a two term function orbimodal distribution gives a good fit as evidence oftwo separate fracturing modes. The reverse might betrue, however. If two separate fracturing modes wereidentified and the two associated distributionfunctions determined, then their merging would yielda compound bimodal fragmentation function. Thismerger would not be a simple matter if the fracturingmodes interact.

The way to go in investigating different fracturingmodes could very well be computational physicswork.1,12 A postulated generic fragmentation modelwith two distinct mechanisms, a crack branching-merging process that creates a scale-invariant sizedistribution in the super fines range and a Poissonprocess that creates an exponential cut-off at systemdependent length scale, yields good agreement withour blasting fines data.2 Still lacking are therelationships between the model parameters and themicroscopic properties of different rock types plustaking the rock mass jointing properly into account.

ACKNOWLEDGEMENTSThe author appreciates the generous help from hiscolleagues in providing data and for discussions on thisdocument, Ingvar Bergqvist of Dyno Nobel, ClaudeCunningham of AEL, Andreas Grasedieck at Montan-Universitt (MU) Leoben, Jan Kristiansen of DynoNobel, Cameron McKenzie of Blastronics, Mats Olssonat Swebrec and Agne Rustan, formerly of LuleUniversity. Alex Spathis of Orica kindly provided hisunpublished manuscript. Professor Peter Moser, MULeoben finally is thanked for his continuing co-operationin all aspects of rock fragmentation.

Part of this work was financed by the EU projectGRD-2000-25224 entitled Less Fines production inaggregate and industrial minerals industry. The LessFines partnership includes, MU Leoben with ballastproducer Hengl Bitustein, Austria, the CGES andCGI laboratories of Armines/ENSMP, France andUniversidad Politcnica de Madrid (UPM) with theexplosives manufacturer UEE and CementosPortland, Spain. The Nordic partners are NordkalkStorugns, Dyno Nobel and SveBeFo.

Swebrec, the Swedish Blasting Research Centre atLule University of Technology, was formed on 1

February 2003 by the blasting group from SveBeFo,Swedish Rock Engineering Research and has sincethen worked on the Less Fines project on SveBeFosbehalf.

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