the swebrec function: linking fragmentation by blasting ...983449/fulltext01.pdf · up with two...

A new three-parameter fragment size distribution functionhas been found that links rock fragmentation by blasting andcrushing The new Swebreccopy function gives excellent fits tohundreds of sets of sieved fragmentation data withcorrelation coefficients of 0middot997 or better (r2gt0middot995) over arange of fragment sizes of 2ndash3 orders of magnitude A five-parameter version reproduces sieved fragmentation curvesall the way into the ndash100 microm range and also handles ball millgrinding data In addition the Swebreccopy function (i) can beused in the KuzndashRam model and removes two of itsdrawbacks ndash the poor predictive capacity in the fines rangeand the upper limit cut-off to block sizes (ii) reduces theJKMRC one-family description of crusher breakagefunctions based on the t10 concept to a minimum and (iii)establishes a new family of natural breakage characteristic(NBC) functions with a realistic shape that connects blastfragmentation and mechanical comminution and offers newinsight into the working of the Steinerrsquos OCS sub-circuits ofmechanical comminution It is suggested that the extendedKuzndashRam model with the Swebreccopy function replacing theRosinndashRammler function be called the KCO model

Finn Ouchterlony is at the Swedish Blasting Research CentreLulearing University of Technology Box 47047 S-10074Stockholm Sweden (Tel +46 8 6922293 Fax +46 86511364 E-mail finnouchterlonyswebrecltuse)

copy 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 fragmentationblasting KuzndashRam model CZM TCM crushing breakagefunction NBC natural breakage characteristics t10

BLAST FRAGMENTATIONIn an area as complicated as rock fragmentation byblasting Cunninghamrsquos KuzndashRam model4513 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 RosinndashRammler cumulativefragment size distribution in Figure 1

( )P x e1 1 2( ) ( )ln x x x x2 n n50 50= - = -- - (1)

The KuzndashRam 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 (1middot1) and the finiteboulder limit is taken care of by a cut-off in theinfinite distribution

The defining equations x50 = h1 (rock mass scaleexplosive 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)1nΓ(1 + 1n)where Γ is the gamma function Its effect is to raise thepredicted amount of fine material in well-gradedmuckpiles ie 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 ndash 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 KuzndashRammodel that corresponds to tensile fracturing and afines part that is derived from the compressivecrushing around a blast-hole The fines distribution isalso of RosinndashRammler 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 ndash1 mm fines tothe blast-hole region In this way the effects ofquantities like the rockrsquos compressive strength blast-hole pressure and VOD enter the prediction equations

In the TCM7 the fragment size distribution is alsomade up of two RosinndashRammler distributions thatrepresent the tensile and compressive fracture modesThe two populations co-exist over the whole range sothe result is a smooth curve (see Fig 2) Again theKuzndashRam 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 modeltwo 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 KuzndashRam model at the price of

Mining Technology (Trans Inst Min Metall A) March 2005 Vol 114 A29DOI 101179037178405X44539

The Swebreccopy 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 concept823

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 case27 Oslash 300-mm diameter100-kg mortar specimens with concentric coloured layerswere shot to produce 2 kg of ndash1 mm fines 1 kg fromthe Oslash 120-mm diameter inner layer and as much from theouter layer The percentage of fines is of course higher inthe inner layer but what matters in practice is the total

amount More evidence to this effect is provided in thework of Moser et al17 and Micklautsch14

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 approachesStill what blasting and crushing fragmentation orcomminution have in common has been neglected

A30 Mining Technology (Trans Inst Min Metall A) March 2005 Vol 114

Ouchterlony The Swebreccopy function linking fragmentation by blasting and crushing

1 Fragment size distribution of blasted hornfels from Mount Coot-tha quarry10 with RosinndashRammler curve fit21 Datarange = 0middot35ndash2000 mm Curve fit parameters x50 = 115middot8 mm n = 0middot572 Coefficient of determination r2 = 0middot9958

2 Comparison of fragment size distributions in CZM9 and TCM7 Note different characters of curves but samelinear behaviour in logP versus logx space in fines range

Steinerrsquos2526 approach to mechanical comminutionis based on the concept that a material which isfractured under lsquopurersquo conditions exhibits a material-specific lsquonatural breakage characteristicrsquo (NBC)Some of the tenets are that

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

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

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

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

kgndash1) created by an OCS is plotted versus the energyconsumed (J kgndash1) the points fall more or less on amaterial specific straight line The slope R (m2 Jndash1) 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 charges16 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)1820

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 feedmaterialrsquos 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 A31


3 Fragmentation curves from crushing and grinding of amphibolite in OCS sub-circuits317

4 Energy register curves315 for limestones and the Hengl amphibolite of Figure 3

principle equivalent to the KuzndashRam model equationwhich 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 thendash500 microm 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 SWEBRECcopy FUNCTIONWhen analysing the data from the Less Finesproject15 it was realised that the following fragmentsize distribution does a very good job of fitting sievedfragmentation data21 The transformation

P(x) = ( )f x1 1 +7 A

with 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 max

b

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 Baringrarp granitic gneiss18

Like the RosinndashRammler 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 parameterUnlike the RosinndashRammler or the CZMTCMfunctions 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 Oslash 51-mm diameter blast-holes on a 1middot8 times 2middot2-mpattern and a specific charge of about 0middot55 kg mndash3 Atthe Baringrarp1820 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 Oslash 38 to 76 mm Themuckpiles were sieved in three steps all of the 25ndash500

mm material and quartered laboratory samples(0middot063ndash22middot4 mm) Figure 9 shows round 4 The 1000-mm value is a boulder counting estimate

The Swebrec function fit is excellent in the range0middot5ndash500 mm The average goodness of fit for the sevenrounds is r2 = 0middot997 plusmn 0middot001 (mean plusmn SD) Theparameter statistics became x50 = 490 plusmn 70 mm xmax =1720 plusmn 440 mm and b = 2middot46 plusmn 0middot45 (Table 1) (Such ahigh x50-value would probably give hard digging in anaggregate quarry but this was a 500-t test blast) Thecorresponding RosinndashRammler fits have a goodness offit of about 0middot98 and the curves start to deviate from thedata from 20 mm fragments and below

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 concept19

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

9 Fragment size distribution for Baringrarp round 4 with best fit Swebrec function21 Data range 0middot5ndash500 mm Curve fitparameters x50 = 459 mm xmax = 1497 mm and b = 2middot238 (Table 1) The value for x = 1000 mm is based onoversize counting not a sieved value

and fitting a RosinndashRammler function gives entirelydifferent results than fitting the Swebrec function (Fig10) The filled symbols denote the data used for thefitting For a typical Swedish aggregate quarry withmarketing problems for ndash4 mm material theRosinndashRammler fit to the +90 mm material predicts0middot3ndash0middot4 of fines whereas the Swebrec function predicts2 which is much closer to the measured value 2middot5

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 max

equiv

50$ $

_ i8 B(3)

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

x x x xmax max

b

50

1b1

=-

_]i

g

or

x x x xmax

b

50 50

1 1b1

=- -

_]i

g(4)

When b rarr 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 1ndash22middot4 mm from Baringrarp round 4 Thisdata set was obtained after quartering of the ndash25 mmfraction from an Extec sizer which sieved all 200-mmmaterial If we know the percentage of the 22middot4-mmfraction and make the guess that xmax asymp 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 ndash22middot4-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 Baringrarp rounds1820

Round Blast-hole x50 xmax b Range r2 Residuals no diameter Oslash (mm) (mm) (mm) (mm) (mm) ( of scale)

1 51 468 1090 1middot778 0middot5ndash500 0middot9966 lt 2middot42 5176 (decoupled) 629 2011 2middot735 0middot5ndash500 0middot9976 lt 0middot93 76 529 2346 3middot189 0middot5ndash500 0middot9969 lt 1middot84 51 459 1497 2middot238 0middot5ndash500 0middot9973 lt 1middot85 38 414 1517 2middot398 0middot5ndash500 0middot9977 lt 2middot26 64 422 2076 2middot651 0middot5ndash500 0middot9977 lt 2middot27 76 511 1509 2middot261 0middot5ndash500 0middot9968 lt 1middot9

10 Comparison of Swebrec and RosinndashRammler 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 18ndash20 and the largest crushed pieceswere 250ndash300 mm

The final dip in the Baringrarp round 4 fragment sizedistribution in the ndash500 microm 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 max

b

50 +_ _i i8 B

a 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 sources21 including

(i) Baringrarp full-scale and model blasts1820

(ii) Less Fines project model blasts on 5 types oflimestone and an amphibolite15

(iii) Norwegian model and full-scale bench blasts indifferent rocks with different explosives11

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



11 Using Baringrarp sample data in range 1ndash22middot4 mm plus estimate of xmax to make Swebrec prediction of x50 and coarsefractions

12 Using 0middot5ndash22middot4 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 = 2middot33

(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 models27

(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 Baringrarp round 4 data with best fit extended Swebrec function21 Data range 0middot075ndash500 mm Curve fit parametersx50 = 459 mm xmax = 1480 mm b = 2middot224 a = 0middot99999812 and c = 2middot0 r2 = 0middot9976 Note magnitude of prefactor(1ndasha) in Equation (5)

Table 2 Curve fit data for crusher product samples at Nordkalk15 xmax free or fixed excluding x = 200 mm points

Sample x50 xmax b Range r2 Residuals number (mm) (mm) (mm) ( of scale)

Free xmax

2-01 75middot9 300 2middot558 8ndash300 0middot9990 lt 1middot92-02 64middot9 508 2middot479 8ndash300 0middot9950 lt 3middot32-03 86middot7 300 2middot211 8ndash300 0middot9997 lt 0middot92-04 48middot5 376 2middot712 8ndash300 0middot9995 lt 1middot22-05 82middot8 300 2middot178 8ndash300 0middot9992 lt 1middot62-06 57middot3 488 2middot681 8ndash300 0middot9987 lt 1middot92-07 84middot7 320 2middot195 8ndash300 0middot9998 lt 0middot62-08 78middot4 300 2middot297 8ndash300 0middot9998 lt 0middot62-09 75middot7 377 2middot218 8ndash300 0middot9985 lt 1middot52-10 86middot8 345 2middot518 8ndash300 0middot9995 lt 1middot32-11 95middot9 300 2middot198 8ndash300 0middot9986 lt 2middot02-12 76middot3 360 2middot142 8ndash300 0middot9991 lt 1middot12-13 64middot9 300 2middot333 8ndash300 0middot9995 lt 1middot0Mean plusmn SD 75middot3 plusmn 13middot2 352 plusmn 72 2middot36 plusmn 0middot20 0middot9989

Fixed xmax

2-01 75middot9 315 2middot625 8ndash300 0middot9989 lt 1middot92-02 66middot0 315 1middot984 8ndash300 0middot9930 lt 3middot22-03 86middot8 315 2middot270 8ndash300 0middot9997 lt 0middot92-04 48middot9 315 2middot497 8ndash300 0middot9993 lt 1middot72-05 82middot8 315 2middot234 8ndash300 0middot9992 lt 1middot62-06 58middot2 315 2middot200 8ndash300 0middot9977 lt 2middot82-07 84middot7 315 2middot175 8ndash300 0middot9998 lt 0middot62-08 78middot4 315 2middot356 8ndash300 0middot9997 lt 0middot72-09 75middot9 315 2middot027 8ndash300 0middot9982 lt 1middot42-10 86middot7 315 2middot400 8ndash300 0middot9994 lt 1middot42-11 96middot1 315 2middot254 8ndash300 0middot9985 lt 2middot02-12 76middot4 315 2middot002 8ndash300 0middot9989 lt 1middot32-13 64middot9 315 2middot386 8ndash300 0middot9994 lt 1middot1Mean plusmn SD 75middot5 plusmn 13middot0 2middot26 plusmn 0middot19 0middot9986

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 RosinndashRammler function

The experience21 is that good-to-excellent fit todifferent kinds of fragmentation data is obtained withcoefficients of determination usually r2 = 0middot995 orbetter over a range of fragment sizes of 2ndash3 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 (ie the size of 50 passing x50) shows themost stable behaviour The maximum fragment sizexmax 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 1ndash4 Only in special cases like model blasting



14 Fragment size distribution for gyratory crusher product 2 (CSS = 1middot5primeprime ) of andesite with Swebrec fit21 Data range0middot425ndash63 mm Curve fit parameters x50 = 35middot6 mm xmax = 68 mm and b = 1middot531 r2 = 0middot9961

15 Fragment size distribution for belt sample 2-04 of Nordkalk limestone21 Swebrec function fit with fixed xmax =315 mm Data range 0middot25ndash300 mm Parameters x50 = 48middot8 mm and b = 2middot451 r2 = 0middot9994 (Table 2)

near the critical burden does the fitting give a value ofb lt 1 When the sieved fragmentation curve becomesRosinndashRammler 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 KuzndashRam 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 logndashlog 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 32middot9 113 2middot441 0middot5ndash100 0middot9976 lt 2middot0Extra dynamite 33middot2 135 2middot635 0middot5ndash100 0middot9969 lt 2middot0Extra dynamite 33middot8 124 2middot437 0middot5ndash100 0middot9970 lt 2middot0Dynamite 62middot9 167 2middot053 0middot5ndash150 0middot9955 lt 3middot1Dynamite 76middot1 168 1middot813 0middot5ndash150 0middot9938 lt 2middot8Dynamite 72middot1 184 1middot976 0middot5ndash150 0middot9936 lt 4middot6

GraniteExtra dynamite 49middot9 143 2middot233 1ndash125 0middot9984 lt 0middot8Extra dynamite 50middot5 136 2middot102 1ndash125 0middot9994 lt 1middot5Extra dynamite 46middot6 142 2middot198 1ndash100 0middot9996 lt 0middot6Dynamite 66middot1 127 1middot657 1ndash125 0middot9996 lt 0middot8Dynamite 74middot2 131 1middot607 1ndash125 0middot9994 lt 1middot6Dynamite 78middot1 133 1middot592 1ndash125 0middot9988 lt 2middot0

GneissExtra dynamite 60middot8 161 2middot465 1ndash125 0middot9990 lt 2middot1Extra dynamite 62middot5 146 2middot279 1ndash125 0middot9988 lt 1middot6Extra dynamite 66middot8 168 2middot537 1ndash125 0middot9958 lt 5middot1Dynamite 95middot3 330 2middot939 1ndash100 0middot9985 lt 0middot9Dynamite 103 464 3middot559 1ndash100 0middot9995 lt 0middot4Dynamite 106 408 3middot261 1ndash100 0middot9989 lt 0middot7

16 Laboratory ball mill data for limestone after 6 min grinding with extended Swebrec function fit21 Data range0middot063ndash3middot36 mm Curve fit parameters x50 = 1middot00 mm xmax = 3middot36 mm b = 1middot010 a = 0middot9911 and c = 1middot753 r2 =0middot9995

curves all the way into the super fines range (x lt 0middot1 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 AGSAG 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 1

with τ = xt and τ le τmax = 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 al19 (their



17 JKMRC breakage function for crusher19 at t10 = 30 with Swebrec function21 Size fraction tx data range0middot025ndash1 Curve fit parameters τ50 = 0middot197 τmax = 1middot0 and b = 2middot431 r2 = 0middot9985

18 Mass passing isolines for constant degrees of size reduction plotted as function of the breakage index t10 Fulllines denote predictions from Equations (7) and (8)

Table 61) may be reduced to the following equation

logt t n100 1 100 1b t

10n

10

= + -_ _]

i ig

8 B 4 (7)

where in this case n is the size reduction ratio When n= 10 eg (log(n) = 1) Equation (7) reduces to thestraight line tn = t10 An approximate expression forb(t10) was obtained from the data fitting21

b = 1middot616 + 0middot02735t10 (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 = 2middot174 valid for t10 = 20It is seen that the variations in the b-value do make adifference The data are well represented by theisolines differing by at most 3ndash4

Excellent fits were also obtained to the data inTables 47 and 49 of Napier-Munn et al19 This is anindication that the whole set of spline functions usedto describe the breakage functions could be replacedby two simple equations ndash 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 Pprime(x)P(x) =constant independent of some parameter that describesthe shift This is not met by the Swebrec function itselfbut the behaviour when x rarr 0

P(x) asymp 1f(x) yields

Pprime(x)P(x) asymp ndashfprime(x)f(x) = lnb x x xmax_ i

which is independent of x50 To retain the meaning ofxmax as the maximum allowable and x50 as the medianfragment size make the substitutions xmax rarr xc andx50 rarr xmax Now xc denotes a characteristic size valuefor the distribution which lies outside the acceptablerange of x-values 0 rarr xmax

Then the following function has NBC properties

ln lnP x x x x xmaxNBC c c

b

=_ _ _i i i8 B (9)

with x x x xmax maxc50

2 1b1

=-

_ 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 xlt 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 chamberwhich 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 stream21

(i) If the entire feed stream passes the chamberthen 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 ndash20-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 Baringrarp 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 KUZndashRAM CONNECTIONIn connection with the KuzndashRam model Equation (3)offers the possibility of simply replacing the originalRosinndashRammler function in Equation (1) with theSwebrec function of Equation (2ab) Thus we arriveat an extended KuzndashRam model (or KCO model seebelow) based on the prediction formulae shown asboxed text (see equation 11andashd)

The factor lng n n2 1 1n1

= +C_ _ _i i i essentiallyshifts the fragment size distribution to smaller valuesof x50 or to predicting more fines24 For expediencecall the original KuzndashRam model with g(n) addedthe shifted KuzndashRam model The shifting factor g(n)could be incorporated into the extended model too ifexperience proves that this is an advantage

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 the

factor 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 KuzndashRamor KCO model overcomes two important drawbacks ofthe previous version ndash the poor predictive capacity in thefines range and the infinite upper limit to block sizes

Use Baringrarp round 4 as an illustration1820 The rockis hard and weakly fissured so we may try A = 13413 tomake things simple For the determination of x50 wefurther need Q = 9middot24 kg q = 0middot55 kg mndash3 and sANFO =70 times 44middot5 = 62middot2 Insertion into Equation (11b)yields x50 = 44middot8 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 Oslash = 51 mm B = 1middot8 m S = 2middot2 m H = 5middot2m Ltot = 4middot2 m or 3middot9 m above grade Lb = Ltot and Lc

= 0 and SD asymp 0middot25 m Insertion into Equation (11c)yields n = 1middot17 The shifting factor g(n) = 0middot659Using the estimate xmax = radic(BmiddotS) asymp 2middot0 m = 2000 mmyields b = 2middot431 which is also close to the value 2middot238given in Table 1

Figure 19 shows the sieved data together with thethree prediction equations in logP versus logx spaceFigure 20 does the same in P versus logx space Sincethe original x50 prediction incidentally was nearly



ln lnP x x x x x1 1 max max

b

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)1nΓ(1+1n) (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 tot

0 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 mndash3)sANFO = explosiversquos weight strength relative to ANFO ()B = blast-hole burden (m) S = spacing (m)Oslashh = 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)

Further the rock mass factor A is given by

A = 0middot06 (RMD+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 angleJPS = 10 (average joint spacing SJ lt 0middot1 m) 20 (0middot1 m-oversize xO) or 50 (gt oversize)JPA = 20 (dip out of face) 30 (strike perpendicular to face) or 40 (dip into face)RDI = rock density influence = 0middot025 times ρ (kg mndash3) ndash 50HF = hardness factor uses compressive strength σc (MPa) and Youngrsquos modulus E (GPa) HF = E3 if E lt 50 and σc5 if E gt 50

exact the shifting adds nothing in this case It maywell in other cases but neither the original nor theshifted KuzndashRam model captures the real behaviourof the fragment size distribution in the fines range aswell as the extended KuzndashRam or KCO model does

Figure 20 focuses the perspective more on thecoarse fractions The slope equivalence between theRosinndashRammler function and the Swebrec function atx50 which is expressed by Equation (3) is clear Thereis only one data point for x gt x50 in Figure 20 Figure9 contains a value for x = 1000 mm The value P(1000mm) = 98middot3 obtained from boulder counting is notvery accurate but it lies closer to the Swebrec functioncurve than to the RosinndashRammler one The P(1000mm) values for the other Baringrarp rounds range from75 to 99

The final judgement as to whether the Swebrecfunction or the RosinndashRammler 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 KuzndashRam model will enhance thetools available to blasting engineers and researchersSince the underlying size distribution is no longer theRosinndashRammler function the name might be changedto the KCO (KuznetsovndashCunninghamndashOuchterlony)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 KuzndashRam CZM and TCMmodels of rock blasting the JKMRC approach todescribing 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 0middot997 or better (r2 gt 0middot995) over a range offragment sizes of two to three orders of magnitudeHundreds of sets of sieved data from crushing andblasting have been analysed with excellent results

The inherent curvature of sieved fragment sizedistributions is captured by the Swebrec functionUsing 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 lt 0middot1 mm)and also of reproducing laboratory ball mill grinding data

The Swebreccopy function further gives the followingadvantages

(i) It can be used in the KuzndashRam model andremoves two of its drawbacks ndash 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 new



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

insight into the working of Steinerrsquos OCS sub-circuits of mechanical comminution

By analogy with the origin of the KuzndashRam name it issuggested that the extended KuzndashRam model be called theKCO (KuznetsovndashCunninghamndashOuchterlony) modelSumming up it may be said that the Swebrec functionhelps give a much less lsquofragmentedrsquo 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 thiscannot a priori be taken as evidence that these threeparameters are physically more relevant It shouldhowever 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 physicswork112 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 data2 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-Universitaumlt (MU) Leoben Jan Kristiansen of DynoNobel Cameron McKenzie of Blastronics Mats Olssonat Swebrec and Agne Rustan formerly of LulearingUniversity Alex Spathis of Orica kindly provided histhen unpublished manuscript Professor Peter MoserMU Leoben finally is thanked for his continuing co-operation in 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 ArminesENSMP France andUniversidad Politeacutecnica de Madrid (UPM) with the

explosivesrsquo manufacturer UEE and CementosPortland Spain The Nordic partners are NordkalkStorugns Dyno Nobel and SveBeFo

Swebrec the Swedish Blasting Research Centre atLulearing University of Technology was formed on 1February 2003 by the blasting group from SveBeFoSwedish Rock Engineering Research and has sincethen worked on the Less Fines project on SveBeForsquosbehalf

REFERENCES1 J A AringSTROumlM B L HOLIAN and J TIMONEN lsquoUniversality

in fragmentationrsquo Phys Rev Lett 2000 84 3061ndash30642 J A AringSTROumlM F OUCHTERLONY R P LINNA and J

TIMONEN lsquoUniversal fragmentation in D dimensionsrsquoPhys Rev Lett 2004 92 245506ndash1 to 4

3 A BOumlHM and R MAYERHOFER lsquoMechanical fragment-ation tests ndash optimized comminution sequence-energyregister functionrsquo Less Fines internal tech rpt no 22EU project GRD-2000-25224 2002

4 C V B CUNNINGHAM lsquoThe KuzndashRam model forprediction of fragmentation from blastingrsquo (eds RHolmberg and A Rustan) Proc 1st Symp on RockFragmentation by Blasting Lulearing University ofTechnology Sweden 1983 439ndash453

5 C V B CUNNINGHAM lsquoFragmentation estimations andthe KuzndashRam model ndash four years onrsquo (eds W LFourney and R D Dick) Proc 2nd Int Symp on RockFragmentation by Blasting Bethel CT SEM 1987475ndash487

6 C V B CUNNINGHAM Personal communication 20037 N DJORDJEVIC lsquoTwo-component model of blast

fragmentationrsquo Proc 6th Int Symp on RockFragmentation by Blasting Symposium Series S21Johannesburg SAIMM 1999 213ndash219

8 C GRUNDSTROM S M KANCHIBOTLA A JANKOVICH andD THORNTON lsquoBlast fragmentation for maximising theSAG mill throughput at Porgera gold minersquo Proc ISEE27th Annu Conf Expl amp Blasting Tech vol 1Cleveland OH ISEE 2001 383ndash399

9 S S KANCHIBOTLA W VALERY and S MORELLlsquoModelling fines in blast fragmentation and its impacton crushing and grindingrsquo (ed C Workman-Davies)Proc Explo 1999 Conf Carlton VIC AusIMM 1999137ndash144

10 T KOJOVIC S MICHAUX and C M MCKENZIE lsquoImpact ofblast fragmentation on crushing and screeningoperations in quarryingrsquo Proc Explorsquo95 Confpublication series no 695 Carlton VIC AusIMM1995 427ndash436

11 J KRISTIANSEN lsquoA study of how the velocity ofdetonation affects fragmentation and the quality offragments in a muckpilersquo Proc Explorsquo95 Confpublication series no 695 Carlton VIC AusIMM1995 437ndash444

12 F KUN and H J HERRMANN lsquoA study of fragmentationprocesses using a discrete element methodrsquo CompMethods Appl Mech Eng 1996 138 3ndash18

13 V M KUZNETSOV lsquoThe mean diameter of the fragmentsformed by blasting rockrsquo Sov Min Sci 1973 9144ndash148

14 A MIKLAUTSCH lsquoExperimental investigation of the blastfragmentation behaviour of rock and concretersquounpublished diploma work Institut fuumlr BergbaukundeBergtechnik und Bergwirtschaft MontanuniversitaumltLeoben Austria 2002



15 P MOSER lsquoLess Fines production in aggregate andindustrial minerals industryrsquo (ed R Holmberg) ProcEFEE 2nd Conf on Explosives amp Blasting TechniquesRotterdam Balkema 2003 335ndash343

16 P MOSER N CHEIMANAOFF R ORTIZ and R

HOCHHOLDINGER lsquoBreakage characteristics in rockblastingrsquo (ed R Holmberg) Proc EFEE 1st Conf onExplosives amp Blasting Techniques RotterdamBalkema 2000 165ndash170

17 P MOSER A GRASEDIECK V ARSIC and G REICHHOLFlsquoCharakteristik der Korngroumlssenverteilung vonSprenghauwerk im Feinbereichrsquo Berg und Huumltten-maumlnnische Monatshefte 2003 148 205ndash216

18 P MOSER M OLSSON F OUCHTERLONY and A

GRASEDIECK lsquoComparison of the blast fragmentationfrom lab-scale and full-scale tests at Baringrarprsquo (ed RHolmberg) Proc EFEE 2nd Conf on Explosives ampBlasting Techniques Rotterdam Balkema 2003449ndash458

19 T J NAPIER-MUNN S MORRELL R D MORRISON and T

KOJOVIC lsquoMineral comminution circuits ndash theiroperation and optimisationrsquo JKMRC MonographSeries in Mining and Mineral Processing BrisbaneQLD JKMRC 1996

20 M OLSSON and I BERGQVIST lsquoFragmentation inquarriesrsquo [in Swedish] Proc Disc Meeting BK 2002Stockholm Swedish Rock Constr Comm 2002 33ndash38

21 F OUCHTERLONY lsquoBend it like Beckham or a wide-range yet simple fragment size distribution for blastedand crushed rockrsquo Less Fines internal tech rpt no 78EU project GRD-2000-25224 2003

22 F OUCHTERLONY lsquoInfluence of blasting on the sizedistribution and properties of muckpile fragments a

state-of-the-art reviewrsquo Report MinFo project P2000-10 Stockholm Swedish Industrial Minerals Asso-ciation 2003

23 N PALEY and T KOJOVIC lsquoAdjusting blasting to increaseSAG mill throughput at the Red Dog minersquo Proc ISEE27th Annu Conf Expl amp Blasting Techn ClevelandOH ISEE 2001 65ndash81

24 A T SPATHIS lsquoA correction relating to the analysis of theoriginal KuzndashRam modelrsquo Int J Fragmentation andBlasting 2005 8 201ndash205

25 H J STEINER lsquoThe significance of the Rittinger equationin present-day comminution technologyrsquo Proc XVIIInt Min Proc Congr Dresden vol I 1991 177ndash188

26 H J STEINER lsquoZerkleinerungstechnische Eigenschaftenvon Gesteinenrsquo Felsbau 1998 16 320ndash325

27 V SVAHN lsquoGeneration of fines in bench blastingrsquoLicentiate thesis Department of Geology publicationA104 Chalmers University of Technology GothenburgSweden 2003

Author

Finn Ouchterlony is a graduate of Chalmers Univ Techn(MSc Engineering Physics) Gothenburg 1966 Royal InstTechn (Dr Techn Applied Mechanics) Stockholm 1980Employment Atlas Copco 1967ndash1984 Swedish DetonicResearch Foundation (SveDeFo Head 1987ndash1991 LulearingUniv Techn (LTU Assistant Professor) 1985ndash1988Yamaguchi University (Professor) Ube Japan 1991ndash1992Swedish Rock Engineering Research (SveBeFo) 1993ndash2003Swedish Blasting Research Centre (Swebrec at LTU) 2003 topresent Current position Professor of Detonics and RockBlasting



complexity The JKMRC prefers the CZM and theyhave used it successfully in a number of fragmentationprojects related to their Mine-to-mill concept823

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 case27 Oslash 300-mm diameter100-kg mortar specimens with concentric coloured layerswere shot to produce 2 kg of ndash1 mm fines 1 kg fromthe Oslash 120-mm diameter inner layer and as much from theouter layer The percentage of fines is of course higher inthe inner layer but what matters in practice is the total

amount More evidence to this effect is provided in thework of Moser et al17 and Micklautsch14

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 approachesStill what blasting and crushing fragmentation orcomminution have in common has been neglected



1 Fragment size distribution of blasted hornfels from Mount Coot-tha quarry10 with RosinndashRammler curve fit21 Datarange = 0middot35ndash2000 mm Curve fit parameters x50 = 115middot8 mm n = 0middot572 Coefficient of determination r2 = 0middot9958

2 Comparison of fragment size distributions in CZM9 and TCM7 Note different characters of curves but samelinear behaviour in logP versus logx space in fines range



















P(x) = ( )f x1 1 +7 A

with f(xmax) = 0 f(x50) = 1 (2a)



b

50_ _i i8 B (2b)

















ln lnn

x x

b

2 2 max

equiv

50$ $

_ i8 B(3)


x x x xmax max

b

50

1b1

=-

_]i

g

or

x x x xmax

b

50 50

1 1b1

=- -

_]i

g(4)















b

50 +_ _i i8 B


c

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






















Free xmax


Fixed xmax



























( ) ln lnP 1 1b

50= +x x x_ _i i8 B 1








logt t n100 1 100 1b t

10n

10

= + -_ _]

i ig

8 B 4 (7)


b = 1middot616 + 0middot02735t10 (8)









b

=_ _ _i i i8 B (9)


2 1b1

=-

_ i











curve is

P x P x1 1 1 NBC= +_ _i i8 B (10)















b

50= +_ _ _i i i8 B( 2 (11a)


1 619 30




0 1

$ $ $+ - +$

_ _i i8 8B B (11c)

































































Author






















P(x) = ( )f x1 1 +7 A

with f(xmax) = 0 f(x50) = 1 (2a)



b

50_ _i i8 B (2b)

















ln lnn

x x

b

2 2 max

equiv

50$ $

_ i8 B(3)


x x x xmax max

b

50

1b1

=-

_]i

g

or

x x x xmax

b

50 50

1 1b1

=- -

_]i

g(4)















b

50 +_ _i i8 B


c

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






















Free xmax


Fixed xmax



























( ) ln lnP 1 1b

50= +x x x_ _i i8 B 1








logt t n100 1 100 1b t

10n

10

= + -_ _]

i ig

8 B 4 (7)


b = 1middot616 + 0middot02735t10 (8)









b

=_ _ _i i i8 B (9)


2 1b1

=-

_ i











curve is

P x P x1 1 1 NBC= +_ _i i8 B (10)















b

50= +_ _ _i i i8 B( 2 (11a)


1 619 30




0 1

$ $ $+ - +$

_ _i i8 8B B (11c)

































































Author
















ln lnn

x x

b

2 2 max

equiv

50$ $

_ i8 B(3)


x x x xmax max

b

50

1b1

=-

_]i

g

or

x x x xmax

b

50 50

1 1b1

=- -

_]i

g(4)















b

50 +_ _i i8 B


c

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






















Free xmax


Fixed xmax



























( ) ln lnP 1 1b

50= +x x x_ _i i8 B 1








logt t n100 1 100 1b t

10n

10

= + -_ _]

i ig

8 B 4 (7)


b = 1middot616 + 0middot02735t10 (8)









b

=_ _ _i i i8 B (9)


2 1b1

=-

_ i











curve is

P x P x1 1 1 NBC= +_ _i i8 B (10)















b

50= +_ _ _i i i8 B( 2 (11a)


1 619 30




0 1

$ $ $+ - +$

_ _i i8 8B B (11c)

































































Author







b

50 +_ _i i8 B


c

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






















Free xmax


Fixed xmax



























( ) ln lnP 1 1b

50= +x x x_ _i i8 B 1








logt t n100 1 100 1b t

10n

10

= + -_ _]

i ig

8 B 4 (7)


b = 1middot616 + 0middot02735t10 (8)









b

=_ _ _i i i8 B (9)


2 1b1

=-

_ i











curve is

P x P x1 1 1 NBC= +_ _i i8 B (10)















b

50= +_ _ _i i i8 B( 2 (11a)


1 619 30




0 1

$ $ $+ - +$

_ _i i8 8B B (11c)

































































Author















Free xmax


Fixed xmax



























( ) ln lnP 1 1b

50= +x x x_ _i i8 B 1








logt t n100 1 100 1b t

10n

10

= + -_ _]

i ig

8 B 4 (7)


b = 1middot616 + 0middot02735t10 (8)









b

=_ _ _i i i8 B (9)


2 1b1

=-

_ i











curve is

P x P x1 1 1 NBC= +_ _i i8 B (10)















b

50= +_ _ _i i i8 B( 2 (11a)


1 619 30




0 1

$ $ $+ - +$

_ _i i8 8B B (11c)

































































Author





























( ) ln lnP 1 1b

50= +x x x_ _i i8 B 1








logt t n100 1 100 1b t

10n

10

= + -_ _]

i ig

8 B 4 (7)


b = 1middot616 + 0middot02735t10 (8)









b

=_ _ _i i i8 B (9)


2 1b1

=-

_ i











curve is

P x P x1 1 1 NBC= +_ _i i8 B (10)















b

50= +_ _ _i i i8 B( 2 (11a)


1 619 30




0 1

$ $ $+ - +$

_ _i i8 8B B (11c)

































































Author





logt t n100 1 100 1b t

10n

10

= + -_ _]

i ig

8 B 4 (7)


b = 1middot616 + 0middot02735t10 (8)









b

=_ _ _i i i8 B (9)


2 1b1

=-

_ i











curve is

P x P x1 1 1 NBC= +_ _i i8 B (10)















b

50= +_ _ _i i i8 B( 2 (11a)


1 619 30




0 1

$ $ $+ - +$

_ _i i8 8B B (11c)

































































Author




curve is

P x P x1 1 1 NBC= +_ _i i8 B (10)















b

50= +_ _ _i i i8 B( 2 (11a)


1 619 30




0 1

$ $ $+ - +$

_ _i i8 8B B (11c)

































































Author































































Author




the swebrec function: linking fragmentation by blasting ...983449/fulltext01.pdf · up with two...

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