prediction of the performance of explosives bench mining

Prediction of the performance of explosivesbench mining

SYNOPSIS

.In

by C. M. LOWNDS*, B.Se. Hons (Rhodes), Ph.D (Cape) (Visitor)

Some of the problems encountered in the calculation of the relative weight strength of explosives are discussed,and a realistic model of the behaviour of explosion products in a borehole is proposed. The heat flow from condensedto gaseous products is calculated as a function of time, and the relationship between the pressure and the volumeof the product gases is found by a consideration of the compression of the rock between radial cracks. Work doneon the burden is assumed to cease when the cracks first appear at the face. Reasonable values of burden and crackvelocities are obtained from the model. Calculations of the relative weight strengths of aluminized mixtures ofammonium nitrate and fuel oil show that the increased density resulting from the addition of aluminium is the mainadvantage to be gained.

SAMEVATTING

Verskeie probleme wat teegekom word in die berekening van relatiewe massasterktes van plofstowwe wordbespreek en 'n realistiese model vir die werkverrigting van die ontploffingsprodukte in die boorgat word voor-gestel. Die warmtestroming vanaf gekondenseerde tot gasprodukte is bereken as 'n funksie van tyd. Die verhoudingtussen die druk en volume van die produkgasse is bepaal deur die drukkrag tussen die straalbarse in aanmerking teneem. Werk wat op die las verrig word, word aangeneem om te staak sodra barse op die werkfront voorkom.Redelike waardes van die las en barssnelhede is verkry van die model. Berekeninge van die relatiewe massasterktesvan mengsels van ammonium nitraat en brand olie wat aluminium bevat, toon dat die hoer digtheid wat verkryword deur die byvoeging van aluminium die vernaamste voordeel is.

INTRODUCTION

Until recent years, the selectionof an explosive for a mining oper-ation was based largely on the costof that explosive - the explosiveselected was the cheapest thatwould deposit rock on the minefloor. More recently, with the con-tinually rising costs of labour andequipment, it has been realizedthat the cheapest explosive doesnot necessarily give the lowestoverall mining costs. This has ledto the development of a number ofnew explosives and blasting agents,notably those in which the 'strength'can be controlled by variation ofthe composition or density. Ex-plosives of varying strength can beused in different parts of a hole andin different positions in the blast,depending on the rock to be broken.

To use these explosives mostprofitably, mine planners and ex-plosives engineers need reliable in-dices of their performance. Thereare several known laboratory andtheoretical methods by which meas-ures of explosive performance can beobtained, but none of the knownmeasures of explosive strength hasgiven entirely reliable indicationsof field performance. For example,the performance of aluminized ex-plosives in the field is disappointingif they are rated according to their

*AE & Cl Ltd, Johannesburg.

maximum theoretical work. Thefinal evaluation of proposed combin-ations of mining method, blastgeometry, and explosive must there-fore be done in the field. Suchtrials are time-consuming and ex-pensive, and need to be kept to aminimum. Manufacturers of ex-plosives are therefore much con-cerned with developing better waysby which the performance of theirproducts in a real situation canreadily be predicted, and thus thenumber and extent of field trialsneeded to establish optimum blast-ing conditions can be reduced.

Since many commercial explosivesproduce condensed products of deto-nation, any attempt to calculate thework potential of an explosiveshould take account of the effect ofthese products on the availableenergy. For example, the aluminaproduced by the detonation of an

explosive with 10 per cent aluminium

contains, immediately after deton-

ation, about 15 per cent of the total

heat of the reaction: the percentage

of heat liberated by the condensed

products during the expansion of the

gases is therefore significant. This

paper presents a model of the be-

haviour of detonation products where

a condensed phase is present, and a

method for the calculation of theenergy transferred to the surround-

ing rock in bench blasting.

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY

THEORETICALCONSIDERA TIONS

The following discussion dealswith established methods of calcula-ting the energy of an explosion, andwith an extension of these theoriesfor the calculation of the energydelivered, in a realistic miningsituation, to the surrounding rock byboth gaseous and condensed deto-nation products.

The thermodynamic treatment ofthe explosion and subsequent ex-pansion is based on the work ofCook!. It is generally assumed byworkers in this field that the ex-plosion state, when the reactionproducts have the density of theunreacted explosive, is characterizedby complete thermal equilibrium.This assumption is open to questionwhen condensed species arise fromthe presence of inert materials inthe explosive, since they mus~ thenbe heated to the explosion tempera-ture by the hot gases in a very shorttime. On the other hand, somecondensed species like alumina areformed in exothermic reactions andtherefore with high temperatures.The thermal-equilibrium approxima-tion is adequate, at least for alumi-nized explosives, and is adopted here.

Mechanical work is done on therock by the gases as they expandfrom the initial explosion state. Ifthe maximum theoretical work is tobe calculated, the lower limit of

FEBRUARY 1975 165

thIs expansIon IS defined by the faHto atmospheric pressure. This isunrealistic because useful work donein breaking rock will cease at somestage before atmospheric pressure isreached. The lower limit of ex-pansion has been given an arbitraryhigher pressure in some cases2. Arealistic lower limit of pressure can,however, be calculated if it isassumed that no useful work isdone on the rock after the gasesstart to escape to the atmospherethrough cracks in the burden. Logic-ally, this lower limit depends on theproperties of both explosive androck. During an expansion withthese limits, the process can withsome justification be assumed to bethermodynamically reversible (i.e.the pressure exerted by the con-fining rock on the gas is nearlyequal to the pressure of the gas).This assumption is less valid whenthe gases are escaping more or lessfreely through cracks in the face,and errors caused by its use in thecalculation of maximum theoreticalwork have therefore been avoided.

Previous analyses of the expansionprocess have assumed it to beadiabatic, with complete heat trans-fer from condensed to gaseousphases!, which means that the rateof heat transfer is effectively infinite.By the assumption of a realisticsurface area of the condensed phase,and by the use of literature valuesfor heat-transfer coefficients, it canbe shown that, with the sort oftimes during which the detonationproducts are confined in the borehole(tens of milliseconds), the assump-tion of thermal equilibrium can besignificantly in error. It is possibleto calculate the amount of heattransferred in the time available ifthe value of the heat-transfer coeffi-cient can be found. The instant-aneous rate of heat flow is given by

dQ=hA LIT

dt '. . . . . .

where Q is the heat transferred fromcondensed to gas phases,

A is the area of the interfacebetween condensed phaseand gas,

h is the heat-transfer coeffi-cient with units of J /K/sper unit area, and

LlT=Tc-Tg is the thermal

166 FEBRUARY 1975

gradient acroSs the Interlaceof condensed phase and gas.

It is theoretically possible to solvethis equation by the separate use ofthe different values of hA applicableto each of the condensed species.However, the data at present avail-able are insufficient. Based on theargument that, under explosion con-ditions, heat transfer is largelydependent on the properties of theinterface between the gas and thecondensed phase, the most import-ant of which is its area, an averagevalue of hA covering all condensedspecies can be found.

If different condensed species areassumed to have similar specificsurface areas (per unit volume),then equation (1) can be applied inthe form

dQ(It=h Vc LIT, . . (2)

where Vc is the volume of con-densed products, and h is theheat-transfer coefficient of the col-lection of condensed species, withunits of J /K/s per unit volume ofcondensed phase.

Approximate methods for findingh Vc and LIT and for finding theintegral of dQ are presented in thenext section.

THERMODYNAMIC BASISFOR CALCULATING THE

STRENGTH OF EXPLOSIVES

(1)

All the data and algorithms neededfor the solution of the equations setout in this section are available inthe form of computer programXPLODE, which was obtained fromthe Explosives Research Labora-tory of Canadian Industries Limited.Definitions

The explosion state exists im-mediately after detonation, whenthe reaction products have the bulkdensity of the unreacted materialand are in thermal equilibrium. Thesubscript E is used to denote theexplosion state.

The initial state differs from theexplosion state only for decoupledcharges, when there is a reduction inpressure due to isothermal expansionof the detonation products to com-pletely fill the borehole (or explosionchamber in the ballistic mortar).The subscript I is used to denotethe initial state.

The final state IS the llmlt of theexpansion process during which use-ful work is performed. The subscriptF is used to denote the final state.

The explosion temperature isfound from

JTE

Qv =298

CvdT

- Qp-P LIV

(3)

. . . . (4)

- (EHOt-EHOt)-298.R Llnprods reacts... (5)

wherc C = molar heat capacity,H = enthalpy, the subscriptsP and V r"fer to constant press-ure and volume respectively, and Llnis the change in the number of gasmoles. For condensed explosives,

Lln=~n;=n,"""

(6)~

subscript i referring to the variousreaction products. An iterative pro-cedure must be adopted to find,simultaneously, the chemical equili-brium composition, represented bythe set (ni) and TE. The heat offormation of the products and theheat capacity integral are calculatedfrom

f~V~T =

f(fEniCVi )dT and

298 298

E Hat = EniHOt.prods i i

The equation for the state of thegases used is that proposed byCook! :

P(V -a(v))=nRT, (7)

where a(v) is a function of volumeonly. This function is evaluatedusing the aVS V data obtained semi-empirically by Cook.

The specific volume V E of thegases in the explosion state isdefined by

VE=(I/e- Wc/ec)/(I- Wc), . (8)where e is the bulk density of the

unreacted explosi ve,ec is the weighted-average

density of the condensedproducts, and

Wc is the mass fraction of thecondensed products.

Equations (7) and (8) are used tocalculate the explosion pressure PE,from which the initial pressure isobtained:


(V-a)E

PI=PE'(V-a)J'.,.. (9)

where, for cylindrical charges inboreholes,

El = TTb2l,b = borehole radius, andl = length of hole section that

originally contained unitmass of explosive.

VI is calculated from equation (6)

with �I in place of E.With either PF or VF specified,

the equation describing the ex-pansion of the gases from states I toF is used to calculate TF and otherdesired properties of the final state.The first law of thermodynamics forthe simplest case of adiabatic, re-versible expansion from I to F withcontinuous thermal equilibration be-tween condensed and gaseous phasesand mechanical work [',ives only

OvdT = PdV . . (10)

orI~OvdT = I: pay. . . . (ll)

Equation (10) is more convenientlywritten in the form

n ~~ dT =P

dV = ~ dV,T T V-a

. . . . (12)

from which

J

" I0

IF

dVn

F; dT = nR

IV - a

.

. . . . (13)

The integrals in (13) can be evaluatedfrom known heat capacity and a( V)data. Equation (13) is easily solvedby iteration for TF if VF or PF isknown. The work done during the

expansion is given by I~OvdT ac-

cording to equation (ll).

ANALYSIS OF HEATTRANSFER

A general statement of the firstlaw, from which equation (10) isderived, is

dE=dQ-dW, (14)where W = mechanical work.If this equation is applied to areversible expansion of the gaseousphase of detonation products, then

dE=Ov, gas dT . . . . . . (15)is the change in internal energy ofthe gas*,

(Footnote; *For the Cook equatIonof state, where a is a function ofvolume only, the total

dE =( ~; )vdT +( ~~-)TdV

reduces to (15). This is alsoimplied in equation (8).)dQ=-hVc. L1T.dt .. (16)

is the heat transferred from con-densed to gaseous products, and

dW=PdV (17)is the mechanical work done.

The definition of the heat transferdQ as a function of time means thattime must be related to two of thethermodynamic parameters. Since at( V,P) function is easily derived, atleast for the ballistic mortar, equa-tion (16) is modified to

dtdQ=-hVc. L1T.

dVdV,

. . . . (18)

and the expansion of the gaseousphase is completely described by

I:PdV = I~Ov,gasdT + hVc

I~ (L1T. :~) dV. . . . (19)

An analytical solution of the lastintegral in (19) was found to beunpractical, and the expansion istherefore analysed by consideringsmall increments of V from VI toVF, and performing numerical in-tegrations to solve (19). Using thismethod, the work integral can beevaluated with sufficient accuracyby the approximation

Pi(Vi-b)a=Pj {Vj-b)a, . . (20)

where lower-case subscripts refer tosmall increments of V,

b=5 X 10-4 m3/kg, anda is a constant for the expansionstage being considered (a > 1). From(20),

If 1. PdV =- 1

[pj(Vj-b)~ -a

-Pi(Vi-b)]. . . (21)Finally, T can be found at anystage (J) of the expansion from thefollowing equations;

L1T= Tc-T

Q = I~dQ

= I~PdV -

I~Ov,gasdT

=I~OcdTc (22)


=Oc. (Tc -'1\ )I J

=Cc. (TE-Tc ),J

where Oc is the heat capacity of thecondensed products.

THE EVALUATION OF THEHEAT -TRANSFER

COEFFICIENT

The best measurements of theperformance of explosives availableto the author were the ballisticmortar powers of certain commercialexplosives* .(Footnote: *Data from Nobel's Ex-plosives Company were used).

The function :~ in equation (18)

is found as follows. If the pendulumis regarded as fixed, the work doneby the gas in expanding from state Ito an intermediate state is equal tothe instantaneous kinetic energy ofthe projectile. Thus

i

PI, VI

tm U2= PdV,. . . (23)

P,V

where m is the mass of the project-ile, and

u is its instantaneous velocity.From the geometry of themortar,

1 dVu=7 (It' . . . . . (24)

where A is the area of the projectileface.

Pressures of 1 kbar and less aretypical of this expansion. An ideal-gas approximation can therefore beused, and, if the process is assumedto occur without significant loss ofheat to the confining metal, then

PVY =PI vI."""

(25)

Equation (23) therefore becomes

m (dV )2 1

2A2 (It =1-y (PV-PIVI),. . . . (26)

from which

dt [ 2A2 ( )]-t

dV = m(l-y) PV -PI VI

. . . . (27)

or, using an alternative expressionof (27),

~= [2A2PIVYJ

dV m(l-y)

(Vl-Y - VII-y)rt . . (28)

FEBRUARY 1975 167

1,7

1,6

1,5

1,4

1,3

1,2

Calculatedwork (MJ/kg)

1,0

0,9

0,8

0,7

0,6

0,5

0,4

168 FEBRUARY 1975

20 30 50Measured

70(re8G)

Fig.I-Calculated versus measured work done in the ballistic mortar. The verticalrange arises from two extremes of heat-transfer coefficient, h


Equation (28) enables :~, and

therefore the integrand of the lastterm in equation (19), to beevaluated at successive steps of Vfrom VI to VF. The work done duringthis expansion can therefore becalculated as outlined above forany heat-transfer coefficient, h.

Four values of h were chosen,and the ballistic mortar powers of17 commercial explosives that pro-duce a variety of condensed prod-ucts were calculated for each h inturn. The results are illustrated inFig. 1. The correlation of measuredwith computed ballistic mortarpowers was found for each h. Theresults of these calculations are pre-sented in Table I, where it is shownthat an approximate value of his 2,5 X 108 W jK per cubic metre ofcondensed products.

TABLE ICORRELATION COEFFICIENTS OF CALCU-

LATED VERSUS MEASURED BALLISTIC

MORTAR WORK FOR DIFFERENT VALUES OF

THE HEAT-TRANSFER COEFFICIENT h

Values of h(WfKfm3)

x 10-8

Correlationcoefficient

0,221,6

1290

0,99560,99680,99720,9968

CALCULA TED FIELDPERFORMANCE

In the application of the aboveheat-transfer coefficient and series ofthermodynamic equations to theexpansion of the products of adetonation in a borehole, a volume-time relationship must be found andthe final state must be specified interms of VF. The model describedbelow is based on that of Johanssonand Persson3.

A fully coupled cylindrical columnof explosive is detonated from thebottom of a borehole. The subse-quent process is analysed aftersuccessive increments of time. It isassumed that the explosive is capableof generating a system of radialcracks. After n time increments ofmagnitude dt, the detonation wavehas proceeded a distance Z up theborehole :

Z=Dndt, (29)where D is the detonation velocity.

The gas confined by the unreactedexplosive and the borehole wallsenters radial cracks at the velocityof sound in the gas:

U=(KjE)', (30)where K is the bulk modulus ofelasticity of the gas and E is itsdensity. Since

K=- VdPdV

. . . . . . (31)

and the approximationPi(Vi-b)a=Pf(Vf-b)a .. (32)

is being used for small changesfrom i to f,

( a VP )~

U=~V-b), (33)

and the distance travelled by thegas in the cracks in time dt isapproximately

( aVP )!

dX=dt E(V-b) . . . . . (34)

The V, P, and E values used inequation (34) are those of thebeginning of the expansion step.Thus, in the first time incrementVE, PE, and EE are used. Thisapproximation becomes more ac-curate as the duration of the timeinterval decreases. Sufficiently smalltime increments were found fromtrial calculations, as was a suitableinitial value of the exponent a.

After some number of time incre-ments, the gas occupies the bore-hole to height Z and radial cracksform to a maximum distance Xm.A volume of rock is therefore beingcompressed by the gas in the cracks.This volume has been calculatedexactly by considering the X-Z

dependence, but sinceddZ

is cons-dX

ttant and

Ttis nearly constant, a

cone of rock with its base at thebottom of the borehole can be con-sidered. The volume of rock undercompression is therefore

VR=; Z (Xm+b)2- VB, . (35)

where V B is the instantaneousvolume of the borehole occupied bythe detonation products. If P is theinstantaneous gas pressure, which isassumed to be constant throughoutthe borehole-crack system, then thedecrease in the volume of the rockis given by

JOURNAL OF THE SOUTH AfRICAN INSTITUTE OF MINING AND METALLURGY

V 'R= VRPjK, (36)where K is the bulk modulus ofcompressibility of the rock. Equa-tions (35) and (36) hold for Xm <spacingj2 in the geometry of thestandard borehole (Fig. 2). From thetheory of elasticity,

K-V 2 (l+fL)-

p E3(1- fL)

, ... (37)

where Vp is the velocity of thecompressive wave in the rock, E isits density, and fL is Poisson's ratiofor the rock. Since fL has a valueclose to 0,25 for most rocks, equa-tion (37) will be used in the form

K=0,56 E Vp2. (38)From equations (35) to (38), the

instantaneous volume of theborehole-crack system is given as afunction of pressure by

V= VRPjK+ VB. (39)The relevant pressure in equation

(39) is found by considering thework-history of the gaseous andcondensed detonation products.

The solution algorithm developedabove can be summarized as follows.Knowing the pressure Pi, volumeVi, gas temperature Ti, solid temp-erature T si, and the previous valueof a at time ti,(1) calculate the maximum crack

length Xm=Xmi+dX;(2) from equation (36), calculate the

volume of rock under com-pression;

(3) change gas temperature and gasvolume simultaneously untilboth equations (39) (describingvolume in terms of final pressureand rock properties) and (19)(the energy balance equation)are satisfied;

(4) calculate new values of Ts fromequation (22), the power a from

a=In(PiJPf)

andIn(Vf-b)j(Vi-b)

proceed to the next increment.

This process is followed from theinitiation of the explosive columnat the bottom of the borehole untilthe gas first appears at the face. Thebreakout angle is assumed to be90°. Before the detonation of theexplosive is complete, (4) abovestepmust include the addition of high-energy material to the system as aresult of the detonation of the nextsection of the column. The averagegas and solid temperatures are

FEBRUARY1975 169

ic - 6,75

1---------

170 FEBRUARY 1975

/°S - 9,0

/~ B=9,0

- ------

E = 13,5

--t_-

>

0

F - 16,9

B - S - 400

C - 300

E - 6OD

F ,. 75D

G ,. 15D

--------

D = Boreho1e diameter - 0,225

Fig. 2-The standard borehole (dimensions in metres)

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING ANI) M~T ALI.URGY

Explosion Final stateExplosive Hole Rock state

diam. typeDens. Al (mm) Press. Temp. Press. Temp. (kK)

(kgjm") (%) (kbar) (kK) (kbar)Gas Solid

0 31 2,80 0,94 1,7510 25 40 3,71 1,00 2,39 3,4718 (1 in) 42 4,41 1,02 2,85 4,27

0 31 2,80 0,93 1,7510 100 40 3,71 1,01 2,39 2,99

1000 18 (4 in) Med. 42 4,41 1,02 2,89 3,93

0 31 2,80 0,93 1,7510 225 40 3,71 1,01 2,40 2,6218 (9 in) 42 4,41 1,02 2,91 3,55

0 31 2,80 0,93 1,7510 450 40 3,71 1,01 2,40 2,4518 (18 in) 42 4,41 1,03 2,94 3,20

0 31 2,80 0,93 1,755 34 3,25 0,99 2,07 2,11

1000 10 225 Med. 40 3,71 1,01 2,40 2,6213 (9 in) 40 3,98 1,02 2,59 2,9918 42 4,41 1,02 2,91 3,55

0 31 2,80 0,61 1,6610 Soft 40 3,71 0,66 2,31 2,5318 42 4,41 0,68 2,83 3,46

1000 0 225 31 2,80 0,93 1,7510 (9 in) Med. 40 3,71 1,01 2,40 2,6218 42 4,41 1,02 2,91 3,55

0 31 2,80 1,33 1,9010 Hard 40 3,71 1,41 2,58 2,8918 42 4,41 1,44 3,15 3,76

800 20 2,80 0,87 1,89900 0 225 Med. 25 2,80 0,90 1,82

1000 (9 in) 31 2,80 0,93 1,75

Volume Time Totalchange to gas workV/IV 0 escape (MJjkg)

(ms)

---6,69 1,2 1,807,22 1,1 1,867,25 1,0 1,80

7,86 4,9 1,808,13 4,4 1,868,17 4,0 1,80

7,84 11 1,808,14 9,9 1,868,21 8,9 1,81

7,83 21 1,808,14 20 1,888,21 18 1,81

7,84 11 1,807,99 10 1,858,14 9,9 1,868,21 9,2 1,868,21 8,9 1,81

10,9 12 1,8411,5 10 1,9611,7 9,3 1,90

7,84 11 1,808,14 9,9 1,868,21 8,9 1,81

6,18 10 1,796,48 9,5 1,866,52 8,3 1,79

7,21 11 1,787,50 11 1,797,82 11 1,80

FEBRUARY 1~7~; 171

accordingly increased, as is the totalmass of material in the system. Anew pressure is therefore calculatedfrom the equation of state. The con-sideration of immediate thermalequilibration of the solids after theaddition of new material can beseen from equations (18) and (22)to be equivalent to a considerationof different masses with differenttemperatures, except for a smallinaccuracy resulting from the de-pendence of mean specific heat ontemperature.

The final volume Vp and thecorresponding values of Pp and Tpare all found from the iteration instep (3) above when Xm has justexceeded burden/cos 450, which isthe breakout distance. All relevantdetails of the expansion of thedetonation products from state Ito state F can therefore be calcu-lated.

CALCULATIONS ANDRESULTS

The explosives considered in thispaper are all oxygen-balanced mix-tures of ammonium nitrate withfuel oil (AN]'O), with fuel oil andaluminium (ALANFO 5, 10, and13), or with aluminium only (ALAN).The oxygen- balanced mixtures ANFOand ALAN contain 6 per cent fueloil and 18 per cent aluminium res-pectively. The explosive densityused was 1000 kg/m3 for mostcalculations, but the effect of densityon the specific work of ANFO wasalso investigated.

A 'standard' borehole modelledon the blast geometry typical ofbench blasting was used in thesimulation and is shown in Fig. 2.The powder factor in this hypo-thetical situation is 0,153 kg/tewith explosive and rock densities of1000 and 2380 kg/m3 respectively.

Blasts of various magnitudes weresimulated by multiplication of allthe linear dimensions of the stand-ard borehole by a constant factor.The powder factor was not changedby this operation. Geometries ofone-ninth, four-ninths, and twicethe standard scale were used toinvestigate the effect of blast scaleon the energy delivered to the rock.This energy is dependent on thetime interval between the initial andfinal states, which relates to thetime required for the furthest ad-vanced gas in the radial cracks toreach the free face.

Details of the relevant thermo-dynamic parameters, the times forgas escape, and the specific work(MJ/kg of explosive) done by anexplosive are listed in Table II forvarious blast geometries, rock types,explosive densities, and aluminiumcontents. The field performance in-

TABLE IICALCULATED THERMODYNAMIC AND RELATED PARAMETERS OF ALUMINIZED ANFO UNDER DIFFERING CONDITIONS OF USE (OXYGEN-

BALANCED MIXTURES)

.JOURNAL OF THE SOUTH AFRICAI'f INSTITI)T~ 9F M'I'fING AI'fD M~TALI,.UR~Y

01018

01018

01000 10

18

01018

05

1000 101318

01018

01000 10

18

01018

Hole Rock F.P.I.diameter type (% ANFO)

(mm)

10025 103

(1 in) 100

100100 103

(4 in) 100

Medium 100225 103

(9 in) 101

100450 104

(18 in) 101

100103

225 103(9 in) Medium 103

100

100Soft 107

103

225 100(9 in) Medium 103

100

100Hard 104

100

100225 Medium 100

(9 in) 100

TABLE IVROCK PROPERTIES

Vp = Velocity Bulk modulusof of

Rock compressive Density (�) elasticitytype wave (kg/m3) (K =

Vp2�) Reference(m/s) (X 1O-1ON/m2)

Soft 3000 2030 1,83 This paper

Medium 4100 2380 4,00 This paper

Hard 5200 2660 7,19 This paper

Limestone 3000 2030 1,83 4

Sandstone 4500 2370 4,80 4

Granite 5200 2660 7,19 4

Marble 6401 2804 11,5 5

Limestone 4267 2563 4,67 5

Granite 5639 2627 8,35 5

172 FEBRUARY 1975

TABLE IIICALCULATED FIELD PERFORMANCE INDICES (F.P.I.) OF ALUMINIZED ANFO UNDER DIFFER-

ING CONDITIONS OF USE (OXYGEN.BALANCED MIXTURES)

Explosivedensity(kg/m3)

Aluminium(%)

800900

10000

dices (F.P.I.) of aluminized ex-plosives are expressed on a massbasis relative to the performance ofANFO under similar conditions,and appear in Table Ill.

Initial comparisons were done ina hypothetical medium-strength rockwith density of 2380 kg/m3 andcompressive-wave velocity of 5,50km/so The effect of rock type on thespecific work of aluminized ANFOwas, however, also investigated byconsideration of the standard bore-hole in hypothetical soft and hardrocks. The properties of these threerock types are listed in Table IV,which gives selected rock propertiesfrom the literature for comparison.An illustration of the effect of rocktype on the F.P.I. of aluminizedexplosives is given in Fig. 3.

As a means of checking the suita-bility of the proposed model, calcu-lations were done on predictedmaximum burden momentum andthus maximum burden velocity, andthe results compared with valuesgiven in the literature.

The maximum burden momentumwas calculated by numerical inte-gration of an expression for themomentum:

MU= J~ PAdT,

where M is the mass of the burdenper borehole,

U is its final velocity,tp is the time for gas escape,P is the instantaneous press-

ure of the confined gases,and

A is the instantaneous areaabove the grade line of thecracks joining adjacentholes.

Burden velocities thus obtainedwere 10 to 14 m/so Experimentallydetermined burden velocities fromthe literature are given in Table V.

It appears from the work ofField and Ladegaard-Pedersen8 thatthe velocity of gas penetration intocracks is close to the velocity of thecrack tip. The effective maximumcrack velocity was therefore ob-tained by dividing the length of thebreakout crack by tp, and was about1300 m/s but dependent on the rocktype. The calculated effective crackvelocities are listed for the threerock types in Table VI, with selected


108

106

F.P.l.

104

(% ANFO)

102

100I

(9

0 5

x

0 x

0

10 2015

% aluminium

Fig. 3-Dependence of field performance index (F.P.I.) of aluminized explosiveson aluminium content for different rock types (standard borehole)

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY FEBRUARY 1975 173

TABLE VBURDEN VELOCITIES

Approximate MaximumExplosive Rock powder burden Reference

factor velocity(kg/te) (m/s)

ANFO HardAluminized Medium 0,15 10-14 This paperANFO Soft

Semi-gelatin Granite 0,25-0,5 11-34 5

Semi-gelatin Marble 0,10-0,23 ll-18 5

ANFO Limestone 0,67 13 5

Decoupled,high-energy Granite 0,16-0,17 7-10 4explosive

GraniteVariety Limestone Variety 2-30 4

Sandstone

Dynamite Granite 1,3 27 6

*( -- Not reported ---------* 10-15 7

--~~-~~-~~

ALANFO 10 Medium rock 1270 0,31-0,23 This paper

ALANFOIO Hard rock 1400 0,27 -0,21 This paper

Decoupleddynamite Granite 950-1080 0,20 6

Azidejstyphnate P.M.M.A. 700 0,32 8

Azide/styphnate Glass 1500 0,30 8

Variety Granite 1200-1700 0,26-0,32 4

--~--

Maximum Useful workthermodynamic (this paper-

work* mean value)(MJ/kg) (MJ/kg)

3,8 1,8

5,2 1,9

6,0 1,8

--~- ---~- -~

*These burden velocities were obtained from high-speed photographs of the movementof burden of about 8 m under actual blasting conditions.

TABLE VIEFFECTIVE CRACK VELOCITIES

--~ ---~ -- ----

ExplosiveCrack

velocity(m/s) Vp

V crackBlastedmaterial

Reference

ALANFO 10- Soft rock-

120J 0~4- -0,30-IThis paper-

TABLE VIICOMPARATIVE MEASURES OF THE 'STRENGTHS' OF ALUMINIZED ANFO

-~~- ~ --~--~-

Aluminium(%)

Heat ofreaction*(MJ/kg)

0 3,8

10 5,4

18 6,6

-~--~-~- -~--

*Calculations done with computer program XPLODE from Canadian Industries Ltd.

174 FEBRUARY 1975

literature values for comparison.The stemming was treated as a

piston with a velocity-dependentfrictional component. The maximumvelocity of the stemming was limitedto 100 m/s, and the stemmingvelocity in the final state was foundto be about 50 m/so

DISCUSSION

The model of the performance ofexplosives has been developed toassess the maximum useful workthat can be done by an explosive ina borehole. Calculations on thisbasis show that less than half of themaximum available energy will beimparted to the rock in a normalblast. This fact alone does notnecessarily invalidate the use ofmaximum theoretical work as ameasure of the comparative strengthsof explosives. However, the fractionof the total energy of an explosionthat appears as useful work issignificantly different for differentexplosives. As can be seen in TableVII, the fraction decreases as thepercentage of aluminium in alumin-ized ANFO increases.

The relative inefficiency predictedfor ALANFO, and presumably forother explosives producing con-densed detonation products, canarise from two effects. These areincomplete heat transfer from solidto gaseous detonation products, andloss of energy as detonation productsescape through cracks in the freeface before delivering all their avail-able energy to the surrounding rock.

In the defined final state of theexpansion, the residual energy in thesolids is given by

ITsER,s=

TFOsdT

or ER,s=Os (Ts-TF),

where 08 is the heat capacity of thesolids at temperature T,

08 is the average heatcapacity of the solids be-tween Ts and TF,

Ts is the temperature of thesolids in the final state,and

TF is the gas temperature inthe final state.

From the calculations, the greatestresidual energy in the solids in thefinal state occurs for ALAN in the


3,8

3,6

3,4

3,2

Temp(kK)

3,0

2,8

2 6

2 4

2,2

Condensed

0 2 6 84

~lme (1115)

Fig. 4--- Temperature of gaseous and condensed phases during expansion (ALAN FO10 in standard borehole)


10 12

FEBRUARY1975 175

Heat contentHeat of Condensed of condensed No. of moles of

Aluminium reaction products products at gaseous(%) (MJ/kg) (mass %) TE (MJ/kg) products/kg

0 3,82 0,0 0,0 43,2

5 4,59 9,5 0,37 39,6

10 5,37 18,9 0,86 36,4

13 5,85 24,6 1,21 34,5

18 6,57 34,1 1,90 31,2

smallest-diameter hole. In this caseTs-TF is 14.20 K and the relevant

Cs is 151 J/K/mo19, which givesES,R=0,72 MJ per kg of explosive.Comparison with the values inTable 11 shows that the residualenergy of the alumina in thissituation amounts to 40 per centof the work done by the expandinggas. By a similar calculation, theresidual energy in the solids pro-duced on detonation of ALAN in thelargest-diameter borehole is only0,13 MJ/kg, or 7 per cent of thespecific work. The greater efficiencyof aluminized explosives in large-diameter boreholes predicted by thetheory and by these calculations is,however, only partly realized; thecalculated specific work of ALAN inthe largest borehole is 1,81 MJ/kg,and in the smallest borehole 1,80MJ/kg. It therefore appears that theloss of energy resulting from theescape of detonation productsthrough cracks in the free face,rather than partial heat transfer,accounts for the predicted in-efficiency of aluminized ANFO.

The calculated temperatures ofgaseous and condensed phases duringthe expansion of the products ofdetonation of ALANFO 10 in astandard borehole are illustrated inFig. 4. The change in slope occurringat 3,7 ms represents the end of thedetonation of the explosive column.Before that stage, reaction productshaving the properties of the ex-plosion state are continually beingadded to the system, with overalltemperature and pressure equil-ibrium being maintained.

An examination of the conditionsexisting in the final state, given inTable 11, shows that the finalpressure and volume of the gasesdiffer markedly for different rocktypes, but are only slightly depend-ent on explosive and blast scale.The role of the rock properties indictating the final pressure can beshown from the calculated perform-ance of ALANFO 10 in the standardborehole. During the expansion ofthe borehole-crack system, the rock,which was originally compressed byhigh gas pressures, relaxes, owing toits elastic properties, as the pressuredecreases. Towards the end of thisprocess, the increased volume causedby penetration of gas into the

176 FEBRUARY 1975

cracks (Fig. 5) is balanced by adecrease in volume owing to com-pression of the gas by the relaxingrock. In this situation an equilibriumgas pressure is reached as shown inFig. 6, the magnitude of which isdependent on the compressibilityof the rock. The final pressure andvolume are therefore largely dictatedby the rock properties.It is evident that the quantity ofuseful work done by the gasesdepends on the magnitude of the gaspressures existing during the earlystages of the expansion. Althoughthese pressures should be high foraluminized explosives because ofthe relatively high temperaturesencountered, the volume of productgas decreases as the aluminiumcontent increases, as shown in TableVIII. The addition of aluminiumtherefore has the advantage ofincreasing the explosion temperatureconsiderably, but the disadvantageof decreasing the number of molesof product gas that can be formedin the detonation of oxygen-balancedexplosives. Consideration of Fig. 3shows that the overall effect of thetwo conflicting factors results in thepredicted performance reaching amaximum and then decreasing asthe aluminium content is increased.

The significance of the pressuresexisting during the early stages ofexpansion on the calculated usefulwork indicate:; the importance ofcorrectly estimating the pressuresexerted on the rock during deto-nation. This analysis makes thesimplifying assumption of instant-aneous pressure equilibrationthroughout the borehole-crack sys-tem during detonation. This may,

however, be an over-simplification.The gases immediately behind the

detonation wave will exert theexplosion pressure on the boreholewalls for a short time. Althoughthe duration of this period is notknown, it can be estimated from aknowledge of the speed of sound inthe gas, from which the rate of gasflow away from the detonation zone,and thus of pressure equilibration,can be assessed. Further, there isevidence to suggest that an inductionperiod precedes the formation andelongation of cracks at the boreholewalP. This induction period implieslonger confinement of the gases inthe borehole itself. It thereforeappears that the initial stages of theexpansion could more profitably beanalysed according to the approachof FavreaulO, which allows theborehole to be treated as an elastic-walled cylinder until the inductionperiod for crack formation is over.When cracks have formed at theborehole wall, the gas pressuredecreases because of the increasedvolume, and the borehole relaxes,reaching its original diameter in theabsence of burden movement. Thisextension of the proposed modelwould increase the calculated relativeperformance of aluminized ANFOby emphasizing the importance ofexplosion pressure on the workdone.

No evidence from the field couldbe found to show that the inclusionof aluminium in ammonium nitrate

- fuel oil explosives increases theirrelative weight strength by morethan 10 per cent. This is in agree-ment with the predictions of thepresent model. Further, the calcu-

TABLE VIIIPROPERTIES OF ALUMINIZED ANFO

(bulk density = 1000 kg/m", oxygen-balanced mixtures)*

*Calculation based on the computer program XPLODE from Canadian Industries Ltd.


Distance(m)

14

12 /10

8

6

4

2

0

0 2 84 6 10 12

Time (rlis)

Fig. S-Maximum distance of gas penetration into radial cracks (ALANFO 10in standard borehole)

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY FEBRUARY 1975 177

Pressur(kbor)

12

Pt =038kbo r)

10 5

8

0

4

Volume(m3 )

6 3

4 2

2 1

0

0 102 .1 6 8

Time (ms)

Fig. 6-Average gas pressure and total volume during expansion (ALANFO 10in standard borehole)

178 FEBRUARY 1975 JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY

lated pressure-time profile of thegaseous phase during expansion is tosome extent validated by the goodagreement between calculated andexperimental burden and crack velo-cities as shown in Tables V and VI.It would appear, therefore, that, inspite of possible shortcomings, thepresent model gives results that areconsistent with those obtained fromother sources.

CONCLUSIONS

Calculations of the performance ofexplosives can be done on the basisof a realistic model of the processesthat occur during and immediatelyafter an explosion in a borehole. Theresulting predictions of relativeweight strength of explosives can bevery different from those obtainedfrom heats of reaction or maximumavailable energy.

The work presented here predictsthat the increase in relative weightstrength resulting from the alumin-izing of ANFO is small, and notlikely to offset the extra cost of thealuminium. It must be realized,however, that the density of alumin-ized ANFO made from the con-ventional porous prill increases byabout 0,9 per cent for Gvery 1 percent of aluminium used to replacethe fuel Oilll. Any apparent benefitgained in practice from the additionof aluminium is possibly caused byincreased density rather than byincreased effective strength per unitmass. A reduction in overall miningcosts can nevertheless result fromthe use of aluminized instead ofstraight ANFO because of the possi-bility of drilling fewer or smallerholes for the higher-densityexplosive, but not from the lowercost of the explosive per unit ofuseful energy. A similar result canbe expected from the use of high-density ANFO where the increaseddensity does not result in any in-crease in effective work per unitmass (as shown in Table Il), butwhere higher borehole loads can beobtained in comparison with thosefor straight ANFO.

Although the density of alumin-ized ANFO increases as the alumin-ium content increases, it appearsfrom this work that too high analuminium content can actuallydetract from the relative weight

strength of the product. This possi-bility requires further theoreticaland field investigation if the mosteffective version of aluminized ANFOis to be found.

Theoretical considerations andsubsequent calculations have shownthat aluminized ANFO is mosteffective in large-diameter boreholesin soft rock.

Although a promising model ofthe performance of explosives in aborehole has been developed, evi-dence from the field must continuallybe gathered so that the model can,if necessary, be improved until thepredictions of the performance ofexplosives in any blast are at leastas reliable as the methods of assessingthe blast.

ACKNOWLEDGEMENTS

The permission of the Manage-ment of AE & Cl Limited to publishthis paper is gratefully ac-knowledged. Much of the work de-scribed is based on methods developedby colleagues in Canadian IndustriesLimited, ICI Australia Limited, andNobel's Explosives Co. Limited, andtheir contribution cannot be over-emphasized.

REFERENCES

1. COOK, M. A. The science of highexplosives. ACS Monograph no. 139.Rheinholdt, 1958.

2. COOK, M. A. How dry mix explosivescan increase costs - even in dryholes. Engng Min. J., Sep. 1971.

3. JOHANSSON, C. H. and PERSSON,P. A. Detonics of high explosives.Acad Press 1970.

4. BERGMANN, O. R., RHWLE, J. W.,and Wu, F. C. Model rock blasting-effect of explosives properties andother variables on blasting results.Int. J. Rock Mech. Min. Sci., vo!. 10.1973. p. 585.

5. ATCHISON, T. C., DUVALL, W. 1.,and PETKOFF, B. Rock breakage inquarry blasting. 4th Symp. on RockMechanics, 1961. p. 163.

6. NOREN, C. M. Blasting experimentsin granite rocks. Colo. School of MinesQ., vo!. 51. 1956. p. 213.

7. LANG, L. C., and FAVREAU, R. F. Amodern approach to open pit blastdesign and analysis. Can. Min.Metall. Bull., Jun. 1972.

8. FIELD, J. E., and LADEGAARD-PEDERSEN, A. Fragmentation prin-ciples in rock blasting. Proc. ThirdEur. Symp. Comminution, 1971.

9. STULL, D. R., PROPHET, H., et al.JANAF thermochemical tables. 2ndEd. NSRDS-NBS 37.

10. FAVREAU, R. F. Generation of strainwaves in rock by an explosion in aspherical cavity. J. Geophys. Res.,vol. 74. 1969. p. 4267.


11. BAUER, A. Current drilling and blast-ing practice in open pit mines. Min.Congo J., vo!. 58. 1972. p. 20.

Contribution to the above paperby A. N. Brown*

The author is to be congratulatedon his interesting and valuablepaper. The approach adopted togain an understanding of this com-plex subject of explosive performanceis to be commended. It serves to fillthe gap in the prediction of perform-ance as an aid to the selection of anexplosive, which so often is aprocess of trial and error. It ishoped that the model will be testedagainst field data, for only then canthe assumptions be validated andmodifications introduced to increaseits usefulness.

The 'total work' column in TableIl and the corresponding F.P.I.column in Table III clearly showthat the weight strength is notgenerally enhanced by the additionof aluminium. However, the bulkstrength is the more importantconsideration in the field, since itdetermines how much energy can bepacked into a blasthole of specificsize. Since blastholes are, in general,more expensive to drill than tocharge, the use of aluminized ANFOoffers specific advantages of loweroverall breaking cost, particularly inthe present climate of rapidly escal-ating costs. It is for this reason thatmetallized slurries, although rela-tively far more expensive, havegained popularity for use in hardrocks, particularly in America.

In many open-pit operations,ANFO is poured from a standardmixing truck into the blastholes sothat the loaded density is generallyaround 0,85 gfcm. Where a dry-liner is used in wet holes, it imposesan even greater limitation on theloaded density. There are so manyinstances where the size of theoperation cannot justify the acqui-sition of mix trucks, and hence theuse of on-site mixed slurries. Asopposed to a far more costly systemusing pre-packed slurry explosives,the use of aluminized ANFO offersa solution to the smaller operator inplacing more energy into the blast-holes. Another interesting alternative

*Consultant

FEBRUARY 1975 179

mentioned in the paper to increaseloaded density is the use of high-density ANFO, which requires themixing of porous and dense prilledammonium nitrate. Because it offersa relatively simple and inexpensivesolution, it is hoped that researchin this direction will forge ahead.

The study of the mechanism ofrock-breaking under the influence ofexplosive charges falls into twodistinct sections: the generation ofthe disrupting forces within theblasthole, and the response of thesurrounding material. The modelvery adequately deals with theformer. However, it assumes that afully coupled cylindrical column ofexplosive is detonated from thebottom and that the gas is confinedby the unreacted explosive. Such asituation does exist in many blast-holes, especially those used in under-ground mining. In the standardtype of bench hole under consider-ation, detonation usually commencesfrom the top through a Cordtexdownline to boosters strategicallyplaced in the column. For simplicityof analysis, the assumption of bot-tom initiation is justifiable because

it would seem that the stemmingvelocity is low enough to allow thefinal state to be reached beforeappreciable movement has takenplace.

There appears to be general ac-ceptance of the fact that the break-ing mechanism in rock blasting isdependent on the combined effect ofthe induced stress wave in the rockaround the blasthole and the sus-tained gas pressure within the blast-hole. The peak stress is a function ofthe velocity of detonation and isconsidered to be the agency re-sponsible for the initiation offractures within the walls of theblasthole. The velocity of detona-tion is a consideration used in theselection of an explosive. For hardcompetent rocks, high-velocity, high-energy explosives give better per-formance, whereas slower-acting,weaker explosives are better suitedto less competent, softer rocks.

It is interesting to note that themodel is primarily concerned withthe sustained gas pressure and thatthe velocity of detonation appearsnot to have been evaluated as afundamental agency. It is pertinent

Institute for publicity and exchangeprograms

The IPEGCP ha" been estab-lished to assist in the publicity andexchange of computer programsassociated with geomechanics prob-lems. In addition, it aims to improvethe standard of documentation ofprograms and highlight techniquesthat permit greater machine in-dependence. Scientists and engineerswho have developed, tested, andfully documented computer pro-grams, especially those related tomathematical modelling, are invitedto submit their problems to theInstitute in the required format.

Details of the required standardsfor programming and documentationmay be obtained from any of themembers of the Sub-Committee thatadvise the Institute. The currentmembership of the Sub-Committeeis as follows: Dr G. D. Aitchison(Chairman), Division of AppliedGeomechanics, C.S.I.R.O., P.O. Box54, Mt. Waverley, Vic., 3154, Aus-tralia; Mr A. E. Furley, GeocompLimited, Eastern Road, Bracknell,

180 FEBRUARY 1975

to the present analysis that in-creased density of the loaded columnwill increase the velocity of deto-nation and also that, in practice, theuse of Cordtex and boosters willeffectively change the detonationcharacteristics of the column ascompared with a uniform columndetonated at one end.

In blasting, the potential chemicalenergy contained in the explosiveis converted into various otherforms. As far as the operator isconcerned, the useful forms are thenew surface energy (fragmentation)and the kinetic energy of the spoil(throw). The operator will assessthe performance of the explosive onthese grounds, principally on thedegree of fragmentation. In themodel, the useful work is related tothe elastic compression of the rocklying within the borehole cracksystem. It would be fortuitous if theuseful work and the field perform-ance index as used in the model canbe correlated with the operator'sassessment of performance. As acomparative measure, the indicesused may well be satisfactory.

of geomechanics computer

Berks., RG12 2UP, England; Mon-sieur J. P. Giroud, Laboratoire dcMecanique des Sols, Instituut deMecanique, B.P. 53, Centre de Tri,38041 Grenoble-Cedex, France; Pro-fessor B. Ladanyi, Department ofMineral Engineering, Ecole Poly-technique, 2500 Avenue Marie-Guy-ard, Montreal 250, Canada; Dr.-Ing.H. Meissner, Institut fur Boden-mechanik und Felsmechanik, Uni-versitat Karlsruhe, Postfach Nr.6380, 7500 Karlsruhe 1, F.R.G.;Professor Za-Chieh Moh, Asian Insti-tute of Technology, P.O. Box 2754,Bangkok, Thailand; Professor R. L.Schiffman, Computing Center, Uni-versity of Colorado, Boulder, Colo.,80302, U.S.A.; Professor E. Togrol,Research Institution for Soil Mech-anics, Teknik Universite, Istanbul,Turkey; Professor C. Viggiani, Insti-tuto di Tecnica delle Fondazionie Costruzioni di Terra (Geotecnica),Universita di Napoli, Via Claudio21, 80125 Napoli, Italy; Dr C. M.Gerrard (Secretary), Division of

Applied Geomechanics, C.S.I.R.O.,P.O. Box 54, Mt. Waverley, Vie.,3149, Australia.

On receipt of the programmaterial the Institute will evaluatethe program documentation, and,where this is of the required stand-ard, the Institute will undertakelimited program evaluation.

At least twice a year, IPEGCPwill publish booklets containing theaccumulated proformas of acceptedprograms and general discussionon programming techniques, pro-gram running, and errors orproblems associated with programs.

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prediction of the performance of explosives bench mining

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