A Hydrodynamics-based Approach for Predicting the Blast 

Damage Zone in Drifting as Demonstrated Using Concrete 

Block Data 

D.R. TESARIK1

1Address correspondence to: D.R. Tesarik, NIOSH-Spokane Research Laboratory, 315 E. Montgomery 

Ave, Spokane, WA 99207, det4@cdc.gov 

& W.A. HUSTRULID2 

2W.A. Hustrulid, NIOSH-Spokane Research Laboratory, 315 E. Montgomery Ave, Spokane, WA 99207, 

bte1@cdc.gov 

ABSTRACT 

Rock falls involving a relatively small amount of material are a leading cause of injuries in underground 

mines in the United States. A contributing factor is unwanted blast damage and over-break. A goal of the 

United States National Institute for Occupational Safety and Health is to reduce the number of ground fallrelated accidents by improving drift driving practices through the introduction of perimeter control blast 

designs. This paper presents a blast design technique based on a hydrodynamic method. Using commonly 

available explosive and geometry data, the theoretical particle velocity is calculated as a function of 

distance away from a cylindrical charge. A calibration factor is introduced to account for actual conditions 

by comparing the predicted values to a set of field data. The calibrated curve can then be applied for design. 

The paper introduces modifications to account for different explosives from those used in a calibration test. 

The application of the approach is demonstrated with respect to a data set from a large-scale concrete block 

test. 

1. INTRODUCTION 

Rock falls are one of the main causes of injuries in U.S. underground mines 

(MSHA 2006). Although the number of such accidents has decreased since reaching a 

peak in 1989, it has remained relatively constant for the past five years. Nearly all of 

these injuries involve relatively small pieces of rock falling from a zone about a halfmeter thick surrounding the excavation (Bauer & Donaldson 1992, Mark & Iannacchione 

2001). The total weight of the rocks in the majority of injury-caused rock falls is less 

than 11 kg (MSHA 2006). The widespread introduction of perimeter control blasting 

techniques in drift driving can potentially significantly reduce the number of these 

injuries. For this to occur, practical drift blast round design procedures including 

perimeter control must be made available to industry in a form which can be easily 

applied. 

Holmberg and Persson (1979) have presented a perimeter control design 

technique based on the peak particle velocity generated by the detonation of a charge. 

Using the relationship between peak particle velocity (PPV), linear charge concentration, 
2

distance from the charge, and the observed extent of damage, one obtains a limiting PPV 

value. This value can then be applied to the design curves to select explosive-hole 

combinations which yield acceptable perimeter damage. This method, known as the 

Holmberg-Persson method, has found quite wide acceptance. Unfortunately, a 

mathematical error was made in the development of the Holmberg-Persson equations 

(Hustrulid and Lu 2002). Iverson et al. (2008) introduced a modification in the 

Holmberg-Persson calculation procedure as a way of at least partially mitigating the 

problem. Blair and Minchinton developed the Scaled Heelen model which is similar in 

form to the Holmberg-Persson model, but uses full waveform superposition to calculate 

particle velocity at a given point (Blair and Minchinton 2006). 

Other approaches for predicting the extent of damage are based on the gas 

pressure generated in the blasthole due to the explosion rather than the shock wave. 

Drukovanyi et al. (1976), for example, used the effect of the quasi-static expansion of 

detonating gases on brittle rock to derive equations for the radii of crushed and cracked 

zones surrounding a cylindrical charge. 

Hustrulid and Johnson (2008) have produced two semi-empirical damage radius 

estimators based on the explosive gas pressure and the explosive energy, respectively. 

The pressure-based estimator can be applied to both fully-coupled and de-coupled 

charges. The importance of the buffer row design is stressed. They contend that if the 

buffer row is properly designed, the detailed design of the perimeter row becomes less 

important to overall perimeter control performance. 

Vlasov (1957), Vlasov and Smirnov (1962), Kuznetsov (1974, 1977), Ill’inskii 

and Yakimov (1980), Ill’inskii and Potashev (1982), Kransov (1981), Nieman (1976, 

1979, 1984a, 1984b, 1986a, 1986b) and others have applied a hydrodynamics-based 

approach to calculate theoretical particle velocity components at a given distance from a 

charge and this is the approach described here. This paper begins with a brief 

presentation of the basic equations pertaining to a cylindrical charge as originally 

presented by Nieman. The paper then continues by indicating how the equations can be 

modified to take into account actual rock behavior, and various explosives. The paper 

concludes by demonstrating the practical application of the approach to field data 

collected during a large concrete block test. 

2. THEORETICAL DEVELOPMENT 

2.1 Basic Assumptions 

 In the development of the equations it is assumed that (Hustrulid 1999): 

The detonation of the charge takes place in a confined space. 

As a result of detonation, the explosive charge is instantaneously transformed into 

a volume of gaseous products of the same size and shape as the space. 
3

In dense, monolithic rocks of high specific gravity and high ultimate rupture 

strength, the gaseous products do not substantially modify the properties of the 

rock prior to the completion of the detonation process. 

After detonation, some average pressure characteristic of the explosive type is 

established in the borehole. 

The medium surrounding the charge is considered to be uniformly deformable 

(linearly elastic) and incompressible. No account is taken of rock microstructure, 

fracturability, strength, or other properties. 

The medium will be considered isotropic with respect to all properties. This 

hypothesis is correct if it is considered that the in-homogeneities present in real 

rocks can, practically speaking, be equalized by the enormous pressure arising in 

the vicinity of the charge. 

The effect of the shock wave will be neglected. The amount of energy carried 

away by the shock wave is small compared to the total energy of the explosive 

and the zone of its action is insignificant. 

Using the principles from hydrodynamics and these assumptions, one can 

calculate the particle velocity as a function of distance away from the charge. In reality, 

there are departures from these assumptions. For example, normal rock masses contain 

joints and other discontinuities which affect the incompressibility assumption. These are 

taken into account through a calibration procedure which will be demonstrated later in 

the paper. 

2.2 Equation Derivation 

The coordinate system shown in Figure 1 is used in developing the general 

expressions for the particle velocity field resulting from detonating a cylindrical charge in 

an infinite medium. As can be seen, the charge is oriented along the z axis and extends 

from z1 to z2. 
4

Fig. 1. Coordinate system for a cylindrical charge in an infinite medium. 

The potential function developed by Nieman as presented in equation (1) will be used, 

which is 

2

1

22

21

2

12

22

2

ln

zzyxzz

zzyxzz

P , 

 

 (1) 

where 

sm

ee

v

q

dP

8

, (2) 

d = diameter of the charge, 

L = length of the charge, 

ρe = density of the explosive, 

qe = energy/mass of the explosive, 

ρm = density of the medium, 

2

2

1

1

ln

LL

LL

vs

, and (3) 

. (4) 
5

Taking derivatives of equation (1) with respect to x, y, and z, yields the velocity 

components u, v, and w, respectively: 

xPu AB CD , (5) 

yPv AB CD , and (6) 

ACPw , (7) 

where 

2

1

22

1

zzyx

A , (8) 

2

1

22

1

1

zzyxzz

B , (9) 

2

2

22

1

zzyx

C , and (10) 

2

2

22

2

1

zzyxzz

D . (11) 

With the x-axis translated (re-positioned) to the mid-length position of the charge, the 

theoretical particle velocity ut (oriented along the x axis) as a function of distance from 

the charge center can be expressed as 

4

8 2

2 L

rrv

q L

u

s

m

ee

t

, (12) 

where 

r = distance along the x axis from the center of the cylindrical charge and 

d

r

r . (13) 

Equation (12) can be written as 

1

t

Iu , (14) 
6

where 

m

qee

I

8

, and (15) 

L

L

rrvs

4

2

2

. (16) 

The geometric terms are all contained in the factor ρ while the rock and explosive 

parameters are contained in the multiplying constant I. 

2.3 Required Modification for Actual Rock Masses 

As indicated, in the derivation of the basic equations it has been assumed that the 

rock mass surrounding the charge is uniformly deformable (linearly elastic) and 

incompressible. No account is taken of rock microstructure, fracturability, strength, or 

other properties. However, rock masses, contain various types of structures on various 

scales and damage in the form of fracture development and extension develops during the 

blasting process. The associated deformations are not elastic except possibly in the far 

field. To obtain the required site specific design equations, a calibration blast is 

performed using an explosive charge of known geometry/properties in the given rock 

mass. The resulting PPV values are measured as a function of distance away from the 

charge. The field data would be expected to have the same form as Equation (14), but 

with a different coefficient (If) and exponent (α). One would write: 

f f

Iu , (17) 

where 

uf = the field measured particle velocity perpendicular to and at mid-length to a single 

cylindrical charge. 

The calibration factor K then becomes 

1

1

I

I

I

I

u

u

K

f f

t

f

. (18) 

2.4 Modifications Based on Different Explosive Type 

If an explosive is chosen for an application different than the one used in the 

calibration tests, one can easily produce a new predictor by modifying the ‘I’ factor. 
7

Assume that ‘I’ applies for the original explosive and ‘I'’ for the new explosive. The 

theoretical particle velocity is then 

'

' I

ut

. (19) 

Solving for ρ in Equation (14) yields: 

t u

I

. (20) 

By substituting this expression for ρ into Equation (19), one finds that the theoretical 

velocity resulting from the different explosive properties is 

t t u

I

I

u

'

'

. (21) 

If it is assumed that field velocities are changed by the same ratio as theoretical velocities 

when the explosive term is changed, a first approximation for the equation for field 

velocity becomes 

f f

I

I

I

u

'

'

. (22) 

3. DEMONSTRATION OF THE APPROACH 

3.1 Introduction 

As part of its overall program to advance cautious blasting principles, NIOSH has 

conducted a series of well-instrumented one-shot blasting tests in large monolithic 

concrete blocks (Iverson et al, 2009). This section illustrates the application of the 

hydrodynamic approach to the interpretation of the data and the development of a design 

curve. 

3.2 Experiment Description 

The concrete block experiment comprises a large concrete block measuring 2.4 m 

wide by 2.4 m long and 1.5 m high, a single blasthole, and an array of strain gage sensors 

placed at different distances away from the charge. The concrete block is intended to 

represent a homogeneous rock mass without structure. The concrete block is shown in 

Figure 2 prior to blasting. 
8

Fig. 2. Photograph of block experiment prior to test showing placement of barrier 

blocks to restrain fly rock. 

The physical properties are listed in Table 1. 

Table I. Properties of the Concrete Block. 

Parameter Value

Cohesion (MPa) 4.9

Internal friction angle (deg) 43

Uniaxial compressive strength (MPa) 21

Tensile strength, Brazilian (MPa) 1.75

Young’s modulus, static (GPa) 13.1

Young’s modulus, dynamic (GPa) 18.6

Poisson’s ratio 0.25

Speed of sound, bar (m/sec) 2800

Speed of sound, block (m/sec) 3100

Density (kg/m3) 2000 

The blasthole (38 mm diameter) was percussion-drilled, nearly horizontal, and 

had a length of 1.8 m. It was collared at block mid-height with a burden of 0.45 m. The 
9

blasthole was charged using Dyno® AP1

1

 Mention of any company or product does not constitute endorsement by the National Institute for 

Occupational Safety and Health. 

(Dyno Nobel 2006). The explosive was tamped 

creating a fully coupled charge 1.2-m long measured from the hole-bottom. The charge 

extended to a distance of 0.6 m from the collar. Table 2 lists the properties for the 

explosive. 

Table II. Explosive properties. 

Parameter Value Units

Explosive density 1.15 g/cm

3

Detonation velocity 4300a m/s

Diameter 31.75 mm

Tamped diameter 38.1 mm

Relative weight strength 0.88

Relative bulk strength 1.00

Energy 775

890

cal/g

cal/cm3

Tamped charge length 1.22 m

Total charge weight 1.60 kg

aMeasured 

The array of strain gages permitted measurement of strain attenuation with 

distance from the charge. Strain gage rosettes were embedded in the concrete at the same 

height as the blasthole. Because the block had been poured prior to deciding on the test 

design, instrument emplacement holes were diamond drilled at the appropriate locations. 

The strain gage rosettes were first bonded to the bottom ends of cured rectangular grout 

bars and the assembly was then grouted in the holes using the same grout. The grout had 

similar acoustic properties as the concrete to prevent wave reflection. The access holes 

were drilled at intervals of 23 cm away from the blasthole. A generalized plan view of the 

block experiment is shown in Figure 3. Although a rectangular grid of three strain gages 

was used at each location, only the results from the center, or radial strain gage were 

used. Each strain gage was connected to one arm of a Wheatstone bridge (quarter bridge 

configuration). The bridge output was connected to a Genesis data recorder manufactured 

by LDS Test and Measurement, Inc. Data were collected at the maximum rate of 1MHz. 
10

Fig. 3. Top view of block showing the locations of the strain gage array and the 

blasthole. 

The measured peak radial strains as a function of distance are given in Table 3. The 

corresponding peak particle velocity (PPV) is calculated using the relationship 

peak cPPV , (23) 

where 

 c = p-wave velocity (m/s) = 3100 m/s,

 εpeak = peak radial strain, and 

 PPV = peak particle velocity (m/s). 

Table III. Peak strain and PPV as a function of distance. 

Distance

(cm)

Peak 

strain

PPV

(m/s)

22.9 .006475 20.1

45.7 .001592 4.9

68.6 .000691 2.1

91.4 .000332 1.0

114.3 .000288 0.9

3.3 Data Analysis Using the Hydrodynamic Approach 

The theoretical velocity equation developed by Nieman is 
Fig. 4. Theoretical and measured velocity versus ρ for block test. 

To plot the experimental results in the same fashion, one must determine the 

corresponding values of ρ for the measured particle velocities. This is done using 

equation (16). 

2

2 L

vsr r

4

L

Since the length (L) and diameter (d) of the charge are, respectively 

L = 1.2m, and 

d = 0.038m, 

one obtains, 

L L = 31.58, d
 

r
 r

r , d 0.038
	


	 L 1 2 L 31.58 1 (31.58)

2 

vs ln ln 8.29 , 2 L 1 2 L 31.58 1 (31.58)

and 
13

 

 

 

2

2

2 r 1

8.29 r31.58 2

2 L

vsr r

4 0.038 0.038 4

L 31.58

or 

.249 32

038.0

399.2

2

r

r . (25) 

At the wall of the hole, r = 0.019 m, the corresponding value of ρ is found using equation 

(25). For this case one obtains, 

 ρ = 0.72 

The theoretical velocity at the wall of the hole is 

 utw 483 .0(483 72 /670) sm

1 1

. (26) 

The calculated values of ρ for the measurement point locations and the associated PPV 

values are given in Table 4. 

Table IV. Calculated values of ρ for measurement point locations and associated 

PPV. 

Distance ρ PPV 

(cm) (m/s) 

22.9 9.29 20.1 

45.7 21.90 4.9 

68.6 39.48 2.1 

91.4 63.10 1.0 

114.3 93.18 0.9 

These experimental values are plotted in Figure 4. As can be seen, the points are well 

described by a straight line on log scales with the equation 

416.1

u f 414 . (27) 

The calculated velocity of 659 m/s at the hole wall is very similar to the theoretical value 

(670 m/s). 

A calibration factor (K) relating the theoretical and experimental velocity curves is 

obtained by dividing equation (14) by equation (27): 

416.0

.1 16

f

t

u

u

K . (28) 
14

The expected velocities (ue) for other charge diameters and lengths can then be obtained 

using 

.1 16

1 416.1

e e

e

I

K

I

u , (29) 

where 

e

e

es ee

e

L

L

rrv

4

2

2

. (30) 

Since the explosive, the charging conditions and the rock mass remains the same, the 

value of I remains constant. 

Design curves of the form PPV versus distance, r, are developed by using 

Equations 29 and 30. Figure 5 is a plot of PPV versus distance for the 1.2-m long fullycoupled cylindrical charge with diameter 38mm. Figure 6 shows the curves for fullycoupled charges 25mm, 38mm, 50mm, 64mm and 76mm in diameter. These data are 

plotted on normal scales in Figure 7. 

Fig. 5. Design curve for 1.2-m long fully-coupled cylindrical charge with diameter 38 

mm.
15

Fig. 6. Design curves for 1.2-m long fully-coupled cylindrical charges using log 

scales. 

Fig. 7. Design curves for 1.2-m long fully-coupled cylindrical charges using normal 

scales. 

3.4 Damage Assessment 
16

Several different techniques were used to assess the extent of blast damage. An 

NX (78mm) diameter diamond drill hole was drilled specifically for this purpose. The 

change in s-wave velocity in the hole wall was observed as a function of distance away 

from the charge using a Micro-Velocity Probe (MVP). One set of results is shown in 

Figure 8. As can be seen, the apparent damage zone is of the order of 50 cm. This value is 

defined as the crushed zone for purposes of this paper. Cross core sonic (p-wave) and 

Brazilian tensile strength tests performed on the recovered core yielded similar damage 

zones. 

Fig. 8. Shear wave velocity as a function of distance from the hole. 

After the blast, several wire-saw cuts oriented normal to the charge axis were 

made through the remaining block. The maximum crack limit was determined by visual 

inspection of the saw cut surfaces. Figure 9 presents the results from one of the cuts. As 

shown, the maximum crack length on this section is 176 cm. In summary, the 

measurements indicate a crush limit of the order of 50 cm measured radially from the 

center of the column charge and a crack limit extending to 176 cm. These damage limits 

are specific to the concrete material and the explosive tested. 
17

Fig. 9. Wire saw section showing the cracking. 

Superimposing the observed damage limits on Figure 5, one finds that the approximate 

limiting PPV limits for crushing and cracking are 4.40 m/s and 0.22 m/s, respectively, as 

shown in Figure 10. 

Fig. 10. PPV versus distance from charge for a 1.2-m long, 38 mm diameter charge 

with the damage limits determined.

These limits are now superimposed on the set of design curves in Figure 11. The crushing 

and cracking limits for the different charge diameters are given in Table 5. 
18

Fig. 11. Design curves with crushing and cracking limits. 

Table V. Crushing and Cracking limits. 

 Charge diameter (mm) Crushing limit (m) Cracking limit (m)

25 0.35 1.38

38 0.50 1.76

50 0.62 2.10

64 0.74 2.40

76 0.82 2.60

4. Summary and Conclusions 

The hydrodynamic approach for predicting the extent of blast damage caused by 

an explosive charge in an idealized rock mass as characterized by the peak particle 

velocity (PPV) has been described. A calibration factor is introduced to account for the 

actual behavior of the explosive in a particular rock mass. This factor is obtained by 

conducting a field test in which particle velocity measurements are made as a function of 

distance away from the charge. Once the calibration factor has been determined, PPV 

versus distance curves can be easily constructed for different explosive charge – hole 

combinations. By observing the extent of damage for a given rock – explosive 

combination and making use of the appropriate PPV versus distance curve, one can 

determine the limiting PPV for a particular damage type. By adding these limits to the 

calibrated PPV versus distance curves, one obtains a very simple approach for developing 

drift round blast designs with a certain maximum extent of perimeter damage. The 
19

practical application of the approach has been demonstrated using data from a single hole 

blast in a large concrete block. 

In summary, the approach appears to be fundamentally sound and quite easy to 

apply. Provided that the calibration test is conducted in a representative section of the 

rock mass, this method could be used to design perimeter charges and improve 

underground mine safety by reducing overbreak. Further calibration tests in rock masses 

containing various joint characteristics and other discontinuities are needed. 

Acknowledgments 

The authors wish to thank Jeff Johnson, Steve Iverson, Carl Sunderman, Ed 

McHugh, Ted Williams, Paul Pierce, Ron Jacksha, Mike Stepan, Rich Rains, and Brian 

Stapleton, NIOSH, SRL for conducting the concrete block and material property tests. 

Mark Johnson, Northwest Contractors, and Troy Knutti and James Zweirs, Dyno Nobel 

conducted the blast. Their work is greatly appreciated. 

References 

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of a blasting program readily accepted by miners. In: Proceedings of the 18th Annual 

Conference on Explosives and Blasting Technique, International Society of Explosive 

Engineers, p. 133-140, Cleveland. 

Blair, D.P. & A. Minchinton (2006) Near-field blast vibration models. In: Proceedings of 

the 8th International Symposia on Rock Fragmentation by Blasting, Asociación Chilena 

de Ingenieros Explosivistas de Chile A.G. (ASIEX), p. 152-159, Santiago. 

Drukovanyi, M.F. (1976) Calculation of fracture zones created by exploding cylindrical 

charges in ledge rocks. Soviet Mining Science, 12(3): 292-295. 

Dyno Nobel Inc. (2006) Explosive Engineer’s Guide. 

Holmberg R. & P.A. Persson (1979) Design of tunnel perimeter blasthole patterns to 

prevent rock damage. In: Proceedings Tunnel '79 (Ed.: Jones, M.J.), p. 280-283, IMM, 

London. 

Hustrulid, W. (1999) Blasting Principles for Open Pit Mining Volume 2 - Theoretical 

foundations. Balkema, Rotterdam. 

Hustrulid, W. & Lu, W. (2002) Some general design concepts regarding the control of 

blast-induced damage during rock slope excavation. In: Proceedings of the Seventh 

International Symposium on Rock Fragmentation by Blasting (Ed.: Wang, X.), Beijing 

Metallurgical Industry Press, Beijing. pp. 595 – 604. 
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Hustrulid W. & J. Johnson (2008) A gas pressure-based drift round blast design 

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during ejection blasting. Soviet Mining Science. 16(2): 107-113. 

Ill’inskii, N.B. & A.V. Potashev (1982) Mathematical modeling of the energy 

characteristics of a charge in pulsed-hydrodynamic models of a throwout explosion. 

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Iverson, S. et al. (2008) Application of the NIOSH-Modified Holmberg-Persson 

Approach to Perimeter Blast Design. In: Proceedings of the 34th Annual Conference on 

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Cleveland. 

Iverson, S.R. et al. (2009) Experimental determination of the extent of blast damage from 

a fully coupled explosive charge and some initial modeling results. Accepted: 9th

International Symposium on Rock Fragmentation by Blasting, Madrid. 

Krasnov, Y.K. (1981) Action of propellants and high explosives on rock. Soviet Mining 

Science. 17(1): 39-44. 

Kuznetsov, V.M. (1974) Hydrodynamic calculation for ejection blasting by long charges. 

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Mark C. & A. Iannacchione (2001) Ground Control Issues for Professionals. In: Mine 

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Metallurgy and Exploration Inc., Colorado. 

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[www.msha.gov/ACCINJ/accinj.htm] and 

[www.msha.gov/STATS/PART50/p50y2k/p50y2k.HTM]. U.S. Department of Labor, 

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Soviet Mining Science, 15(5): 480-485. 
 

 

 

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APPENDIX A - Limiting Damage Distance 

If a relationship between the degree of damage and PPV has been established for 

the particular rock mass using some technique, it is of interest to be able to easily 

determine the corresponding extent of damage. To do this, one begins by writing 

Equation (16), 

2

r r

2 L

vs 4 . 

, (16) 

L

in the form 

 

2 2

4 L 2 L

r r 0 . 4

2 

vs

 (A1) 

Equation (A1) is a bi-quartic equation that can be solved by substituting R for 

r

2 . One obtains 

 

2 2 2 L L R R 0 . 42 

vs

 (A2) 

The two roots of Equation (A2) can be found by using the quadratic equation: 

21 
22

2

4

4 16

2

2

2 4

1

s

v

LLL

R (A3) 

and 

2

4

4 16

2

2

2 4

2

s

v

LLL

R , (A4) 

where 

R1 , R2

are the roots of Equations (A3) and (A4). 

The four roots of Equation (A1) are: 

 

1 Rr 1 , 

 

 

 

 (A5) 

r 2 R1 , (A6) 

3 Rr 2 , and (A7) 

r 4 R2 . (A8) 

Equation (A5) is used for this application because Equation (A6) produces a negative 

number and Equations (A7) and (A8) produce imaginary numbers. The appropriate 

equation is 

5.0

2

4

4 16

2

2

2 4

1

s

v

LLL

r . (A9) 

In practice, one determines the ρlim corresponding to the particular PPV value, PPVlim by 

using the crushing or cracking limit in Equation (17) and taking the inverse, which gives 

1

lim

lim

f

I

PPV

, (A10) 
 

 

 

 

 

where PPVlim is the crushing or cracking limit. 

One then substitutes ρlim for ρ in Equation (A9) to determine r

1 lim . The extent of damage 

rlim corresponding to PPVlim is then 

r lim r . 1 lim d (A11) 

To demonstrate the use of this technique using the properties and explosive for the 

concrete block test, the limiting PPV for crushing, 4.45 m/s, is substituted into Equation 

(A10): 

 lim 414 

4.45 416 .1

1 

24.56 . (A12) 

For
	

L = 1.2 m,
	

d = 0.038 m, and
	

vs = 8.29, 

Equation (A9) becomes 

 

31.582 31.584 4 x 24.562 

x 31.582 

4 16 8.29 

5.0

r1lim 13.1 

2 (A13) 

Multiplying r1lim by the charge diameter, 

r 0.038 m 13.1 0.5m lim (A14) 

which is the limiting distance corresponding to the limiting crushing velocity. 

Applying the same technique for d = 76 mm yields 0.83 m for rlim. The calibration 

method yielded 0.82 m for rlim for this case. 

23