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RECENT DEVELOPMENTS IN DETONATIONMODELING CONDENSED PHASE MEDIA

Martin Braithwaite Imperial College, London UK

40th UKELG Meeting 40th UKELG Meeting -- 25th Anniversary25th AnniversaryCardiff University 19th September 2007Cardiff University 19th September 2007

Gas Phase vs Condensed Phase Detonations

simple thermodynamics

cell structure can be measured

role of turbulence

no mass transfer control

near CJ conditions

supercritical fluids

heterogeneity, hot spot control

VOD < VODCJ

limited characterization

pronounced shock front curvature

Condensed Phase Detonations

mining, quarrying, seismic applications

demolition, tunneling

safety eg FCMO, transport

defence

security & improvised devices

EPSRC Condensed Phase Reactive Flow Networkhttp://www.dcmt.cranfield.ac.uk/dmas/cdc/condphase

Modeling of the Detonation (and Rarefaction) Processes for Commercial (Non-Ideal, heterogeneous) Explosives in Rock

Simple Theoretical Approach

Validation using AMR Finite Volume code

Pressure History at Detonation Product RockInterface for Shock, Detonation (and Rarefaction)

Ammonium Nitrate Based Bulk Explosives ANFO, Emulsions

Heterogeneous, gas sensitized

Large critical diameter (detonation) & VOD < Ideal VOD

Limited characterization studies for most explosives

eg separate oil and oxidiser phasesporous prill, chemical gassing

ex critical diameter > 30 mmsreaction zone (DDZ) > 10 mms

Typically, unconfined VOD vs charge diameterDensity and thermodynamic parametersLimited Shock Front Curvature data (unconfined)Particle, droplet size distributions

Confinement RockCompression, crack propagation,fragmentationand muckpile formation

Non-isotropic

Variable properties strength, composition, density

Acoustic velocity < VOD This study

Detonation

Steady state , axisymmetric

Homogeneous media

Simple EoS, thermodynamics and rate law for boththe explosive, detonation products and confinement

nb major aim of this work is to ascertain where fluid mechanicsapproximations in quasi-1-D theories are valid thereforenot a test for EoS or thermodynamics

Detonation Velocity greater than sound speedof confining media

(i) Are the fluid dynamics in a quasi-one dimensional theory adequate to approximately describe the detonation process for central axis and product-confinement interface in the context of rock blasting ?

(ii) What resolution is required in a finite volumecomputer simulation study such that the DDZpredictions are invariant with grid size ?

(iii) Is some order of DSD theory as opposed toWood Kirkwood slightly divergent flow appropriateto the rock blasting problem ?

0

1

2

3

4

5

6

7

0 0.02 0.04 0.06 0.08 0.1 0.12

unconfined80% dry80% wetconf approxconf correct

Plot of VOD (km/s) vs inverse diameter (1/mm)

Data Validation Preview Scaling

WarningsDefaults

DATA FILE

BMS Vis.

JointStats

BMS

Interface DLL

BLO-UP

JKSimBlast

Hybrid StressBlast Model

Vixen

Condensed Phase Detonations - Model

Conservation relations (Euler equations)MassMomentumEnergy

Thermodynamic EoS all phasesP, V, T, E relationsMixing rules (finite reaction zones)

Rate Expression lumped

Initial and Boundary Conditions

Cobra solution of 2Cobra solution of 2--D Reactive Euler EquationsD Reactive Euler Equations

simplified rate expression

same EoS explosive, detonation products and confinement

open tube (rear boundary condition)

run to steady state detonation

Cobra Analysis of ANFO Rate Stick

(all simplifications (rate and EoS) necessary to reduce run timesto > 2 weeks on Linux based PCs)

Cobra Analysis ApproachCobra Analysis ApproachANFO Rate StickANFO Rate Stick

simple pseudo-polytropic EoS explosive, products and confinement

simplified rate expression

( ) ( )2

0 1 2 20 0

1 :1

Pe Q

= + = + +

( )32

ref

1 1PwP

=

0.523.252.934.73.2

0.213.673.019.31.6

0.153.883.0518.70.8

0.124.113.1037.30.4

0.094.243.1274.50.2

0.054.323.131490.1

LambdashkPshockGPa

VOD(km/s)

Points/lDDZ(mm)

Effect of grid spacing/ no of points in DDZ on predicted VOD, shock pressure and extent of reaction at the shock front

Cobra Analysis Comparison Cobra Analysis Comparison -- ANFO Rate StickANFO Rate Stick

0.8736100H0.7462100G0.6480.0013 - air100F0.9018100E0.8294100D0.8370.8200C0.7780.8150B0.6520.8100A

D/DCJConfinementDensity g/ccDiameter (mm)Case

Blue shockRed sonic locusGreen - contact

100 mm 0.8 g/cc

150 mm 0.8 g/cc

200 mm 0.8 g/cc

100 mm 4 g/cc

100 mm 8 g/cc

100 mm 0.8 g/cc

150 mm 0.8 g/cc

200 mm 0.8 g/cc

100 mm 4.0 g/cc

100 mm 8.0 g/cc

100 mm ANFO charge in 2 g/cc confinement Pressure profile

100 mm ANFO charge in 2 g/cc confinement - Density Profile

100 mm ANFO charge in 2 g/cc confinement Radial Velocity Profile

100 mm ANFO charge in 2 g/cc confinement Reaction Extent Profile

Quasi- 1-D Analysis for General EoS and Rate Law

Shock

Reaction Zone

n

is normal direction to shockn

is tangential direction to shock

& are fluid velocities normal and perpendicular to shocknu u

Steady Euler Equations in curvilinear coordinates

( )( )

( ) ( )

2

terms1

1 terms

terms

terms

where terms involve derivatives and : Reaction rate and EoS, are given by

W=W ,P, & , , . and

n nnn

nn

nn

n

u Du un n nKu Pun n

e Pn n

uu Wn

u

e e P D

+ + = +

+

= +

=

= +

= are the local total curvature and normal component of detonation speed. , and correspond to pressure, density and extent of reaction respectively.

P

Quasi-1-D approximation

radial derivative and velocity terms are assumed negligible

remaining partial derivatives become full

Dn normal velocity becomes VOD

Analogous to WK central stream-tube but without unknown divergence term

Q1D Equations in curvilinear coordinates final

( )

2

2 2

2 200

0

22

( )(1 )

( )(1 )

1 1, ,2 2

/

/

n n n

n

n

n n n

n

n

n

n n

u u DWu nd

dn c uu u DW

du ndn c ud Wdn u

PPe P u e D

P e ecP

e eP

+

+ =

+

+ =

=

+ + = + +

= =

3

3.4

3.8

4.2

4.6

0 100 200 300 400 500 600 700

D (k

m/s)

Rshk (mm)Axial Detonation velocity vs axial radius of curvature,

Q1D predictions are given as a solid line and the data from COBRA runs as circles

2.5

3

3.5

4

4.5

5

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Dn

(km

/s)

(1/mm)kkkkkkkkkkkkkkkkkk

nD

Normal Detonation Speed vs curvature (ex COBRA) for a range of charge diameter and confinement densitycf Q1D theory (dotted line)

Conclusions

High resolution AMR simulations required ~ 0.1 mm

Reasonable to good agreement Q1D and COBRAfor central axis

Disadvantage of Wood Kirkwood unknown axial divergence parameter

DSD for non-ideal explosives in heavy confinement eg rock ?

Limited impact of DDZ directly on rock

Acknowledgements

SponsorsAEL, De Beers, Debswana, Dyno Nobel, Placer Dome,Codelco, Rio Tinto, Anglo Chile, Sandvik Tamrock.