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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.