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 ρ ρλ γ γ γ γγ ρ ρ ρ

= + − = + +−

( )32

ref

1 1PwP

λτ⎛ ⎞

= −⎜ ⎟⎝ ⎠

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/cc

Diameter (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

ξ

n

is normal direction to shockn

is tangential direction to shockξ

& are fluid velocities normal and perpendicular to shocknu uξ

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

ξ

ρρρ ξ

ξρρ ξ

ρ

ξ

ξξ

ρ λ ρ λ

Κ +∂ ∂+ = − +

∂ ∂ +

∂ ∂= − +

∂ ∂∂ ∂

− =∂ ∂∂

= +∂

∂∂

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

P ρ λ

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

ρσρ

ρσ

λ

ρ λρ ρ

ρ ρ

σλ

Κ +−

+ Κ= −

−Κ +

−+ Κ=

−

=

+ + = + +

⎡ ⎤⎛ ⎞∂ ∂⎛ ⎞= −⎢ ⎥ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠⎣ ⎦∂ ∂⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

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 κ−

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.