The influence of an air gap on the response of an explosive to - - PowerPoint PPT Presentation

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The influence of an air gap on the response of an explosive to spigot impact

Bolaji Adesokan Yani Berdeni Ben Collyer Andrew Crosby Duncan Joyce Andrew Lacey Davide Michieletto David Nigro Hilary Ockendon John Ockendon Rosalind Oglethorpe Charlotte Page Richard Purvis Michael Tsardakas Bristol, 19 April 2013

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Outline

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1 Introduction

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2 No air leakage

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3 Air leakage

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4 Explosion energetics

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5 Conclusion

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Background

Concerned with high explosive violent response (HEVR) and when (or if) this occurs in low velocity situations. Using the spigot test with a confined explosive and velocities of approximately 50m/s. With no air gap there are no explosions; with an air gap there are sometimes explosions. Why? What is the sensitivity?

Explosive Spigot r = 3.5 cm 2.54 cm 1.3 cm

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Problem statement

u Explosive Spigot L = 7cm u Explosive Spigot L Air (p0)

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Shockwave speeds

Spigot movement causes shockwaves, travelling to the explosive and back. Right-travelling shockwave: c2n+1 = √ (2γv2

n − (γ − 1)c2 2n) ((γ − 1)v2 n + 2c2 2n)

(γ + 1)2v2

n

Left-travelling shockwave: c2n = √( 2γ(u + wn)2 − (γ − 1)c2

2n−1

) ( (γ − 1)(u + wn)2 + 2c2

2n−1

) (γ + 1)2(u + wn)2

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Shockwave movement

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Shockwave speed

cn ∼ n n ∼ (t⋆ − t)−1 , t⋆: time at impact

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Pressure ratio over n

pn ∼ n7

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Figure : Log10 of pressure ratio vs time

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Lubrication theory

u Explosive Spigot d d L Air (p0)

Lubrication theory (d ≪ L) leads to qn = (p2

n − p2 0) d3

24TnµLnR .

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Effect of Leakage

Strong dependance on d.

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Maximum Pressure

Conservation of mass: (Eε)(εleπr2

e)γ = Po(πr2 glg)γ

Upper bound (maximum pressure): P = 2.1MPa

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Phase change

δ-phase more sensitive than β-phase to impacts Reaction-diffusion equation ∂T ∂t = ∂2T ∂x2 − q cpT0 ∂δ ∂t Mass fraction ∂δ ∂t = (1 − δ)(t0qZ)e−(E/RT0)/T

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Phase change

Diffusion length z0 = O(10−6) m Grain size ≈ 50 × 10−6 m

0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 z / z0 Proportion of δ-phase

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Explosion Energetics

Simple Arrhenius reaction model: dα dt = A(1 − α) exp ( − E RT ) Proportion of reaction complete α ∈ [0, 1] Temperature T Activation energy E Molar gas constant R Reaction rate A

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1D model

Flux Q for t ∈ [0, δ] to model heat generation by spigot

Q

x = 0 x = L T = T0

Energy equation: ρcp ∂T ∂t = λ∂2T ∂x2 + ρQdα dt Dimensionless form: ∂u ∂ˆ t ≈ ∂2u ∂ˆ x2 + B exp(u)

T = T0(1 + ϵu) where ϵ ≡ (RT0)/E ≪ 1, B ≡ ρQAL2E

λT 2

0 R

exp ( − E

RT0

)

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Example behaviours

Temperature u(0) at heated end of 1D model

2 4 6 8 10 δ ˆ 0.5 1 1.5 u(0) t ut = uxx + B eu (B = 0.5, δ ˆ = 0.1) Q = 12.0 Q = 12.5 Q = 13.0 Q = 15.0

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Energy input

Realistic values for 1D model: B ≈ 2 × 10−10, ˆ δ ≈ 10−7 Gives critical energy ≈ 1J Estimated energy bounds: Minimum: 3J – required to overcome yield stress and deform explosive Maximum: 2000J – energy of bullet Possible resolutions: Diffusion more effective in 3D Actual reaction kinetics are more complicated

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Other thoughts

Model for a reaction wave travelling into the explosive: Single reaction First order No mass diffusion Issues with this model: A cut-off “ignition” temperature (Tig) is needed as speed has sensitive dependence on Tig. Explosive is actually multi-step and includes gas-phase

• reactions. Some equations have been written down.

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Other thoughts

Possible alternative mechanism for encouraging HEVR: High pressure air reduces escape of intermediate gas reactants ⇒ higher reaction rate High pressure air reduces escape of hot products, increasing local temperature ⇒ higher reaction rate

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Compression of Explosive Material

Model as thermo-elasto-plastic material but this is tricky even in 1-d Boundary condition could be no airgap: u, T given on piston x = Ut with airgap: p, T given on x = 0 from the gas dynamic model until yield stress is reached. Then surface of material will start to move and problem is coupled

x t x=Ut plasticity elastic deformation yield point

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Other thoughts

Consumptions of explosive? Is pv = nRT ‘not bad’? Effects of air loss along spigot sides on shock waves. Possible Mach stems? Effects of asymmetric placing of spigot. Lies on tube horizontally Would it be good to do range of experiments with bigger escape rings - gap between spigot and hole?

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Thank you for your attention. Questions?