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

. 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Outline . 1 Introduction . . 2 No air leakage . . 3 Air leakage . . 4 Explosion energetics . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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? Spigot 1.3 cm Explosive 2.54 cm r = 3.5 cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Problem statement Spigot Spigot L u L = 7 cm u Air ( p 0 ) Explosive Explosive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Shockwave speeds Spigot movement causes shockwaves, travelling to the explosive and back. Right-travelling shockwave: √ n − ( γ − 1) c 2 n + 2 c 2 (2 γv 2 2 n ) (( γ − 1) v 2 2 n ) c 2 n +1 = ( γ + 1) 2 v 2 n Left-travelling shockwave: √( ( γ − 1)( u + w n ) 2 + 2 c 2 2 γ ( u + w n ) 2 − ( γ − 1) c 2 ) ( ) 2 n − 1 2 n − 1 c 2 n = ( γ + 1) 2 ( u + w n ) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Shockwave movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Shockwave speed c n ∼ n n ∼ ( t ⋆ − t ) − 1 , t ⋆ : time at impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Pressure ratio over n p n ∼ n 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure : Log 10 of pressure ratio vs time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lubrication theory Spigot L u d d Lubrication theory ( d ≪ L ) leads to q n = ( p 2 n − p 2 0 ) d 3 24 T n µL n R . Air ( p 0 ) Explosive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Effect of Leakage Strong dependance on d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Maximum Pressure Conservation of mass: e ) γ = P o ( πr 2 ( Eε )( εl e πr 2 g l g ) γ Upper bound (maximum pressure): P = 2 . 1MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Phase change δ -phase more sensitive than β -phase to impacts Reaction-diffusion equation ∂t = ∂ 2 T ∂T q ∂δ ∂x 2 − c p T 0 ∂t Mass fraction ∂δ ∂t = (1 − δ )( t 0 qZ )e − ( E/RT 0 ) /T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Phase change Proportion of δ -phase 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 z / z 0 0.8 Diffusion length 1 z 0 = O (10 − 6 ) m 1.2 Grain size ≈ 50 × 10 − 6 m 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

1D model Flux Q for t ∈ [0 , δ ] to model heat generation by spigot T = T 0 Q x = L x = 0 Energy equation: ∂t = λ∂ 2 T ∂T ∂x 2 + ρ Q dα ρc p dt Dimensionless form: t ≈ ∂ 2 u ∂u x 2 + B exp( u ) ∂ ˆ ∂ ˆ ( ) B ≡ ρ Q AL 2 E − E T = T 0 (1 + ϵu ) where ϵ ≡ ( RT 0 ) /E ≪ 1, exp . . . . . . . . . . . . . . . . . λT 2 RT 0 0 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Example behaviours Temperature u (0) at heated end of 1D model u t = u xx + B e u (B = 0.5, δ ˆ = 0.1) 10 Q = 12.0 Q = 12.5 Q = 13.0 8 Q = 15.0 6 u(0) 4 2 0 ˆ δ 0 0.5 1 1.5 t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Energy input ˆ B ≈ 2 × 10 − 10 , δ ≈ 10 − 7 Realistic values for 1D model: 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 ( T ig ) is needed as speed has sensitive dependence on T ig . Explosive is actually multi-step and includes gas-phase reactions. Some equations have been written down. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 t x=Ut plasticity elastic deformation yield point . . . . . . . . . . . . . . . . . . x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .