Reactive Burn Model Research Articles

Verification and validation of detonation-shock-dynamics relations for explosives described by general equation of state and chemical reaction models

Detonation shock dynamics is a powerful method to model the behaviour of High Explosives (HE). However in order to use this method, the underlying relationship between the local radius of curvature and the detonation speed must be known. Previous work has developed methods to calculate this effect using simple, single-step Arrhenius and polytropic gas, models for the chemical reaction and the equation of state, respectively. In recent years, more complex models for both reaction rates and equations of state have been developed which show better agreement with experimental data than these simple models, especially when considering condensed phase explosives.. This work presents the governing equations for solving these problems in a way that is generalised to use arbitrary equations of state as well as reaction models which may have more than a single step and multiple product species. This implementation is verified against exact solutions, demonstrating that the equations were implemented properly. The verified algorithm is then validated against experimental data and high fidelity simulations, showing that it is able to make accurate predictions in a regime where the underlying assumptions of the governing equations are valid. This approach has many applications: from creating equivalent detonation shock dynamics models for existing reactive burn calibrations for HE; to developing new functional forms and calibrations of reactive burn models for condensed phase high explosives.

Multi-scale modeling of shock initiation of a pressed energetic material III: Effect of Arrhenius chemical kinetic rates on macro-scale shock sensitivity

Multi-scale predictive models for the shock sensitivity of energetic materials connect energy localization (“hotspots”) in the microstructure to macro-scale detonation phenomena. Calculations of hotspot ignition and growth rely on models for chemical reaction rates expressed in Arrhenius forms; these chemical kinetic models, therefore, are foundational to the construction of physics-based, simulation-derived meso-informed closure (reactive burn) models. However, even for commonly used energetic materials (e.g., HMX in this paper) there are a wide variety of reaction rate models available. These available reaction rate models produce reaction time scales that vary by several orders of magnitude. From a multi-scale modeling standpoint, it is important to determine which model best represents the reactive response of the material. In this paper, we examine three global Arrhenius-form rate models that span the range of reaction time scales, namely, the Tarver 3-equation, the Henson 1-equation, and the Menikoff 1-equation models. They are employed in a meso-informed ignition and growth model which allows for connecting meso-scale hotspot dynamics to macro-scale shock-to-detonation transition. The ability of the three reaction models to reproduce experimentally observed sensitivity is assessed by comparing the predicted criticality envelope (Walker–Wasley curve) with experimental data for pressed HMX Class V microstructures. The results provide a guideline for model developers on the plausible range of time-to-ignition that are produced by physically correct Arrhenius rate models for HMX.

Physically evocative meso-informed sub-grid source term for energy localization in shocked heterogeneous energetic materials

Reactive burn models for heterogeneous energetic materials (EMs) must account for chemistry as well as microstructure to predict shock-to-detonation transition (SDT). Upon shock loading, the collapse of individual voids leads to ignition of hotspots, which then grow and interact to consume the surrounding material. The sub-grid dynamics of shock-void interactions and hotspot development are transmitted to macro-scale SDT calculations in the form of a global reactive “burn model.” This paper presents a physically evocative model, called meso-informed sub-grid source terms for energy localization (MISSEL), to close the macro-scale governing equations for calculating SDT. The model parameters are explicitly related to four measurable physical quantities: two depending on the microstructure (the porosity ϕ and average pore size D¯void), one depending on shock–microstructure interaction (the fraction of critical voids ξcr), and the other depending on the chemistry (the burn front velocity Vhs). These quantities are individually quantifiable using a small number of rather inexpensive meso-scale simulations. As constructed, the model overcomes the following problems that hinder the development of meso-informed burn models: (1) the opacity of more sophisticated surrogate/machine-learning approaches for bridging meso- and macro-scales, (2) the rather large number of high-resolution mesoscale simulations necessary to train machine-learning algorithms, and (3) the need for calibration of many free parameters that appear in phenomenological burn models. The model is tested against experimental data on James curves for a specific class of pressed 1,3,5,7-tetranitro-1,3,5,7-tetrazoctane materials. The simple, evocative, and fast-to-construct MISSEL model suggests a route to develop frameworks for physics-informed, simulation-derived meso-informed burn models.

Post-detonation fireball modeling: Validation of freeze out approximations

A numerical simulation study was performed to examine the post-detonation reaction processes produced by the detonation of a 12 mm diameter hemispherical pentaerythritol tetranitrate (PETN) explosive charge. The simulations used a finite rate detailed chemical reaction model consisting of 59 species and 368 reactions to capture post-detonation reaction processes including air dissociation from Mach 19+ shock waves that initially break out of the PETN charge, reactions within the detonation products during expansion, and afterburning when the detonation products mix with the shock heated air. The multi-species and thermodynamically complete Becker–Kistiakowsky–Wilson real-gas equation of state is used for the gaseous phase to allow for the mixing of reactive species. A recent simplified reactive burn model is used to propagate the detonation through the charge and allow for detailed post-detonation reaction processes. The computed blast, shock structures, and mole fractions of species within the detonation products agree well with experimental measurements. A comparison of the simulation results to equilibrium calculations indicates that the assumption of a local equilibrium is fairly accurate until the detonation products rapidly cool to temperatures in the range of 1500–1900 K by expansion waves. Below this range, the computed results show mole fractions that are nearly chemically frozen within the detonation products for a significant portion of expansion. These results are consistent with the freeze out approximation used in the blast modeling community.

Explosive dispersal of particles in high speed environments

In this paper, we present the results of the explosive dispersal of particles in high-speed environments. We carry out Euler–Lagrange numerical simulations of a source at quiescent ambient conditions as well as moving at Mach numbers of 3 and 6. Particle volume fractions of 0%, 1%, and 4.5% are presented. The detonation profile is computed with the Jones–Wilkins–Lee equation of state using a reactive burn model. Non-static cases provide a framework to consider the effect of a bow shock and pre-existing high-speed flow conditions on the dispersal process. We also compute averages of both static and dynamic pressures, as well as impulse density histories on virtual probe planes to characterize the momentum of the flow and particles that would deposit on a target. Results suggest that the presence of the particles can have a substantial effect on the pressure average of the virtual target planes.

Physically Evocative Meso-Informed Burn Model: Topology of Evolving Hotspot Fields

Reactive burn models for heterogeneous energetic material initiation must account for chemistry as well as microstructure to predict shock-to-detonation transition. Upon transient (strong) shock loading, the collapse of individual voids leads to ignition of hotspots, which then grow and interact to completely consume the surrounding material. This paper shows that a length scale measuring the perimeter of the evolving burn front in a field of hotspots, in combination with a topological function/form factor for the hotspot field, is a key input to a framework for constructing a meso-informed reaction rate model. The approach suggested in this paper provides a route to constructing reaction rate models with only two quantities of interest, viz., the burn front perimeter and burn front velocity, which are physically evocative of the underlying hotspot physics and can be quantified individually using meso-scale simulations. It overcomes the problem of calibrating many free parameters that appear in phenomenological burn models and therefore suggests a route to developing general frameworks for simulation-derived meso-informed burn models.

Reactive burn model calibration using high-throughput initiation experiments at sub-millimeter length scales

A first-of-its-kind model calibration was performed using Sandia National Laboratories’ high-throughput initiation (HTI) experiment for two types of vapor-deposited explosive films consisting of hexanitrostilbene (HNS) or pentaerythritol tetranitrate (PETN). These films exhibit prompt initiation, and they reach steady detonation at sub-millimeter length scales. Following prior work on HNS, we test the hypothesis of approximating these explosive films as fine-grained homogeneous solids with simple Arrhenius kinetics burn models. The model calibration process is described herein using a single-step as well as a two-step Arrhenius rate law, and it consists of systematic parameter sampling leading to a reduction in the model degrees of freedom. Multiple local minima are observed; results are given for seven different optimized parameter sets. Each model set is further evaluated in a two-dimensional simulation of the critical failure thickness for a sustained detonation. Overall, the two-step Arrhenius kinetics model captures the observed behavior for HNS; however, neither model produces a good fit to the PETN data. We hypothesize that the HTI results for PETN correspond to a heterogeneous response, owing to the smaller reaction zone of PETN compared to HNS (i.e., it does not homogenize the fine-grained hot spots as well). Future work should consider using the ignition and growth model for PETN, as well as other reactive burn models such as xHVRB, AWSD, PiSURF, and CREST.

Investigating growth to detonation in vapor-deposited hexanitrostilbene and pentaerythritol tetranitrate films using high-throughput methods

A high-throughput experimental setup was used to characterize initiation threshold and growth to detonation in the explosives hexanitrostilbene (HNS) and pentaerythritol tetranitrate (PETN). The experiment sequentially launched an array of laser-driven flyers to shock samples arranged in a 96-well microplate geometry, with photonic Doppler velocimetry diagnostics to characterize flyer velocity and particle velocity at the explosive–substrate interface. Vapor-deposited films of HNS and PETN were used to provide numerous samples with various thicknesses, enabling characterization of the evolution of growth to detonation. One-dimensional hydrocode simulations were performed with reactions disabled to illustrate where the experimental data deviate from the predicted inert response. Prompt initiation was observed in 144 μm thick HNS films at flyer velocities near 3000 m/s and in 125 μm thick PETN films at flyer velocities near 2400 m/s. This experimental setup enables rapid quantification of the growth of reactions in explosive materials that can reach detonation at sub-millimeter length scales. These data can subsequently be used for parameterizing reactive burn models in hydrocode simulations, as discussed in Paper II [D. E. Kittell, R. Knepper, and A. S. Tappan, J. Appl. Phys. 131, 154902 (2022)].

Verification of a Specialized Hydrodynamic Simulation Code for Modeling Deflagration and Detonation of High Explosives

Abstract A specialized hydrodynamic simulation code has been developed and verified for the simulation of one-dimensional unsteady problems involving the detonation and deflagration of high explosives. To model all the relevant physical processes in these problems, a code is required to simulate compressible hydrodynamics, unsteady thermal conduction, and chemical reactions with complex rate laws. Several verification exercises are presented which test the implementation of these capabilities. The code also requires models for physics processes such as equations of state and conductivity for pure materials and mixtures as well as rate laws for chemical reactions. Additional verification tests are required to ensure that these models are implemented correctly. Though this code is limited in the types of problems it can simulate, its computationally efficient formulation allows it to be used in calibration studies for reactive burn models for high explosives. This study demonstrates how a series of verification tests can be used to ensure that the various physics processes needed to simulate complex phenomenon can be tested to ensure that they are correctly implemented.

Experimental observations of shock-wave-induced bubble collapse and hot-spot formation in nitromethane liquid explosive

We studied the collapse of individual helium gas bubbles in the homogeneous explosive nitromethane (NM) to investigate effects of hot-spot formation on the detonation process. A bubble was injected into a NM sample, and a shock wave from an explosive detonator compressed the bubble, creating a localized hot spot. We measured shock and detonation wave speeds with optical velocimetry, and we used a high-speed camera to image the shock propagation and the pre- and post-bubble collapse processes. An infrared camera image showed the residual radiance temperature distribution after the bubble collapse, and an optical fiber pyrometer measured the time-resolved thermal radiance. We measured the optical spectra of light emitted from detonating NM without a bubble and from a collapsing bubble in shocked, undetonated NM. We estimated temperatures of the detonation fronts and of the hot spots formed by bubble collapse. To study the incipient detonation process, we performed all bubble collapse experiments at pressures below the threshold for creating a sustained detonation. Where the bubble collapsed, we observed an opaque, thermally emissive region believed to be chemical reaction products. Chemical reactions in NM can be produced with lower shock pressures (∼1 GPa) when a helium bubble is present than without a bubble (∼10 GPa). We used hydrodynamic modeling to predict shock wave propagation, extent of chemical reaction, and subsequent temperature rise from the collapsing bubble. Simulations using a temperature-dependent Arrhenius burn model gave much better results than reactive burn models that depend only on pressure and density.

Experimental and modeling analysis of detonation in circular arcs of the conventional high explosive PBX 9501

We examine the diffraction dynamics of a two-dimensional (2D) detonation in a circular arc of the conventional HMX-based, high performance, solid explosive PBX 9501, for which the detonation reaction zone length scale is estimated to be of the order of 100–150 µm. In this configuration, a steady propagating detonation will develop, sweeping around the arc with constant angular speed. We report on results from three PBX 9501 arc experiments, exploring the variation in linear speed on the inner and outer arc surfaces for the steady wave along with the structure of the curved detonation front, as a function of varying inner surface radius and arc thickness. Comparisons of the properties of the motion of the steady wave for each arc configuration are then made with a spatially-distributed PBX 9501 reactive burn model, calibrated to detonation performance properties in a 2D planar slab geometry. We show that geometry-induced curvature of the detonation near the inner arc surface has a significant effect on the detonation motion even for conventional high explosives. We also examine the detonation driving zone structure for each arc case, and thus the subsonic regions of the flow that determine the influence of the arc geometry on the detonation propagation. In addition, streamline paths and reaction progress isolines are calculated. We conclude that a common approximation for modeling conventional high explosive detonation, wherein the shock-normal detonation speed is assumed equal to the Chapman–Jouguet speed, can lead to significant errors in describing the speed at which the detonation propagates.

A multiphase Lagrangian discontinuous Galerkin hydrodynamic method for high-explosive detonation physics

We present a new multidimensional multiphase Lagrangian discontinuous Galerkin (DG) hydrodynamic method that supports programmed burn and reactive burn high-explosive (HE) detonation models. A modal DG approach is used to evolve fields relevant to conservation laws. These fields are approximated by Taylor series polynomials of varying degree. The HE reaction mass fraction is represented using a nodal quantity. A new limiting approach is presented along with new special limiting conditions for the reactive burn model. The HE models are applied to individual quadrature points in the DG elements to better resolve the physics with the modal method. In the programmed burn model, this means that reaction times are calculated for each point separately. The Scaled Unified Reactive Front (SURF) reactive burn model uses a lead shock pressure to determine the reaction rate at the quadrature points, with a new lead shock pressure detector that works with the Taylor series polynomials. The temporal evolution of the governing equations is achieved with the total variation diminishing Runge–Kutta (TVD RK) time integration method. The robustness of the new limiting approach is tested using 1D and 2D strong shock gamma-law gas problems. The accuracy of the point-wise programmed burn model is tested using a 1D gamma-law gas verification problem and a 2D radial problem. The implementation of the SURF reactive burn model within the DG method is tested using a 1D shock-to-detonation problem simulating the behavior of PBX-9501 and a 2D confined slab problem simulating the behavior of PBX-9502, both represented by the Davis equations of state with a multiphase modeling method. All 2D results use quadratic elements, which have faces that can bend.

Numerical Considerations in the Modeling of a High Explosive Cylinder Experiment Using an ALE Continuum Mechanics Code

A series of experiments involving the detonation of PBX 9501 encased in a copper cylinder are modeled with the objective of evaluating a proposed set of phenomenological parameters for the Wescott–Stewart–Davis reactive burn model. The numerical analysis is conducted using the Los Alamos continuum mechanics code FLAG. Numerical considerations pertaining to various aspects of modeling the experiments using FLAG are discussed. It is shown that use of the proposed set of phenomenological parameters results in predictions of free-surface velocity that match empirically measured velocities reasonably well.

Shock initiation of the HMX-based explosive PBX 9012: Experiments, uncertainty analysis, and unreacted equation-of-state

Shock initiation experiments have been carried out on the polymer-bonded explosive PBX 9012 [nominally 90% octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazine (HMX), 10% vinylidene–hexafluoropropylene copolymer (Viton A) by weight] in order to provide calibration data for the explosive’s unreacted equation-of-state (EOS) and shock-to-detonation transition for reactive burn rate calibration. The input pressures covered the range of 1.86–4.43 GPa. This provided run-to-detonation depths ranging from >22.3mm for the lowest pressure shot to 4.91 mm at the highest pressure. The relative shock sensitivity of PBX 9012 is compared to other HMX-explosives in terms of the Pop-plot, showing that the studied explosive is more sensitive than other similar HMX-based counterparts (with notable exceptions). The uncertainty in the shock velocity determinations from the shock tracker measurements are also investigated, yielding new uncertainty measures in the generated Hugoniot data and run-to-detonation coordinates. Finally, the unreacted equation-of-state is determined using a linear Us–up Mie–Grüneisen relation and the Davis reactants EOS analytical form, the latter being more suited for reactive burn modeling.

Unified Approach for Meso-Informed Burn Models in Shocked Energetic Materials

Reactive hydrocode calculations of shock-to-detonation transition in heterogeneous energetic (HE) materials need to be closed with burn models. One requirement of burn models is to supply a macroscale control volume with an energy release rate due to chemical reactions that reflects the subgrid physics of hotspot ignition and growth. Energy localization at hotspots delivers chemical energy at a rate , where is the heat of reaction and is a meso-scale energy localization time scale. This energy deposition rate is much larger than nominal Arrhenius-form chemical energy deposition rates in a homogeneous sample subjected to the same shock loading. To develop meso-informed energy deposition rate models, this paper identifies a meso-scale energy release time scale that is common to burn models based on the hotspot ignition and growth concept. The identification of a common time scale allows for a unified microstructure-aware, physics-based reactive burn model; high-fidelity meso-scale numerical simulations are used to construct a surrogate model for . The surrogate model is shown to capture the effects of microstructural parameters on and can be used for meso–macro coupling in a multiscale model to predict the sensitivity of HE materials.

An Eulerian multimaterial framework for simulating high-explosive aquarium tests

Aquarium tests of cylindrical high-explosive charges provide optical data of the detonation front velocity and shape, propagation of the shock wave in the surrounding water, and expansion rates of the detonation products behind the front. Data from aquarium experiments is often used for calibration of reactive burn models based on phenomenological equations of state (EOS) and reaction rate laws. This paper presents a multimaterial numerical modeling framework to solve the 2D axisymmetric reactive Euler equations for high-explosive aquarium tests, in particular for ammonium nitrate - fuel oil (ANFO) explosives. An extension of the Ghost Fluid Method (GFM) is used to handle the dynamic material interfaces for the ANFO explosion products, the charge-confining material (polymethyl methacrylate PMMA), and the surrounding water. This study analyzes the sensitivity of calculations (both computational efficiency and numerical accuracy) to different algorithms for the material interface models including the original GFM versus Riemann solver-based strategies. A novel method for defining the left and right states in the interfacial Riemann problem eliminates the need for sorting or nodal interpolation during the projection along the material interface. Numerical tests indicate that populating the interface node values using the Riemann solution mitigate the overheating error observed in steady-state calculations. Solution convergence and computational efficiency are explored as a function of the spatial and temporal order of the schemes. Results from the computational model with analytical equations of state and fitted reaction rate parameters show very good quantitative agreement with experimentally observed detonation front velocity, reaction products expansion, and shock wave propagation in the surrounding water for a cylindrical ANFO charge encased in PMMA. The proposed modeling framework, in conjunction with experimental tests, provides a reliable tool to assess equations of state and reaction rate expressions for reactive burn models of confined high explosives.

WGT: A mesoscale-informed reactive burn model

Polymer bounded explosives (PBXs) consist of energetic crystals coated with a polymer binder. These materials exhibit a highly heterogeneous microstructure. The initiation of the detonation phenomenon in PBXs is believed to be generated at the microstructure scale through hotspots. Hence, many of the explosives properties (initiation, desensitization, etc.) are understood as a direct consequence of their microstructure. Mesoscale modeling directly addresses the physics of hotspot formation. Unfortunately, high computational cost prevents their use on laboratory-sized and large scale experiments. In practice, continuum-scale models remain mandatory. We describe a new reactive burn model, named WGT, aiming to represent, at the continuum scale, some of the complexity of the PBX’s microstructure. The initiation regime is driven by the shock temperature and results from surrogate modeling of the kinetics of a heterogeneous nucleation and growth model. The other regimes follow the formulation of the WHS2D2 reactive burn model and are driven by the local temperature. This model was calibrated on experimental results for PBX 9502 available in the literature, such as detonation velocity–curvature laws, Pop-plot data, or embedded electromagnetic particle velocity gauges. The model was also tested against desensitization and propagation data.

Prediction of Probabilistic Detonation Threshold via Millimeter‐Scale Microstructure‐Explicit and Void‐Explicit Simulations

AbstractWe present an approach and relevant models for predicting the probabilistic shock‐to‐detonation transition (SDT) behavior and Pop plot (PP) of heterogeneous energetic materials (HEM) via mesoscopic microstructure‐explicit (ME) and void explicit (VE) simulations at the millimeter (mm) sample size scale. Although the framework here is general, the particular material considered in this paper is pressed Octahydro‐1,3,5,7‐tetranitro‐1,2,3,5‐tetrazocine (HMX). To systematically delineate the effects of material heterogeneities, four material cases are considered. These cases are homogeneous material, material with granular microstructure but no voids, homogeneous material with voids, and material with both granular microstructure and voids. Statistically equivalent microstructure sample sets (SEMSS) are generated and used. Eulerian hydrocode simulations explicitly resolve the material heterogeneities, voids, and the coupled mechanical‐thermal‐chemical processes. In particular, it is found that both microstructure and voids strongly influence the SDT behavior and PP. The effects of different combinations of microstructure heterogeneity and voids on the SDT process and PP are quantified and rank‐ordered. The overall framework uses the Mie–Grüneisen equation of state and a history variable reactive burn model (HVRB). A novel probabilistic representation for quantifying the PP is developed, allowing the calculation of (1) the probability of observing SDT at a given combination of shock pressure and run distance, (2) the run‐distance to detonation under a given combination of shock pressure and prescribed probability, and (3) the shock pressure required for achieving SDT at a given run distance with a prescribed probability. The results are in agreement with general trends in experimental data in the literature.

Numerical modelling of steady detonations with the CREST reactive burn model

Watt et al. [J Eng Math 75(1):1–14, 2012] have shown that one can obtain good results for the propagation of detonation waves in cylindrical charges by assuming that the post-shock streamlines are straight. In this paper, we compare this Straight Streamline Approximation (SSA) to high-resolution Direct Numerical Simulations (DNS) for different models of explosives. We find that the SSA is less accurate for realistic explosion models than it is for polytropic equations of state with power-law reaction rates.

Detonation diffraction in a circular arc geometry of the insensitive high explosive PBX 9502

We describe the details of an unconfined insensitive high explosive (PBX 9502) circular arc section experiment, in which, after a transient period, a detonation sweeps around the arc with constant angular speed. The arc section is sufficiently wide that the flow along the centerline of the arc section remains two-dimensional. Data includes time-of-arrival diagnostics of the detonation along the centerline inner and outer arc surfaces, which is used to obtain the angular speed of the steadily rotating detonation. We also obtain the lead shock shape of the detonation as it sweeps around the arc. Reactive burn model simulations of the PBX 9502 arc experiment are then conducted to establish the structure of the detonation driving zone, i.e. the region enclosed between the detonation shock and flow sonic locus (in the frame of the steady rotating detonation). It is only the energy released in this zone which determines the speed at which the steady detonation sweeps around the arc. We show that the sonic flow locus of the detonation driving zone largely lies at the end of, or within, the fast reaction stage of the PBX 9502 detonation, with the largest section of the detonation driving zone lying close to the inner arc surface. We also demonstrate that the reactive burn model provides a good prediction of both the angular speed of the detonation wave and the curved detonation front shape.