A two-phase model for aluminized explosives on the ballistic and brisance performance

The performance of aluminized high explosives is considered by varying the aluminum (Al) mass fraction in a heterogeneous mixture model. Since the time scales of the characteristic induction and combustion of high explosives and Al particles differ, the process of energy release behind the leading detonation wave front occurs over an extended period of time. For simulating the performance of aluminized explosives with varying Al mass fraction, HMX (1,3,5,7-tetrahexmine-1,3,5,7-tetrazocane) is considered as a base explosive when formulating the multiphase conservation laws of mass, momentum, and energy exchanges between the HMX product gases and Al particles. In the current study, a two-phase model is utilized in order to determine the effects of the Al mass fraction in a condensed phase explosive. First, two types of confined rate stick tests are considered to investigate the detonation velocity and the acceleration ability, which refers to the radial expansion velocity of the confinement shell. The simul...


I. INTRODUCTION
In this study, we consider an aluminized HMX including the afterburning of Al particles at a high temperature and pressure condition provided by the detonation of HMX.The Al particles are often mixed with high explosives because of the high heat of combustion, which is approximately 5-6 times higher than the base explosive during its aerobic reaction.The shortcoming of adding Al to the explosives is the incomplete combustion of Al particles at high aluminum content.Accordingly, the effect of the Al mass fraction in the aluminized explosive on the blast performance has been considered, and in particular, the maximum blast ability of the aluminized explosive has been observed at an optimal Al mass fraction.This observation was reported in the AA evaluation tests and measuring dent depth tests.Grishkin et al. 1 and Trzcinski et al. 2 conducted a series of confined rate stick tests comprising HMX (1,3,5,7-tetrahexmine-1,3,5,7-tetrazocane) or RDX (1,3,5-trinitro-1,3,5-triazinane) as a base explosive for evaluating AA and reported the existence of an optimal Al mass fraction, as shown in Fig. 1(a).This observation was also confirmed from the results of a dent depth test conducted by Jouet et al. 3 According to the experimental results shown in Fig. 1(b), the maximum dent depth of porous HMX is at an optimal Al mass fraction.In addition, a decrease of the dent depth was observed with the increase of the Al mass fraction after reaching the optimal Al mass fraction, which is a similar tendency to that observed in the AA evaluation study.
Extensive studies for the combustion of Al particles in combustible or reactive gases and in high explosives have been reported in the literature.In particular, Chinnayya et al. 4 and Fedina and Fureby 5 reported progress in modeling and simulation of the combustion of aluminum in solid explosives.And also, numerical analysis of the combustion of Al particles in gas flow has been conducted with a twophase model, which was developed to interpret the behavior of diluted particles in a gas suspension. 6The model involves conservation laws of mass, momentum, and energy, including interactions between two phases: gas and diluted particles.The physical quantities of interaction (mass, momentum, and energy) have been prescribed by the phenomenological model of Veyssiere and Khasainov. 7n this study, the two-phase model, formerly developed for gaseous detonation, is reconstructed in order to analyze the effects of the Al mass fraction on the blast performance of the aluminized HMX.The mass fraction of 0%-25% is considered for comparison with the available experimental data.In addition, a series of plate dent tests is considered for an additional numerical investigation to evaluate the brisance ability of the aluminized HMX.spatial integration and time integration, respectively.The model separates the base explosive and Al particles in a multiphase medium based on the following numerical assumptions: (i) the volume of particles in the condensed medium is ignored, (ii) the interactions between particles are ignored, (iii) both the HMX and the distributed particulates are treated as a continuum within each space, and (iv) the sound speed and pressure of the diluted particle phase are ignored.
Therefore, as shown in Fig. 2, the heterogeneous material is treated as two homogeneous media, with each phase treated separately as a continuum space.However, unlike the case of a gas suspension containing Al particles, the densities of the Al particles and HMX are similar and the initialization of the diluted densities therefore becomes somewhat important.The diluted densities in each space are calculated in a simple form as shown in Eqs. ( 1) and ( 2), which is based on the known value of the mass ratio of Al to the total high explosive (a), the density of the HMX (q HMX ), the density of Al (q Al ), and the volume fraction (V a ) occupied by the Al particles for the high explosive Even though the assumptions of the model can deviate from reality when the Al mass fraction is increased, the model has been adopted as one promising means to handle the difficulty of two-phase (condensed) process by making use of the Eulerian based mathematical formulation.

A. Governing equations
Based on the conservation laws with two distinct matter interaction effects considered, the governing equations of the two-phase model in axisymmetric cylindrical (u ¼ 1) and rectangular (u ¼ 0) coordinates are constructed in the following equations: 8 S ¼ q; q v 1 ;q v 2 ; q E HMX ; q k; r; r u 1 ; r u 2 ; r E Al ½ > ; (4) Here, q; v; E HMX ; k; r; u; E Al ; P; T; are the diluted HMX density (kg/m 3 ), HMX product gas velocity (m/s), specific total energy of HMX (J/kg), mass fraction of the product, diluted Al particle density (kg/m 3 ), Al particle velocity (m/s), specific total energy of Al (J/kg), pressure (Pa), and temperature (K), respectively.To preserve the mass, momentum, and energy of the entire system, RHS (the source term) is composed of the rate of mass generation or Al consumption _ r (kg/m 3 s), the momentum exchange rate _ f (kg/m 2 s 2 ), and the rate of heat exchange due to convection _ Q H (J/m 3 s).The physical quantities of the interaction between the two phases follow the formula in Eqs. ( 9)- (15)._ w HMX is the reaction rate of HMX (s À1 ).Then, the burning time t b of an individual particle follows the empirical law 9 which considers the amount of oxidizing species for an Al particle.For ignition of Al particles, we define the film  16), which is derived from curve fitting with the experiment 10 Nu ¼ 2:0 þ 0:459Re 0:55 Pr 0:33 ; (15) where d p is the diameter of the initial Al particle (m), / is the volume fraction of gaseous components able to oxidize aluminum, q Al is the Al density (kg/m 3 ), C D is the drag coefficient of the Al particle, T Al is the solid phase Al temperature (K), and T HMX is the HMX temperature (K).The temperature of Al is calculated based on the thermodynamic identity of typical bulk scale Al, and is determined from Eq. ( 17).Here, c p ¼ 910 J/kg K, and T b ¼ 2743 K are the specific heat of Al and the boiling temperature of Al (K), respectively.The melting temperature of Al (T m ) varies with pressure 11 as implemented in the calculation The conductivity (k g ), Nusselt number (Nu), Prandtl number (Pr), and viscosity (l) of the burned explosive properties are computed from temperature curve fitting by the NASA Chemical Equilibrium with Applications (CEA) code. 12

B. Reaction model and EOS of high explosives
To simulate the detonation of a condensed energetic substance, the reaction model and EOS are adapted from the Ignition and Growth (I&G) model and the Jones-Wilkins-Lee equation of state (JWL EOS) as shown in the following equations: It is assumed that both constituents, namely unreacted solid HMX and reacted product gases, are in pressure and temperature equilibrium in the reaction zone.Further indepth descriptions of the separate JWL EOS are given in Ref. 13.The parameters of 95% HMX with 5% estane and BDNPA/F [50/50 wt.% eutectic mixture of BDNPA (bis

III. RESULTS AND DISCUSSION
The objective of this study is to evaluate the effect of the Al mass fraction in aluminized HMX on the detonation and blast performance.First, we consider two types of confined rate stick tests (0%-25% Al with fixed particle size of 7 lm).The two main features of interest in the rate stick analysis include: (1) the decrease in detonation velocity and (2) the explosive work of expansion form, AA, with increasing Al mass fraction in aluminized HMX.To measure the detonation velocity with varying Al mass fractions, the geometry condition (D¼ 20 mm and L ¼ 100 mm), as shown in Fig. 3(a), is set for the test, which is the same as that for the experiment. 15The confiner material is copper 16 of 2 mm thickness.The geometry condition of the confined rate stick (D ¼ 20 mm and L ¼ 120 mm), as shown in Fig. 3(b), is set for the test of AA, which is also the same geometry condition as that in the experiment. 1 The confiner material is steel 17 of 5 mm thickness.The plate dent test is considered for the additional numerical investigation to measure the power of the aluminized HMX at the end of the rate stick, the objective of which is similar to that of the rate stick test for evaluating AA. Figure 4(a) shows the experimental setup for the plate dent test. 18Even though the length of the rate stick is 203 mm in the experimental setup, numerically, a length of 40 mm is sufficient to develop a steady detonation wave for the aluminized HMX or HMX.Therefore, the computational domain is shown in Fig. 4(b).As shown in Figs. 3 and 4, the computational domains are divided into three distinct components: an aluminized HMX, metal, and a void.The boundary between the components is partitioned using a level-set method, and the interface is represented by a solid line as a zero level-set.Also, the boundary conditions near the interface are determined by the Ghost Fluid Method (GFM).Further in-depth descriptions of the level-set method and GFM are given in Ref. 19.In addition, the treatment of the void and metal region is detailed in Ref. 17.

A. Non-aluminized HMX characterization in an infinite diameter rate stick
Prior to investigating the effect of the Al mass fraction on the performance of aluminized HMX, the characterization of a non-aluminized HMX is considered.The initial density of the charge is q HMX ¼ 1835 kg/m 3 and the stick is assumed to be infinitely large.We then selected a mesh size of 1/ 10 mm, which is sufficient to simulate the detonation phenomenon of HMX, which has a reaction zone length of 0.33 6 0.05 mm. 20Thus, the mesh size is 1/10 mm for all simulations conducted in this study.
Figure 5 presents the relevant pressure, temperature, and mass fraction of the products under the steady-state of detonation.The temperature of the product gases of HMX is calculated based on the method given in Ref. 21.Here, the von Neumann spike pressure (P ZND ), Chapman-Jouguet (CJ) pressure (P CJ ), temperature (T CJ ), and reaction zone length (X R ) are compared with the reference data 20,22,23 and are summarized in Table II, where the equilibrium values are obtained from running the Cheetah code. 23 Confined rate stick test with a copper confiner: Detonation velocity of aluminized HMX After the detonation of the base high explosive (95% HMX with 5% estane and BDNPA/F) is quantitatively verified, the effect of the Al mass fraction on the detonation  In addition, as the combustion of Al particles occurs behind the detonation front of the base explosive, the pressure increase is observed with increasing Al mass fraction, which results in the reduction in the pressure decay rate.From this result, it can be inferred that the burning of Al particles can compensate the loss of pressure at the front of detonation.In other words, including Al particles in the high explosive formulations can reduce the detonation velocity but improve explosive performance since the instantaneous combustion of the Al particles in the product gases occurs early enough to support the impact.The effect of the Al mass fraction on the blast performance will be treated in Secs.III B and III C.
A comparison between the experimental results 15 and the numerical results for the detonation velocity of aluminized HMX is shown in Fig. 7.The experimental data have three Al mass fractions (5%, 15%, and 25%), while the present work was conducted with five mass fractions, with an additional two Al mass fractions of 10% and 20%.As confirmed in the experimental data in Fig. 7, the detonation velocity decreases as the Al mass fraction increases.This tendency has been reported in the experimental treatises  with various aluminized explosives containing micro-size Al particles. 2,15,24The two main factors responsible for the decrease in detonation velocity with increasing Al mass fraction are considered to be: (1) the mass of the HMX explosive in the rate stick is reduced in all of the explosives as the increase of Al particles is contained, thus reducing the energy of the detonation of the base explosive and (2) the loss of the momentum and energy starting from the leading shock front occurs because of the interaction of Al particles with the detonation products.These two factors are considered in the present model.The predicted detonation velocity therefore decreases with the Al mass fraction, and the error in comparison to the experimental measurements is less than 650 m/s. Figure 8 represents the two-dimensional cylindrical flow field via pressure of a confined rate stick for the 25% Al mass fraction.The detonation velocity is measured to be 8596 m/s, which is similar to the experimental value of 8563 m/s.
C. Confined rate stick test with a steel confiner: Acceleration ability of aluminized HMX Numerical analysis of the AA evaluation of aluminized HMX is conducted using the steel confined rate stick of diameter, D ¼ 20 mm and length, L ¼ 120 mm, shown in Fig. 3(b).The radial expansion velocity of the shell was recorded in the midsection of the rate stick in the experiment. 1 Similarly, as shown in Fig. 9, the numerical radial expansion velocity of the shell was recorded at the middle point in the shell of the midsection of the rate stick.
In this study, the melting temperature of Al is varied with pressure according to pressure data given by. 11Figure 10 shows the comparison of the pressure dependence of the melting temperature.Shown is for 15% Al mass fraction, and the results are the same for tested 5% and 25% cases.There is a discrepancy in the temperature of aluminum; however, the pressure is in excellent agreement regardless of how the melting temperature is implemented.Thus, the choice of melting temperature does not influence the resulting detonation pressure and acceleration ability.
Figure 11 shows the two-dimensional flow field via the density and corresponding shadowgraph for a steel confined rate stick test with the Al mass fraction of 15%, which has the radial expansion velocity of 1257 m/s.The expansion velocity of the shell is recorded when the expansion ratio of the shell reaches about 2.5.
The experiment was conducted with two Al mass fractions (15% and 20%), while the present work was conducted with five mass fractions, with an additional three Al mass fractions of 5%, 10%, and 25%.
As shown in Table III, the experimental expansion velocity of the steel confinement shell for the non-aluminized HMX is 1150 m/s, 1 while the value of 1193 m/s is of the present numerical simulation for the non-aluminized HMX, which difference has the error in 3.76% compared to the experimental result.In order to compare more clearly the increase in the radial velocity of the confinement shell with increasing Al mass fraction between the experiment and the numerical results, we define the AA as the normalized radial  velocity of the confinement shell, which is the velocity represented as a normalized velocity divided by the experimental or numerical result of 0% as shown in Eq. ( 21).So, the normalized velocity of non-aluminized HMX for both experimentally and numerically is unity.Figure 12 shows a comparison of the AA with varying Al mass fraction between the experiment 1 and present numerical results, Despite the limited number of experimental measurements, one can infer that the AA increases as the Al mass fraction increases.The AA then decreases with further increasing Al mass fraction.That is, as the Al mass fraction increases, the AA of aluminized HMX reaches its maximum value at an optimal Al mass fraction, not particularly at a higher mass fraction.
In this study, the product gases of the base explosive are used as the oxidizer for Al particle reactions.Therefore, as the mass fraction of Al particles increases, the amount of oxidizer required for burning the Al particles is reduced.According to Eqs. ( 2) and ( 9), as the mass fraction of Al particles increases, the Al combustion per unit volume increases due to the increase in the diluted density of Al particles.With increasing mass fraction of Al, the rate of burning of Al particles is reduced since the volume fraction of gaseous components able to oxidize Al, comprised of CO, CO 2 , and H 2 O in the products gases of HMX decreases according to Eq. (10).The volume fraction of gaseous components able to oxidize Al is calculated by the Amagat's law with molar fractions of CO, CO 2 , and H 2 O obtained from the thermochemical equilibrium code. 23In addition, according to Eq. ( 16), the incomplete combustion of Al particles due to a deficiency in the oxidizer leads to a decrease in the exothermic energy release from the Al reaction.
Thus, the explosive performance is affected by the complex interaction between the product gases of the base explosive and the Al particles.Thus, by combining an optimum Al  mass fraction within the base explosive, one can seek a maximum performance.In the experimental results, it is expected that the maximum AA would be obtained at an Al mass fraction of about 10%-15%.The increase in the radial velocity of the confinement shell of the experimental data of the 15% Al mass fraction is 5.2% higher than that of the non-aluminized HMX.Also, the numerical results show a similar tendency of AA with increasing Al mass fraction, the maximum increase of which is 5.3% in AA at around an Al mass fraction of 15%-20%.
However, the experimental results show that the rate of reduction of AA is relatively rapid with the increase in the Al mass fraction after the optimal condition compared with the present numerical results.The reason for the rapid reduction in the AA is difficult to determine because the experiment did not give the exact density and the amount of measured data is insufficient.Therefore, if the experimental results are sufficient according to the Al mass fraction and the accurate compositions are known, the information on the aluminum reaction would be accurately implemented.
Although the experimental results and the numerical results show differences of the optimal Al mass fraction and the reduction rate for the AA, the maximum increase in AA and the tendency of AA with varying Al mass fraction can be simulated.
Figure 13 depicts the AA of aluminized HMX with varying Al mass fraction using the radial expansion velocitytime profiles of the Al mass fractions of 0%, 15%, and 25%.After the expansion velocity has stabilized, the velocities are 1193 m/s, 1257 m/s, and 1219 m/s at mass fractions of 0%, 15%, and 25%, respectively.Moreover, we can confirm that, at the beginning of acceleration, the middle component of the steel shell oscillates and accelerates.This is due to the oscillating pressure wave caused by the detonation shock in the steel shell, as shown in Fig. 11.

D. Plate dent test: Brisance ability of aluminized HMX
To measure the brisance ability of the aluminized HMX, the plate dent test is considered.The height of the witness steel plate is 50.8 mm and the diameter of the rate stick is 41.3 mm, enabling the development of the detonation wave with the approximately infinite radius for the numerical and experimental tests. 18In order to closely observe the dent shape, the two-dimensional calculations are illustrated in three-dimensions by rotating along the center axis.
Prior to analyzing the effect of the Al mass fraction on the depth of the dent, initial calculation was first carried out with a 95% HMX with no Al for quantitative comparison with the experiment. 18The experimental result of the dent depth is 10.5 mm.
Figure 14 shows the propagation of detonation for the non-aluminized HMX, viewed facing the witness plate, with the density contour at 4 ls.As shown in Fig. 15, the progression of the dent shape by the transmitted detonation wave to the steel plate is shown by the radial deviatoric stress contour inside the plate and the pressure iso-surface.In addition, the depth of the dent increases continuously with time.Figure 16, however, shows the dent depth of the numerical results versus time, confirming that when the time exceeds 50 ls after shock initiation, the depth of the dent is stabilized.This result shows the depth of the dent as 11 mm and an error of 4.76% compared to the experimental result, 18 as shown in Table IV.
We then conducted a series of plate dent tests with varying Al mass fractions.The following is an example of aluminized HMX with 5% Al mass fraction.Similar to the results shown in Fig. 15 for the non-aluminized HMX, the radial deviatoric stress contours are shown in Fig. 17.After the depth of the dent has stabilized, it is confirmed to be 11.5 mm.
The depths of dent with increasing Al mass fraction are shown in Fig. 18.For the Al mass fraction of 0%-30%, the depth of dent increases with the Al mass fraction.Then, at an Al mass fraction of around 20%, the maximum depth of dent is observed.The maximum increase in the depth of dent is 13.7 mm, which shows an increase of 24.5% compared to the numerical result of 0% Al.After confirming the effect of the Al mass fraction, the tendency of the plate dent test results follows the observation of the AA with varying Al mass fraction, as anticipated with the investigations previously explained earlier.However, the optimal mass fractions for the AA and the increase in the depth of dent differ.The difference in the results obtained from end and radial performance might be related to the time at which the target receives energy from the detonation products.In other words, the time exerted on the witness plate is greater than the confinement shell because the direction of the detonation propagation is facing the witness plate.As a result, the maximum increase in the depth of the dent is higher than the AA.

IV. CONCLUSION
The present work suggests that the effects of the Al mass fraction on the detonation velocity and the blast performance of aluminized explosives can be described by using a two-phase model.In the simulation, the aluminized HMX with varying mass fractions of Al (5%-30%) is considered, and the decrease in the detonation velocity is confirmed with increasing Al mass fraction.Furthermore, the radial expansion velocity of the confining shell is shown to increase and to reach the maximum at an optimal Al mass fraction.A series of plate dent test results also revealed that the tendency of dent depths with varying Al mass fractions expectedly follows the experimentally available data.

FIG. 1 .
FIG. 1. Experimental tendency of the performance of aluminized explosives according to the Al mass fraction: (a) AA of HMX and RDX based aluminized explosives 1,2 and (b) brisance ability of porous HMX based aluminized explosives.3

FIG. 2 .
FIG. 2. Structure of the heterogeneous solid explosive mixture model with Al particles.

FIG. 4 .
FIG. 4. Experimental and numerical setup of the plate dent test: (a) experimental setup (D ¼ 41.3 mm, L ¼ 203 mm) 18 and (b) numerical setup of twodimensional cylindrical coordinates (D ¼ 41.3 mm, L ¼ 40 mm) depicted with a black solid line for the steel, and a purple solid line for the explosive.

FIG. 7 .
FIG. 7. Detonation velocity versus Al mass fraction in 2 mm confined copper, 20 mm diameter rate stick experiments, 15 and present simulation.

FIG. 10 .
FIG. 10.Comparison of the pressure dependence on Tm with constant Tm being 934 K for 15% Al mass fraction.

FIG. 14 .
FIG. 14. Density contour of the propagation of detonation for the nonaluminized HMX at 4 ls after shock initiation.

FIG. 15 .FIG. 17 .
FIG. 15.Radial deviatoric stress contours and pressure iso-surfaces of the deformation at the top of the witness plate for the non-aluminized HMX at (a) 6 ls, (b) 8 ls, (c) 10 ls, and (c) 12 ls after shock initiation.

TABLE I .
JWL EOS parameters of 95% HMX at an initial density of q HMX ¼ 1835 kg/m 3 .

TABLE II .
ZND and CJ conditions of 95% HMX.

TABLE III .
AA of the experiment and present work for the non-aluminized HMX.
FIG. 12. AA versus Al mass fraction in 5 mm confined copper, 20 mm diameter rate stick experiments, 1 and present simulation.