Research on the quasi-isentropic driving model of aluminized explosives in the detonation wave propagation direction

Hongfu Wng , Yn Liu , Fn Bi , Cho He , Yingling Xu , Qing Zhou ,Chun Xio , Fenglei Hung

a State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing,100081, China

b China Research and Development Academy of Machinery Equipment, Beijing,100089, China

c Beijing Institute of Technology Chongqing Innovation Center, Chongqing, 401120, China

Keywords:Aluminized explosive Flyer plate experiment Quasi-isentropic theoretical model Al reaction Driving characteristics

ABSTRACT Taking CL-20(Hexanitrohexaazaisowurtzitane)-based aluminized explosives with high gurney energy as the research object,this research experimentally investigates the work capability of different aluminized explosive formulations when driving metal flyer plates in the denotation wave propagation direction.The research results showed that the formulations with 43 μm aluminum (Al) powder particles (The particle sizes of Al powder were in the range of 2～43 μm) exhibited the optimal performance in driving flyer plates along the denotation wave propagation direction.Compared to the formulations with Al powder 13 μm, the formulations with Al powder 2 μm delivered better performance in accelerating metal flyer plates in the early stage, which, however, turned to be poor in the later stage.The CL-20-based explosives containing 25% Al far under-performed those containing 15% Al.Based on the proposed quasi-isentropic hypothesis, relevant isentropy theories, and the functional relationship between detonation parameters and entropy as well as Al reaction degree, the characteristic lines of aluminized explosives in accelerating flyer plates were theoretically studied,a quasi-isentropic theoretical model for the aluminized explosive driving the flyer plate was built and the calculation methods for the variations of flyer plate velocity, Al reaction degree, and detonation product parameters with time and axial positions were developed.The theoretical model built is verified by the experimental results of the CL-20-based aluminized explosive driving flyer plate.It was found that the model built could accurately calculate the variations of flyer plate velocity and Al reaction degree over time.In addition,how physical parameters including detonation product pressure and temperature varied with time and axial positions was identified.The action time of the positive pressure after the detonation of aluminized explosives was found prolonged and the downtrend of the temperature was slowed down and even reversed to a slight rise due to the aftereffect reaction between the Al powder and the detonation products.

In this paper, the quasi-isentropic theoretical model for the aluminized explosive driving the flyer plate is based on the characteristic line analysis, so the detonation products fall into four zones.Hence, subscripts I, II, III, and IV in physical parameter symbols represent which zone the physical parameter of the detonation product belongs to.

1.Introduction

Symbols interpretation of symbols D detonation velocity cH sound velocity in the detonation products on the C-J plane uH particle velocity of the detonation products on the C-J plane ρH density of the detonation products on the C-J plane PH pressure of the detonation products on the C-J plane TH temperature of the detonation products on the C-J plane cp,1ti/ cp,2ti sound velocities in the detonation products before/after the Al reactions at ti cp,mti sound velocity in the detonation products undergoing isentropic flow at ti+1 after the Al reactions have completed in ti ～ti+1 up,1ti/up,2ti particle velocities of the detonation products before/after the Al reactions at ti up,mti particle velocity of the detonation products undergoing isentropic flow at ti+1 after the Al reactions have completed in ti ～ti+1 ρp,1ti/ρp,2ti densities of the detonation products before and after the Al reactions at ti ρp,mti density of the detonation products undergoing isentropic flow at ti+1 after the Al reactions have completed in ti ～ti+1 Pp,1ti/Pp,2ti pressures of the detonation products before/after the Al reactions at ti Pp,mti pressure of the detonation products undergoing isentropic flow at ti+1 after the Al reactions have completed in ti ～ti+1 Tp,1ti/Tp,2ti temperatures of the detonation products before and after the Al reactions at ti Tp,mti temperature of the detonation products undergoing isentropic flow at ti+1 after the Al reactions have completed in ti ～ti+1 Dp,i propagation velocity of the right-traveling disturbance in the detonation products in ti ～ti+1 cp,0 sound velocity in the detonation products in 0 ～t1(cH and cp,1t1 are the initial and final states of cp,0)cp,i sound velocity in the detonation products in ti ～ti+1(cp,2ti and cp,mti are the initial and final states of cp,i)up,0 particle velocity of the detonation products in 0 ～t1(uH and up,1t1 are the initial and final states of up,0)up,i particle velocity of the detonation products in ti ～ti+1(up,2ti and up,mti are the initial and final states of up,i)ρp,0 density of the detonation products in 0 ～t1 (ρH and ρp,1t1 are the initial and final states of ρp,0)ρp,i density of the detonation products in ti ～ti+1 (ρp,2ti and ρp,mti are the initial and final states of ρp,i)Pp,0 pressure of the detonation products in 0 ～t1 (PH and Pp,1t1 are the initial and final states of Pp,0)Pp,i pressure of the detonation products in ti ～ti+1 (Pp,2ti and Pp,mti are the initial and final states of Pp,i)Tp,0 temperature of the detonation products in 0 ～t1 (TH and Tp,1t1 are the initial and final states of Tp,0)Tp,i temperature of the detonation products in ti ～ti+1(Tp,2ti and Tp,mti are the initial and final states of Tp,i)

In high-explosive research,high yield is always one of the main goals to be pursued.With a theoretical detonation velocity of 9500-9600 m/s [1], hexanitrohexaazaisowurtzitane, also called CL-20 with the formula C5H6N12O12, is the highest-energy-density single-compound explosive that has been applied around the world[2].However, the excessively fast energy release of CL-20 accelerates the attenuation of its shock wave pressure.Hence,for CL-20,it is necessary to add some metal particles so that the energy release time can be extended.Currently, Al, as a flammable high-energy material, is commonly applied to high explosives for military uses so as to improve their performance.As such,aluminized explosives account for a large proportion of metal-added explosives[3-7].CL-20-based aluminized explosives are of great research and application potential in fields such as high-energy ammunition, shaped charges, and rocket propellants [8].Al powder characterized by a high thermal value, once added to an explosive, can increase the explosive yield from two aspects: on the one hand, Al powder reaction releases a large amount of heat, which increases the detonation heat and thus augments the total energy of the explosive;and on the other hand, the Al reaction changes the energy release process of the detonation,extending the energy release time while boosting the work capacity of the explosive[9-11].

At present, it is of particular importance to understand the energy release characteristics of CL-20-based aluminized explosives,while the driving characteristics of an explosive is considered the important manifestation of its energy release structure.Hence,it is in urgent need to identify the effect of Al powder properties (particle size and content) on the explosive's capacity of driving work.Meanwhile, a theoretical model for the driving of aluminized explosives and the flow characteristics of denotation products is proposed, and the physical mechanism of aluminized explosives doing work in driving flyer plates is developed.Moreover, the relationship between the rules of Al reaction and the movement rules of the driven object is schematically constructed.These efforts are of great significance in optimizing the design of CL-20-based aluminized explosive formulations, perfecting the structure of explosive energy release,and thus boosting the yield of explosives[4,6,10-14].

In order to explore the mechanism of Al powder reaction in aluminized explosive detonation products, the dynamic response of Al particles is systematically analyzed.Detonation environments are typically characterized by high temperature,high pressure,and high-velocity motion.Regarding the effects of the detonation wave and high-velocity fluid on Al particles,Bdzil et al.[15]calculated the motion processes of particles made of different materials such as Al,tantalum, steel and resin in explosive detonation products using the simulation technology of fluid dynamics.The simulation results showed that the effect of explosive detonation on the motion state of the particles fell into two stages, namely the stage where the detonation accelerated the particles,which was mainly affected by detonation products and particle material impedance, and the stage where the detonation products dragged the particle surface.Frost et al.[16]observed and derived the motion trajectory of metal particles of different sizes during explosive detonation by combining radiography and two-phase flow theory.For metal particles in a small size,they would penetrate the wave front under the continuous driving effect of the denotation wave and the resultant products.However, they had a decelerating velocity and ended up with re-sinking into the detonation products as a result of the resistance of the air.In the case of metal particles with a large size, the particle velocity would always be lower than the velocity of the shock wave front.Wang et al.[17] simulated the impact of HMX detonation wave on Al particles and built a dynamic computational model for Al particles under the effect of the detonation wave.Upon the detonation, Al particles firstly horizontally underwent the squeezing effect of the detonation wave, which caused particle shape to exhibit horizontal compression, and then horizontal stretching due to the rarefaction wave.In the longitudinal direction, the particles exhibited a continuous stretching state.

Wang [18] investigated the breaking response of the Al oxide shell on the surface of the Al powder under the impact of the detonation wave.Fig.1 shows the process of the detonation wave impacting Al particles.As can be seen, the detonation wave began to impact Al particles with an amplitude of more than a dozen GPa,which led to their deformation due to the difference in mechanical properties between alumina and Al.Under the impact,the alumina went to invalidation and breaking, while the Al core underwent plastic deformation due to the squeezing.By extracting the motion velocities of the detonation products and Al core, Pei [19] found that there was a velocity difference between detonation products and Al and the velocity of the products was higher,resulting in the stripping of the alumina shell and facilitating the contact between the Al core and detonation products to react.As the explosive velocity increased, the velocity difference between the Al core and detonation products gradually enlarged.For 2-100 μm Al powder,the velocity they could reach was approximately 59%-83% of the velocity of the detonation product gases.In addition, for the Al powder of different micon sizes that has different alumina shell thicknesses,it took a short time for the temperature of their Al core surface(inner surface of the alumina shell)to rise to the gasification point of the Al, and the delay time was in nanosecond order.Therefore, the Al core surface in the detonation products had sufficient time to absorb the heat.

With respect to the reaction mechanism of Al powder in detonation products produced by aluminized explosives, Cook M.A.et al.[20] proposed a secondary reaction theory for aluminized explosives.According to the theory, only a small amount of Al powder, or even no powder, involves the reaction within the detonation wave front, and the absorption of heat reduces the energy that can promote detonation wave propagation.After the interface acoustic wave, most of the Al powder reacted with the detonation products, from which the energy released could enhance the work-by-expansion capacity of the products.Kato et al.[21] measured the detonation pressure from aluminized explosives using a pressure sensor and found that for Al particles with a size of 8 μm,they began to react in the wake of about 2.5 μs after the leading shock wave, which raised the pressure of the detonation products.Pahl et al.[22] measured the detonation velocity,pressure, and temperature of an aluminized explosive, concluding that there was secondary energy release after some time following the detonation,which suggested the reaction of the Al powder after a period of time following the detonation wave front.

Fig.1.Deformation and breaking processes of Al core and alumina shell:(a)0.7 ns;(b)1.2 ns; (c) 2.5 ns; (d) 3.4 ns?

Scholars added the secondary reaction of Al powder to the equation of state of detonation products to form the reaction models of Al powder in the detonation aftereffect environment[23,24], Baker et al.[25] added one item to the Jones-Wilkins-Lee(JWL) equation of state, which was used to reflect the Al reaction in aluminized explosives and describe the expansion of explosive products.Zhou et al.[26] calculated the time spans of Al powder's heat transfer,melting,and gasification using the temperature field after aluminized explosive detonation,and built an Al powder shell breaking gasification and combustion model for the core-shell structure, as shown in Fig.2.The combustion process of Al particles in the detonation environment was divided into three stages:(1) Al particles having a rising temperature and the alumina shell beginning to break and melt;(2)the Al core getting exposed to the detonation products, which caused a reaction between them and the resultant energy release;and(3)the layer-by-layer gasification of the Al core until the end of the reaction.

The aftereffect reaction of Al powder in detonation products can significantly increase the detonation heat of explosives and thus boost their working capacity.Yin et al.[27] systematically studied the detonation heat and the product composition and proportions of RDX-based explosives containing different mass fractions of Al powder.It was found that the amount of the detonation heat released increased with the content of Al powder and reached the maximum when the content was 40%,while the specific volume of the detonation always decreased with the growing amount of the Al powder added.By analyzing the detonation products of RDXbased aluminized explosives, Hu et al.[28] found that the content of alumina in the detonation products no longer increased as the Al powder content in the explosive exceeded 30%.Wuhyun et al.[29]found that the decrease in the detonation velocity is confirmed with increasing Al mass fraction for the aluminized HMX with varying mass fractions of Al (5%-30%), and the driving metal velocity is shown to increase and to reach the maximum at an optimal Al mass fraction.Li et al.[30] recorded the detonation heat of different aluminized explosive formulations in environments with varying oxygen contents using an isothermal calorimeter.The results revealed that the detonations of the aluminized explosives in the vacuum,air,and pure oxygen environments delivered growing detonation heat in order, and that in both the vacuum and air environments,the explosives containing fine Al particles(12.43 μm in the median diameter) produced a smaller amount of detonation heat than those containing coarse Al particles (74.14 μm in the median diameter).Jin et al.[31]measured the quasi-static pressure in an enclosed space after the detonation of HMX/HTPB 88/12 explosives without Al powder particles, and HMX/Al/HTPB 53/35/12 explosives with 13 and 130 μm Al powder particles separately.According to their research results, the quasi-static pressure produced by HMX-based explosives with Al powder was 1.24 times that produced by those without, and the explosive formulation containing fine Al powder particles delivered higher quasi-static pressure.

Fig.2.Shell broken, gasification and combustion model of Al powder.

Metal flyer plate experiments are considered one of the standard methods for evaluating the work capacity of explosives.Manner et al.[6]conducted flyer plate experiments on HMX-based aluminized explosives and revealed that the flyer plate driven by HMX-based aluminized explosives had a velocity 31% higher than that driven by HMX-based explosives containing lithium fluoride(LiF).TAO et al.[32] captured the velocity of the metal flyer plate accelerated by PETN-based aluminized explosives using a Fabry Perot interferometer and determined the reaction rate and response degree of explosives separately with 5 and 18 μm Al particles 3 μs after the detonation from the experimental results.According to their findings, when the Al content was 5%-10%, the time for the complete reaction of Al powder was 1.5 μs, and the energy released was 18%-22% higher than that released by pure PETN explosives.Chen et al.[4,33,34] performed metal flyer plate experiments on several RDX-based explosives containing Al particles with a diameter ranging from tens of nanometers to tens of microns, in which the acceleration time and capacity of these explosives were compared.By comparing with LiF explosives, it was found that the Al powder with a large particle size had a slow rate of energy release but the lasting time of the release was long.Moreover,for the Al powder with a small particle size,the time for the reaction was found to move up, with that in nanometer considerably boosting the work capacity of the explosive.In addition to the influence of Al powder activity on the detonation velocity of explosives [35], Gogulya et al.[36] carried out flyer plate acceleration experiments using HMX-based explosives containing 15%nanometer Al powder.According to their experimental results,the acceleration of flyer plates was linearly correlated with the activity of Al powder and the content of active Al was the primary factor affecting the work capacity and detonation heat of the explosive.Liu et al.[37] discovered that for the CL-20-based explosive containing 15% Al powder, the change in Al particle size,when in the range of 2～50 μm,posed an insignificant effect on the explosive's performance in accelerating flyer plates.It was also found that the effect of Al particle size changes on the acceleration performance was enhanced as the content of Al was reduced from 15% to 5%.

In the research on the work mode of aluminized explosives when driving flyer plates along the detonation wave propagation direction,establishing a doing-work model serves as an important theoretical basis for predicting and evaluating the properties of aluminized explosives in energy release, driving, and damage effects.However, for aluminized explosives that are different from ideal ones, CJ and ZND models are not applicable.Most of the Al powder does not react until product expansion following the detonation wave propagation.The expansion of explosive gas products is not an isentropic process, which leads to the inapplicability of many EoSs that are used for the detonation products of ideal explosives.What's more, compared to ideal explosives,aluminized ones have a slow energy release process consisting of a fast and a slow stage, which results in a long time scale for their reaction kinetics.As such, new formula descriptions are required for the acceleration-by-driving model that characterizes the energy release rate.

Aiming to identify the energy release characteristics and driving work capacity of CL-20-based aluminized explosives in the detonation aftereffect stage,the metal flyer plate experiment is carried out, this research identifies the effects of Al powder particle size and content on the performance of the explosives in accelerating flyer plates along the detonation wave direction.Furthermore, the relationship between the reaction rate of the Al powder in the detonation products and the motion law of the driven flyer plate is analyzed.Based on the quasi-isentropic assumptions, isentropic theory, and the functional relationship between detonation physical parameters and entropy as well as Al reactivity, a quasiisentropic theoretical model for the aluminized explosive driving the flyer plate is developed according to characteristic line principles.An Al powder combustion sub-model is introduced into the theoretical model developed so as to identify the variations of flyer plate velocity, Al reaction degree, and detonation parameters over time.The theoretical model is verified using flyer plate experiments on different CL-20-based aluminized explosive formulations.In the end,with the model,the variations of detonation products’physical parameters such as pressure and temperature with time and axial positions during driving plate are determined.

2.CL-20-based aluminized explosive driving flyer plate experiment

2.1.Flyer plate experiment settings

Considering that the research goal is to identify the effects of Al powder particle size and content on the driving performance of CL-20-based aluminized explosives in the detonation wave propagation direction, the components in the explosive formulations,excluding Al powder, are set to have the same particle size and proportion.Since the driving effect of explosives is dependent on energy release,it is necessary to ensure the same charge density in the explosive formulations.Four formulations are designed,namely three having the same Al powder content of 15% but different particle sizes including 2,13,and 43 μm and one having a content of 25%and a particle size of 13 μm.The Al particles in the former three formulations are in spherical shape.

In order to more intuitively analyze the effects of the added Al powder on driving flyer plates along the detonation wave direction,lithium fluoride (LiF) that can maintain chemical inertia in explosive products is added as a comparison so that two CL-20-based formulations with 15% and 25% LiF contents are designed.The component proportions of the six formulations are shown in Table 1.The charge densities are uniformly set to be 1.888 g/cm3,a reasonable value for existing charging process, with the ratios of the actual to the theoretical charge density being in the range of 94%～100%.The finished grains for the six formulations are shown in Fig.3.

The designed flyer plate experimental system driven by CL-20-based Al/LiF-contained explosives is shown in Fig.4, and the assembling schematic diagram is shown in Fig.5.During the experiment, the explosive plane wave lens is first detonated by a detonator to produce a planar detonation wave, which willdetonate the booster pellet and generate a stronger detonation wave so as to ignite the CL-20-based Al/LiF explosive samples to be analyzed.Moreover,the detonation products of the explosive plane wave lens initiate the probe to emit signals due to ionization,thereby starting the displacement interferometer system for any reflector (DISAR) at the measuring point.As the detonation products push the flyer plate forward, the DISAR records the axial velocity of the center of the flyer plate.The specific materials and dimensions of the experimental devices are shown in Table 2.

Table 1 The component proportions of the measured CL-20-based explosives.

Fig.3.Finished grains of the six CL-20-based formulations.

Fig.4.Flyer plate experiment device driven by the CL-20-based explosives.

Detonator; ②Explosive plane wave lens; ③Trigger probe;④Pellet jacket; ⑤Booster trigger; ⑥Tested explosive; ⑦Metal flyer plate;⑧Confinement sleeve; ⑨DISAR.

2.2.Experimental result analysis

With the above experimental methods,the effects of Al powder particle size and content on CL-20-based explosive accelerating flyer plates in the detonation wave propagation direction are identified,as shown in Fig.6.The flyer plate started to jump at 12 μs following the detonation and continuously accelerated.The flyer plates driven by CL-20-based aluminized explosives had lower takeoff and early-stage velocities than those driven by CL-20-based LiF explosives.Therefore, the driving performance of aluminized explosive formulations was poor in the early stage of energy release.The LiF kept inert as CL-20 released energy, which suggested that little or even no Al powder in the aluminized explosive reacted in the detonation stage.During the detonation, Al powder became gasified with the constant absorption of heat, which,therefore,further reduced the energy release of CL-20 in the early stage.This is especially true for Al powder particles with a small size, which have a large specific surface area and a fast heat absorption rate as a result.

With sufficient heat absorption and gasification, the Al powder in CL-20-based aluminized explosives underwent oxidation reactions, accompanied by energy release, with the detonation products of CL-20,which gave a higher driving velocity than CL-20-based LiF explosives.The driving velocities of No.1, 2, 3, and 5 formulations exceeded those of LiF explosives after 3.84,3.88,3.13 and 5.85 μs from the plate jump,respectively.The flyer plate,driven by detonation products,continuously accelerated,and the velocity at 22 μs was defined as the maximum, with the time for accelerating to 99% vpmaxas tp.As can be seen from Fig.6, when the Al content is 15%, the explosive containing 2 μm Al powder particles accelerates fastest in the early stage (the flyer plate velocity from 0 to 99% vpmax).The explosives with 2 and 43 μm Al powder delivered similar acceleration effects in the early stage.Despite the poorest acceleration effect in the early stage, the explosive containing 25% Al, if seen from another perspective, is the formulation that can accelerate the flyer plate for the longest time.

Fig.5.Schematic diagram of the flyer plate experiment device for the CL-20-based explosives: ①

Table 2 The materials and dimensions of the experimental devices.

Fig.6.Comparative analysis of acceleration effects of Al/LiF formulations:(a)Comparison between No.1(15%/2 μm/Al)and 4(15%/LiF)formulations;(b)Comparison between No.2(15%/13 μm/Al)and 4 formulations;(c)Comparison between No.3(15%/43 μm/Al)and 4 formulations;(d)Comparison between No.5(25%/13 μm/Al)and 6(25%/LiF)formulations.

As illustrated in Fig.6,under the same Al content of the CL-20-based explosives, the acceleration plate effects of No.1-3 formulations are mainly affected by the initial takeoff velocity of the flyer plate in the period of 12～13 μs.Specifically, the No.3 formulation containing 43 μm Al powder particles gives a takeoff velocity significantly higher than the No.1 and 2 formulations, which deliver roughly the same plate velocity in the period.In the period of 13～14 μs, all the formulations achieved almost the same plate velocity.From the acceleration state after 14 μs, the CL-20-based formulation with 43 μm Al powder particles is found to prominently outperform those with 2 and 13 μm Al powder particles in accelerating flyer plates along the detonation wave direction,while the 2 μm Al powder formulation is slightly better than the 13 μm one as a whole.In the case of the same powder particle size, the formulations containing 25%Al far under-perform those containing 15% Al.

2.3.Al reaction rate measurement based on the flyer plate experiment

The energy released by an aluminized explosive to drive a flyer plate include not only the detonation energy from the matrix explosive but also the energy from the Al powder involved burning reaction.However, for LiF explosives, the driving energy is completely from the detonation of the matrix explosive since LiF keeps inert and does not participate in the burning reaction.From the motion law of flyer plates, the variations of the Al reaction degree in aluminized explosives with time in the process of detonation products doing work on the flyer plate are identified, as shown in Eq.(1)[37].The principle behind the equation is that the useful work done by the energy released from the Al reaction can be obtained by the flyer plate's kinetic energy from the aluminized explosive minus that from the LiF explosive; then based on the efficiency of the energy released by the explosive in accelerating the flyer plate along the detonation wave propagation direction,the degree of reaction of the Al powder when the detonation products do work on the plate can be determined.

where m is the mass of the metal flyer plate,η the efficiency of the explosive accelerating the flyer plate, QAlthe heat from the Al reaction, which is set to be 20.126 kJ/g, α the mass fraction of Al powder in aluminized explosives, m1the mass of the aluminized explosive under test,vAl(t)the velocity m of the flyer plate at time t driven by an aluminized explosive,and vLiF(t)the velocity m of the flyer plate at time t driven by a LiF explosive.Using the flyer plate experimental results and substituting vAl(t)and vLiF(t)in Eq.(1),we can get the degree of Al reaction λ(t).By repeating the above process, the variations of the degree of Al reaction with time can be obtained.

The driving efficiency of Al/LiF-contained explosives η is dependent on matrix explosive content and experimental settings.Hence, under the same experimental conditions, aluminized explosives deliver the same acceleration effect on flyer plates as LiFcontained ones of the same component proportions (That is, the LiF content therein is equal to the Al content in the aluminized explosive, other components being exactly the same).By calculating the driving efficiency of LiF explosives, we can get the efficiency of the corresponding aluminized explosive η.The specific calculation method is shown in Eq.(2).Based on the experimental data,the efficiencies of CL-20-based explosives containing 15%and 25% Al are calculated to be 0.2163 and 0.1987, respectively.

where,Ekis the kinetic energy of the flyer plate driven by the tested LiF explosive when the maximum velocity is reached, and Qexpis the total internal energy contained in the LiF explosive.

Fig.7.Variations of the Al reaction degree with time during driving the flyer plates with the different aluminized explosive formulations.

With the theoretical calculation method of the Al reaction degree based on flyer plate experimental results,the variations of the Al reaction degree in the No.1, 2, 3, and 5 aluminized explosive formulations with time are obtained,as shown in Fig.7.In the flyer plate experiments,the aluminized explosives were detonated at 12 μs on the inner surface of the plate,which made the plate to jump.The Al powder in No.1,2,3,and 5 formulations began to react with the detonation products of the matrix explosive at 15.84, 15.88,15.13 and 17.85 μs,respectively.As can be seen,a large particle size of Al powder facilitates the initiation of the Al reaction in the detonation aftereffect stage.Compared to CL-20-based explosives containing 15%Al,those containing 25%Al take a long preparation time to occur the Al-oxygen reaction following the detonation.Within the effectively recorded time of the flyer plate experiment,the No.3 formulation containing 15% Al with a particle size of 43 μm always maintains the highest degree of Al reaction.The No.1 formulation containing 2 μm Al powder particles showed a larger degree of Al reaction before 20.30 μs following the detonation than the No.2, in which the particle size of the Al powder was 13 μm.However, as the 13 μm Al powder particles sped up their reaction rate,the Al reaction degree of the No.2 gradually exceeded that of the No.1.The CL-20-based explosives with 25% Al powder underperformed those with 15% Al powder all the way in the degree and rate of Al reaction.According to the Al-consuming standard of the explosives containing 15%Al powder,the degree of Al reaction was calculated to be 7.22%at 24.91 μs,a value still relatively small.

As can be seen from Fig.7, under the condition of the same Al content,a large Al particle size means a small specific surface area,which leads to a small amount of heat absorbed during the detonation.Consequently, this facilitates the heat acting on the places,where there are a relatively small quantity of Al particles, hence initiating an Al powder reaction in a short period of time.Once the Al reaction occurs, although Al powder with a small particle size have a large specific surface area that contributes to improve the reaction rate, the after-burning oxidation reaction of single Al particle has poor persistence, resulting in the unsatisfactory potential for further improvement in the rate of subsequent reaction.

From the acceleration process of flyer plates and the Al reaction degree in the aforementioned CL-20-based explosives formulations with different Al powder characteristics, several findings are obtained.Specifically, compared to the formulations with Al powder 13 μm, the formulation with Al powder 2 μm performs more remarkably in the early stage,but the performance turns to be weak in the latter stage in accelerating metal flyer plates along the denotation wave propagation direction.The formulations with Al powder 43 μm drives the flyer plate most strongly.Under the same Al particle size, a higher Al content absorbs more heat during the detonation and the action area of the heat following the detonation is more scattered.This retards the after-burning reaction of Al powder,resulting in a lowered rate of reaction.The high density of Al powder facilitates the dispersion and distribution of energy release, contributing to the flyer plate accelerating in a faster and longer-lasting manner along the axial direction.

2.4.Characteristic parameters of the CL-20-based aluminized explosives

Based on the measurement and calculation methods of the detonation parameters of aluminized explosives [38-40], The detonation velocities and pressures of the No.1, 2, 3, and 5 CL-20-based aluminized explosive formulations are obtained using the experimental system shown in Fig.8.During the experiment, the explosive lens and booster produce a strong planar shock wave to detonate the tested explosives.The tested explosives consist of an upper and a lower 50 mm×φ50 mm grain.On both the upper-end face of the upper grain and the lower-end face of the lower grain is an ionization probe so that the detonation velocity D of the aluminized explosive formulations can be calculated according to the time interval between the signals of the two probes.Meanwhile, a copper-manganese piezoresistive pressure sensor is embedded between the two grains.When the plane lens is detonated,the probe fixed on the bottom surface of the lens initiates the pulsed constant current source to power the pressure sensor so that the signal of local pressure can be collected and then recorded with an oscilloscope, as shown in Fig.9.From Eq.(3), the collected detonation pressure signals are converted to pressure values so that the detonation pressure PHcan be determined.

Fig.8.Detonation pressure test experiment.

Fig.9.Original detonation pressure signal.

where U0is the baseline voltage value of the constant current source recorded by the oscilloscope and ΔU is the voltage change value at both ends of the sensor.

The detonation velocities and pressures of the No.1,2,3,and 5 aluminized explosive formulations have been measured experimentally.The initial temperature after detonation is calculated using EXPLO5.Since the Al powder in the tested formulations has not yet reacted with the detonation products at the time, only its physical processes such as phase changes in the presence of detonation products is considered in the calculation,to the neglect the effect of Al particle size on the processes.The physical parameters of the formulations are shown in Table 3.

3.Quasi-isentropic theoretical model for the aluminized explosive driving the flyer plate

3.1.Quasi-isentropic theoretical model assumptions

The schematic diagram of an aluminized explosive's detonation products driving a flyer plate is shown in Fig.10.As can be seen,the detonation wave propagates to the right at a velocity D, of which following the Chapman-Jouget (C-J) plane there is a rarefaction wave that spreads in the same direction.The first rarefaction wave following the detonation wave also propagates at a velocity D.Due to the high temperature and high pressure characteristics of the detonation products,the burning rate of Al powder is fast,forming a pressure gradient in the products and thus enhancing rarefaction disturbance.Upon arrival at the right end face of the aluminized explosive, the first rarefaction wave drives the flyer plate to move axially.During the motion process, while the rarefaction wave drives the flyer plate,reflection occurs in the inner wall of the plate.The reflected left-traveling rarefaction wave passes into the detonation products.Here, a quasi-isentropic theoretical model is constructed, in which the x-axis is vertical to the right-traveling rarefaction wave front.

3.1.1.Quasi-isentropic assumption: aluminized explosive reaction process

In the flyer plate experiment,the left end face of the explosive is the detonation surface, which is set as the original point of the x axis.The positive direction of the x-axis is the lengthwise direction of the explosive,as shown in Fig.11.The whole experimental process can be divided into two stages according to the occurrence of the aftereffect Al reaction.The before-occurrence stage covers the detonation process(i.e.,the 0 ～L/D stage shown in Fig.11,in which L is the length of the aluminized explosive, respectively) and the post-detonation preparation process of Al reaction.At t =t1,the Al powder enters the aftereffect reaction stage, which can be further divided into several smaller time intervals.Moreover,

(1) In ti～ti+1, the Al powder that reacts in this interval completely reacts with the detonation products in a single instant at ti.

(2) In ti～ti+1, upon the completion of the Al reaction at ti, the products keep the isentropic flow in this time interval.

(3) In ti～ti+1, Al powder completes the reaction instantly at ti,the physical property parameters of the reaction products are changed from up,1ti,cp,1ti,Pp,1ti,ρp,1ti,Tp,1ti,dp,1tito up,2ti,cp,2ti,Pp,2ti,ρp,2ti,Tp,2ti,dp,2ti, respectively.During the reaction time of Al powder, the flyer plate does not move.After the completion of the reaction, the products undergo isentropicflow, with which the flyer plate moves.At ti+1following isentropic expansion, the physical property parameters of the reaction products turn into up,mti,cp,mti,Pp,mti,ρp,mti,Tp,mti,dp,mti.

Table 3 Characteristic parameters of the CL-20-based aluminized explosive formulations.

Fig.10.Schematic diagram of the aluminized explosive driving the flyer plate.

(4) The effects of the experimental system and heat exchange with surroundings on entropy are not considered.After the reaction of Al powder,the cumulative effect of the tiny time intervals causes the entropy of the system to change significantly.

(5) Al powder is evenly distributed in the aluminized explosives.During the detonation aftereffect reaction stage, detonation product gases absorb heat energy on average;

(6) The experimental process shall be divided into as many time intervals as possible so that the intervals are sufficiently tiny to ensure the rationality of the above assumptions.

Note: The subscript p in the parametric symbol represents the physical parameter in the quasi-isentropic theoretical model for an aluminized explosive driving a flyer plate.A symbol with different subscripts mti-1and 1tirepresent the same physical parameter,i.e.,up,mti-1,cp,mti-1,Pp,mti-1,ρp,mti-1,Tp,mti-1,dp,mti-1are equal to up,1ti,cp,1ti,Pp,1ti,ρp,1ti,Tp,1ti,dp,1ti, respectively.The use of two different subscripts to represent the same physical parameter contributes to the description of the physical and chemical processes within the time intervals, as well as the derivation and expansion of models and formulas

3.1.2.Quasi-isentropic assumption: physical parameter

proportional relationship between adjacent tiny time intervals

The physical parameter proportional relationship of detonation products between adjacent tiny time intervals is established.The Al powder in aluminized explosives does not react during the detonation process, thereby playing a role in diluting the explosive.As such,the EoS of the explosive in the detonation process is similar to that of an ideal gas:

Al powder and detonation products undergo an oxidation reaction.By referring to literature, the Noble-Abel EOS is used to describe the contribution of Al reaction to product pressure [41]:

The pressure of aluminized explosive detonation products is partially from the detonation reaction of ideal explosive components and partially from the oxidation reaction of Al powder:

Fig.11.Quasi-isentropic assumption for the aluminized explosive driving the flyer plate.

where P,ρ and T are the pressure, density, and temperature of the detonation products; a and b are the initial mas fractions of Al powder and ideal explosive; R is the gas constant, R = R0/ M, in which R0is the universal gas constant,and M is the molar mass of the detonation products;λ and w are the degree of Al reaction and the mole number of the detonation products in a unit volume;and Λ is an empirical constant.

For the sake of subsequent derivation,it is necessary to write Eq.(6) in the form of the isentropic Eq.(7), in which A(s) depends on the initial entropy constant, γ=cp/cv(cpis the constant-pressure specific heat and cvis the constant-volume specific heat) is the isentropic exponent.

where λi+1and λ1are the degrees of Al reaction in t1～ti+1and 0 ～t1,respectively,and λ1=0;B(λi+1)is the coefficient related to the degree of Al reaction in t1～ti+1,which can also be expressed as Bi+1; Ap(λi+1) is the coefficient related to the entropy and the degree of Al reaction in t1～ti+1, which can also be expressed as Ap,i+1,describing the effective factors of the effect of Al reaction on physical parameters in t1～ti+1.

Based on the reaction process described in Section 3.1.1, the Al reaction at tiin ti～ti+1causes the changes of Aiand γ,which affect the sound velocity in detonation products at ti.Hence, assume cp,2ti/cp,1ti= (Ap,i+1/Ap,i)αp, in which αpis the correlation coefficient of Al reaction causing γ changes - that is, the extracted influencing factor of an Al reaction on aluminized explosives’isentropic exponent.

Since λiis a monotonically increasing function over time and the pressure of the detonation products decreases with their expansion,the temperature of the detonation products,influenced by the heat released by Al reaction, rises with the progressing degree of the reaction.Meanwhile, γ>1, so Bi

According to the isentropic formulas of two adjacent time intervals ti-1～tiand ti～ti+1;

Knowing that Al powder has completely reacted at tiin ti～ti+1and in the rest of the time of the interval, the disturbance is constantly spread toward the detonation product at Dp,i=up,i+cp,ialong the direction of fluid flow,we have up,i=Dp,i/(γ+1)from the equations of the Hugoniot curve and Rayleigh line.As such,cp,i=γup,i= γDp,i/(γ + 1), so the proportional relationship between up,2tiand up,1tiat tiat the junction of these two adjacent time intervals can be obtained:

3.1.3.Quasi-isentropic assumption: model of aftereffect reaction between Al powder and detonation products

According to the molecular dynamics theory and the gasification theory for condensed matters,the reaction process of Al powder in an explosive detonation environment characterized by high temperature, high pressure,and strong fluidity is analyzed so that the thermodynamical function of the Al powder gasification rate is obtained [18,19,26].From the thermodynamical function, the degree of reaction λiof Al powder in t1～tiis defined as

The calculation object of the quasi-isentropic theoretical model is Formulations 1,2,3 and 5 designed to drive copper flyer plates.In Eq.(14), k is the Bolzmann constant, mathe molecular mass of Al with a value of 4.34 × 10-23g, Evthe enthalpy of Al particle gasification with a value of 300 kJ/mol,R the universal gas constant,ρathe density of Al with a value of 2.7 g/cm3,rathe molecular radius of Al with a value of 1.82×10-10m,and D0the initial diameter of Al particles, whose value is set to be 2 μm, 13 μm, and 43 μm,respectively.Regarding the value of ambient temperature Tafor the Al reaction, the following analysis is made.

When the degree of Al powder reaction in the detonation products is calculated using the Al particle combustion model in Eq.(14), the initial value Ta0of the ambient temperature Tafor the Al reaction is commonly selected as the detonation temperature of the matrix explosive TH[18,19,26].Subsequent temperature changes can be calculated by the Kast average heat capacity [42,43] equation.While calculating the ambient temperature of Al particles,this method only considers the detonation heat from the matrix explosive and the heat released from Al powder combustion,to the neglect of the effect of the surroundings on the detonation products.From the flyer plate experimental results in Section 2,it can be seen that the Al powder in the CL-20-based aluminized explosive occurs to react after 3-4 μs following the detonation of the matrix explosive.Specifically, the Al powder absorbs heat from the C-J plane to form a hot spot in the detonation products.After the Al powder starts the aftereffect reaction, the detonation gas temperature on the surface of Al particles is no longer the detonation temperature.Hence, in this case, it is inaccurate to select the detonation temperature THas the initial ambient temperate Ta0for the aftereffect reaction of the Al powder.As such,when calculations are conducted based on the quasi-isentropic theoretical model,the initial ambient temperature Ta0has to be fit for calibration so that its value conforms to the change laws of flyer plate motion and Al powder reaction.Meanwhile, considering the effect of the rarefaction wave,it is unreasonable to calculate the ambient temperature changes of Al particles only by the Kast average heat capacity equation.While considering the conditions of the detonation products accelerating a metal target plate, the quasi-isentropic theoretical model allows deriving the physical temperature change law of the detonation products.It is assumed that Al powder is evenly distributed in the aluminized explosive, and in the detonation aftereffect reaction stage, the detonation product gases absorb energy on average when being heated.The molar specific heat capacity of the detonation products is also assumed to be only dependent on temperature difference in a high temperature environment [42,43], so the change in the ambient temperature on the surface of Al particles is equal to the overall physical temperature change of the detonation products, i.e., ΔTa=(Tp,mti-1- Tp,mt1).The ambient temperature on the surface of Al particles can be more accurately given by Eq.(15).

3.2.Characteristic line analysis of zones I and II

Based on the assumptions in the quasi-isentropic theoretical model,the characteristic line analysis for the flyer plate experiment is shown in Fig.12.The left end face of the aluminized explosive is the detonation surface, which is set as the original point of the x axis.After the detonation surface of the aluminized explosive is ignited,the Al powder does not react at t

Fig.12.Analysis of the characteristic lines of the aluminized explosive used for driving the flyer plate.

According to whether or not the Al powder begins to react and the flow characteristics of rarefaction waves,the physical parameters in the driving are analyzed, as shown in Fig.13.Zones I and II correspond to the stage where Al powder reacts with the detonation products,covering both the detonation process(0 ～L/D stage in Fig.13)and the post-detonation Al reaction preparation process(L/D ～t1stage in Fig.13).In the process of the detonation, the acoustic wave propagates so fast that the disturbed medium has no time to transfer its heat to surroundings.Hence, the propagation process of the acoustic wave can be regarded as an isentropic process, defined as Eq.16(a).

Eq.16(b)is the isentropic equation of ideal fluids,in which A(s)is a constant dependent on the initial entropy, γ is the isentropic exponent with a value of 3.Eqs.16(c) and 16(d) are the continuity and momentum equations of a fluid in the case of isentropic flow,respectively.By solving the equation set, Eq.(17) can be obtained;that is, the state (or disturbance) defined by (u+c) propagates along the direction of the fluid flow(i.e,along the positive direction of the x axis) at velocity dx/dt =u+c.However,the state defined by(u-c)propagates along the reverse direction of the fluid flow at velocity dx/dt = u- c.

where c is the sound velocity in the detonation products, u the particle velocity of the detonation products, t the time, and x the position to be analyzed on the x-axis.

In 0 ～t1,before the detonation wave arrives at the right end of the aluminized explosive following the detonation at the left end,i.e.,under the condition t ≤L/D,there is a right-traveling centered rarefaction wave.At t = L/D, the distance traveled by the detonation wave is L, and upon arrival at the flyer plate, a left-traveling rarefaction wave occurs due to the reflection and propagates into the detonation products.The characteristic equation of the lefttraveling rarefaction wave is

Fig.13.Analysis of the physical parameters of the aluminized explosive driving the flyer plate.

The left-traveling wave front propagates at velocity(u0-c0)=cons=-D/2 in the area disturbed by the right-traveling rarefaction wave,so the time it arrives at the x section is

where ρp,0,IIand Pp,0,IIare the density and pressure of the detonation products in Zone II in 0 ～t1, respectively.

In Zone II, from Eq.(6), we have TH/Tp,0,II= (PH/Pp,0,II)•(B1/B1)•(ρp,0,II/ρH).Hence,

3.3.Flyer plate velocity before t1

While the detonation products of an aluminized explosive undergo axial one-dimensional flow, the detonation wave and the subsequent right-traveling rarefaction wave in the products chase after and drive the flyer plate to move.Upon arrival at the internal surface of the plate, the right-traveling rarefaction wave will produce a left-traveling rarefaction wave due to reflection, which propagates into the detonation products.Since the reflected lefttraveling rarefaction wave is always behind the flyer plate, it poses no effect on the motion of the latter[44].Hence,the equation of motion of the right-traveling rarefaction wave is utilized to solve the velocity of the flyer plate.First of all, when the products drive the flyer plate before t1, from Newton's second law of motion, we have

where S0is the cross-sectional area of the flyer plate, Pf,0is the pressure of the detonation products on the inner wall of the flyer plate in 0 ～t1, M is the mass of the flyer plate, and vf,0is the velocity of the flyer plate in 0 ～t1.

From the sound velocity Eq.16(a) and the isentropic Eq.16(b),we have Pf,0/PH=(cf,0/cH)3.Using the CJ parameters cHand PHon the detonation wave front, Eq.(28) can be written as

where m is the mass of the aluminized explosive,m =S0ρeL;cf,0is the sound velocity in the detonation products at the inner wall surface of the flyer plate in 0 ～t1.

In 0 ～t1, the right-traveling rarefaction wave propagates at velocity up,0+cp,0and the velocities of the waves along the characteristic lines remain unchanged at up,0+cp,0.When it catches up with the flyer plate, reflection will occur, lowering the particle velocity of the detonation products instantly from up,0to the velocity at the inner wall surface of the flyer plate uf,0, the sound velocity from cp,0to the velocity at the inner wall surface of the flyer plate cf,0.Hence,

Since the detonation products are closely against the inner wall of the flyer plate,the particle velocity of the detonation products at the inner wall uf,0is equal to the velocity of the flyer plate vf,0,i.e.,dx/dt = vf,0= uf,0.Substituting this relational expression in Eq.(31), we have

By substituting Eq.(36) in Eq.(29) and then integrating it, the relationship between flyer plate velocity and time in L/D ～t1can be obtained:

3.4.Flyer plate velocity after t1

3.4.1.Flyer plate velocity in t1～t2

In t1～t2, when the detonation products drive the flyer plate,from Newton's second law of motion, we have

where Pf,1is the pressure of the detonation products on the inner wall of the flyer plate in t1～t2and vf,1is the velocity of the flyer plate in t1～t2.

The right-traveling rarefaction wave propagates at up,1+cp,1in t1～t2,catches up with the flyer plate and reflects off it at t1.At the moment, the particle velocity of the detonation products immediately decreases from up,1to uf,1, the particle velocity of the products at the inner wall surface of the flyer plate, and the sound velocity cp,1also immediately drops to cf,1,the sound velocity in the products at the inner wall surface.Hence,

Substituting cf,1in Eq.(43)and carrying out the integration,we can get the relationship between the velocity of the inner wall surface of the flyer plate and time during the isentropic flow of the detonation products in t1～t2:

where vf,0(t1)is the calculated vf,0by substituting t =t1in Eq.(37).

From Eq.(50),the variations of the flyer plate velocity with time in t1～t2can be acquired.Hence,integrating vf,1with respect to t,we have the distance traveled by the flyer plate in t1～t2.Therefore,the distance Zp,2traveled by the flyer plate in L/D ～t2can be obtained:

3.4.2.Flyer plate velocity in ti～ ti+1(i ≥1)

In a similar way, we can derive the sound velocity in the detonation products at the inner wall surface of the flyer plate during isentropic flow in ti～ti+1,cf,iand the velocity of the flyer plate vf,i(the particle velocity of the detonation products at the inner wall of the flyer plate), as shown in Eqs.(52) and (53), respectively.

From Eq.(53),the variations of the flyer plate velocity with time in ti～ti+1can be identified.By integrating vf,iwith respect to t to work out the distance traveled by the flyer plate in ti～ti+1, we have the distance traveled by the flyer plate in L/D ～ti+1, Zp,i+1:

3.5.Analysis of the physical parameters in zones III and IV

Fig.14.Analysis of the detonation product parameters as the aluminized explosive driving the flyer plate.

Zp,1is defined as Eq.(38).By substituting cp,1t1,II,up,1t1,IIand Zp,1in Eq.(57),we can get Pp,1(x)and Qp,1(x).In t1～t2in Zone III,using the obtained Pp,1(x) and Qp,1(x), the sound velocity in and particle velocity of the detonation products after the completeness of the Al reaction can be solved,as shown in Eq.(59).As t =t1,cp,1,IIIis the sound velocity in the detonation products cp,2t1,IIIat t1,while up,1,IIIis the particle velocity of the detonation products up,2t1,IIIat t1after the Al reaction in t1～t2.As t = t2, cp,1,IIIand up,1,IIIare the sound velocity in and particle velocity of the detonation products undergoing isentropic flow at t2,cp,mt1,IIIand up,mt1,III,after the Al reaction has completed in t1～t2.

In the case that i ≥2, in ti～ti+1of Zone III, using the derived Zp,i,Qp,i(x) and Pp,i(x), we can determine the particle velocity of and sound velocity in the detonation products after the completeness of the Al reaction, as shown in Eq.(66).As t = ti, cp,i,IIIand up,i,IIIare the sound velocity in and particle velocity of the detonation products at ti,cp,2ti,IIIand up,2ti,III,respectively,after the Al reaction in ti～ti+1.As t =ti+1,cp,i,IIIand up,i,IIIare the sound velocity in and particle velocity of the detonation products undergoing isentropic flow at ti+1, cp,mti,IIIand up,mti,III, respectively, after the Al reaction has completed in ti～ti+1.The density ρp,mti,III,pressure Pp,mti,III,and temperature Tp,mti,IIIof the detonation products undergoing isentropic flow at ti+1after the Al reaction has completed in ti～ti+1can be obtained.

According to the characteristic line analysis of Zone I in Section 3.2, when t1～t2in Eq.(22), cp,0,Iand up,0,Iare the sound velocity cp,1t1,Iin and particle velocity up,1t1,Iof the detonation products before the Al reaction at t1in 0 ～t1of Zone I respectively:

In 0 ～t1of Zone I, from Eq.(6), we have TH/Tp,1t1,I=(PH/Pp,1t1,I)•(B1/B1)•(ρp,1t1,I/ρH).Hence, before the Al reaction at t1, the temperature of the detonation products Tp,1t1,Iis

In the case that i ≥2, using the derived Fp,i(x) and Up,i(x), we can determine the particle velocity and sound velocity in the detonation products in ti～ti+1of Zone IV after the completeness of the Al reaction,as shown in Eq.(78).As t =ti,cp,i,IVand up,i,IVare equal to the sound velocity in and particle velocity of the detonation products at ti,cp,2ti,IVand up,2ti,IV,after the Al reaction in ti～ti+1.As t = ti+1, cp,i,IVand up,i,IVare equal to the sound velocity in and particle velocity of the detonation products undergoing isentropic flow at ti+1,cp,mti,IVand up,mti,IV,after the Al reaction has completed in ti～ti+1.The density ρp,mti,IV, pressure Pp,mti,IV, and temperature Tp,mti,IVof the detonation products that have undergone isentropic flow at ti+1after the Al reaction has completed in ti～ti+1can be obtained.

3.6.Comparisons between the calculated results of the quasiisentropic theoretical model and experimental results

3.6.1.Parameter fitting and result verification of the quasiisentropic theoretical model

The results of the fly plate experiments in Section 2 are calculated using the quasi-isentropic theoretical model built.The parameters of the charge and the dimensions of the flyer plate are shown in Table 2, the physical parameters of the CL-20-based aluminized explosive formulations are shown in Table 3.The moment when the velocity of the flyer plate driven by an aluminized explosive formulation exceeds that driven by the corresponding LiF explosive formulation is the time t1when the Al powder begins to react.For Formulations 1,2,3 and 5,the values of t1are 3.84,3.88,3.13 and 5.85 μs,respectively.The particle sizes of Al powder have a small effect on t1.The t1values of the CL-20-based explosives containing 15% Al are in the range of 3～4 μs, and if the particle sizes of the Al powder are between 2 and 13 μm,the value of t1is around 3.80 μs.For the CL-20-based explosives containing 25% Al, their t1values are approximately 6 μs.The two modules of the quasi-isentropic theoretical model, namely Al not reacting and Al reacting, are utilized to determine flyer plate acceleration laws and detonation product parameters in the two time intervals before and after t1.

Fig.15.Verified calculation for Formulation 1 based on the quasi-isentropic theoretical model: (a) Comparison between the calculated flyer plate velocity with the quasiisentropic model and the experimental results; (b) Comparison between the calculated flyer plate displacement with the quasi-isentropic model and the experimental results; (c) Comparison between the calculated Al reaction degree with the quasiisentropic model and the experimental results.

Fig.16.Verified calculation for Formulation 2 based on the quasi-isentropic theoretical model: (a) Comparison between the calculated flyer plate velocity with the quasiisentropic model and the experimental results; (b) Comparison between the calculated flyer plate displacement with the quasi-isentropic model and the experimental results; (c) Comparison between the calculated Al reaction degree with the quasiisentropic model and the experimental results.

Fig.17.Verified calculation for Formulation 3 based on the quasi-isentropic theoretical model: (a) Comparison between the calculated flyer plate velocity with the quasiisentropic model and the experimental results; (b) Comparison between the calculated flyer plate displacement with the quasi-isentropic model and the experimental results; (c) Comparison between the calculated Al reaction degree with the quasiisentropic model and the experimental results.

Using the quasi-isentropic theoretical model, the theoretical motion law of flyer plates(red solid lines in Fig.15(a)-18(a))and the degree of Al reaction(red dot lines in Fig.15(c)-18(c))are fitted and calculated according to the flyer plate motion laws(black solid lines in Fig.15(a)-18(a)) and the degree of Al reaction (blue square dot lines in Fig.15(c)-18(c)) obtained from the experimental results.Table 4 presents the fitted initial ambient temperature Ta0at which the Al powder begins to react and influencing factor αpof Al reaction on the isentropic exponent.

Fig.18.Verified calculation for Formulation 5 based on the quasi-isentropic theoretical model: (a) Comparison between the calculated flyer plate velocity with the quasiisentropic model and the experimental results; (b) Comparison between the calculated flyer plate displacement with the quasi-isentropic model and the experimental results; (c) Comparison between the calculated Al reaction degree with the quasiisentropic model and the experimental results.

Using the quasi-isentropic theoretical model and taking the data in Tables 2-4 as model parameters, the velocity and displacement of flyer plates are calculated.The calculated and experimental results are found in good agreement, with a maximum error not exceeding 5%.Meanwhile, the calculated Al reaction degrees and variation trends of the four aluminized explosive formulations are closer to the experimental results.Hence,the use of the same set of model parameters allows acquiring both the acceleration law of the flyer plate and the degree of Al reaction, which verifies the good accuracy and self-consistency of the quasi-isentropic theoretical model.While considering the effect of the rarefaction waves on thetemperature of the detonation products as well as the initial ambient temperature of the Al reaction, the theoretical model substitutes the Al particle surface ambient temperature Tathat is more in line with the real state in Eq.(14), thereby boosting the calculation accuracy.

Table 4 Initial ambient temperature Ta0 and influencing factor αp of Al reaction for the different aluminized explosive formulations.

Regarding the flyer plate velocity and Al reaction degree,there is a gap when using the quasi-isentropic theoretical model compared to metal flyer plate tests.The reasons are as follows: First, in this paper,the total model combines the macroscopic driving process of the detonation products and the microscopic reaction process of the Al powder,and the error mainly comes from the association of the two processes.Second, the effect of the reflected wave inside the metal plate is not considered in the quasi-isentropic model.Overall, however, the quasi-isentropic model can describe the driving process of the detonation products and Al reaction degree of the aluminized explosives with an acceptable error (less than 5%), which indicates that this model can effectively be applied to non-isentropic flow problems of the detonation products of the aluminized explosives.

3.6.2.Detonation product parameter analysis based on the proposed quasi-isentropic theoretical model

Considering the entire process of the detonation products of the aluminized explosive driving a flyer plate, the quasi-isentropic theoretical model is employed to analyze the variations of the detonation product parameters with time.Using the model parameters in Tables 2-4, the pressures and temperatures of the detonation products whose initial positions are 30 and 50 mm away from the detonation surface (the position of the detonation surface is 0 mm) are calculated respectively.The detonation products whose initial position is 30 mm lie in the middle of the products, and the detonation products whose initial position is 50 mm lie at the ends of the products and move closely against the inner surface of the flyer plate.These two typical detonation products are analyzed, and the results are shown in Figs.19-22.

As can be seen in Figs.19-22, the product pressure of the aluminized explosives decreases rapidly at the initial acceleration stage of the flyer plate.However, with the start of the Al reaction,the pressure of the products whose initial position is 30 mm attenuates slowly with time, while for the products whose initial position is 50 mm, the pressure tends to be near a specific value.The temperature of the detonation products drops fast in the first half,but with the constant energy release from the Al reaction,the temperature of the product whose initial position is 30 mm no longer drops with time but instead slightly rises, while for the products whose initial position is 50 mm, the temperature finally remains near a specific temperature value.The No.3 formulation exhibits the fastest Al reaction rate,a slow drop rate of pressure and temperature during the dropping stage, and the high rate of temperature rise during the rising stage.Due to the aftereffect reaction between the Al powder and the detonation products, aluminized explosives exhibit prolonged action time of positive pressure and a temperature rise after detonation.Moreover, the aftereffect reaction maintains the temperature of the detonation products at over 1500°C for dozens of microseconds [45].Hence, aluminized explosives are characterized by a high post-detonation temperature and long duration of thermal effect.Although Al powder absorbs the heat on the CJ plane to lower the front energy used to support the propagation of the detonation wave and thus reduces the detonation pressure and temperature to a certain extent,it extends the effect of over-pressure on doing work and prolongs the high temperature stage of the detonation products.As such,the addition of Al powder boosts the aftereffect of explosives.

Fig.19.Variations of the product pressure and temperature with time for Formulation 1 based on the quasi-isentropic theoretical model:(a)The pressure and temperature of the detonation products of the aluminized explosives at 30 mm; (b) The pressure and temperature of the detonation products of the aluminized explosives at 50 mm.

Fig.20.Variations of the product pressure and temperature with time for Formulation 2 based on the quasi-isentropic theoretical model:(a)The pressure and temperature of the detonation products of the aluminized explosives at 30 mm; (b) The pressure and temperature of the detonation products of the aluminized explosives at 50 mm.

Fig.21.Variations of the product pressure and temperature with time for Formulation 3 based on the quasi-isentropic theoretical model:(a)The pressure and temperature of the detonation products of the aluminized explosives at 30 mm; (b) The pressure and temperature of the detonation products of the aluminized explosives at 50 mm.

Fig.22.Variations of the product pressure and temperature with time for Formulation 5 based on the quasi-isentropic theoretical model:(a)The pressure and temperature of the detonation products of the aluminized explosives at 30 mm; (b) The pressure and temperature of the detonation products of the aluminized explosives at 50 mm.

4.Conclusions

In this paper, an experiment of explosive axially driving metal flyer plate was designed to analyze the effects of Al powder sizes and contents on the driving performance of CL-20-based aluminized explosive formulations along the detonation wave propagation direction.The reaction laws of Al powder after detonation for different CL-20-based aluminized explosive formulations were obtained, and the energy output structures of the aluminized explosive formulations were given.Based on the characteristic line theories of aluminized explosive driving flyer, considering the reaction characteristics of Al powder during the driving process of detonation products, coupled with the isentropic theories of ideal explosives and the non-isentropic aftereffect reaction of Al powder,the effect of the Al reaction on flyer motion laws during the detonation aftereffect stage was studied, and the functional relationships between the parameters of detonation products and entropy and Al reactivity were explored.A quasi-isentropic theoretical model of aluminized explosive driving flyer was established.Compared to other conventional models, the quasi-isentropic theoretical model of the flyer driven by aluminized explosives correlates the movement laws of metal flyer and the reaction characteristics of Al powder, and constructs the variation laws of the physical parameters such as density,pressure,and temperature of detonation products with time and spatial location.

(1) The accelerating performance of the different CL-20-based aluminized explosive formulations in driving metal flyer plates along the detonation wave propagation direction was investigated.The formulations with 43 μm Al powder particles were found to obviously outperform those with 2 and 13 μm in the acceleration effect, and meanwhile, the formulations with 2 μm Al powder particles were slightly better than those with 13 μm as a whole.It was also shown that the formulations containing 25% Al have worse driving performance than those containing 15%Al in the detonation wave propagation direction.The variations of the Al reaction degree with time were obtained according to the flyer plate experimental results.Specifically,the formulation containing 15% 43 μm Al powder particles maintained the highest degree of Al reaction; the formulation containing 15% 2 μm Al powder particles had a larger degree of Al reaction in the period 0～5.74 μs following the detonation than that with 15%13 μm Al powder particles,but after the period,its degree of Al reaction was gradually surpassed by the latter's; the formulations containing 25% Al showed a lower degree of Al reaction than those containing 15%.

(2) Based on the quasi-isentropic assumptions, characteristic line principles, and isentropy theory, a quasi-isentropic theoretical model for the aluminized explosive driving the flyer plate was constructed.By separating the Al reaction and expansion driving in each calculated time interval,the model formed a “two-stage” quasi-isentropic assumption and combined the non-isentropic exponential relationships between physical parameters before and after the Al reaction with the isentropy theory in the isentropic expansion process.In this way, the action mechanism of an aluminized explosive's detonation products in axially driving flyer plates was revealed, and the variations of the flyer plate velocity and Al reaction degree with time were identified.Moreover,how the detonation products'physical parameters including sound velocity, particle velocity, density, pressure and temperature changed over time and axial positions was investigated during accelerating the flyer plate.

(3) The accuracy and self-consistency of the quasi-isentropic theoretical model built were verified using the flyer plate experimental results of the four CL-20-based aluminized explosive formulations.Meanwhile,with the model,how the pressure and temperature of the detonation products at different initial positions varied with time in the process of driving flyer plate with the different CL-20-based aluminized explosive formulations was calculated.It was found that the energy released from the reaction between the Al powder and the detonation products prolonged the action time of the positive pressure after the detonation, slowed down the dropping of the temperature and even reversed the dropping trend to a slight rise.Compared to the detonation products in the middle of the products, those closely against the inner wall of the flyer plate had fast declining pressure and temperature,then followed by the tendency to a constant value with the progress of the Al reaction.The addition of Al powder to explosives contributes to lengthening the doingwork-by-pressure effect and the thermal effect of the explosives, thereby remarkably improving the detonation aftereffect of the explosives.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This paper is supported by National Natural Science Foundation of China (Grant No.11872120).