Numerical simulation on the pressure wave in a 30 mm electrothermal-chemical gun with the discharge rod plasma generator

Yn-jie Ni , Yong Jin ,*, Gng Wn , Bo-ming Li ,b Ntionl Key Lbortory of Trnsient Physics, Nnjing University of Science nd Technology, Nnjing 210094, Jingsu, Chin

b China Academy of Ordnance,100089, Beijing, China

Keywords:Electrothermal-chemical launch Pressure wave Plasma Solid propellant Electric energy ratio

ABSTRACT An axisymmetric two-dimensional(2D)internal ballistic model including the transient burning rate law is used to simulate the 30 mm electrothermal-chemical (ETC) launch with the discharge rod plasma generator(DRPG). The relationship between the pressure wave and the initial parameters,such as input electric power,discharging timing sequence,loading density and propellant web thickness,is researched through the change of initial parameters in the model. In the condition of synchronous discharging, the maximum of the pressure wave can be controlled while the ratio of the input electric energy to the propellant chemical energy (electric energy ratio) is less than 0.11. If the electric energy ratio is larger than 0.11, the maximum of the pressure wave increases rapidly with the electric energy ratio. With the increasing of the electric energy ratio,the change of the first negative amplitude value can be ignored.In the condition of timing sequence discharging, the allowed input electric energy ratio to control the pressure wave is proportional to the current pulse duration. At the high electric energy ratio, the maximum of the pressure wave is inverse proportional to the current pulse duration.The pressure wave increases with the increasing of the loading density. But the allowed electric energy ratio to control the pressure wave and the variation trend of the first negative amplitude wave value doesn't change.During the discharging of the DRPG, the influence of changing propellant web thickness in ETC launch can be ignored.

1. Introduction

Electrothermal-chemical (ETC) launch is considered to be an effective way to control the pressure and increase the projectile muzzle velocity. Different plasma generators, such as capillary plasma generator (CPG), piccolo plasma generator (PPG) and discharge rod plasma generator (DRPG), are designed for the ETC launchers [1e3]. The enhanced gas generator rate of propellants with the plasma were verified in experiments[4],the burning rate law is improved by adding an enhanced gas generator rate(EGGR)coefficient [5e7].

The pressure wave (the pressure of the breech-face minus the pressure of the base-initial-position) in the conventional launch increases with the loading density. The pressure wave can be decreased by changing the ignition method, the structure of the chamber and the loading structure of the propellant[8e10].In the ETC launch,the ignition time of the propellant can be reduced[11],and the combustion of the propellant can be enhanced [5]. In addition to the charging parameters,the pressure wave in the ETC launch can also be influenced by the electric parameters.

In this paper, an axisymmetric two-dimensional interior ballistics model of the solid propellant ETC gun (2D-IB-SPETCG) is used to simulate the 30 mm ETC launch. The 2D-IB-SPETCG model includes: pulse forming network (PFN) discharge model [12,13],DRPG model[14]and 2D two-phase flow model[15,16].Depending on simulation results,the electric energy ratio and discharge timing sequences are used to analyze the influence of electric parameters(input electric power, discharging timing sequence) and charging parameters (loading density, propellant web thickness) on the pressure wave.

2. Physical model

Interior ballistics involves flow field components of a continuous(gas)and a discrete(solid propellant)nature.We consider that a single solid propellant is surrounded by the flow of the gas,so the solid propellant is considered to be pseudo fluid. On a sufficiently small scale of resolution in both space and time,the components of the flow are represented by the balance equations for a multicomponent mixture describing the conservation of mass, momentum and energy. These equations require a number of constitutive laws for closure, including interphase drag, state equation, intergranular stresses, interphase transfer of energy and so on. All parameters are the function of time, axial and radial distance.

2.1. Multiphase governing equations

where p is the pressure, Dzis the axial interphase drag, uigis the axial velocity of the ignition powder,uplis the axial velocity of the plasma,Dris the radial interphase drag, vigis the radial velocity of the ignition powder,vplis the radial velocity of the plasma,and Rsis the intergranular stress.

The energy equation for the gas takes the form,

The balance equations for mass (gas or solid propellant) are given by,

where rgis the density of the gas,f is the porosity, ugis the axial velocity of the gas,vgis the radial velocity of the gas,mpis the gas generation rate of the solid propellant per volume, migis the gas generation rate of the ignition powder per volume,mplis the mass rate of the plasma per volume, rpis the density of the solid propellant,upis the axial velocity of the solid propellant,and vpis the radial velocity of the solid propellant.

The balances of momentum for the gas and solid propellant take the form,where e is the energy of the gas,e?egt(t)/2,egis the internal energy of the gas,Apis the surface area of the solid propellant per volume, qcvis the convention heat transfer(qcv?hcv(Tg-Tp), hcvis the convention heat transfer coefficient, Tgis the gas temperature,Tpis the surface temperature of the propellant),qradis the radiation heat transfer (qrad?εps(-), εpis the radiation absorption coefficient,s is the Stefan-Boltzmann constant),epis the enthalpy of the solid propellant,Eigis the enthalpy of the ignition powder,Eplis the enthalpy of the plasma.

The gas generation rate of the solid propellant per volume,mp,is given in the following form,

where j is the relative burned volume of the solid propellant.

2.2. Model of discharge rode plasma generator

The structure diagram of the DRPG is shown in Fig.1.The DRPG is consisted of a periodic array of solid (copper) and plasma(polyethylene)conductors.The DRPG can be considered as a couple of series plasma generators.The DRPG inserted into the chamber is surrounded by the solid propellant. While the radial exits for the plasma are free for the flow of the gas and propellant in the chamber.The exit area decreases with the increasing of the porosity of the propellant. The plasma in the DRPG is deeply influenced by the environment in the chamber. It is considered that “flowback”high pressure gas may lead to the current interruption of the DRPG.

Fig.1. The structure diagram of the DRPG.

In order to simulate the parameters in the DRPG, the working process can be simplified into two stages. At the beginning of the ignition,the pressure of the plasma is much higher than that in the chamber. Then the plasma diffuses in the chamber and the solid propellant is ignited. The influence of the environment in the chamber can be neglected. It is regarded as the “flowout” stage which is quite similar to the CPG working process. With the combustion of the solid propellant, the gas pressure increases and the diffusion of the plasma is hindered.When the gas pressure is high enough,the gas in the chamber begins to flow back into the DRPG.It is regarded as the “flowback” stage. It makes the equivalent resistance of the DRPG increase rapidly. Finally, the current interruption occurs.

The DRPG is considered to be connected in series by small plasma generators with a couple of solid and plasma conductor.Depending on the model of CPG,the zero-dimensional(0D)model of the small plasma generator is built. The conservations of mass,momentum and energy during the “flowout” stage are given by

where rplis the average density of the plasma, Seis the exit area,Vcapis the volume of the small plasma generator, rais the mass ablation density rate,pplis the average pressure of the plasma,peis the gas pressure at the exit,eplis the specific internal energy of the plasma, J is the current density,splis the electrical conductivity of the plasma.

During the “flowback” stage, the influence of the solid propellant is neglected. Only the gas in the chamber flows back into the DRPG and the“flowback”gas mixes with the plasma which makes the equivalent resistance increase. The conservations of mass,momentum and energy are now changed to

where vpl<0, during the “flowback” stage.

3. Simulation of the conventional and ETC launch

Fig. 2 is the structure diagram of the 30 mm ETC gun with the DRPG and PFN.The PFN contains four modules which can be used independently as a system circuit.Each module contains a 1220 mF capacitor, a 40 mH inductance, a high power switch, a crowbar circuit and a surge protection resistor.The chamber volume is 356 cm3and the length of the barrel is 2.75 m. The standard projectile of 72 g was fired with 4/7 high-nitrogen propellant in the experiment.The 4/7 high-nitrogen propellant is a homogeneous single-base propellant mainly containing nitrocellulose.

The 30 mm ETC launch with 220 g 4/7 high-nitrogen propellant is simulated.The four modules are charged at 5 kV in the condition of synchronous discharging. The simulated pressure wave curve is shown in Fig.3.There are three characteristic values in the pressure wave: the first positive amplitude value, the first negative amplitude value and the second positive amplitude value.In Fig.3,point 1 represents the first positive amplitude value of the pressure wave,point 2 represents the first negative amplitude value of the pressure wave and point 3 represents the second positive amplitude value of the pressure wave.

In the ETC launch, the internal ballistic process can be influenced by both the input electric energy of the DRPG and the propellant chemical energy.The electric energy ratio,he,is the ratio of the input electric energy to the propellant chemical energy.

Fig. 2. Structure diagram of the 30 mm ETC gun with DRPG.

Fig. 3. The simulated pressure wave curve of the ETC launch.

where Eplis the input electric energy of the DRPG,U(t)is the voltage of the DRPG, I(t) is the current of the DRPG, f is the propellant impetus,u is the propellant charge in the chamber,k is the specific heat ratio.

4. Analysis of the simulation results

The initial parameters,such as input electric power,discharging timing sequence,loading density and propellant web thickness,are changed in the 2D-IB-SPETCG. The influence of the electric parameters and charging parameters on the pressure wave are analyzed.

4.1. The influence of the electric parameters

In order to analyze the influence of the electric parameters,the charging parameters are not changed (the loading density is 0.618 g/cm3), only the discharge voltage changes from 3 kV to 10 kV. The discharge timing sequences of the four modules are shown in Table 1. Timing #1 is the synchronous discharge timing,and timing #2 and #3 are the timing sequence discharge timings.

The simulated current curves of the DRPG at the voltage of 10 kV in the condition of different discharge timing sequences are shown in Fig. 4. The discharge time of the DRPG in the condition of different discharge timing sequences is 0.83 ms,1.08 ms and 1.6 ms.The maximum of the current decreases from 147 kA(timing#1)to 109 kA (timing #3) with the increasing of the discharge time.

Fig. 5 shows the pressure wave curves of different discharging voltage in the condition of synchronous discharge (timing #1).With the increasing of the discharging voltage, the first positive amplitude value increases,while the first negative amplitude value stays.

Table 1 The discharge timing sequences.

Fig. 4. The simulated current curves of the DRPG at the voltage of 10 kV in the condition of different discharge timing sequences.

Fig. 5. The pressure wave curves of different discharging voltage in the condition of synchronous discharge.

The pressure of the breech-face,the pressure of the base-initialposition and the projectile position at the discharge voltage of 5 kV and 10 kV are shown in Fig. 6. Before the moving of the projectile,the projectile is at the position where L?0. At the moment the projectile leaving the barrel,the projectile is at the position where L?2.75 m.At the beginning of the ignition,the plasma exists near the area of the DRPG,and flows from the center axis to the wall.The gas generator rate of the propellant in this area increases with the electric power. When the discharge voltage increases from 5 kV to 10 kV, the maximum of the breech-face pressure increases from 331 MPa to 433 MPa, the maximum of the base-initial-position pressure increases from 306 MPa to 408 MPa. The first positive amplitude value of the pressure wave increases with the discharging voltage. At the discharge voltage of 5 kV, the projectile starts to move at 0.34 ms.When the discharge voltage increases to 10 kV,the projectile starts to move at 0.25 ms.And the time of the projectile moving in the barrel decreases from 2.36 ms to 2.09 ms.

In the condition of synchronous discharge, the first negative amplitude value appears at about 0.5 ms,when the DRPG is at the end of the “steady flow-out” stage or during the “oscillation flowout” stage [14]. The influence of the changing of the discharging voltage on the first negative amplitude value can be ignored.Depending on the pressure wave curves in Fig.5,the maximum of the pressure wave is equal to the first positive amplitude value or the second positive amplitude value.

Fig. 6. The pressure and the projectile position at the discharge voltage of 5 kV and 10 kV.

Different discharge timing sequences are used, and the discharge voltage changes from 3 kV to 10 kV. The first negative amplitude value and maximum of the pressure wave are shown in Fig. 7. In the condition of the timing sequence discharge, the first negative amplitude value decreases with the increasing of the electric energy ratio. When the first negative amplitude value appears,the DRPG is still during the“steady flow-out”stage[14].The gas generator rate of the propellant near the breech increases with the electric power. It makes the first negative amplitude value decrease with the increasing of the electric energy ratio.

In the ETC launch,the maximum of the pressure wave is decided by the positive amplitude value of the pressure wave. With the change of the electric energy ratio, the maximum of the pressure wave at the loading density of is 0.618 g/cm3is greater than 37 MPa.If the maximum increases within 18% (the maximum is less than 45 MPa),the pressure wave is thought to be controlled.In order to control the pressure wave,the allowed electric energy ratio in the condition of timing#1 is less than 0.11,the allowed electric energy ratio in the condition of timing#2 is less than 0.25,and the allowed electric energy ratio in the condition of timing#3 is less than 0.27.

Fig. 7. The peak values of the pressure wave in the condition of different discharge timing sequences.

4.2. The influence of the charging parameters

In order to analyze the influence of the charging parameters,the discharge voltage changes from 3 kV to 10 kV in the condition of synchronous discharge.The loading density of the propellant or the propellant web thickness is changed.

In order to analyze the influence of the loading density,different loading densities of the propellant is chosen:0.562 g/cm3,0.618 g/cm3and 0.674 g/cm3. The first negative amplitude value and maximum of the pressure wave are shown in Fig.8.The maximum of the pressure wave at the loading density of 0.562 g/cm3is greater than 27 MPa. The maximum of the pressure wave at the loading density of 0.672 g/cm3is greater than 50 MPa. In order to control the pressure wave (the maximum increases within 18%), the allowed electric energy ratio is less than 0.11. The influence of the loading density on the allowed electric energy ratio can be ignored.

In order to analyze the influence of the propellant web thickness, the propellant web thickness is changed from 0.46 mm to 0.54 mm.The loading density of the propellant is 0.618 g/cm3,and the discharge voltage is 4 kV and 6 kV.The pressure wave curves of the different propellant web thickness at the discharge voltage of 4 kV and 6 kV are shown in Fig. 9 and Fig.10.

The first positive amplitude value at the discharge voltage of 4 kV is 32 MPa, the first positive amplitude value at the discharge voltage of 6 kV is 40 MPa. The first positive amplitude value decreases 10 MPa, when the propellant web thickness changes from 0.46 mm to 0.54 mm.

From Fig.9 and Fig.10,we know that the influence of the electric parameters on the pressure wave is much larger than the influence of the propellant web thickness before 0.75 ms. After the working period of the DRPG, the influence of the propellant web thickness increases,but the change of the maximum of the pressure wave can be ignored.

5. Conclusion

An axisymmetric two-dimensional(2D)internal ballistic model is built to simulate the pressure wave of the 30 mm electrothermalchemical (ETC) launch with the discharge rod plasma generator(DRPG).The relationship between the pressure wave and the initial parameters, such as input electric power, discharging timing sequence, loading density and propellant web thickness, is researched through the change of initial parameters in the model.

Fig. 8. The peak values of the pressure wave with the different loading density.

Fig. 9. The pressure wave curves of the different propellant web thickness at the discharge voltage of 4 kV.

Fig. 10. The pressure wave curves of the different propellant web thickness at the discharge voltage of 6 kV.

In the condition of synchronous discharge,the maximum of the pressure wave can be controlled while the electric energy ratio is less than 0.11. If the electric energy ratio is larger than 0.11, the maximum of the pressure wave increases rapidly with the electric energy ratio. With the increasing of the electric energy ratio, the change of the first negative amplitude wave value can be ignored.In the condition of timing sequence discharge, the allowed input electric energy ratio to control the pressure wave is proportional to the current pulse duration. At the high electric energy ratio, the maximum of the pressure wave is inverse proportional to the current pulse duration.

The pressure wave increases with the loading density. But the allowed electric energy ratio to control the pressure wave stays,and the variation trend of the first negative amplitude wave value doesn't change.During the discharge of the DRPG,the influence of changing propellant web thickness in 30 mm ETC launch can be ignored.