Run-duo Cao, Xiao-bing Zhang

School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, China

Keywords:Interior ballistics Mobile chamber Design optimization

ABSTRACT A launching system with novel structure using optimization method is investigated to improve the muzzle velocity of guns in this article. This system has two tandem chambers of which the front one is ignited first and the other is ignited after a while.The launching process of this novel system is modelled and a series of different schemes are simulated,to discover the interior ballistic performance of this novel launching system. In order to obtain the optimal loading conditions, an optimization model combined with the combustion model is established. The optimal schemes can improve the muzzle velocity by 20.6% without changing the parameters of barrel. It means that this novel launch system could improve the interior ballistics performance significantly and it still has considerable potential to be ameliorated.?2019 The Authors.Production and hosting by Elsevier B.V.on behalf of China Ordnance Society.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Since guns were invented, nearly all guns applied a launching principle in which the projectile is pushed by high-pressure gas generated from burning propellants in a chamber. However, this launching principle has an inherent deficiency that, with the motion of the projectile in chamber,there is a pressure gradient along the gun barrel because the expansion of gas is lower than the projectile.Thus,the pressure behind the projectile is always lower than the pressure at the breech. For the interior ballistics of guns,the muzzle velocity of the projectile depends on the projectile base pressure,but the breech pressure is limited by the barrel design and materials,so that the guns could not get a higher muzzle velocity in the conventional launching method. So far, many basic methods such as enhancing the impetus of propellant,increasing the mass of propellant, extending the length of gun barrel, have been investigated but fail to solve this problem.

In order to improve the muzzle velocity of guns, designers worldwide have invented and developed many novel methods by adopting some novel technologies.For example,a kind of traveling charge technique to maintain higher pressure have been tested for decades[1e3],in this method,the propellant is combusting behind the projectile constantly, which pushes the projectile more efficiently. And some electrothermal launchers are also proposed by some researchers [4e6], in which the propellant in the chamber reacts with the plasma generated by the pulse power and it releases much energy to give the projectile a high velocity. Comparatively,the electromagnetic launch is developed more deeply, and some prototypes have been developed in America by Dr. Fair [7e9] and some other countries [10]. This method can provide a sustaining acceleration during the complete interior ballistics process so that the projectile could get a high muzzle velocity. In addition, some other launching systems were also proposed, such as ram accelerator [11], rarefaction wave gun [12,13] and so on. These launching methods all applied advanced theory to improve the muzzle velocity, however, these methods also have disadvantages which are not able to be solved at present. For example, the traveling charge system requires liquid propellants or solid propellant with high burning rate, whose combustion is difficult to control. As for the electromagnetic launcher, it has a high requirement for electrical power system so it is not convenient to bring to service in the short term. In addition, the electrothermal launchers and rarefaction wave gun have been developed for many years but few achievements have been made.

In this paper a mobile chamber launching system (MCLS) is proposed, of which the combustion chamber is divide into two chambers, separated by a piston, with one chamber fixed and the other moving. Although the similar structure has been proposed[14],the ignition order and motion of projectile for this method in this paper are both different.For this method,the propellants in the mobile chamber are ignited first and then generate gas to drive the projectile. After several milliseconds, the propellants in the fixed chamber are ignited and the piston starts moving under the pressure in both chambers. In the wake of the piston moving, the mobile chamber expands slowly relatively so that there are two pressure peaks in the mobile chamber and the pressure at the base of projectile drops more slowly than that in a conventional gun.Thus,the propellent gas does more work on the projectile along the barrel so the projectile can attain a higher muzzle velocity. Moreover, this launching system could be applied into existing guns because it avoids too much modification on the guns. Considering the difference in launching process between proposed method and conventional launching method, the modified physical and mathematical models are established in section 2. Because this novel launching system is an original structure for guns, the design experience for the conventional guns is for reference only but not able to obtain the best design scheme. Thus, the optimization design combines the interior ballistic model and optimization model is necessary. Some discussions about selecting variable and the design optimization applying a standard particle swarm optimization(SPSO)algorithm is described in section 3.The results and discussion about the launching process are expressed in section 4.This paper is for the purpose of providing a new approach to improve the muzzle velocity on the conventional guns. The simulation results indicate that this method is effect.

2. Model establishment and simulations

The schematic of the mobile chamber launching system is shown in Fig.1, which includes three main parts: a fixed chamber including the grain propellant and igniter tube, a mobile chamber consisting of a piston, stick propellant and grain propellant, and a projectile. For the proposed system, the igniter tube connects between the breech and the piston, which can avoid the piston moving earlier.As similar with the conventional launching system,a lumped parameter model proposed by Zhang [15], which is a useful and robust model for the interior ballistic cycle, is used to simulate this novel launching process.In this model,propellants in each chamber are assumed to be ignited simultaneously and uniformly, while their combustion is assumed to take place in a smoothly varying, well-stirred mixture, and the burning rate is determined by the instantaneous, space-mean chamber pressure.The state equation for gunpowder gas is Nobel-Abel equation given in Ref.15. The engraving process of projectile is ignored. In the process of combustion, the impetus of the propellant, the specific ratio and the covolume are constant.

2.1. The model of launching process

The launching process of this novel method can be divided into 4 stages and each stage is analyzed below. The schematic of launching process is shown in Fig. 3 (a)~(f). In the mathematical model of launching process, subscript “1” and “2” represent the parameters of fixed chamber and mobile chamber respectively,and subscript “c” and “p” represent the parameters of piston and projectile respectively.

1. Combustion model of propellants

For the propellants in the two chambers, a common parameter model is applied,which is described in Eq.(1).In this article,a 19-perforation rosette propellant is used, whose cross-section schematic is shown in Fig. 2.

Fig.1. Schematic of the mobile chamber launching system.

Fig. 2. 19-perforation rosette propellant.

2. The first stage of launching process

The first stage,which is shown in Fig.3(a),begins with ignition of the propellant in the mobile chamber and ends when the propellant in the fixed chamber is ignited. In this stage, the projectile moves ahead and other parts are immobilized. The state equation in mobile chamber is shown as Eq.(2)and the energy conservation equation in mobile chamber can be described as Eq. (3).

According to Eqs. (2) and (3), the pressure in the mobile chamber can be obtained by Eq. (4). Besides, the motion of piston and projectile are described as Eq. (5) and pressure in fixed chamber remains unchanged in this stage as Eq. (6).

2 The second stage

Several milliseconds after the projectile starts moving,which is called delay time(td),this stage starts when the propellants in fixed chamber begin burning and ends when the piston begins to move,which is shown in Fig.3(b).In this stage,the pressure in the fixed chamber is less than that in the mobile chamber so that it is not able to push the piston forward. Because the projectile is still moving individually while the other parts are kept still, the expression of pressure in the mobile chamber and the motion of projectile are both same as previous as Eq.(4)and Eq.(5).However,the pressure in the fixed chamber changes as Eq. (3), which is similar to the combustion of propellants in a closed bomb vessel.

3 The third stage

Fig. 3. Schematic of the launching process.

When the piston starts moving, which means the pressure in the fixed chamber is large enough to push the piston forward and the motion of piston can be described as Eq. (8). In this stage, the projectile is driven only by the pressure in the mobile chamber and its motion is described as Eq. (9).

Meanwhile,based on the law of conversation of energy and state equation of gas,the state in fixed chamber and mobile chamber can be described as Eq. (10) and Eq. (11).

Thus,the pressure in the fixed chamber and the mobile chamber can be obtained by Eq. (12) and Eq. (13).

This stage finishes when the projectile leaves the gun barrel as shown in Fig.3(d).So far,the main process of launching is over and the muzzle velocity can be obtained.

4 The last stage

After the projectile has left the gun barrel, the main process of the interior ballistic cycle has been finished.However,because the piston is still moving, this stage ends when the piston leaves the gun barrel. In this stage, the pressures in the two chambers drive the piston and the outflow of gas in the mobile chamber is assumed to be adiabatic. It could be seen as another projectile launching from the gun. Therefore, the pressure in both chambers could be calculated as Eq.(14)and Eq.(15),which are given by Ref.[16],the motion of piston is described as Eq. (16) and Eq. (17).

where,p2;outand toutrepresent the pressure in the mobile chamber and the time at the moment of the projectile leaving the gun barrel.

Thus, the interior ballistic process of this method can be simulated by Eq. (1) to Eq. (17).

Considering the pressure distribution in two chambers, the motion equation of piston (9) is calculated by the pressure at the piston base (pb1) and the pressure of the projectile base (pb2). In order to obtain them, we assume that the propellant gas in both chambers obey the Lagrange hypothesis[15].Because of the muzzle velocity is over 600 m/s, for which the classic The Lagrange pressure gradient model is not reasonable [17,18], a modified Lagrange pressure gradient model given by Ref.[19]is adopted.This model is modified by a series of experiment and the results show that it applied to for some guns of which the muzzle velocity is excess of 1700 m/s. Thus, this model is suit for the proposed method in this paper.The detailed process of this method is presented as follows:

2.2. Numerical simulations

After establishing the launch model, some numerical simulations are proceeded to investigate if this novel method could improve the interior ballistic performance.The original parameters of a 130-mm cannon and propellant are listed in Table 1.And some additional parameters for mobile chamber are listed in Table 2,which are inspired by the previous work and the conventional launching system.

Table 1 The original parameters of a 130-mm cannon and propellant.

Table 2 Additional parameters of propellant for MCLS.

The simulation results are shown in Figs.4e7,Figs.4 and 5 show the pressure history and velocity history of different methods.Due to the limit of the pressure in fixed chamber,the pressure in mobile chamber cannot attain a high level, and vpis lower than v0before the piston starting moving. However, the piston starts moving while the p1exceeds p2so that the volume of mobile chamber expands slowly and p2can hold a high level and drop more slowly than p0. Figs. 6 and 7 is the pressure and velocity to the projectile displacement of different methods.Fig.6 shows that the maximum of p0appears while the piston moves about 1 m,and the maximum of p1appears at the breech of gun. The simulation results indicate that two methods can obtain similar muzzle velocity, even if p0is higher than p2in the initial stage.However, the parameters of the mobile chamber in this condition is just obtained by experience,the pressure in mobile chamber is too low to do enough work on the projectile. Thus,it is necessary to obtain the best scheme by using the design optimization to achieve full potential of this launching system adequately.

3. The design optimization

According to the model established above, the shape of the mobile chamber should be designed carefully to meet the requirements. For the mobile chamber, the most important parameters are its volume and charge mass,which is similar to the fixed chamber.Besides,the mass of piston and the delay time,which are new variables involved in interior ballistics process, also influence the performance. Therefore, some schemes are simulated to investigate the relationship between these parameters and interior ballistic preference. These simulations also provide us some experience on choosing value ranges of each variables.

3.1. Analyses of some variables

Fig. 4. Pressure history of different methods.

Fig. 5. Velocity history of different methods.

Fig. 6. Pressure to the projectile displacement of different methods.

Fig. 7. Velocity to the projectile displacement of different methods.

At first, this article discusses about effects of the piston mass(mh), delay time (td) and the propellants thickness (2e2) in the mobile chamber on muzzle velocity, and the results are shown in Figs. 8e10. It is important to note that while one variable is considered, some necessary changes of the propellants' shape are taken to ensure an equal maximum pressure in the mobile chamber every time, but the mass of charge is unchanged. In Fig. 8, muzzle velocity (vg) declines and the maximum pressure rises with the mass of piston increasing,because the lighter the piston is,it can be accelerated more easily, so that the pressure in mobile chamber could be maintained to work on the projectile more efficiently.However, with the mcincreasing, the piston needs a higher pressure to be push forward and more energy of propellant is converted to the velocity of the piston, so that the velocity of projectile declines but the maximum pressure in fixed chamber(P1max)rises.As for the influence of the delay time, Fig. 9 shows that both vgand P1maxdecline with the tdbeing extended. If the delay time were shorter, pressure peaks in two chambers would appear at the similar time so the piston was difficult to move to enlarge the fixed chamber and the P1maxwas at a very high level. If the delay time were too long, the propellant in the fixed chamber does not have enough time to burn out, so the projectile fails to obtain a high velocity. In Fig.10, vgand P1maxboth decline with 2e2increasing,because with thinner 2e2, the propellants generate gas more quickly, leading to larger P1max.

Fig. 8. vg and P1max with changes of mc/mp.

Fig. 9. vg and P1max with changes of td.

Fig.10. vg and P1max with changes of 2e2.

Fig.11. Pressure history with different mh/mp.

Fig.12. Pressure history with different td.

Fig.13. Pressure history with different 2e2.

Moreover, in order to investigate the influences of these variables on pressure, more schemes are simulated to observe the changing process of pressure in the two chambers,and the results are shown in Figs.11e13. It is worthy to note that in each simulation, only the corresponding variable is changed and other parameters of the charge are unchanged. Fig. 11 shows that the maximum pressure in the fixed chamber rises along with increasing mh/mp, but the second pressure peak in the mobile chamber changes on the contrary.Because with larger mh,it needs a higher p2to give piston sufficient acceleration to shrink the mobile chamber relatively. As for p2, because before piston starts moving,the volume of mobile chamber does not change,so it does not change obviously.Similarly,Fig.12 shows the influence of tdon p1and p2,it is obvious that when the tdis short,the pressure peak in the fixed chamber appears early so that the piston moves early and p1declines slowly. When tdis too long, the piston starts moving later to shrink the mobile chamber and p1is nearly not affected.As for 2e2, p2rises more quickly while 2e2is thinner, which means it needs a larger p1to drive the piston so that pressures in both chambers are affected by 2e2,as it is shown in Fig.13.Considering the safety during the launching process, a too thin thickness of propellants in the mobile chamber should be avoided.

The previous simulations and analyses have shown the significant influence on the interior ballistic performance due to the mass of piston, the delay time and thickness of extra propellants.Moreover, different combinations of them will lead to more variations. Because there is no similar design before, and this novel launching system involves many multidisciplinary variables of charges and canon structure in both chambers.It is hard to obtain a superior design scheme based on experience. Therefore, using an optimization method is an ideal way to predict the overall performance of an interior ballistic design. Our group has successfully applied some optimization algorithms into the charge design and other complex design optimizations [20,21]. Thus, similar approaches are used to solve the design optimization of the proposed launching system in this article as follows.

3.2. Optimization mathematical model

Generally, the optimization problem involves objective, design variables and contains.

1) Design objective

The mobile launching system is applied to improve the muzzle velocity,so the only design objective is muzzle velocity of projectile(vg).

2) Design variables

To satisfy the design requirements, it is important to choose some appropriate design variables.Considering the above analyses on the proposed method, it is known that if the structural parameters of gun are pre-determined, only the volume of mobile chamber(Vm),mass of piston(mc),and delay time of the ignition in fixed chamber (td) can be modified. In addition，the main characteristic parameters of the propellants in both chamber, which include web thickness(2e1(2)),perforation diameter(d1(2)),and the charge mass (u1(2)), can be redesigned. Therefore, all these nine parameters should be regarded as design variables in the design optimization. The structural parameters of guns and some design variables are shown in Fig.14.

3) Constraints

Considering the particularity of the proposed launching system,in order to insure the launching safety and tactical requirements,some special and necessary constraints are listed as follow, which include maximum pressure and charge density.

At first, because of the limit of the gun barrel's strength, the pressure in both chambers(Pm1(2))must be lower than a maximum pressure(Pmax),of which the pressure distribution along the barrel have been considered. Furthermore, the charge density (D1e2T)which is calculated by Eq.(19)should be limited as follows,because it has a great effect on the launching safety.

According to the previous analyses, an optimization mathematical model applied in this article could be described as follows:

To achieve the optimization of this novel launching system, an enhanced SPSO algorithm [22] is applied in this article. Using a 130 mm cannon as an example, the combustion model for multiperforated propellant and the lumped parameter interior ballistic model for mobile chamber are firstly combined to establish the optimization model, and then SPSO is applied to find the optimal result. The flowchart of optimization design is shown in Fig.15.

4. Optimization results

Here,the first aim is to obtain the optimal scheme of the design variables and the second aim is to validate whether the mobile launching system improves the interior ballistic performance significantly. The values of the original parameters of a 130-mm cannon and a 19-perforation rosette propellant have been listed in Table 1. As for the optimization model, according to the result and discussion above, the lower bounds (LB) and upper bounds(UB) of each variable are given in Table 3. The values of each variable are inspired by the simulation results above,which ensure the maximum pressure in chamber not exceeding the design limit.

After 500 iterations,a converged result is obtained and listed in Table 3,and the convergence process is illustrated in Fig.16(a)-(c).It is obvious that after hundreds of iterations, all variables and objective remain unchanged and the muzzle velocity changes from 975 m/s to 1026 m/s, which means the optimization algorithm is useful. Moreover, each variable has an obvious change at the objective's every improvement, which means the interior ballistics performance of this novel launching system depends on these variables and our choices about these variables are appropriate.

Then the parameterized model is applied to simulate the launch process of the optimum scheme and results are listed in Table 4.The muzzle velocity of this novel method can attain 1026 m/s, which has an improvement of 20.6%. And the maximum pressure in two chambers are in the safe range. These results indicate that our proposed mobile launching system can improve the muzzle velocity significantly.

In order to explain why this novel launching system could improve the performance of the interior ballistics system, Fig. 17(a)-(b) show the pressure and projectile velocity variation over time and projectile displacement. Compared with the parameters obtained by experience in Tables 1 and 2, 2e2and u2increases evidently leading that the energy in mobile chamber is higher and the propellant burns slowly so that it releases gas in a long time.Besides, the mass of piston is much lighter so that it can be accelerated more easily.

Fig.14. Parameters of cannon and projectile.

Fig.15. The flowchart of optimization design.

Table 3 The Lower Bound, Upper Bound and Optimum results of each variable.

Fig. 17 (a)e(b) show that when the propellant in the mobile chamber begins burning,p2raises quickly due to the charge density is pretty high. Considering the delay time, the p2is higher than p1even pass the first pressure peak. However, after the ignition in fixed chamber, the p1climb more fast and exceed the p2, then the piston starts moving and obtains a large acceleration. Fig. 17 (b)shows that the piston moves even faster than piston in a short period. Thus, the volume of the mobile chamber to expand slowly and even shrink so that the pressure in the mobile chamber can achieve the second pressure peak, which is higher than first one.For the piston adjusting two chambers dynamically in the effect of the p1and p2,Fig.17(a)shows that after about 8.2 ms,the pressure in both chambers are equal,which means the velocity of the piston stays unchanged as shown in Fig.17(b) so that energy of the propellant in two chambers both work on the projectile. Because the pressure in the mobile chamber remains at a high level, the projectile can obtain energy effectively. As shown in Fig.17(c), the p2does more work on projectile than p0after the projectile moving 2 m, so that the projectile can obtain a higher muzzle velocity by the proposed method. Fig.17 (b) show that the v1increases more quickly than v0in the most of launching process,the projectile can maintain a large acceleration in the whole time so that the proposed method can obtain a higher muzzle velocity. These results indicate that without increase the maximum pressure in chamber,the proposed method can improve the muzzle velocity obviously.

Fig.16. The convergence process of each variable.

Table 4 The original parameters of a 125 mm gun and propellant.

Fig.17. The launch process of different launch method.

5. Discussion

Although our optimal design scheme could improve the muzzle velocity from 851 m/s to 1026 m/s,this novel launching system still has potential for perfection. For example, the simulation above is based on an existing cannon, whose fixed chamber and barrel are unchangeable, the propellants in two chambers are the same in shape and composition. If the fixed chamber or barrel could be redesigned and different propellant are applied considering the different requirement of two chambers, this novel launching system might achieve a much higher muzzle velocity. Besides, the mass of the mobile chamber also influences the acceleration in the initial state, and it may decrease by applying some new materials.

For this proposed method, some methods are adopted to deal with new problems. Before the piston starting moving, a highstrength igniter tube is applied to prevent the piston moving earlier. Besides, the stick propellant and multi-point ignition are used to ensure the ignition in mobile chamber uniform as far as possible. In addition, there are also some problems existing with this system.For example,there are two pressure peaks in the barrel and their position is after that in the conventional method and some additional measures should be applied to ensure the launching safety.On the other hand,owing to the shorter effective movement and much higher velocity, the movement time is short so the proposed method requires a high-burning rate propellant.

6. Conclusions

In this article, a novel mobile chamber launching system is proposed, with its interior ballistics model being established and the launching process being simulated.Compared with other novel launching methods,such as electromagnetic gun,rarefaction wave gun and so on, the proposed method can be applied into the existing guns without too many modifications for guns, so it is a low-cost system and suitable for being equipped in a short time.

Then a design optimization for the novel launching system using the SPSO is applied to obtain an optimum scheme. In order to evaluate its performance, it is compared with an original scheme and an optimum scheme in conventional method.The results show that the novel method could improve the muzzle velocity by 20.6%,while the maximum pressure in chamber has no significant change.Owing to much potential for the mobile launching system,it could be a very worthwhile technology to improve the interior ballistic performance. Although the result is satisfactory, this paper only provides this new approach and adopts a simplified interior ballistic model to simulate the launching process. For the purpose of describing launching process in more detail,some complex model can be applied in the future work.