A GAS DYNAMICS BASED METHOD FOR ARTILLERY INTERIOR BALLISTICS DEDICATED TO NEW PROPULSIVE CHARGES DESIGN

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1 A GAS DYNAMICS BASED METHOD FOR ARTILLERY INTERIOR BALLISTICS DEDICATED TO NEW PROPULSIVE CHARGES DESIGN LIVIU MATACHE MARIUS VALERIU CÎRMACI MATEI ADRIAN NICOLAE ROTARIU NISTORAN GEORGETA DIANA Faculty of Mechatronics and Integrated Armament Systems Military Technical Academy Bucharest, George Coşbuc Avenue, ROMANIA Abstract: New propulsive charges for artillery systems require, very often, an inverse approach. The designers start with the limitations of the existing gun. Firstly, an interior ballistic code must be developed (if it doesn t exist) using, as initial conditions, the existing barrel geometry, the ballistic powder in use, the specific projectiles, all the loading conditions and igniting systems of the fielded weapons. The main goals in ballistic design is to fulfil an imposed initial velocity of the projectile without exceeding the limit pressure value inside the barrel. A good control of the propelland combustion is mandatory in this achievement. In order to do it, the propellant s chemical composition and shape play an important role. Things evolve towards combinations between different propellants, each one with specific geometry and composition, which lead to different combustion velocities. As a consequence, the combustion gases will be produced with various rates, according to projectile movement down the barrel and free volume left behind it. Temperature is an important parameter, as every chemical composition of the ballistic propellants is associated with a very high combustion temperature. Particular homogenous compositions, as triple base propellants have been made to accomplish the design requirements but also to reduce the overall temperature in order to increase the barrel life. A good start in finding the solution of our concerns may be represented by imagining ballistic powders with various burn rates inside the powder element. A combination of two or more types of such powders may give the required charge. In this way it is possible to maintain the high pressure level inside the gun barrel, in order to transfer more energy to the projectile. As in the work-energy theorem, the net work applied on an object causes a change in the kinetic energy of the object. In this paper we want to show a general method for interior ballistic evaluation of artillery gun systems. The method is based on Fluent software. An application is made around the 76 mm field cannon. For a high fidelity in simulations, a fine knowledge of the combustion parameters is required. They influence the thermodynamic behavior of combustion gases and they also are considered entry data for the direct problem in interior ballistic methods. In our case, these parameters are calculated using a Matlab code. The code covers a set of thermochemical equations. Keywords: gas dynamic, equivalent model, Fluent, numerical simulation, interior ballistic, propellant 274

2 1. Overview It is of utmost importance for the new propulsive charges to be less sensitive, to have higher energetic performance and to limit barrel erosion [1]. An acceptable solution is combining several layers of powders (co-layered propellants) [1-2] a method patented by US Patent 4,581,998 (1986) by A.W. Horst et al. Although it is a time-consuming and difficult manufacturing method, it turned out to be very useful in improving the performance of the weapon systems due to the increasing momentum of the gases in the projectile. A relatively new method is the one proposed by TNO [1-2], called co-extrusion propellants. This method has the following advantages [1-2]: it increases the weapon system performances, it increases progressively the burning for propellant elements without interior channel or with one channel, it decreases barrel erosion and improves propellants initiation (e.g. LOVA-type propellants), it allows the possibility of achieving a large number of geometries, leading to multiple applications. Figure 1 shows two types of propellant elements made using this method a Double-based propellant b LOVA-type propellant Fig. 1 Two types of propellant elements obtained by co-extrusion [1-2] In Figure 2 there are shown the curves of pressure and velocity variation versus time for two types of propellant elements: one conventional, and one obtained by the method of combining several layers of propellants (co-layered propellants) [2]. 275

3 Fig. 2 Variation of pressure and velocity versus time for two types of propellant elements As one may notice, the pressure curve corresponding to the powder obtained by the colayered propellants method, presents the particularity of a plateau at the level of maximum pressure, which leads to improvements in weapon system performances. Based on the data above mentioned, in the present paper, a method for solving the fundamental problem of interior ballistics has been developed using numerical simulation. 2. The method In order to design and implement a new energetic material to equip an existing weapon system, an iterative working method has been achieved, that has as input the ballistic and thermo-gas-dynamic properties of the propellant, its geometry, its size, the type of the powder (conventional powder obtained by the method of combining multiple powder layers), etc. The solving method achieved is based on a gas-dynamic equivalent model, the parameters calculation being carried out using numerical calculation, namely the finite volume method. Taking into account the fact that, in case of weapon systems, the work field is quite large and given that a solution is required in a reasonable time, the issue has been approached by transposing it into a 2D axial symmetrical model. Generalizing, the propellant elements from the propelling charge were considered to be 9/7 type, i.e. cylindrical elements with 7 channels. These elements cannot be designed in a realaxially symmetric model, which is why a model of equivalence between the real propellant and the one used in numerical calculation has been built. Since it has been considered that all the real elements of powder have the same weight, every element of real powder has been associated with a toroidal element, equivalent in terms of weight with the real element. Since the rifled barrel cannot be effectively represented in a 2D axial-symmetric model, the problem has been formulated for a barrel having an average diameter. Moreover, an equivalent model has been considered for the muzzle brake of the weapon system, given that the real brake reveals no axial symmetry. 276

4 The condition of equivalence for the considered muzzle brake is given by the real brake shape and size and by the areas through which gases are discharged into the environment in the real case. The values of the input data are either determined experimentally, or calculated based on certain experimental or theoretical calculation, taking into account the recommendations from the literature [3-4]. In addition, working hypotheses for solving the fundamental problem by various methods are not completely preserved. Thus, the hypotheses are as follows: a. plastic deformation of the rotating band in the barrel ridges is considered progressive, as it occurs in reality, and not instantaneously; b. the mechanical work due to the projectile rotation is taken into account by a coefficient of fictitious mass; c. the composition of the propellant gases does not change, which allows the force and covolume (f and α) to be considered constant values; d. mechanical work for the elastic deformation of the barrel and the thermal energy that is lost due to the gas leakage between the rotating band and the barrel are not taken into account; e. losses due to heat transfer to the barrel wall and to the projectile body are taken into account; f. each powder element has the same weight; g. the law of propellant burning rate is expressed by u u p ; 1 h. the propellant ignition is performed using a priming charge that has the same weight as the real one, but which generates only gaseous products; i. the barrel blow-back is not considered 3. The simulation environment 3.1 Initial data The finite volume method can be used to solve a wide range of issues: compressible gas dynamics (Euler equations), applications in aerodynamics, astrophysics, detonics and related fields where shockwaves are involved. For the proposed issue it is one of the most suitable approaches. This method is applied in FLUENT software, a powerful tool for fluid dynamics calculus. The software provides a comprehensive finite volume network flexibility, including the ability to solve flow problems using unstructured networks, which can be easily generated in complex geometries. The types of networks are 2D triangular and quadrilateral, 3D tetrahedral / hexagon / pyramid / polyhedral elements and hybrid networks. FLUENT also allows local modification of the network based on flow solutions. In order to solve the proposed problem, there were used the following capabilities provided by this software [5]: adoption and use of a "real" gas, taking into account the real powder gases covolume and that the parameters of gases generated into the barrel are a temperature function ("user-define function"); 277

5 possibility of introducing variable boundary conditions through direct access on certain variables via C program ("user-define function"); possibility of modeling real movements that occur at the muzzle barrel during firing (combustion of powder elements, movement of the powder elements in the barrel, projectile motion in and outside the barrel, etc.) using the "dynamic mesh" option; possibility of collecting data from various points of the computing field and writing them into external files via C program, in order to validate the theoretical model; possibility to obtain values mediation for each cell in the computing field via C program, for the flow parameters, for comparison purposes with certain computing models; using a heat transfer model for the barrel and the projectile wall; assessment of certain sizes (mass flow, gas pressure on different surfaces, amount of heat, etc.), in the eye of a strict control of the formulated model. 3.2 Complementary data In solving the fundamental problem of interior ballistics, the following physical and chemical characteristics are required (in addition to data on the barrel weapon system, projectile, propelling charge weight, priming charge weight): reaction temperature of the priming powder; reaction temperature of the propelling powder; burning rate coefficient for the priming powder; exponent and coefficient of burning rate for the propelling powder; force and covolume of the propelling powder; average molecular weight of the powder gases; specific heats of the powder gases versus temperature; dynamic viscosity of the powder gases versus temperature; thermal conductivity of powder gases versus temperature; local speed of the sound in the powder gases versus temperature Most ballistics calculation models based on fluid dynamics require additional thermodynamic functions to be obtained from the equation of state. They are intended, in particular, to the combustion modeling with high-level software, such as FLUENT, but can be adapted to define the input data into any other code of fluid dynamics computation (CFD - Computational Fluid Dynamics) [6]. In order to simulate the interior ballistics processes, the models require the description of the thermodynamic behavior of gases resulted from powder combustion. In order to assess the parameters that describe the thermodynamic behavior of the combustion gases, a Matlab code has been used. It is based on the thermochemical calculation model. Its results are entry data in the direct problem-solving method of interior ballistics. The equation of state for real gases describes with enough accuracy the burning gases state at high temperatures and densities inside the gun and inside the ballistic bomb. 278

6 3.3 Underlying equations for the proposed method As noted above, the problem-solving mode is based on the finite volume method using FLUENT software. The program has the possibility to solve the problem in two different ways: based on pressure or based on density. We have chosen this working method because the problem is particularly complex. The pressure-based solver, due to its conception design, provides a faster resolution of the problem. The major disadvantage of the pressure-based working method implemented within FLUENT software is that it does not contain an integrated definition for the real gas. This led to the achievement of a mathematical model. Its means are intended to give a correct solution for a real gas flow both inside the barrel and in the environment after the projectile exit. The equations used in the formulation of the method and that were included in the FLUENT software [5] are presented below. The equation of conservation of mass and equation of continuity for a 2D axial-symmetric geometry, has the following form: v v v r S t x x r m r x - axial coordinate; r - radial coordinate; vx - velocity in the axial direction; vr - velocity in the radial direction; ρ density of the combustion products; S m r, where (1) - source providing the continuous phase mass to be dispersed in the secondary phase, if necessary (e.g., phase changes). The equations of motion and momentum conservation, axially and radially, are: 1 1 ( v ) ( rv v ) ( rv v ) x x x r x t r x r r p 1 v 2 1 v v x x r r 2 ( v) r F ; x x r x dx 3 r r dr dx (2) 1 1 p 1 v v r x ( v ) ( rv v ) ( rv v ) r r x r r r t r x r r r r x dx dr 2 1 v 2 2 r v v r z r 2 v 2 v F. 2 r r r dr 3 r 3 r r, Fx, Fr axial and radial volume forces; - kinematic viscosity of the combustion products. The equation of energy conservation is represented as: 279

7 t k x x E v E p r v E p c p x t T Pr x t (3) where: E is the total energy; 1 r r r 1 u ( ) r k ij eff r r ( ij ) eff is the viscous heating; c p t T Pr x t ru( ) ij eff, Pr t is the number of Prandtl. Due to the fact that the generated powder gases show high pressures and temperatures, the ideal gas law does not describe the situation with accuracy. As a consequence, there will be used the real gases law of state: 1 p RT, where (4) p - powder gas pressure; ρ - gas density; α gas covolume; R - gas constant; T - gas temperature. For the law of the burning rate, the equation used is ua u p 1, where (5) ν - burning rate exponent; u1 - burning rate coefficient For the projectile motion, there has been considered the following equation: a F m (6) The system of equations presented has introduced a number of 6 equations with 6 unknowns: v x v, r, p, T, ρ and a. 4. Model simulation. Equivalent model and discretization The equivalent physical model of a cannon system designed in the view of addressing the interior ballistic cycle modeling is illustrated in Figure

8 Fig. 3 The cannon equivalent physical model used in numerical modeling 2 D case The equivalent physical model developed is based on the following considerations (equivalence conditions): the volume of the loading chamber of the equivalent model is the same as the real one; the volume of the barrel for the equivalent model is the same as the real one, i.e. the equivalent barrel section area is equal to the real-section area of the rifled barrel and the barrel length of the model is the same with the barrel real length; the gases exhaust surfaces areas of the muzzle brake must have the same values; physical and chemical properties of the equivalent powder and the generated gas are the same as in the real case; propelling powder mass of the proposed model is the same with real propelling powder mass; the density of the equivalent powder is the same with the real one; the mass of the equivalent powder element must be equal to the average real powder elements mass (real powder element-mediated), thus leading to the same number of powder elements in both equivalent and in real case; the combustion products flow inside the powder element is the same, both in equivalent and in real powder; The fraction of burnt powder of an element has to be the same in both cases. According to the working hypothesis no. 6, the law of burning rate for the propelling powder is u u p expressed as 1. Based on the equivalence conditions mentioned above and making u the assumption that the exponent of burning rate remains constant, 1 e has been determined. The system of equations previously defined, which includes continuous field variables such as velocities, pressure, temperature, density, is submitted to meshing transformation. In this process, the continuous fields are replaced with discrete fields of values defined in the volume control centers. The FLUENT software operates with the discrete set of these measures. The solutions accuracy is conditioned by the meshing network of the field. A sophisticated network computing leads to higher accuracy, instead of involving a greater computational effort. The analyzed model has been meshed using triangular elements as shown in Figure

9 Fig. 4 Meshing the analyzed equivalent model - detail in the projectile In the case of the weapon system under analysis, there has been used a dynamic mesh network. Taking into account the fact that the weapon internal configuration is variable, the mesh allows the domain transformation following the manner in which the powder blast evolves. The transport movement and interactions (elastic type collisions) of the powder elements as a consequence of the projectile movement allow the area occupied by the real combustion products to be filled with finite volumes with controlled dimensions. 5. Results Numerical simulation of interior ballistics was performed for a 76 mm cannon, equipped with two types of propelling charge: a 9/7 conventional one, and a co-extruded one which contains both 9/7 powder and a powder with higher burning vivacity, arranged as a circular crown inside the conventional powder while maintaining the same charge density. By solving the problem of interior ballistics, the determination of flow variation for relevant parameters is allowed, as shown further. Variations in status and kinematic parameters of powder gases at a subsequent time to the achievement of the maximum pressure in the barrel for conventional powder are given in Figure

10 Fig. 5 Distribution of the pressure and the velocity at t = 3,2 ms Pressure distribution Velocity distribution a conventional powder b co-extruded powder In Figure 6 the plotted curves of the barrel pressure and the projectile velocity versus time are shown in the two cases considered. Figure 6. Variation of the pressure at the projectile base and the projectile velocity versus time a Pressure variation versus time b Velocity variation versus time 283

11 6. Conclusions The work presented in this paper is based on how to build equivalence between the real weapon system (geometry, shape, muzzle brake characteristics, projectile shape, type of powder etc.) and the one used in numerical simulation. Further, we analyze the general method for solving the fundamental problem of interior ballistics for any type of medium, namely for a large caliber weapon system whose powder elements show simple or multi-channel features. An equivalent real gas model has been defined in relation to the gas generated by the deflagration of the powder when firing. The way to achieve a product is greatly shortened if experimental data are correlated to the data obtained by numerical simulation in order to optimize the product. At this point the method presented in this paper shows its advantage. It allows obtaining design data in short time, at no additional cost, in order to optimize the propelling charge and the entire system. References [1] SCHOLTES, G. Novel techniques for improved munitions development, 44 th annual Gun and Missile System Conference, April 2009 [2] ZEBREGS, M, van DRIEL,C, Experimental set-up and results of the process of co-extruded perforated gun propellants, IM/EM Symposium, Tucson, AZ, May 2009 [3] FITT, A.D., CROWLEY, A.B., ASTON, J, TORO, E, Contrasting Numerical Methods for Two- Dimensional Two-Phase Internal Ballistics Test Problems, Proceedings of the 11 th International Symposium on Ballistics, Vol.1, Royal Military Academy, Brussels, Belgium, May 1989 [4] VASILE, T, Balistica Interioară a gurilor de foc, Vol I, Ed. ATM, 1993 [5] ***FLUENT User Guide Fluent Inc. Centerra Resource Park 10 Cavendish Court Lebanon, NH FLUENT 6.3 User s Guide PathScale Corporation [6] P.G. BAER, J.M. FRANKLE, The Simulation of Interior Ballistic Performance of Guns by Digital Computer Program, U.S. Army Ballistic Research Lab., Aberdeen Proving Ground, MD, Dec Acknowledgement: This paper is financially supported within the project entitled Horizon Doctoral and Postdoctoral Studies: Promoting the National Interest through Excellence, Competitiveness and Responsibility in the Field of Romanian Fundamental and Applied Scientific Research, contract number POSDRU/159/1.5/S/ This project is cofinanced by European Social Fund through Sectorial Operational Program for Human Resources Development Investing in people! 284